Group #50304511 — subgroups

Subgroups — C₂×C₃×C₅×C₇×C₁₃×D₇ #50304511

Order 38220.

208 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×C₁₃×D₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅×C₇²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₇×C₁₃×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅×C₇²×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₇×C₁₃×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×C₇×C₁₃×D₇	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₅×C₁₃×D₇	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃×C₅×C₇×C₁₃	—	7	1	no	maximal
10	C₃×C₇×C₁₃×D₇	C₂×C₅	1	2	yes	direct product factor
11	C₂×C₃×C₇²×C₁₃	C₂×C₅	1	1	yes	characteristic, complemented
12	C₅×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₅×C₇×D₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃×C₅×C₁₃×D₇	C₂×C₇	1	2	yes	direct product factor
15	C₂×C₃×C₅×C₇×C₁₃	—	2	3	no	—
16	C₂×C₃×C₅×C₇×C₁₃	—	7	2	no	—
17	C₂×C₃×C₅×C₇×C₁₃	D₇	1	1	yes	characteristic, direct product factor
18	C₂×C₃×C₅×C₇×C₁₃	C₂×C₇	1	1	yes	characteristic, complemented
19	C₂×C₇×C₁₃×D₇	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
20	C₃×C₇²×C₁₃	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
21	C₂²×C₅×C₇×C₁₃	—	7	1	no	—
22	C₂×C₅×C₁₃×D₇	C₃×C₇	1	1	yes	characteristic, direct product factor
23	C₂×C₃×C₅×C₇²	C₂×C₁₃	1	1	yes	characteristic, complemented
24	C₃×C₅×C₇×D₇	C₂×C₁₃	1	2	yes	direct product factor
25	C₃×C₅×C₇×C₁₃	—	2	3	no	—
26	C₃×C₅×C₇×C₁₃	C₂×D₇	1	1	yes	characteristic, direct product factor
27	C₃×C₅×C₇×C₁₃	C₂²×C₇	1	1	yes	characteristic, complemented
28	C₂×C₇²×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
29	C₇×C₁₃×D₇	C₂×C₃×C₅	1	2	yes	direct product factor
30	C₂²×C₃×C₇×C₁₃	—	7	1	no	—
31	C₂×C₃×C₁₃×D₇	C₅×C₇	1	1	yes	characteristic, direct product factor
32	C₂×C₅×C₇×D₇	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₅×C₇×C₁₃	—	2	3	no	—
34	C₂×C₅×C₇×C₁₃	—	7	2	no	—
35	C₂×C₅×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic, direct product factor
36	C₂×C₅×C₇×C₁₃	C₂×C₃×C₇	1	1	yes	characteristic, complemented
37	C₅×C₁₃×D₇	C₂×C₃×C₇	1	2	yes	direct product factor
38	C₂²×C₃×C₅×C₁₃	—	7	1	no	Hall
39	C₃×C₅×C₇²	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
40	C₇²×C₁₃	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₇×D₇	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
42	C₂×C₃×C₇×C₁₃	—	2	3	no	—
43	C₂×C₃×C₇×C₁₃	—	7	2	no	—
44	C₂×C₃×C₇×C₁₃	C₅×D₇	1	1	yes	characteristic, direct product factor
45	C₂×C₃×C₇×C₁₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented
46	C₃×C₁₃×D₇	C₂×C₅×C₇	1	2	yes	direct product factor
47	C₂×C₅×C₇²	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
48	C₅×C₇×D₇	C₂×C₃×C₁₃	1	2	yes	direct product factor
49	C₅×C₇×C₁₃	—	2	3	no	—
50	C₅×C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic, direct product factor
51	C₅×C₇×C₁₃	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
52	C₂²×C₃×C₅×C₇	—	7	1	no	—
53	C₂×C₃×C₅×D₇	C₇×C₁₃	1	1	yes	characteristic, direct product factor
54	C₂×C₃×C₅×C₁₃	—	7	2	no	—
55	C₂×C₃×C₅×C₁₃	C₇×D₇	1	1	yes	characteristic, direct product factor
56	C₂²×C₇×C₁₃	—	7	1	no	—
57	C₂×C₁₃×D₇	C₃×C₅×C₇	1	1	yes	characteristic, direct product factor
58	C₂×C₃×C₇²	C₂×C₅×C₁₃	1	1	yes	characteristic, complemented
59	C₃×C₇×D₇	C₂×C₅×C₁₃	1	2	yes	direct product factor
60	C₃×C₇×C₁₃	—	2	3	no	—
61	C₃×C₇×C₁₃	C₂×C₅×D₇	1	1	yes	characteristic, direct product factor
62	C₃×C₇×C₁₃	C₂²×C₅×C₇	1	1	yes	characteristic, complemented
63	C₂²×C₅×C₁₃	—	7	1	no	Hall
64	C₅×C₇²	C₂²×C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
65	C₂×C₃×C₅×C₇	—	2	3	no	—
66	C₂×C₃×C₅×C₇	—	7	2	no	—
67	C₂×C₃×C₅×C₇	C₁₃×D₇	1	1	yes	characteristic, direct product factor
68	C₂×C₃×C₅×C₇	C₂×C₇×C₁₃	1	1	yes	characteristic, complemented
