Group #44473016 — subgroups

Subgroups — C₂×C₃×C₇×C₁₇×D₂₁ #44473016

Order 29988.

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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×C₇×C₁₇×D₂₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₇²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×D₇×C₁₇	—	3	1	no	maximal
4	C₂×C₇×C₁₇×D₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×C₁₇×D₂₁	C₂×C₃	1	2	yes	direct product factor
7	C₃×C₇×D₇×C₁₇	—	3	2	no	—
8	C₂×C₃×C₇²×C₁₇	—	2	1	no	—
9	C₂×C₃×C₇²×C₁₇	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃×C₇×C₁₇	—	7	1	no	maximal
12	C₂×C₃×C₁₇×D₂₁	C₇	1	1	yes	maximal, characteristic, direct product factor
13	C₂×C₇×D₇×C₁₇	—	3	1	no	Hall
14	C₃×C₇²×C₁₇	—	2	1	no	—
15	C₃×C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented
16	C₃×C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented
17	C₃×C₁₇×D₂₁	C₂×C₇	1	2	yes	direct product factor
18	C₃×D₃×C₇×C₁₇	—	7	2	no	—
19	C₂×C₃²×C₇×C₁₇	—	2	3	no	—
20	C₂×C₃²×C₇×C₁₇	D₇	1	1	yes	characteristic, complemented
21	C₂×C₃²×C₇×C₁₇	C₂×C₇	1	1	yes	characteristic, complemented
22	C₂×C₃×C₇×D₂₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₇²×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented
24	C₇×D₇×C₁₇	—	3	2	no	—
25	C₂²×C₃×C₇×C₁₇	—	21	1	no	—
26	C₂×C₁₇×D₂₁	C₃×C₇	1	1	yes	characteristic, direct product factor
27	C₂×D₃×C₇×C₁₇	—	7	1	no	—
28	C₂×C₃×D₇×C₁₇	—	3	1	no	—
29	C₃²×C₇×C₁₇	—	2	3	no	—
30	C₃²×C₇×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented
31	C₃²×C₇×C₁₇	C₂²×C₇	1	1	yes	characteristic, complemented
32	C₂×C₃²×C₇²	C₂×C₁₇	1	1	yes	characteristic, complemented
33	C₃×C₇×D₂₁	C₂×C₁₇	1	2	yes	direct product factor
34	C₇²×C₁₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₇×C₁₇	—	2	6	no	—
36	C₂×C₃×C₇×C₁₇	—	2	1	no	—
37	C₂×C₃×C₇×C₁₇	—	2	1	no	—
38	C₂×C₃×C₇×C₁₇	—	2	3	no	—
39	C₂×C₃×C₇×C₁₇	—	2	3	no	—
40	C₂×C₃×C₇×C₁₇	—	21	2	no	—
41	C₂×C₃×C₇×C₁₇	D₃×C₇	1	1	yes	characteristic, complemented
42	C₂×C₃×C₇×C₁₇	C₃×D₇	1	1	yes	characteristic, complemented
43	C₂×C₃×C₇×C₁₇	D₂₁	1	1	yes	characteristic, direct product factor
44	C₂×C₃×C₇×C₁₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
45	C₁₇×D₂₁	C₂×C₃×C₇	1	2	yes	direct product factor
46	D₃×C₇×C₁₇	—	7	2	no	—
47	C₃×D₇×C₁₇	—	3	2	no	—
48	C₂×C₃×D₃×C₁₇	—	7	1	no	Hall
49	C₂×C₇×D₂₁	C₃×C₁₇	1	1	yes	characteristic, direct product factor
