Subgroups — #11729136

Order 896.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#500	C₂	1	1	yes	maximal, characteristic
2	#89471	C₂	1	1	yes	maximal, characteristic
3	#89287	C₂	1	4	yes	maximal
4	C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₇×C₄⋊Q₈	—	2	1	no	—
6	C₂×Dic₂₈	—	2	1	no	—
7	C₂×Dic₂₈	C₂²	1	2	yes	—
8	#6432	—	2	1	no	—
9	C₇×#107	—	2	1	no	—
10	C₇×#107	C₂²	1	2	yes	—
11	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₁₆	C₂²	1	2	yes	—
13	#2241	—	7	1	no	maximal, Sylow
14	C₂×C₇×Q₈	—	4	1	no	—
15	C₂×Dic₁₄	—	2	2	no	—
16	C₂×C₇×C₈	D₄	1	1	yes	characteristic
17	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₇×C₄⋊C₄	—	2	2	no	—
19	C₇×C₄⋊C₄	—	4	1	no	—
20	C₄²×C₇	D₄	1	1	yes	characteristic
21	C₄⋊Dic₇	—	4	1	no	—
22	C₄⋊Dic₇	—	4	1	no	—
23	Dic₂₈	—	2	4	no	—
24	Dic₂₈	—	4	2	no	—
25	C₇⋊C₁₆	—	2	2	no	—
26	#500	D₇	1	1	yes	characteristic
27	#494	—	7	1	no	—
28	#358	—	7	4	no	—
29	C₄×C₁₆	—	7	1	no	—
30	C₇×Q₈	—	8	2	no	—
31	C₂×C₄×C₇	—	4	2	no	—
32	C₂×C₄×C₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₇	—	4	2	no	—
36	Dic₁₄	—	4	2	no	—
37	Dic₁₄	—	4	2	no	—
38	C₇×C₈	SD₁₆	1	2	yes	—
39	C₇×C₈	#41	1	2	yes	—
40	C₂×Q₁₆	—	7	2	no	—
41	C₂×Q₁₆	—	14	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	—	14	1	no	—
44	C₂×C₁₆	—	7	2	no	—
45	#107	—	2	1	no	—
46	#107	C₂×D₇	1	2	yes	—
47	C₄×C₈	C₂×D₇	1	1	yes	characteristic
48	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
49	C₄×C₇	—	8	2	no	—
50	C₄×C₇	SD₃₂	1	2	yes	—
51	C₄×C₇	Q₃₂	1	2	yes	—
52	C₄×C₇	#124	1	1	yes	characteristic
53	C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₇	—	8	2	no	—
55	C₂×Q₈	—	4	1	no	—
56	C₂×Q₈	—	14	2	no	—
57	Q₁₆	—	14	4	no	—
58	Q₁₆	—	28	2	no	—
59	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
60	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	28	1	no	—
64	C₄⋊C₄	—	28	1	no	—
65	C₄²	C₂²⋊D₇	1	1	yes	characteristic
66	C₁₆	—	14	2	no	—
67	C₂×C₇	#486	1	1	yes	characteristic
68	C₂×C₇	C₂×SD₃₂	1	1	yes	characteristic
69	C₂×C₇	C₂×Q₃₂	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	Q₈	—	28	2	no	—
72	Q₈	—	28	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	28	2	no	—
75	C₂×C₄	#1141	1	1	yes	characteristic
76	C₂×C₄	#1141	1	1	yes	characteristic
77	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
78	C₈	#1143	1	2	yes	—
79	C₈	#1161	1	2	yes	—
80	C₇	#2241	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×#1141	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	56	2	no	—
84	C₄	#6401	1	2	yes	—
85	C₄	#6403	1	2	yes	—
86	C₄	C₂×#1143	1	1	yes	characteristic
87	C₄	#6510	1	1	yes	characteristic
88	C₂	#89860	1	1	yes	characteristic
89	C₂	C₂×#6401	1	1	yes	characteristic
90	C₂	C₂×#6403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#500	C₂
socle, prime radical	C₂²×C₇	C₂×D₈
hypercenter	#500	D₇
hypocenter	C₇	#2241
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
80. C₇	#2241	semidirect

Maximal subgroups

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

subgroup	index
C₇×#500	2
#89471	2
#89287	2
C₄×C₇⋊C₁₆	2
#2241	7