69	C₃×C₅×D₇	C₂×C₇×C₁₃	1	2	yes	direct product factor
70	C₂×C₇×D₇	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
71	C₃×C₅×C₁₃	C₂×C₇×D₇	1	1	yes	characteristic, direct product factor, Hall
72	C₂×C₇×C₁₃	—	2	3	no	—
73	C₂×C₇×C₁₃	—	7	2	no	—
74	C₂×C₇×C₁₃	C₃×C₅×D₇	1	1	yes	characteristic, direct product factor
75	C₂×C₇×C₁₃	C₂×C₃×C₅×C₇	1	1	yes	characteristic, complemented
76	C₁₃×D₇	C₂×C₃×C₅×C₇	1	2	yes	direct product factor
77	C₂²×C₃×C₁₃	—	7	1	no	Hall
78	C₃×C₇²	C₂²×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
79	C₂²×C₅×C₇	—	7	1	no	—
80	C₂×C₅×D₇	C₃×C₇×C₁₃	1	1	yes	characteristic, direct product factor
81	C₂×C₅×C₁₃	—	7	2	no	—
82	C₂×C₅×C₁₃	C₃×C₇×D₇	1	1	yes	characteristic, direct product factor
83	C₃×C₅×C₇	—	2	3	no	—
84	C₃×C₅×C₇	C₂×C₁₃×D₇	1	1	yes	characteristic, direct product factor
85	C₃×C₅×C₇	C₂²×C₇×C₁₃	1	1	yes	characteristic, complemented
86	C₂×C₇²	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic, complemented
87	C₇×D₇	C₂×C₃×C₅×C₁₃	1	2	yes	direct product factor
88	C₇×C₁₃	—	2	3	no	—
89	C₇×C₁₃	C₂×C₃×C₅×D₇	1	1	yes	characteristic, direct product factor
90	C₇×C₁₃	C₂²×C₃×C₅×C₇	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₇	—	7	1	no	—
92	C₂×C₃×D₇	C₅×C₇×C₁₃	1	1	yes	characteristic, direct product factor
93	C₂×C₃×C₁₃	—	7	2	no	—
94	C₂×C₃×C₁₃	C₅×C₇×D₇	1	1	yes	characteristic, direct product factor
95	C₂×C₅×C₇	—	2	3	no	—
96	C₂×C₅×C₇	—	7	2	no	—
97	C₂×C₅×C₇	C₃×C₁₃×D₇	1	1	yes	characteristic, direct product factor
98	C₂×C₅×C₇	C₂×C₃×C₇×C₁₃	1	1	yes	characteristic, complemented
99	C₅×D₇	C₂×C₃×C₇×C₁₃	1	2	yes	direct product factor
100	C₅×C₁₃	C₂×C₃×C₇×D₇	1	1	yes	characteristic, direct product factor, Hall
101	C₂²×C₃×C₅	—	7	1	no	Hall
102	C₂²×C₁₃	—	7	1	no	Hall
103	C₇²	C₂²×C₃×C₅×C₁₃	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₃×C₇	—	2	3	no	—
105	C₂×C₃×C₇	—	7	2	no	—
106	C₂×C₃×C₇	C₅×C₁₃×D₇	1	1	yes	characteristic, direct product factor
107	C₂×C₃×C₇	C₂×C₅×C₇×C₁₃	1	1	yes	characteristic, complemented
108	C₃×D₇	C₂×C₅×C₇×C₁₃	1	2	yes	direct product factor
109	C₃×C₁₃	C₂×C₅×C₇×D₇	1	1	yes	characteristic, direct product factor, Hall
110	C₅×C₇	—	2	3	no	—
111	C₅×C₇	C₂×C₃×C₁₃×D₇	1	1	yes	characteristic, direct product factor
112	C₅×C₇	C₂²×C₃×C₇×C₁₃	1	1	yes	characteristic, complemented
113	C₂×C₃×C₅	—	7	2	no	—
114	C₂×C₃×C₅	C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor
115	C₂²×C₇	—	7	1	no	—
116	C₂×D₇	C₃×C₅×C₇×C₁₃	1	1	yes	characteristic, direct product factor
117	C₂×C₁₃	—	7	2	no	—
118	C₂×C₁₃	C₃×C₅×C₇×D₇	1	1	yes	characteristic, direct product factor
119	C₃×C₇	—	2	3	no	—
120	C₃×C₇	C₂×C₅×C₁₃×D₇	1	1	yes	characteristic, direct product factor
121	C₃×C₇	C₂²×C₅×C₇×C₁₃	1	1	yes	characteristic, complemented
122	C₂²×C₅	—	7	1	no	Hall
123	C₃×C₅	C₂×C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor, Hall
124	C₂×C₇	—	2	3	no	—
125	C₂×C₇	—	7	2	no	—
126	C₂×C₇	C₂×C₃×C₅×C₇×C₁₃	1	1	yes	characteristic, complemented
127	C₂×C₇	C₃×C₅×C₁₃×D₇	1	1	yes	characteristic, direct product factor
128	D₇	C₂×C₃×C₅×C₇×C₁₃	1	2	yes	direct product factor
129	C₁₃	C₂×C₃×C₅×C₇×D₇	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×C₃	—	7	1	no	Hall
131	C₂×C₅	—	7	2	no	—
132	C₂×C₅	C₃×C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor
133	C₇	—	2	3	no	—
134	C₇	C₂²×C₃×C₅×C₇×C₁₃	1	1	yes	characteristic, complemented
135	C₇	C₂×C₃×C₅×C₁₃×D₇	1	1	yes	characteristic, direct product factor
136	C₂×C₃	—	7	2	no	—
137	C₂×C₃	C₅×C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor
138	C₅	C₂×C₃×C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²	—	7	1	no	Sylow
140	C₃	C₂×C₅×C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor, Sylow
141	C₂	—	7	2	no	—
142	C₂	C₃×C₅×C₇×C₁₃×D₇	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×C₇×C₁₃×D₇