50	C₂×C₃×C₇×D₇	—	3	1	no	—
51	C₂²×C₇×C₁₇	—	21	1	no	—
52	C₂×D₇×C₁₇	—	3	1	no	—
53	C₃²×C₇²	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
54	C₃×C₇×C₁₇	—	2	6	no	—
55	C₃×C₇×C₁₇	—	2	1	no	—
56	C₃×C₇×C₁₇	—	2	1	no	—
57	C₃×C₇×C₁₇	—	2	3	no	—
58	C₃×C₇×C₁₇	—	2	3	no	—
59	C₃×C₇×C₁₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
60	C₃×C₇×C₁₇	C₂×D₃×C₇	1	1	yes	characteristic, complemented
61	C₃×C₇×C₁₇	C₂×D₂₁	1	1	yes	characteristic, direct product factor
62	C₃×C₇×C₁₇	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
63	C₂×C₃²×C₁₇	C₇×D₇	1	1	yes	characteristic, complemented
64	C₃×D₃×C₁₇	—	7	2	no	—
65	C₂×C₃×C₇²	—	2	1	no	—
66	C₂×C₃×C₇²	D₃×C₁₇	1	1	yes	characteristic, complemented
67	C₂×C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
68	C₇×D₂₁	C₂×C₃×C₁₇	1	2	yes	direct product factor
69	C₃×C₇×D₇	—	3	2	no	—
70	C₂×C₃×D₂₁	C₇×C₁₇	1	1	yes	characteristic, direct product factor
71	C₂×C₃×D₃×C₇	—	7	1	no	—
72	C₂×C₇×C₁₇	—	2	3	no	—
73	C₂×C₇×C₁₇	—	21	2	no	—
74	C₂×C₇×C₁₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented
75	C₂×C₇×C₁₇	C₃×D₂₁	1	1	yes	characteristic, direct product factor
76	D₇×C₁₇	—	3	2	no	—
77	C₂²×C₃×C₁₇	—	21	1	no	—
78	C₂×D₃×C₁₇	—	7	1	no	—
79	C₂×C₇×D₇	—	3	1	no	Hall
80	C₃²×C₁₇	C₂×C₇×D₇	1	1	yes	characteristic, complemented, Hall
81	C₃×C₇²	—	2	1	no	—
82	C₃×C₇²	C₂×D₃×C₁₇	1	1	yes	characteristic, complemented
83	C₃×C₇²	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented
84	C₂×C₃²×C₇	—	2	3	no	—
85	C₂×C₃²×C₇	D₇×C₁₇	1	1	yes	characteristic, complemented
86	C₂×C₃²×C₇	C₂×C₇×C₁₇	1	1	yes	characteristic, complemented
87	C₃×D₂₁	C₂×C₇×C₁₇	1	2	yes	direct product factor
88	C₃×D₃×C₇	—	7	2	no	—
89	C₇×C₁₇	—	2	3	no	—
90	C₇×C₁₇	C₂×C₃×D₃×C₇	1	1	yes	characteristic, complemented
91	C₇×C₁₇	C₂×C₃×D₂₁	1	1	yes	characteristic, direct product factor
92	C₂×C₃×C₁₇	—	2	1	no	—
93	C₂×C₃×C₁₇	—	21	2	no	—
94	C₂×C₃×C₁₇	C₃×C₇×D₇	1	1	yes	characteristic, complemented
95	C₂×C₃×C₁₇	C₇×D₂₁	1	1	yes	characteristic, direct product factor
96	D₃×C₁₇	—	7	2	no	—
97	C₂×C₇²	C₃×D₃×C₁₇	1	1	yes	characteristic, complemented
98	C₇×D₇	—	3	2	no	—
99	C₂²×C₃×C₇	—	21	1	no	—
100	C₂×D₂₁	C₃×C₇×C₁₇	1	1	yes	characteristic, direct product factor
101	C₂×D₃×C₇	—	7	1	no	—
102	C₂×C₃×D₇	—	3	1	no	—
103	C₂²×C₁₇	—	21	1	no	Hall
104	C₃²×C₇	—	2	3	no	—
105	C₃²×C₇	C₂×D₇×C₁₇	1	1	yes	characteristic, complemented
106	C₃²×C₇	C₂²×C₇×C₁₇	1	1	yes	characteristic, complemented
