Subgroups — #11729583

Order 896.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#533	C₂	1	1	yes	maximal, characteristic
2	#89697	C₂	1	2	yes	maximal
3	#89342	C₂	1	4	yes	maximal, complemented
4	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
5	C₇×#125	C₄	1	2	yes	—
6	C₇×C₂²⋊Q₈	—	2	2	no	—
7	C₄×C₇×Q₈	—	2	2	no	—
8	C₇×#117	C₄	1	2	yes	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
11	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
12	C₄×C₇⋊C₈	—	2	2	no	—
13	#1544	—	7	1	no	maximal, Sylow
14	C₂×C₇×Q₈	—	2	2	no	—
15	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₇	D₄	1	2	yes	—
17	C₇×C₄⋊C₄	—	2	4	no	—
18	C₇×C₄⋊C₄	—	2	2	no	—
19	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
20	C₇×C₂²⋊C₄	—	2	2	no	—
21	C₄²×C₇	—	2	2	no	—
22	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₇	D₄	1	2	yes	—
24	C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₇⋊C₈	—	2	4	no	—
26	#533	D₇	1	1	yes	characteristic
27	#431	—	7	2	no	—
28	#332	—	7	4	no	—
29	C₂³×C₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₇×Q₈	—	4	4	no	—
31	C₂×C₄×C₇	—	2	2	no	—
32	C₂×C₄×C₇	—	2	4	no	—
33	C₂×C₄×C₇	—	2	4	no	—
34	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
38	C₇⋊C₈	—	4	4	no	—
39	C₄⋊Q₈	C₂×D₇	1	2	yes	—
40	#125	Dic₇	1	2	yes	—
41	C₂²⋊Q₈	—	2	2	no	—
42	C₄×Q₈	—	2	2	no	—
43	#117	Dic₇	1	2	yes	—
44	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
45	C₄⋊C₈	—	7	4	no	—
46	C₂²⋊C₈	—	14	2	no	—
47	C₄×C₈	—	14	2	no	—
48	C₂²×C₇	—	2	1	no	—
49	C₂²×C₇	—	2	2	no	—
50	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₇	—	2	1	no	—
52	C₄×C₇	—	2	2	no	—
53	C₄×C₇	—	4	4	no	—
54	C₄×C₇	#101	1	2	yes	—
55	C₄×C₇	#135	1	4	yes	—
56	C₂×Q₈	—	2	2	no	—
57	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
59	C₂×C₈	—	14	4	no	—
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
63	C₂²⋊C₄	—	2	2	no	—
64	C₄²	—	2	2	no	—
65	C₄²	C₂×Dic₇	1	1	yes	characteristic
66	C₄²	C₂²⋊D₇	1	2	yes	—
67	C₄²	C₂²×D₇	1	1	yes	characteristic
68	C₂×C₇	—	4	1	no	—
69	C₂×C₇	C₂×#101	1	1	yes	characteristic
70	C₂×C₇	#416	1	2	yes	—
71	C₂³	C₂²⋊Dic₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
77	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
78	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
80	C₈	—	28	4	no	—
81	C₇	#1544	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	4	no	—
88	C₄	#6410	1	2	yes	—
89	C₄	#6513	1	4	yes	—
90	C₂	—	4	1	no	—
91	C₂	C₂×#6410	1	1	yes	characteristic
92	C₂	#90005	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#533	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₇	C₂²⋊C₄
cosocle	C₄²×C₇	C₂³
hypercenter	#533	D₇
hypocenter	C₇	#1544
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
3. #89342	C₂	semidirect
81. C₇	#1544	semidirect

Maximal subgroups

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

subgroup	index
C₇×#533	2
#89697	2
#89342	2
#1544	7