C₂

yes

maximal, direct product factor

C₂×C₃×C₅×C₇²×C₁₃

C₂

yes

maximal, characteristic, complemented

C₂×C₅×C₇×C₁₃×D₇

C₃

yes

maximal, characteristic, direct product factor, Hall

C₃×C₅×C₇²×C₁₃

C₂²

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₃×D₇

C₅

yes

maximal, characteristic, direct product factor, Hall

C₅×C₇×C₁₃×D₇

C₂×C₃

yes

direct product factor

C₂×C₅×C₇²×C₁₃

C₂×C₃

yes

characteristic, complemented

C₂×C₃×C₅×C₁₃×D₇

C₇

yes

maximal, characteristic, direct product factor

C₂²×C₃×C₅×C₇×C₁₃

—

maximal

C₃×C₇×C₁₃×D₇

C₂×C₅

yes

direct product factor

C₂×C₃×C₇²×C₁₃

C₂×C₅

yes

characteristic, complemented

C₅×C₇²×C₁₃

C₂²×C₃

yes

characteristic, complemented, Hall

C₂×C₃×C₅×C₇×D₇

C₁₃

yes

maximal, characteristic, direct product factor, Hall

C₃×C₅×C₁₃×D₇

C₂×C₇

yes

direct product factor

C₂×C₃×C₅×C₇×C₁₃

—

C₂×C₃×C₅×C₇×C₁₃

—

C₂×C₃×C₅×C₇×C₁₃

D₇

yes

characteristic, direct product factor

C₂×C₃×C₅×C₇×C₁₃

C₂×C₇

yes

characteristic, complemented

C₂×C₇×C₁₃×D₇

C₃×C₅

yes

characteristic, direct product factor, Hall

C₃×C₇²×C₁₃

C₂²×C₅

yes

characteristic, complemented, Hall

C₂²×C₅×C₇×C₁₃

—

C₂×C₅×C₁₃×D₇

C₃×C₇

yes

characteristic, direct product factor

C₂×C₃×C₅×C₇²

C₂×C₁₃

yes

characteristic, complemented

C₃×C₅×C₇×D₇

C₂×C₁₃

yes

direct product factor

C₃×C₅×C₇×C₁₃

—

C₃×C₅×C₇×C₁₃

C₂×D₇

yes

characteristic, direct product factor

C₃×C₅×C₇×C₁₃

C₂²×C₇

yes

characteristic, complemented

C₂×C₇²×C₁₃

C₂×C₃×C₅

yes

characteristic, complemented

C₇×C₁₃×D₇

C₂×C₃×C₅

yes

direct product factor

C₂²×C₃×C₇×C₁₃

—

C₂×C₃×C₁₃×D₇

C₅×C₇

yes

characteristic, direct product factor

C₂×C₅×C₇×D₇

C₃×C₁₃

yes

characteristic, direct product factor, Hall

C₂×C₅×C₇×C₁₃

—

C₂×C₅×C₇×C₁₃

—

C₂×C₅×C₇×C₁₃

C₃×D₇

yes

characteristic, direct product factor

C₂×C₅×C₇×C₁₃

C₂×C₃×C₇

yes

characteristic, complemented

C₅×C₁₃×D₇

C₂×C₃×C₇

yes

direct product factor

C₂²×C₃×C₅×C₁₃

—

Hall

C₃×C₅×C₇²

C₂²×C₁₃

yes

characteristic, complemented, Hall

C₇²×C₁₃

C₂²×C₃×C₅

yes

characteristic, complemented, Hall

C₂×C₃×C₇×D₇

C₅×C₁₃

yes

characteristic, direct product factor, Hall

C₂×C₃×C₇×C₁₃

—

C₂×C₃×C₇×C₁₃

—

C₂×C₃×C₇×C₁₃

C₅×D₇

yes

characteristic, direct product factor

C₂×C₃×C₇×C₁₃

C₂×C₅×C₇

yes

characteristic, complemented

C₃×C₁₃×D₇

C₂×C₅×C₇

yes

direct product factor

C₂×C₅×C₇²

C₂×C₃×C₁₃

yes

characteristic, complemented

C₅×C₇×D₇

C₂×C₃×C₁₃

yes

direct product factor

C₅×C₇×C₁₃

—

C₅×C₇×C₁₃

C₂×C₃×D₇

yes

characteristic, direct product factor

C₅×C₇×C₁₃

C₂²×C₃×C₇

yes

characteristic, complemented

C₂²×C₃×C₅×C₇

—

C₂×C₃×C₅×D₇

C₇×C₁₃

yes

characteristic, direct product factor

C₂×C₃×C₅×C₁₃

—

C₂×C₃×C₅×C₁₃

C₇×D₇

yes

characteristic, direct product factor

C₂²×C₇×C₁₃

—

C₂×C₁₃×D₇

C₃×C₅×C₇

yes

characteristic, direct product factor

C₂×C₃×C₇²

C₂×C₅×C₁₃

yes

characteristic, complemented

C₃×C₇×D₇

C₂×C₅×C₁₃

yes

direct product factor

C₃×C₇×C₁₃

—

C₃×C₇×C₁₃

C₂×C₅×D₇

yes

characteristic, direct product factor

C₃×C₇×C₁₃

C₂²×C₅×C₇

yes

characteristic, complemented

C₂²×C₅×C₁₃

—

Hall

C₅×C₇²

C₂²×C₃×C₁₃

yes

characteristic, complemented, Hall

C₂×C₃×C₅×C₇

—

C₂×C₃×C₅×C₇

—

C₂×C₃×C₅×C₇

C₁₃×D₇

yes

characteristic, direct product factor

C₂×C₃×C₅×C₇

C₂×C₇×C₁₃

yes

characteristic, complemented

C₃×C₅×D₇

C₂×C₇×C₁₃

yes

direct product factor

C₂×C₇×D₇

C₃×C₅×C₁₃

yes

characteristic, direct product factor, Hall

C₃×C₅×C₁₃

C₂×C₇×D₇