107	C₃×C₁₇	—	2	1	no	—
108	C₃×C₁₇	C₂×C₃×C₇×D₇	1	1	yes	characteristic, complemented
109	C₃×C₁₇	C₂×C₇×D₂₁	1	1	yes	characteristic, direct product factor
110	C₇²	C₂×C₃×D₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₃×C₇	—	2	6	no	—
112	C₂×C₃×C₇	—	2	1	no	—
113	C₂×C₃×C₇	—	2	1	no	—
114	C₂×C₃×C₇	—	2	3	no	—
115	C₂×C₃×C₇	—	2	3	no	—
116	C₂×C₃×C₇	—	21	2	no	—
117	C₂×C₃×C₇	C₃×D₇×C₁₇	1	1	yes	characteristic, complemented
118	C₂×C₃×C₇	D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
119	C₂×C₃×C₇	C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
120	C₂×C₃×C₇	C₂×C₃×C₇×C₁₇	1	1	yes	characteristic, complemented
121	D₂₁	C₂×C₃×C₇×C₁₇	1	2	yes	direct product factor
122	C₃×D₇	—	3	2	no	—
123	D₃×C₇	—	7	2	no	—
124	C₂×C₃×D₃	—	7	1	no	Hall
125	C₂×C₁₇	—	21	2	no	—
126	C₂×C₁₇	C₃×C₇×D₂₁	1	1	yes	characteristic, direct product factor
127	C₂²×C₇	—	21	1	no	—
128	C₂×D₇	—	3	1	no	—
129	C₃×C₇	—	2	6	no	—
130	C₃×C₇	—	2	1	no	—
131	C₃×C₇	—	2	1	no	—
132	C₃×C₇	—	2	3	no	—
133	C₃×C₇	—	2	3	no	—
134	C₃×C₇	C₂×C₃×D₇×C₁₇	1	1	yes	characteristic, complemented
135	C₃×C₇	C₂×D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
136	C₃×C₇	C₂×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
137	C₃×C₇	C₂²×C₃×C₇×C₁₇	1	1	yes	characteristic, complemented
138	C₂×C₃²	C₇×D₇×C₁₇	1	1	yes	characteristic, complemented
139	C₃×D₃	—	7	2	no	—
140	C₁₇	C₂×C₃×C₇×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₇	—	2	3	no	—
142	C₂×C₇	—	21	2	no	—
143	C₂×C₇	C₃×D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
144	C₂×C₇	C₃×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
145	D₇	—	3	2	no	—
146	C₂²×C₃	—	21	1	no	—
147	C₂×D₃	—	7	1	no	—
148	C₃²	C₂×C₇×D₇×C₁₇	1	1	yes	characteristic, complemented, Sylow
149	C₇	—	2	3	no	—
150	C₇	C₂×C₃×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
151	C₇	C₂×C₃×D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
152	C₂×C₃	—	2	1	no	—
153	C₂×C₃	—	21	2	no	—
154	C₂×C₃	C₃×C₇×D₇×C₁₇	1	1	yes	characteristic, complemented
155	C₂×C₃	C₇×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
156	D₃	—	7	2	no	—
157	C₂²	—	21	1	no	Sylow
158	C₃	—	2	1	no	—
159	C₃	C₂×C₇×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
160	C₃	C₂×C₃×C₇×D₇×C₁₇	1	1	yes	characteristic, complemented
161	C₂	—	21	2	no	—
162	C₂	C₃×C₇×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₇×C₁₇×D₂₁

