Subgroups — C₇×#13556 #13190772

Order 1792.

290 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₇×#1400	C₂	1	4	yes	maximal, complemented
4	C₂×C₇×#131	C₄	1	1	yes	characteristic
5	C₇×#504	—	2	2	no	—
6	C₇×#504	—	2	2	no	—
7	C₂×C₇×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₇×C₁₆	C₄	1	2	yes	—
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₇×M₆₄	—	2	2	no	—
12	C₇×M₆₄	C₂²	1	4	yes	—
13	C₂×C₇×C₃₂	—	2	2	no	—
14	#13556	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂×C₇×#41	C₈	1	1	yes	characteristic
16	C₇×#131	—	2	2	no	—
17	C₇×#131	—	2	2	no	—
18	C₂×C₇×M₁₆	C₈	1	1	yes	characteristic
19	C₂×C₇×M₁₆	C₈	1	2	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₇×M₃₂	—	2	2	no	—
23	C₇×M₃₂	—	2	4	no	—
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₂×C₇×C₁₆	—	2	2	no	—
26	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×C₁₆	D₄	1	2	yes	—
29	C₂×C₇×C₁₆	D₄	1	2	yes	—
30	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
31	C₇×C₃₂	—	2	4	no	—
32	C₂×#504	C₂×C₇	1	1	yes	characteristic
33	C₂×M₆₄	C₂×C₇	1	2	yes	complemented
34	#1400	C₂×C₇	1	4	yes	complemented
35	C₇×#41	—	2	2	no	—
36	C₇×#41	—	2	2	no	—
37	C₂×C₇×Q₈	C₁₆	1	1	yes	characteristic
38	C₂×C₇×D₄	C₁₆	1	2	yes	—
39	C₂×C₇×D₄	C₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
42	C₇×M₁₆	—	2	2	no	—
43	C₇×M₁₆	—	2	4	no	—
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
47	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₇×C₈	M₁₆	1	2	yes	—
50	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₇×C₁₆	—	2	2	no	—
52	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
53	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
54	C₇×C₁₆	C₂×D₄	1	2	yes	—
55	C₂×#131	C₄×C₇	1	1	yes	characteristic
56	#504	—	2	2	no	—
57	#504	—	2	2	no	—
58	C₂×M₃₂	C₄×C₇	1	1	yes	characteristic
59	C₂×M₃₂	C₂²×C₇	1	2	yes	—
60	C₂²×C₁₆	C₄×C₇	1	2	yes	—
61	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
62	M₆₄	—	2	2	no	—
63	M₆₄	C₂²×C₇	1	4	yes	—
64	C₂×C₃₂	—	2	2	no	—
65	C₂³×C₇	C₂×C₁₆	1	2	yes	—
66	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
67	C₇×Q₈	—	2	2	no	—
68	C₇×D₄	—	2	4	no	—
69	C₇×D₄	—	2	2	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	—	2	2	no	—
72	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
73	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
74	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇	M₃₂	1	2	yes	—
76	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
77	C₇×C₈	—	2	2	no	—
78	C₇×C₈	C₂²⋊C₈	1	2	yes	—
79	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
81	C₂×#41	C₇×C₈	1	1	yes	characteristic
82	#131	—	2	2	no	—
83	#131	—	2	2	no	—
84	C₂×M₁₆	C₇×C₈	1	1	yes	characteristic
85	C₂×M₁₆	C₇×C₈	1	2	yes	—
86	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
87	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
88	M₃₂	—	2	2	no	—
89	M₃₂	—	2	4	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
93	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
94	C₂×C₁₆	C₇×D₄	1	2	yes	—
95	C₂×C₁₆	C₇×D₄	1	2	yes	—
96	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
97	C₃₂	—	2	4	no	—
98	C₂²×C₇	—	2	2	no	—
99	C₂²×C₇	—	4	1	no	—
100	C₂²×C₇	—	4	2	no	—
101	C₂²×C₇	C₂²⋊C₁₆	1	2	yes	—
102	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
103	C₄×C₇	—	2	2	no	—
104	C₄×C₇	C₂²⋊C₁₆	1	2	yes	—
105	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	C₇×C₁₆	1	1	yes	characteristic
110	C₂×D₄	C₇×C₁₆	1	2	yes	—
111	C₂×D₄	C₇×C₁₆	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
113	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
119	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
120	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
121	C₂×C₈	C₇×M₁₆	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
123	C₁₆	—	2	2	no	—
124	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
125	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
126	C₁₆	C₂×C₇×D₄	1	2	yes	—
127	C₂×C₇	—	2	1	no	—
128	C₂×C₇	—	2	2	no	—
129	C₂×C₇	—	4	2	no	—
130	C₂×C₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
131	C₂³	C₂×C₇×C₁₆	1	2	yes	—
132	C₂³	C₂×C₇×C₁₆	1	1	yes	characteristic
133	Q₈	—	2	2	no	—
134	D₄	—	2	4	no	—
135	D₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₇×C₂²⋊C₈	1	2	yes	—
139	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
140	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
141	C₂×C₄	C₇×M₃₂	1	2	yes	—
142	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	C₇×C₂²⋊C₈	1	2	yes	—
145	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
146	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
147	C₇	#13556	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₇×C₂²⋊C₁₆	1	2	yes	—
152	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	C₇×C₂²⋊C₁₆	1	2	yes	—
155	C₄	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
156	C₄	C₂×C₇×M₃₂	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×C₇×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₁₆	C₂²⋊C₄
commutator	C₂²	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₂×C₂²⋊C₁₆
prime radical	C₂×C₇×#41	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₁₆ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₇×M₆₄	C₂	semidirect
3. C₇×#1400	C₂	semidirect
14. #13556	C₇	direct (N × H)
33. C₂×M₆₄	C₂×C₇	semidirect
34. #1400	C₂×C₇	semidirect
147. C₇	#13556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#504	2
C₂×C₇×M₆₄	2
C₇×#1400	2
#13556	7

Subgroups — C7×#13556 #13190772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13556 #13190772