yes

characteristic, direct product factor, Hall

C₂×C₇×C₁₃

—

C₂×C₇×C₁₃

—

C₂×C₇×C₁₃

C₃×C₅×D₇

yes

characteristic, direct product factor

C₂×C₇×C₁₃

C₂×C₃×C₅×C₇

yes

characteristic, complemented

C₁₃×D₇

C₂×C₃×C₅×C₇

yes

direct product factor

C₂²×C₃×C₁₃

—

Hall

C₃×C₇²

C₂²×C₅×C₁₃

yes

characteristic, complemented, Hall

C₂²×C₅×C₇

—

C₂×C₅×D₇

C₃×C₇×C₁₃

yes

characteristic, direct product factor

C₂×C₅×C₁₃

—

C₂×C₅×C₁₃

C₃×C₇×D₇

yes

characteristic, direct product factor

C₃×C₅×C₇

—

C₃×C₅×C₇

C₂×C₁₃×D₇

yes

characteristic, direct product factor

C₃×C₅×C₇

C₂²×C₇×C₁₃

yes

characteristic, complemented

C₂×C₇²

C₂×C₃×C₅×C₁₃

yes

characteristic, complemented

C₇×D₇

C₂×C₃×C₅×C₁₃

yes

direct product factor

C₇×C₁₃

—

C₇×C₁₃

C₂×C₃×C₅×D₇

yes

characteristic, direct product factor

C₇×C₁₃

C₂²×C₃×C₅×C₇

yes

characteristic, complemented

C₂²×C₃×C₇

—

C₂×C₃×D₇

C₅×C₇×C₁₃

yes

characteristic, direct product factor

C₂×C₃×C₁₃

—

C₂×C₃×C₁₃

C₅×C₇×D₇

yes

characteristic, direct product factor

C₂×C₅×C₇

—

C₂×C₅×C₇

—

C₂×C₅×C₇

C₃×C₁₃×D₇

yes

characteristic, direct product factor

C₂×C₅×C₇

C₂×C₃×C₇×C₁₃

yes

characteristic, complemented

C₅×D₇

C₂×C₃×C₇×C₁₃

yes

direct product factor

100

C₅×C₁₃

C₂×C₃×C₇×D₇

yes

characteristic, direct product factor, Hall

101

C₂²×C₃×C₅

—

Hall

102

C₂²×C₁₃

—

Hall

103

C₇²

C₂²×C₃×C₅×C₁₃

yes

characteristic, complemented, Sylow

104

C₂×C₃×C₇

—

105

C₂×C₃×C₇

—

106

C₂×C₃×C₇

C₅×C₁₃×D₇

yes

characteristic, direct product factor

107

C₂×C₃×C₇

C₂×C₅×C₇×C₁₃

yes

characteristic, complemented

108

C₃×D₇

C₂×C₅×C₇×C₁₃

yes

direct product factor

109

C₃×C₁₃

C₂×C₅×C₇×D₇

yes

characteristic, direct product factor, Hall

110

C₅×C₇

—

111

C₅×C₇

C₂×C₃×C₁₃×D₇

yes

characteristic, direct product factor

112

C₅×C₇

C₂²×C₃×C₇×C₁₃

yes

characteristic, complemented

113

C₂×C₃×C₅

—

114

C₂×C₃×C₅

C₇×C₁₃×D₇

yes

characteristic, direct product factor

115

C₂²×C₇

—

116

C₂×D₇

C₃×C₅×C₇×C₁₃

yes

characteristic, direct product factor

117

C₂×C₁₃

—

118

C₂×C₁₃

C₃×C₅×C₇×D₇

yes

characteristic, direct product factor

119

C₃×C₇

—

120

C₃×C₇

C₂×C₅×C₁₃×D₇

yes

characteristic, direct product factor

121

C₃×C₇

C₂²×C₅×C₇×C₁₃

yes

characteristic, complemented

122

C₂²×C₅

—

Hall

123

C₃×C₅

C₂×C₇×C₁₃×D₇

yes

characteristic, direct product factor, Hall

124

C₂×C₇

—

125

C₂×C₇

—

126

C₂×C₇

C₂×C₃×C₅×C₇×C₁₃

yes

characteristic, complemented

127

C₂×C₇

C₃×C₅×C₁₃×D₇

yes

characteristic, direct product factor

128

D₇

C₂×C₃×C₅×C₇×C₁₃

yes

direct product factor

129

C₁₃

C₂×C₃×C₅×C₇×D₇

yes

characteristic, direct product factor, Sylow

130

C₂²×C₃

—

Hall

131

C₂×C₅

—

132

C₂×C₅

C₃×C₇×C₁₃×D₇

yes

characteristic, direct product factor

133

C₇

—

134

C₇

C₂²×C₃×C₅×C₇×C₁₃

yes

characteristic, complemented

135

C₇

C₂×C₃×C₅×C₁₃×D₇

yes

characteristic, direct product factor

136

C₂×C₃

—

137

C₂×C₃

C₅×C₇×C₁₃×D₇

yes

characteristic, direct product factor

138

C₅

C₂×C₃×C₇×C₁₃×D₇

yes

characteristic, direct product factor, Sylow

139

C₂²

—

Sylow

140

C₃

C₂×C₅×C₇×C₁₃×D₇

yes

characteristic, direct product factor, Sylow

141

C₂

—

142

C₂

C₃×C₅×C₇×C₁₃×D₇

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅×C₇×C₁₃	D₇
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₅×C₇×C₁₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇²×C₁₃	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₅×C₇×C₁₃