C₂

yes

maximal, direct product factor

C₂×C₃²×C₇²×C₁₇

C₂

yes

maximal, characteristic, complemented

C₂×C₃×C₇×D₇×C₁₇

—

maximal

C₂×C₇×C₁₇×D₂₁

C₃

yes

maximal, characteristic, direct product factor

C₃²×C₇²×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₇×C₁₇×D₂₁

C₂×C₃

yes

direct product factor

C₃×C₇×D₇×C₁₇

—

C₂×C₃×C₇²×C₁₇

—

C₂×C₃×C₇²×C₁₇

D₃

yes

characteristic, complemented

C₂×C₃×C₇²×C₁₇

C₂×C₃

yes

characteristic, complemented

C₂×C₃×D₃×C₇×C₁₇

—

maximal

C₂×C₃×C₁₇×D₂₁

C₇

yes

maximal, characteristic, direct product factor

C₂×C₇×D₇×C₁₇

—

Hall

C₃×C₇²×C₁₇

—

C₃×C₇²×C₁₇

C₂×D₃

yes

characteristic, complemented

C₃×C₇²×C₁₇

C₂²×C₃

yes

characteristic, complemented

C₃×C₁₇×D₂₁

C₂×C₇

yes

direct product factor

C₃×D₃×C₇×C₁₇

—

C₂×C₃²×C₇×C₁₇

—

C₂×C₃²×C₇×C₁₇

D₇

yes

characteristic, complemented

C₂×C₃²×C₇×C₁₇

C₂×C₇

yes

characteristic, complemented

C₂×C₃×C₇×D₂₁

C₁₇

yes

maximal, characteristic, direct product factor, Hall

C₂×C₇²×C₁₇

C₃×D₃

yes

characteristic, complemented

C₇×D₇×C₁₇

—

C₂²×C₃×C₇×C₁₇

—

C₂×C₁₇×D₂₁

C₃×C₇

yes

characteristic, direct product factor

C₂×D₃×C₇×C₁₇

—

C₂×C₃×D₇×C₁₇

—

C₃²×C₇×C₁₇

—

C₃²×C₇×C₁₇

C₂×D₇

yes

characteristic, complemented

C₃²×C₇×C₁₇

C₂²×C₇

yes

characteristic, complemented

C₂×C₃²×C₇²

C₂×C₁₇

yes

characteristic, complemented

C₃×C₇×D₂₁

C₂×C₁₇

yes

direct product factor

C₇²×C₁₇

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

D₃×C₇

yes

characteristic, complemented

C₂×C₃×C₇×C₁₇

C₃×D₇

yes

characteristic, complemented

C₂×C₃×C₇×C₁₇

D₂₁

yes

characteristic, direct product factor

C₂×C₃×C₇×C₁₇

C₂×C₃×C₇

yes

characteristic, complemented

C₁₇×D₂₁

C₂×C₃×C₇

yes

direct product factor

D₃×C₇×C₁₇

—

C₃×D₇×C₁₇

—

C₂×C₃×D₃×C₁₇

—

Hall

C₂×C₇×D₂₁

C₃×C₁₇

yes

characteristic, direct product factor

C₂×C₃×C₇×D₇

—

C₂²×C₇×C₁₇

—

C₂×D₇×C₁₇

—

C₃²×C₇²

C₂²×C₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

C₂×C₃×D₇

yes

characteristic, complemented

C₃×C₇×C₁₇

C₂×D₃×C₇

yes

characteristic, complemented

C₃×C₇×C₁₇

C₂×D₂₁

yes

characteristic, direct product factor

C₃×C₇×C₁₇

C₂²×C₃×C₇

yes

characteristic, complemented

C₂×C₃²×C₁₇

C₇×D₇

yes

characteristic, complemented

C₃×D₃×C₁₇

—

C₂×C₃×C₇²

—

C₂×C₃×C₇²

D₃×C₁₇

yes

characteristic, complemented

C₂×C₃×C₇²

C₂×C₃×C₁₇

yes

characteristic, complemented

C₇×D₂₁

C₂×C₃×C₁₇

yes

direct product factor

C₃×C₇×D₇

—

C₂×C₃×D₂₁

C₇×C₁₇

yes

characteristic, direct product factor

C₂×C₃×D₃×C₇

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

C₃×D₃×C₇

yes

characteristic, complemented

C₂×C₇×C₁₇

C₃×D₂₁

yes

characteristic, direct product factor

D₇×C₁₇

—

C₂²×C₃×C₁₇

—

C₂×D₃×C₁₇

—

C₂×C₇×D₇

—

Hall

C₃²×C₁₇

C₂×C₇×D₇

yes

characteristic, complemented, Hall

C₃×C₇²

—

C₃×C₇²

C₂×D₃×C₁₇

yes

characteristic, complemented

C₃×C₇²

C₂²×C₃×C₁₇

yes

characteristic, complemented

C₂×C₃²×C₇

—

C₂×C₃²×C₇

D₇×C₁₇

yes

characteristic, complemented

C₂×C₃²×C₇

C₂×C₇×C₁₇

yes

