Subgroups — #11725321

Order 896.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89460	C₂	1	1	yes	maximal, characteristic
2	C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1131	—	2	1	no	—
5	C₄×D₂₈	C₄	1	1	yes	characteristic
6	C₂×C₇×C₁₆	—	2	1	no	—
7	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	(C₂×D₇)⋊C₈	—	2	1	no	—
9	C₂×C₇⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
11	#1330	—	7	1	no	maximal, Sylow
12	C₂×D₂₈	C₈	1	1	yes	characteristic
13	C₂×C₄×D₇	—	2	1	no	—
14	C₂×C₇×C₈	D₄	1	1	yes	characteristic
15	C₂×C₇×C₈	D₄	1	1	yes	characteristic
16	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
17	(C₂×D₇)⋊C₄	—	2	1	no	—
18	C₄⋊Dic₇	C₈	1	1	yes	characteristic
19	C₂×C₇⋊C₈	—	2	1	no	—
20	#1131	—	4	1	no	—
21	#1131	—	4	1	no	—
22	C₇×C₁₆	—	2	2	no	—
23	C₇⋊C₁₆	—	4	1	no	—
24	#436	—	7	1	no	—
25	C₄⋊C₁₆	—	7	1	no	—
26	#346	D₇	1	1	yes	characteristic, complemented
27	C₂²×D₇	—	2	1	no	—
28	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₇	M₁₆	1	1	yes	characteristic
31	C₂×Dic₇	—	2	1	no	—
32	D₂₈	—	4	1	no	—
33	C₄×D₇	—	4	1	no	—
34	C₄×D₇	—	4	1	no	—
35	C₇×C₈	—	2	1	no	—
36	C₇×C₈	D₈	1	1	yes	characteristic
37	C₇×C₈	SD₁₆	1	1	yes	characteristic
38	C₇⋊C₈	—	4	1	no	—
39	C₂×M₁₆	—	14	1	no	—
40	C₄×D₄	—	7	1	no	—
41	C₂×C₁₆	—	2	1	no	—
42	C₂×C₁₆	—	14	1	no	—
43	C₄⋊C₈	—	7	1	no	—
44	C₂²⋊C₈	—	14	1	no	—
45	C₄×C₈	C₂×D₇	1	1	yes	characteristic
46	C₂²×C₇	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×D₇	—	4	1	no	—
48	C₂×D₇	—	4	1	no	—
49	C₂×D₇	—	4	1	no	—
50	C₄×C₇	—	2	1	no	—
51	C₄×C₇	—	2	1	no	—
52	C₄×C₇	#102	1	1	yes	characteristic
53	C₄×C₇	#104	1	1	yes	characteristic
54	Dic₇	—	4	1	no	—
55	C₂×D₄	—	7	1	no	—
56	C₂²×C₄	—	14	1	no	—
57	M₁₆	—	28	1	no	—
58	M₁₆	—	28	1	no	—
59	C₂×C₈	—	14	1	no	—
60	C₂×C₈	D₂₈	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
62	C₄⋊C₄	—	7	1	no	—
63	C₂²⋊C₄	—	14	1	no	—
64	C₄²	C₄×D₇	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	C₁₆	—	28	1	no	—
67	C₂×C₇	D₄⋊C₈	1	1	yes	characteristic
68	C₂×C₇	#349	1	1	yes	characteristic
69	C₂×C₇	#350	1	1	yes	characteristic
70	D₇	—	8	1	no	—
71	C₂³	—	14	1	no	—
72	D₄	—	28	1	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	28	1	no	—
75	C₂×C₄	—	28	1	no	—
76	C₂×C₄	C₈×D₇	1	1	yes	characteristic
77	C₂×C₄	#1131	1	1	yes	characteristic
78	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	28	1	no	—
81	C₈	#1132	1	1	yes	characteristic
82	C₈	D₅₆	1	1	yes	characteristic
83	C₇	#1330	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	28	1	no	—
85	C₂²	—	28	1	no	—
86	C₂²	—	28	1	no	—
87	C₂²	(C₂×D₇)⋊C₈	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	28	1	no	—
91	C₄	#6379	1	1	yes	characteristic
92	C₄	#6395	1	1	yes	characteristic
93	C₂	—	56	1	no	—
94	C₂	#89254	1	1	yes	characteristic
95	C₂	#89309	1	1	yes	characteristic
96	C₂	#89310	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇)⋊C₄
commutator	C₂×C₄×C₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₇
Fitting	C₇×#346	C₂
socle	C₂²×C₇	C₂²⋊C₈
prime radical	C₂×D₂₈	C₈
cosocle	C₄×C₇×C₈	C₂²
hypercenter	#346	D₇
hypocenter	C₇	#1330
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₇×#346	C₂	semidirect
3. C₄⋊(C₇⋊C₁₆)	C₂	semidirect
26. #346	D₇	semidirect
83. C₇	#1330	semidirect

Maximal subgroups

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

subgroup	index
#89460	2
C₇×#346	2
C₄⋊(C₇⋊C₁₆)	2
#1330	7