subgroup

G/H

center, hypercenter

C₂×C₃×C₅×C₇×C₁₃

D₇

commutator, cosocle, hypocenter

C₇

C₂²×C₃×C₅×C₇×C₁₃

Frattini

C₁

Fitting, socle

C₂×C₃×C₅×C₇²×C₁₃

C₂

LCS

G ▷ C₇

UCS

C₁ ◁ C₂×C₃×C₅×C₇×C₁₃

N	H ≅ G/N	type
1. C₃×C₅×C₇×C₁₃×D₇	C₂	direct (N × H)
2. C₂×C₃×C₅×C₇²×C₁₃	C₂	semidirect
3. C₂×C₅×C₇×C₁₃×D₇	C₃	direct (N × H)
4. C₃×C₅×C₇²×C₁₃	C₂²	semidirect
5. C₂×C₃×C₇×C₁₃×D₇	C₅	direct (N × H)
6. C₅×C₇×C₁₃×D₇	C₂×C₃	direct (N × H)
7. C₂×C₅×C₇²×C₁₃	C₂×C₃	semidirect
8. C₂×C₃×C₅×C₁₃×D₇	C₇	direct (N × H)
10. C₃×C₇×C₁₃×D₇	C₂×C₅	direct (N × H)
11. C₂×C₃×C₇²×C₁₃	C₂×C₅	semidirect
12. C₅×C₇²×C₁₃	C₂²×C₃	semidirect
13. C₂×C₃×C₅×C₇×D₇	C₁₃	direct (N × H)
14. C₃×C₅×C₁₃×D₇	C₂×C₇	direct (N × H)
17. C₂×C₃×C₅×C₇×C₁₃	D₇	direct (N × H)
18. C₂×C₃×C₅×C₇×C₁₃	C₂×C₇	semidirect
19. C₂×C₇×C₁₃×D₇	C₃×C₅	direct (N × H)
20. C₃×C₇²×C₁₃	C₂²×C₅	semidirect
22. C₂×C₅×C₁₃×D₇	C₃×C₇	direct (N × H)
23. C₂×C₃×C₅×C₇²	C₂×C₁₃	semidirect
24. C₃×C₅×C₇×D₇	C₂×C₁₃	direct (N × H)
26. C₃×C₅×C₇×C₁₃	C₂×D₇	direct (N × H)
27. C₃×C₅×C₇×C₁₃	C₂²×C₇	semidirect
28. C₂×C₇²×C₁₃	C₂×C₃×C₅	semidirect
29. C₇×C₁₃×D₇	C₂×C₃×C₅	direct (N × H)
31. C₂×C₃×C₁₃×D₇	C₅×C₇	direct (N × H)
32. C₂×C₅×C₇×D₇	C₃×C₁₃	direct (N × H)
35. C₂×C₅×C₇×C₁₃	C₃×D₇	direct (N × H)
36. C₂×C₅×C₇×C₁₃	C₂×C₃×C₇	semidirect
37. C₅×C₁₃×D₇	C₂×C₃×C₇	direct (N × H)
39. C₃×C₅×C₇²	C₂²×C₁₃	semidirect
40. C₇²×C₁₃	C₂²×C₃×C₅	semidirect
41. C₂×C₃×C₇×D₇	C₅×C₁₃	direct (N × H)
44. C₂×C₃×C₇×C₁₃	C₅×D₇	direct (N × H)
45. C₂×C₃×C₇×C₁₃	C₂×C₅×C₇	semidirect
46. C₃×C₁₃×D₇	C₂×C₅×C₇	direct (N × H)
47. C₂×C₅×C₇²	C₂×C₃×C₁₃	semidirect
48. C₅×C₇×D₇	C₂×C₃×C₁₃	direct (N × H)
50. C₅×C₇×C₁₃	C₂×C₃×D₇	direct (N × H)
51. C₅×C₇×C₁₃	C₂²×C₃×C₇	semidirect
53. C₂×C₃×C₅×D₇	C₇×C₁₃	direct (N × H)
55. C₂×C₃×C₅×C₁₃	C₇×D₇	direct (N × H)
57. C₂×C₁₃×D₇	C₃×C₅×C₇	direct (N × H)
58. C₂×C₃×C₇²	C₂×C₅×C₁₃	semidirect
59. C₃×C₇×D₇	C₂×C₅×C₁₃	direct (N × H)
61. C₃×C₇×C₁₃	C₂×C₅×D₇	direct (N × H)