characteristic, complemented

C₃×D₂₁

C₂×C₇×C₁₇

yes

direct product factor

C₃×D₃×C₇

—

C₇×C₁₇

—

C₇×C₁₇

C₂×C₃×D₃×C₇

yes

characteristic, complemented

C₇×C₁₇

C₂×C₃×D₂₁

yes

characteristic, direct product factor

C₂×C₃×C₁₇

—

C₂×C₃×C₁₇

—

C₂×C₃×C₁₇

C₃×C₇×D₇

yes

characteristic, complemented

C₂×C₃×C₁₇

C₇×D₂₁

yes

characteristic, direct product factor

D₃×C₁₇

—

C₂×C₇²

C₃×D₃×C₁₇

yes

characteristic, complemented

C₇×D₇

—

C₂²×C₃×C₇

—

100

C₂×D₂₁

C₃×C₇×C₁₇

yes

characteristic, direct product factor

101

C₂×D₃×C₇

—

102

C₂×C₃×D₇

—

103

C₂²×C₁₇

—

Hall

104

C₃²×C₇

—

105

C₃²×C₇

C₂×D₇×C₁₇

yes

characteristic, complemented

106

C₃²×C₇

C₂²×C₇×C₁₇

yes

characteristic, complemented

107

C₃×C₁₇

—

108

C₃×C₁₇

C₂×C₃×C₇×D₇

yes

characteristic, complemented

109

C₃×C₁₇

C₂×C₇×D₂₁

yes

characteristic, direct product factor

110

C₇²

C₂×C₃×D₃×C₁₇

yes

characteristic, complemented, Sylow

111

C₂×C₃×C₇

—

112

C₂×C₃×C₇

—

113

C₂×C₃×C₇

—

114

C₂×C₃×C₇

—

115

C₂×C₃×C₇

—

116

C₂×C₃×C₇

—

117

C₂×C₃×C₇

C₃×D₇×C₁₇

yes

characteristic, complemented

118

C₂×C₃×C₇

D₃×C₇×C₁₇

yes

characteristic, complemented

119

C₂×C₃×C₇

C₁₇×D₂₁

yes

characteristic, direct product factor

120

C₂×C₃×C₇

C₂×C₃×C₇×C₁₇

yes

characteristic, complemented

121

D₂₁

C₂×C₃×C₇×C₁₇

yes

direct product factor

122

C₃×D₇

—

123

D₃×C₇

—

124

C₂×C₃×D₃

—

Hall

125

C₂×C₁₇

—

126

C₂×C₁₇

C₃×C₇×D₂₁

yes

characteristic, direct product factor

127

C₂²×C₇

—

128

C₂×D₇

—

129

C₃×C₇

—

130

C₃×C₇

—

131

C₃×C₇

—

132

C₃×C₇

—

133

C₃×C₇

—

134

C₃×C₇

C₂×C₃×D₇×C₁₇

yes

characteristic, complemented

135

C₃×C₇

C₂×D₃×C₇×C₁₇

yes

characteristic, complemented

136

C₃×C₇

C₂×C₁₇×D₂₁

yes

characteristic, direct product factor

137

C₃×C₇

C₂²×C₃×C₇×C₁₇

yes

characteristic, complemented

138

C₂×C₃²

C₇×D₇×C₁₇

yes

characteristic, complemented

139

C₃×D₃

—

140

C₁₇

C₂×C₃×C₇×D₂₁

yes

characteristic, direct product factor, Sylow

141

C₂×C₇

—

142

C₂×C₇

—

143

C₂×C₇

C₃×D₃×C₇×C₁₇

yes

characteristic, complemented

144

C₂×C₇

C₃×C₁₇×D₂₁

yes

characteristic, direct product factor

145

D₇

—

146

C₂²×C₃

—

147

C₂×D₃

—

148

C₃²

C₂×C₇×D₇×C₁₇

yes

characteristic, complemented, Sylow

149

C₇

—

150

C₇

C₂×C₃×C₁₇×D₂₁

yes

characteristic, direct product factor

151

C₇

C₂×C₃×D₃×C₇×C₁₇

yes

characteristic, complemented

152

C₂×C₃

—

153

C₂×C₃

—

154

C₂×C₃

C₃×C₇×D₇×C₁₇

yes

characteristic, complemented

155

C₂×C₃

C₇×C₁₇×D₂₁

yes

characteristic, direct product factor

156

D₃

—

157

C₂²

—

Sylow

158

C₃

—

159

C₃

C₂×C₇×C₁₇×D₂₁

yes

characteristic, direct product factor

160

C₃

C₂×C₃×C₇×D₇×C₁₇

yes

characteristic, complemented

161

C₂

—

162

C₂

C₃×C₇×C₁₇×D₂₁

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇×C₁₇	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂²×C₃×C₇×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇²×C₁₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃×C₇×C₁₇

