Subgroups — #27159032

Order 12416.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261866	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261865	C₂	1	2	yes	maximal, complemented
3	#20261864	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15690909	C₂	1	2	yes	maximal, complemented
5	#20261856	C₂	1	1	yes	maximal, characteristic
6	#15690909	C₂²	1	1	yes	characteristic
7	#15690909	C₂²	1	2	yes	—
8	#15690909	C₂²	1	1	yes	characteristic
9	#15690907	C₄	1	1	yes	characteristic
10	#15690906	C₄	1	2	yes	—
11	#15690905	C₂²	1	1	yes	characteristic
12	C₂×#13095523	C₂²	1	2	yes	—
13	#15690897	C₄	1	1	yes	characteristic
14	#13095523	C₂×C₄	1	2	yes	—
15	#13095523	C₂×C₄	1	1	yes	characteristic
16	#13095523	C₂³	1	1	yes	characteristic
17	#13095521	C₈	1	1	yes	characteristic
18	#13095520	C₈	1	2	yes	—
19	#13095519	C₂×C₄	1	1	yes	characteristic
20	C₂×#11715897	C₂×C₄	1	2	yes	—
21	#13095510	C₈	1	1	yes	characteristic
22	D₄×C₉₇	C₁₆	1	1	yes	characteristic
23	C₂²⋊D₉₇	C₁₆	1	2	yes	—
24	C₂×Dic₉₇	C₂×C₈	1	2	yes	—
25	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
26	Dic₁₉₄	C₁₆	1	1	yes	characteristic
27	#11715897	C₂×C₈	1	2	yes	—
28	#11715897	C₂×C₈	1	1	yes	characteristic
29	#11715897	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₉₇	C₂×C₁₆	1	2	yes	—
31	C₂×D₉₇	C₂×C₁₆	1	1	yes	characteristic
32	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
33	Dic₉₇	C₂×C₁₆	1	2	yes	—
34	Dic₉₇	C₂²×C₈	1	1	yes	characteristic
35	C₂×C₉₇	—	2	2	no	—
36	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₉₇	—	2	1	no	—
38	#2258	—	97	1	no	maximal, Sylow
39	C₉₇	#2258	1	1	yes	characteristic, complemented, Sylow
40	#504	—	97	1	no	—
41	M₆₄	—	97	2	no	—
42	M₆₄	—	97	1	no	—
43	C₂×C₃₂	—	97	2	no	—
44	C₂×C₃₂	—	97	1	no	—
45	#131	—	97	1	no	—
46	M₃₂	—	97	2	no	—
47	M₃₂	—	97	1	no	—
48	C₂×C₁₆	—	97	2	no	—
49	C₂×C₁₆	—	97	1	no	—
50	C₃₂	—	97	1	no	—
51	C₃₂	—	97	2	no	—
52	C₃₂	—	97	1	no	—
53	#41	—	97	1	no	—
54	M₁₆	—	97	2	no	—
55	M₁₆	—	97	1	no	—
56	C₂×C₈	—	97	2	no	—
57	C₂×C₈	—	97	1	no	—
58	C₁₆	—	97	1	no	—
59	C₁₆	—	97	2	no	—
60	C₁₆	—	97	1	no	—
61	Q₈	—	97	1	no	—
62	D₄	—	97	2	no	—
63	D₄	#13095525	1	1	yes	characteristic
64	C₂×C₄	—	97	2	no	—
65	C₂×C₄	—	97	1	no	—
66	C₈	—	97	1	no	—
67	C₈	—	97	2	no	—
68	C₈	—	97	1	no	—
69	C₂²	—	97	1	no	—
70	C₂²	C₂×#13095525	1	2	yes	—
71	C₄	—	97	1	no	—
72	C₄	—	97	2	no	—
73	C₄	C₂×#13095525	1	1	yes	characteristic
74	C₂	—	2	2	no	—
75	C₂	—	194	1	no	—
76	C₂	C₂²×#13095525	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#13095525
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	D₄×C₉₇	C₁₆
prime radical	#13095510	C₈
cosocle	#13095523	C₂³
hypercenter	D₄	#13095525
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #20261866	C₂	semidirect
2. #20261865	C₂	semidirect
3. #20261864	C₂	semidirect
4. C₂×#15690909	C₂	semidirect
39. C₉₇	#2258	semidirect

Maximal subgroups

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

subgroup	index
#20261866	2
#20261865	2
#20261864	2
C₂×#15690909	2
#20261856	2
#2258	97