62. C₃×C₇×C₁₃	C₂²×C₅×C₇	semidirect
64. C₅×C₇²	C₂²×C₃×C₁₃	semidirect
67. C₂×C₃×C₅×C₇	C₁₃×D₇	direct (N × H)
68. C₂×C₃×C₅×C₇	C₂×C₇×C₁₃	semidirect
69. C₃×C₅×D₇	C₂×C₇×C₁₃	direct (N × H)
70. C₂×C₇×D₇	C₃×C₅×C₁₃	direct (N × H)
71. C₃×C₅×C₁₃	C₂×C₇×D₇	direct (N × H)
74. C₂×C₇×C₁₃	C₃×C₅×D₇	direct (N × H)
75. C₂×C₇×C₁₃	C₂×C₃×C₅×C₇	semidirect
76. C₁₃×D₇	C₂×C₃×C₅×C₇	direct (N × H)
78. C₃×C₇²	C₂²×C₅×C₁₃	semidirect
80. C₂×C₅×D₇	C₃×C₇×C₁₃	direct (N × H)
82. C₂×C₅×C₁₃	C₃×C₇×D₇	direct (N × H)
84. C₃×C₅×C₇	C₂×C₁₃×D₇	direct (N × H)
85. C₃×C₅×C₇	C₂²×C₇×C₁₃	semidirect
86. C₂×C₇²	C₂×C₃×C₅×C₁₃	semidirect
87. C₇×D₇	C₂×C₃×C₅×C₁₃	direct (N × H)
89. C₇×C₁₃	C₂×C₃×C₅×D₇	direct (N × H)
90. C₇×C₁₃	C₂²×C₃×C₅×C₇	semidirect
92. C₂×C₃×D₇	C₅×C₇×C₁₃	direct (N × H)
94. C₂×C₃×C₁₃	C₅×C₇×D₇	direct (N × H)
97. C₂×C₅×C₇	C₃×C₁₃×D₇	direct (N × H)
98. C₂×C₅×C₇	C₂×C₃×C₇×C₁₃	semidirect
99. C₅×D₇	C₂×C₃×C₇×C₁₃	direct (N × H)
100. C₅×C₁₃	C₂×C₃×C₇×D₇	direct (N × H)
103. C₇²	C₂²×C₃×C₅×C₁₃	semidirect
106. C₂×C₃×C₇	C₅×C₁₃×D₇	direct (N × H)
107. C₂×C₃×C₇	C₂×C₅×C₇×C₁₃	semidirect
108. C₃×D₇	C₂×C₅×C₇×C₁₃	direct (N × H)
109. C₃×C₁₃	C₂×C₅×C₇×D₇	direct (N × H)
111. C₅×C₇	C₂×C₃×C₁₃×D₇	direct (N × H)
112. C₅×C₇	C₂²×C₃×C₇×C₁₃	semidirect
114. C₂×C₃×C₅	C₇×C₁₃×D₇	direct (N × H)
116. C₂×D₇	C₃×C₅×C₇×C₁₃	direct (N × H)
118. C₂×C₁₃	C₃×C₅×C₇×D₇	direct (N × H)
120. C₃×C₇	C₂×C₅×C₁₃×D₇	direct (N × H)
121. C₃×C₇	C₂²×C₅×C₇×C₁₃	semidirect
123. C₃×C₅	C₂×C₇×C₁₃×D₇	direct (N × H)
126. C₂×C₇	C₂×C₃×C₅×C₇×C₁₃	semidirect
127. C₂×C₇	C₃×C₅×C₁₃×D₇	direct (N × H)
128. D₇	C₂×C₃×C₅×C₇×C₁₃	direct (N × H)
129. C₁₃	C₂×C₃×C₅×C₇×D₇	direct (N × H)
132. C₂×C₅	C₃×C₇×C₁₃×D₇	direct (N × H)
134. C₇	C₂²×C₃×C₅×C₇×C₁₃	semidirect
135. C₇	C₂×C₃×C₅×C₁₃×D₇	direct (N × H)
137. C₂×C₃	C₅×C₇×C₁₃×D₇	direct (N × H)
138. C₅	C₂×C₃×C₇×C₁₃×D₇	direct (N × H)
140. C₃	C₂×C₅×C₇×C₁₃×D₇	direct (N × H)
142. C₂	C₃×C₅×C₇×C₁₃×D₇	direct (N × H)

