Subgroups — #11725317

Order 896.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#346	C₂	1	1	yes	maximal, characteristic
2	#89256	C₂	1	1	yes	maximal, characteristic
3	#89250	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×C₁₆	C₄	1	2	yes	—
5	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₇⋊C₁₆	C₄	1	2	yes	—
7	C₄×C₇⋊C₈	C₄	1	2	yes	—
8	#1313	—	7	1	no	maximal, Sylow
9	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
11	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₇⋊C₈	—	2	2	no	—
13	C₇×C₁₆	C₈	1	4	yes	complemented
14	C₇⋊C₁₆	—	2	2	no	—
15	#346	—	7	1	no	—
16	#346	D₇	1	1	yes	characteristic
17	#322	—	7	1	no	—
18	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₇	M₁₆	1	2	yes	—
20	C₇×C₈	C₂×C₈	1	2	yes	—
21	C₇×C₈	M₁₆	1	2	yes	—
22	C₇⋊C₈	—	4	2	no	—
23	C₂×C₁₆	—	7	2	no	—
24	C₂×C₁₆	Dic₇	1	2	yes	—
25	C₄×C₈	—	7	2	no	—
26	C₄×C₈	C₂×D₇	1	1	yes	characteristic
27	C₂²×C₇	#97	1	1	yes	characteristic
28	C₄×C₇	—	2	2	no	—
29	C₄×C₇	C₄×C₈	1	1	yes	characteristic
30	C₄×C₇	#97	1	1	yes	characteristic
31	C₂×C₈	—	14	2	no	—
32	C₂×C₈	C₄×D₇	1	1	yes	characteristic
33	C₂×C₈	C₂×Dic₇	1	1	yes	characteristic
34	C₄²	C₄×D₇	1	1	yes	characteristic
35	C₁₆	—	14	2	no	—
36	C₁₆	C₇⋊C₈	1	4	yes	complemented
37	C₂×C₇	#322	1	1	yes	characteristic
38	C₂×C₇	#347	1	2	yes	—
39	C₂×C₄	#1131	1	2	yes	—
40	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
41	C₈	—	28	2	no	—
42	C₈	#1131	1	2	yes	—
43	C₈	C₂×C₇⋊C₈	1	2	yes	—
44	C₇	#1313	1	1	yes	characteristic, complemented, Sylow
45	C₂²	#6389	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄×C₇⋊C₈	1	1	yes	characteristic
48	C₄	#6389	1	1	yes	characteristic
49	C₂	#89250	1	1	yes	characteristic
50	C₂	#89307	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₇
commutator	C₄×C₇	C₄×C₈
Frattini	C₄×C₈	C₂×D₇
Fitting	C₇×#346	C₂
socle, prime radical	C₂²×C₇	#97
cosocle	C₄×C₇×C₈	C₂²
hypercenter	#346	D₇
hypocenter	C₇	#1313
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
13. C₇×C₁₆	C₈	semidirect
36. C₁₆	C₇⋊C₈	semidirect
44. C₇	#1313	semidirect

Maximal subgroups

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

subgroup	index
C₇×#346	2
#89256	2
#89250	2
#1313	7