subgroup

G/H

center, hypercenter

C₂×C₃×C₇×C₁₇

D₂₁

commutator, cosocle, hypocenter

C₃×C₇

C₂²×C₃×C₇×C₁₇

Frattini

C₁

Fitting, socle

C₂×C₃²×C₇²×C₁₇

C₂

LCS

G ▷ C₃×C₇

UCS

C₁ ◁ C₂×C₃×C₇×C₁₇

N	H ≅ G/N	type
1. C₃×C₇×C₁₇×D₂₁	C₂	direct (N × H)
2. C₂×C₃²×C₇²×C₁₇	C₂	semidirect
4. C₂×C₇×C₁₇×D₂₁	C₃	direct (N × H)
5. C₃²×C₇²×C₁₇	C₂²	semidirect
6. C₇×C₁₇×D₂₁	C₂×C₃	direct (N × H)
9. C₂×C₃×C₇²×C₁₇	D₃	semidirect
10. C₂×C₃×C₇²×C₁₇	C₂×C₃	semidirect
12. C₂×C₃×C₁₇×D₂₁	C₇	direct (N × H)
15. C₃×C₇²×C₁₇	C₂×D₃	semidirect
16. C₃×C₇²×C₁₇	C₂²×C₃	semidirect
17. C₃×C₁₇×D₂₁	C₂×C₇	direct (N × H)
20. C₂×C₃²×C₇×C₁₇	D₇	semidirect
21. C₂×C₃²×C₇×C₁₇	C₂×C₇	semidirect
22. C₂×C₃×C₇×D₂₁	C₁₇	direct (N × H)
23. C₂×C₇²×C₁₇	C₃×D₃	semidirect
26. C₂×C₁₇×D₂₁	C₃×C₇	direct (N × H)
30. C₃²×C₇×C₁₇	C₂×D₇	semidirect
31. C₃²×C₇×C₁₇	C₂²×C₇	semidirect
32. C₂×C₃²×C₇²	C₂×C₁₇	semidirect
33. C₃×C₇×D₂₁	C₂×C₁₇	direct (N × H)
34. C₇²×C₁₇	C₂×C₃×D₃	semidirect
41. C₂×C₃×C₇×C₁₇	D₃×C₇	semidirect
42. C₂×C₃×C₇×C₁₇	C₃×D₇	semidirect
43. C₂×C₃×C₇×C₁₇	D₂₁	direct (N × H)
44. C₂×C₃×C₇×C₁₇	C₂×C₃×C₇	semidirect
45. C₁₇×D₂₁	C₂×C₃×C₇	direct (N × H)
49. C₂×C₇×D₂₁	C₃×C₁₇	direct (N × H)
53. C₃²×C₇²	C₂²×C₁₇	semidirect
59. C₃×C₇×C₁₇	C₂×C₃×D₇	semidirect
60. C₃×C₇×C₁₇	C₂×D₃×C₇	semidirect
61. C₃×C₇×C₁₇	C₂×D₂₁	direct (N × H)
62. C₃×C₇×C₁₇	C₂²×C₃×C₇	semidirect
63. C₂×C₃²×C₁₇	C₇×D₇	semidirect
66. C₂×C₃×C₇²	D₃×C₁₇	semidirect
67. C₂×C₃×C₇²	C₂×C₃×C₁₇	semidirect
68. C₇×D₂₁	C₂×C₃×C₁₇	direct (N × H)
70. C₂×C₃×D₂₁	C₇×C₁₇	direct (N × H)
74. C₂×C₇×C₁₇	C₃×D₃×C₇	semidirect
75. C₂×C₇×C₁₇	C₃×D₂₁	direct (N × H)
80. C₃²×C₁₇	C₂×C₇×D₇	semidirect
82. C₃×C₇²	C₂×D₃×C₁₇	semidirect
83. C₃×C₇²	C₂²×C₃×C₁₇	semidirect
85. C₂×C₃²×C₇	D₇×C₁₇	semidirect
86. C₂×C₃²×C₇	C₂×C₇×C₁₇	semidirect
87. C₃×D₂₁	C₂×C₇×C₁₇	direct (N × H)
90. C₇×C₁₇	C₂×C₃×D₃×C₇	semidirect
91. C₇×C₁₇	C₂×C₃×D₂₁	direct (N × H)
94. C₂×C₃×C₁₇	C₃×C₇×D₇	semidirect
95. C₂×C₃×C₁₇	C₇×D₂₁	direct (N × H)
97. C₂×C₇²	C₃×D₃×C₁₇	semidirect
100. C₂×D₂₁	C₃×C₇×C₁₇	direct (N × H)
105. C₃²×C₇	C₂×D₇×C₁₇	semidirect
106. C₃²×C₇	C₂²×C₇×C₁₇	semidirect
108. C₃×C₁₇	C₂×C₃×C₇×D₇	semidirect
109. C₃×C₁₇	C₂×C₇×D₂₁	direct (N × H)
110. C₇²	C₂×C₃×D₃×C₁₇	semidirect
117. C₂×C₃×C₇	C₃×D₇×C₁₇	semidirect
118. C₂×C₃×C₇	D₃×C₇×C₁₇	semidirect
119. C₂×C₃×C₇	C₁₇×D₂₁	direct (N × H)
120. C₂×C₃×C₇	C₂×C₃×C₇×C₁₇	semidirect
121. D₂₁	C₂×C₃×C₇×C₁₇	direct (N × H)
126. C₂×C₁₇	C₃×C₇×D₂₁	direct (N × H)
134. C₃×C₇	C₂×C₃×D₇×C₁₇	semidirect
135. C₃×C₇	C₂×D₃×C₇×C₁₇	semidirect
136. C₃×C₇	C₂×C₁₇×D₂₁	direct (N × H)
137. C₃×C₇	C₂²×C₃×C₇×C₁₇	semidirect
138. C₂×C₃²	C₇×D₇×C₁₇	semidirect
140. C₁₇	C₂×C₃×C₇×D₂₁	direct (N × H)
143. C₂×C₇	C₃×D₃×C₇×C₁₇	semidirect
144. C₂×C₇	C₃×C₁₇×D₂₁	direct (N × H)
148. C₃²	C₂×C₇×D₇×C₁₇	semidirect
150. C₇	C₂×C₃×C₁₇×D₂₁	direct (N × H)
151. C₇	C₂×C₃×D₃×C₇×C₁₇	semidirect
154. C₂×C₃	C₃×C₇×D₇×C₁₇	semidirect
155. C₂×C₃	C₇×C₁₇×D₂₁	direct (N × H)
159. C₃	C₂×C₇×C₁₇×D₂₁	direct (N × H)
160. C₃	C₂×C₃×C₇×D₇×C₁₇	semidirect
162. C₂	C₃×C₇×C₁₇×D₂₁	direct (N × H)

