Subgroups — #27158753

Order 12416.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095523	C₂	1	2	yes	maximal
2	C₈×#11715897	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095523	C₄	1	4	yes	—
4	C₄×#11715897	C₄	1	1	yes	characteristic
5	C₄×#11715897	C₂²	1	1	yes	characteristic
6	C₈×Dic₉₇	C₄	1	1	yes	characteristic
7	#13095523	—	2	4	no	—
8	C₂×#11715897	C₈	1	2	yes	—
9	C₂×#11715897	C₂×C₄	1	2	yes	—
10	C₂×#11715895	C₈	1	1	yes	characteristic
11	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₉₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
14	C₂×Dic₉₇	C₄²	1	1	yes	characteristic
15	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
16	#11715897	—	2	2	no	—
17	#11715897	M₁₆	1	4	yes	—
18	C₈×C₉₇	C₁₆	1	2	yes	complemented
19	#11715895	C₁₆	1	2	yes	complemented
20	C₂²×C₉₇	C₄×C₈	1	1	yes	characteristic
21	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
22	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
23	Dic₉₇	#97	1	2	yes	—
24	Dic₉₇	M₃₂	1	2	yes	—
25	C₂×C₉₇	#322	1	1	yes	characteristic
26	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
27	C₂×C₉₇	#346	1	1	yes	characteristic
28	#1312	—	97	1	no	maximal, Sylow
29	C₉₇	#1312	1	1	yes	characteristic, complemented, Sylow
30	C₄×C₁₆	—	97	2	no	—
31	C₈²	—	97	1	no	—
32	C₂×C₁₆	—	97	4	no	—
33	C₄×C₈	—	97	1	no	—
34	C₄×C₈	—	97	1	no	—
35	C₄×C₈	—	97	1	no	—
36	C₂×C₈	—	97	2	no	—
37	C₂×C₈	—	97	1	no	—
38	C₂×C₈	—	97	2	no	—
39	C₂×C₈	#11715906	1	1	yes	characteristic
40	C₄²	—	97	1	no	—
41	C₁₆	—	194	4	no	—
42	C₂×C₄	—	97	1	no	—
43	C₂×C₄	—	97	1	no	—
44	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
45	C₈	—	97	4	no	—
46	C₈	—	97	2	no	—
47	C₈	—	194	2	no	—
48	C₈	#13095523	1	2	yes	complemented
49	C₂²	C₄×#11715906	1	1	yes	characteristic
50	C₄	—	97	2	no	—
51	C₄	—	97	2	no	—
52	C₄	C₂×#13095523	1	1	yes	characteristic
53	C₄	#15690905	1	1	yes	characteristic
54	C₂	#20261819	1	1	yes	characteristic
55	C₂	C₄×#13095523	1	1	yes	characteristic
56	C₂	#20261830	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#11715906
commutator	C₂×C₉₇	C₄×C₁₆
Fitting	C₂×C₈×C₉₇	C₈
socle, prime radical	C₂²×C₉₇	C₄×C₈
cosocle	C₄×#11715897	C₂²
hypercenter	C₂×C₈	#11715906
hypocenter	C₉₇	#1312
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
18. C₈×C₉₇	C₁₆	semidirect
19. #11715895	C₁₆	semidirect
29. C₉₇	#1312	semidirect
48. C₈	#13095523	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095523	2
C₈×#11715897	2
#1312	97