H ≅ G/N

type

1. C₃×C₅×C₇×C₁₃×D₇

C₂

direct (N × H)

2. C₂×C₃×C₅×C₇²×C₁₃

C₂

semidirect

3. C₂×C₅×C₇×C₁₃×D₇

C₃

direct (N × H)

4. C₃×C₅×C₇²×C₁₃

C₂²

semidirect

5. C₂×C₃×C₇×C₁₃×D₇

C₅

direct (N × H)

6. C₅×C₇×C₁₃×D₇

C₂×C₃

direct (N × H)

7. C₂×C₅×C₇²×C₁₃

C₂×C₃

semidirect

8. C₂×C₃×C₅×C₁₃×D₇

C₇

direct (N × H)

10. C₃×C₇×C₁₃×D₇

C₂×C₅

direct (N × H)

11. C₂×C₃×C₇²×C₁₃

C₂×C₅

semidirect

12. C₅×C₇²×C₁₃

C₂²×C₃

semidirect

13. C₂×C₃×C₅×C₇×D₇

C₁₃

direct (N × H)

14. C₃×C₅×C₁₃×D₇

C₂×C₇

direct (N × H)

17. C₂×C₃×C₅×C₇×C₁₃

D₇

direct (N × H)

18. C₂×C₃×C₅×C₇×C₁₃

C₂×C₇

semidirect

19. C₂×C₇×C₁₃×D₇

C₃×C₅

direct (N × H)

20. C₃×C₇²×C₁₃

C₂²×C₅

semidirect

22. C₂×C₅×C₁₃×D₇

C₃×C₇

direct (N × H)

23. C₂×C₃×C₅×C₇²

C₂×C₁₃

semidirect

24. C₃×C₅×C₇×D₇

C₂×C₁₃

direct (N × H)

26. C₃×C₅×C₇×C₁₃

C₂×D₇

direct (N × H)

27. C₃×C₅×C₇×C₁₃

C₂²×C₇

semidirect

28. C₂×C₇²×C₁₃

C₂×C₃×C₅

semidirect

29. C₇×C₁₃×D₇

C₂×C₃×C₅

direct (N × H)

31. C₂×C₃×C₁₃×D₇

C₅×C₇

direct (N × H)

32. C₂×C₅×C₇×D₇

C₃×C₁₃

direct (N × H)

35. C₂×C₅×C₇×C₁₃

C₃×D₇

direct (N × H)

36. C₂×C₅×C₇×C₁₃

C₂×C₃×C₇

semidirect

37. C₅×C₁₃×D₇

C₂×C₃×C₇

direct (N × H)

39. C₃×C₅×C₇²

C₂²×C₁₃

semidirect

40. C₇²×C₁₃

C₂²×C₃×C₅

semidirect

41. C₂×C₃×C₇×D₇

C₅×C₁₃

direct (N × H)

44. C₂×C₃×C₇×C₁₃

C₅×D₇

direct (N × H)

45. C₂×C₃×C₇×C₁₃

C₂×C₅×C₇

semidirect

46. C₃×C₁₃×D₇

C₂×C₅×C₇

direct (N × H)

47. C₂×C₅×C₇²

C₂×C₃×C₁₃

semidirect

48. C₅×C₇×D₇

C₂×C₃×C₁₃

direct (N × H)

50. C₅×C₇×C₁₃

C₂×C₃×D₇

direct (N × H)

51. C₅×C₇×C₁₃

C₂²×C₃×C₇

semidirect

53. C₂×C₃×C₅×D₇

C₇×C₁₃

direct (N × H)

55. C₂×C₃×C₅×C₁₃

C₇×D₇

direct (N × H)

57. C₂×C₁₃×D₇

C₃×C₅×C₇

direct (N × H)

58. C₂×C₃×C₇²

C₂×C₅×C₁₃

semidirect

59. C₃×C₇×D₇

C₂×C₅×C₁₃

direct (N × H)

61. C₃×C₇×C₁₃

C₂×C₅×D₇

direct (N × H)

62. C₃×C₇×C₁₃

C₂²×C₅×C₇

semidirect

64. C₅×C₇²

C₂²×C₃×C₁₃

semidirect

67. C₂×C₃×C₅×C₇

C₁₃×D₇

direct (N × H)