H ≅ G/N

type

1. C₃×C₇×C₁₇×D₂₁

C₂

direct (N × H)

2. C₂×C₃²×C₇²×C₁₇

C₂

semidirect

4. C₂×C₇×C₁₇×D₂₁

C₃

direct (N × H)

5. C₃²×C₇²×C₁₇

C₂²

semidirect

6. C₇×C₁₇×D₂₁

C₂×C₃

direct (N × H)

9. C₂×C₃×C₇²×C₁₇

D₃

semidirect

10. C₂×C₃×C₇²×C₁₇

C₂×C₃

semidirect

12. C₂×C₃×C₁₇×D₂₁

C₇

direct (N × H)

15. C₃×C₇²×C₁₇

C₂×D₃

semidirect

16. C₃×C₇²×C₁₇

C₂²×C₃

semidirect

17. C₃×C₁₇×D₂₁

C₂×C₇

direct (N × H)

20. C₂×C₃²×C₇×C₁₇

D₇

semidirect

21. C₂×C₃²×C₇×C₁₇

C₂×C₇

semidirect

22. C₂×C₃×C₇×D₂₁

C₁₇

direct (N × H)

23. C₂×C₇²×C₁₇

C₃×D₃

semidirect

26. C₂×C₁₇×D₂₁

C₃×C₇

direct (N × H)

30. C₃²×C₇×C₁₇

C₂×D₇

semidirect

31. C₃²×C₇×C₁₇

C₂²×C₇

semidirect

32. C₂×C₃²×C₇²

C₂×C₁₇

semidirect

33. C₃×C₇×D₂₁

C₂×C₁₇

direct (N × H)

34. C₇²×C₁₇

C₂×C₃×D₃

semidirect

41. C₂×C₃×C₇×C₁₇

D₃×C₇

semidirect

42. C₂×C₃×C₇×C₁₇

C₃×D₇

semidirect

43. C₂×C₃×C₇×C₁₇

D₂₁

direct (N × H)

44. C₂×C₃×C₇×C₁₇

C₂×C₃×C₇

semidirect

45. C₁₇×D₂₁

C₂×C₃×C₇

direct (N × H)

49. C₂×C₇×D₂₁

C₃×C₁₇

direct (N × H)

53. C₃²×C₇²

C₂²×C₁₇

semidirect

59. C₃×C₇×C₁₇

C₂×C₃×D₇

semidirect

60. C₃×C₇×C₁₇

C₂×D₃×C₇

semidirect

61. C₃×C₇×C₁₇

C₂×D₂₁

direct (N × H)

62. C₃×C₇×C₁₇

C₂²×C₃×C₇

semidirect

63. C₂×C₃²×C₁₇

C₇×D₇

semidirect

66. C₂×C₃×C₇²

D₃×C₁₇

semidirect

67. C₂×C₃×C₇²

C₂×C₃×C₁₇

semidirect

68. C₇×D₂₁

C₂×C₃×C₁₇

direct (N × H)

70. C₂×C₃×D₂₁

C₇×C₁₇

direct (N × H)

74. C₂×C₇×C₁₇

C₃×D₃×C₇

semidirect

75. C₂×C₇×C₁₇

C₃×D₂₁

direct (N × H)

80. C₃²×C₁₇

C₂×C₇×D₇

semidirect

82. C₃×C₇²

C₂×D₃×C₁₇

semidirect

83. C₃×C₇²

C₂²×C₃×C₁₇

semidirect

85. C₂×C₃²×C₇

D₇×C₁₇

semidirect

86. C₂×C₃²×C₇

C₂×C₇×C₁₇

semidirect

87. C₃×D₂₁

C₂×C₇×C₁₇

direct (N × H)

90. C₇×C₁₇

C₂×C₃×D₃×C₇

semidirect

91. C₇×C₁₇

C₂×C₃×D₂₁

direct (N × H)

94. C₂×C₃×C₁₇

C₃×C₇×D₇

semidirect

95. C₂×C₃×C₁₇

C₇×D₂₁

direct (N × H)

97. C₂×C₇²

C₃×D₃×C₁₇

semidirect

100. C₂×D₂₁

C₃×C₇×C₁₇

direct (N × H)

105. C₃²×C₇

C₂×D₇×C₁₇

semidirect

106. C₃²×C₇

C₂²×C₇×C₁₇

semidirect

108. C₃×C₁₇

C₂×C₃×C₇×D₇

semidirect

109. C₃×C₁₇

C₂×C₇×D₂₁

direct (N × H)

110. C₇²

C₂×C₃×D₃×C₁₇

semidirect

117. C₂×C₃×C₇

C₃×D₇×C₁₇

semidirect

118. C₂×C₃×C₇

D₃×C₇×C₁₇

semidirect

119. C₂×C₃×C₇

C₁₇×D₂₁

direct (N × H)

120. C₂×C₃×C₇

C₂×C₃×C₇×C₁₇

semidirect

121. D₂₁

C₂×C₃×C₇×C₁₇

direct (N × H)

126. C₂×C₁₇

C₃×C₇×D₂₁

direct (N × H)

134. C₃×C₇

C₂×C₃×D₇×C₁₇

semidirect

135. C₃×C₇

C₂×D₃×C₇×C₁₇

semidirect

136. C₃×C₇

C₂×C₁₇×D₂₁

direct (N × H)

137. C₃×C₇

C₂²×C₃×C₇×C₁₇

semidirect

138. C₂×C₃²

C₇×D₇×C₁₇

semidirect

140. C₁₇

C₂×C₃×C₇×D₂₁

direct (N × H)

143. C₂×C₇

C₃×D₃×C₇×C₁₇

semidirect

144. C₂×C₇

C₃×C₁₇×D₂₁

direct (N × H)

148. C₃²

C₂×C₇×D₇×C₁₇

semidirect

150. C₇

C₂×C₃×C₁₇×D₂₁

direct (N × H)

151. C₇

C₂×C₃×D₃×C₇×C₁₇

semidirect

154. C₂×C₃

C₃×C₇×D₇×C₁₇

semidirect

155. C₂×C₃

C₇×C₁₇×D₂₁

direct (N × H)

159. C₃

C₂×C₇×C₁₇×D₂₁

direct (N × H)

160. C₃

C₂×C₃×C₇×D₇×C₁₇

semidirect

162. C₂

C₃×C₇×C₁₇×D₂₁

direct (N × H)

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

subgroup	index
C₃×C₇×C₁₇×D₂₁	2
C₂×C₃²×C₇²×C₁₇	2
C₂×C₃×C₇×D₇×C₁₇	3
C₂×C₇×C₁₇×D₂₁	3
C₂×C₃×D₃×C₇×C₁₇	7
C₂×C₃×C₁₇×D₂₁	7
C₂×C₃×C₇×D₂₁	17

subgroup

index

C₃×C₇×C₁₇×D₂₁

C₂×C₃²×C₇²×C₁₇

C₂×C₃×C₇×D₇×C₁₇

C₂×C₇×C₁₇×D₂₁

C₂×C₃×D₃×C₇×C₁₇

C₂×C₃×C₁₇×D₂₁

C₂×C₃×C₇×D₂₁

Subgroups — C₂×C₃×C₇×C₁₇×D₂₁ #44473016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C3×C7×C17×D21 #44473016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×C₁₇×D₂₁ #44473016