68. C₂×C₃×C₅×C₇

C₂×C₇×C₁₃

semidirect

69. C₃×C₅×D₇

C₂×C₇×C₁₃

direct (N × H)

70. C₂×C₇×D₇

C₃×C₅×C₁₃

direct (N × H)

71. C₃×C₅×C₁₃

C₂×C₇×D₇

direct (N × H)

74. C₂×C₇×C₁₃

C₃×C₅×D₇

direct (N × H)

75. C₂×C₇×C₁₃

C₂×C₃×C₅×C₇

semidirect

76. C₁₃×D₇

C₂×C₃×C₅×C₇

direct (N × H)

78. C₃×C₇²

C₂²×C₅×C₁₃

semidirect

80. C₂×C₅×D₇

C₃×C₇×C₁₃

direct (N × H)

82. C₂×C₅×C₁₃

C₃×C₇×D₇

direct (N × H)

84. C₃×C₅×C₇

C₂×C₁₃×D₇

direct (N × H)

85. C₃×C₅×C₇

C₂²×C₇×C₁₃

semidirect

86. C₂×C₇²

C₂×C₃×C₅×C₁₃

semidirect

87. C₇×D₇

C₂×C₃×C₅×C₁₃

direct (N × H)

89. C₇×C₁₃

C₂×C₃×C₅×D₇

direct (N × H)

90. C₇×C₁₃

C₂²×C₃×C₅×C₇

semidirect

92. C₂×C₃×D₇

C₅×C₇×C₁₃

direct (N × H)

94. C₂×C₃×C₁₃

C₅×C₇×D₇

direct (N × H)

97. C₂×C₅×C₇

C₃×C₁₃×D₇

direct (N × H)

98. C₂×C₅×C₇

C₂×C₃×C₇×C₁₃

semidirect

99. C₅×D₇

C₂×C₃×C₇×C₁₃

direct (N × H)

100. C₅×C₁₃

C₂×C₃×C₇×D₇

direct (N × H)

103. C₇²

C₂²×C₃×C₅×C₁₃

semidirect

106. C₂×C₃×C₇

C₅×C₁₃×D₇

direct (N × H)

107. C₂×C₃×C₇

C₂×C₅×C₇×C₁₃

semidirect

108. C₃×D₇

C₂×C₅×C₇×C₁₃

direct (N × H)

109. C₃×C₁₃

C₂×C₅×C₇×D₇

direct (N × H)

111. C₅×C₇

C₂×C₃×C₁₃×D₇

direct (N × H)

112. C₅×C₇

C₂²×C₃×C₇×C₁₃

semidirect

114. C₂×C₃×C₅

C₇×C₁₃×D₇

direct (N × H)

116. C₂×D₇

C₃×C₅×C₇×C₁₃

direct (N × H)

118. C₂×C₁₃

C₃×C₅×C₇×D₇

direct (N × H)

120. C₃×C₇

C₂×C₅×C₁₃×D₇

direct (N × H)

121. C₃×C₇

C₂²×C₅×C₇×C₁₃

semidirect

123. C₃×C₅

C₂×C₇×C₁₃×D₇

direct (N × H)

126. C₂×C₇

C₂×C₃×C₅×C₇×C₁₃

semidirect

127. C₂×C₇

C₃×C₅×C₁₃×D₇

direct (N × H)

128. D₇

C₂×C₃×C₅×C₇×C₁₃

direct (N × H)

129. C₁₃

C₂×C₃×C₅×C₇×D₇

direct (N × H)

132. C₂×C₅

C₃×C₇×C₁₃×D₇

direct (N × H)

134. C₇

C₂²×C₃×C₅×C₇×C₁₃

semidirect

135. C₇

C₂×C₃×C₅×C₁₃×D₇

direct (N × H)

137. C₂×C₃

C₅×C₇×C₁₃×D₇

direct (N × H)

138. C₅

C₂×C₃×C₇×C₁₃×D₇

direct (N × H)

140. C₃

C₂×C₅×C₇×C₁₃×D₇

direct (N × H)

142. C₂

C₃×C₅×C₇×C₁₃×D₇

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×C₇×C₁₃×D₇	2
C₂×C₃×C₅×C₇²×C₁₃	2
C₂×C₅×C₇×C₁₃×D₇	3
C₂×C₃×C₇×C₁₃×D₇	5
C₂×C₃×C₅×C₁₃×D₇	7
C₂²×C₃×C₅×C₇×C₁₃	7
C₂×C₃×C₅×C₇×D₇	13

subgroup

index

C₃×C₅×C₇×C₁₃×D₇

C₂×C₃×C₅×C₇²×C₁₃

C₂×C₅×C₇×C₁₃×D₇

C₂×C₃×C₇×C₁₃×D₇

C₂×C₃×C₅×C₁₃×D₇

C₂²×C₃×C₅×C₇×C₁₃

C₂×C₃×C₅×C₇×D₇

Subgroups — C₂×C₃×C₅×C₇×C₁₃×D₇ #50304511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C3×C5×C7×C13×D7 #50304511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×C₇×C₁₃×D₇ #50304511