Subgroups — #27156685

Order 12416.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261631	C₂	1	2	yes	maximal
2	#20260761	C₂	1	1	yes	maximal, characteristic
3	#15690884	—	2	2	no	—
4	C₂×#13095495	C₂²	1	1	yes	characteristic
5	#15690769	C₄	1	1	yes	characteristic
6	#15690739	—	2	2	no	—
7	C₂²×C₈×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	4	2	no	—
9	#13095495	—	4	1	no	—
10	C₂×#11715895	D₄	1	2	yes	—
11	C₄⋊Dic₉₇	—	2	1	no	—
12	C₂²⋊Dic₉₇	C₈	1	2	yes	—
13	C₄×Dic₉₇	—	2	1	no	—
14	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₉₇	—	2	1	no	—
16	C₂×C₈×C₉₇	—	4	1	no	—
17	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	1	no	—
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
22	C₂×Dic₉₇	—	2	2	no	—
23	C₈×C₉₇	—	4	1	no	—
24	#11715895	—	2	2	no	—
25	C₂²×C₉₇	—	2	1	no	—
26	C₂²×C₉₇	—	4	1	no	—
27	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₉₇	—	2	1	no	—
29	C₄×C₉₇	#99	1	1	yes	characteristic
30	C₄×C₉₇	#101	1	1	yes	characteristic
31	Dic₉₇	—	4	2	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	—	4	1	no	—
34	C₂×C₉₇	#324	1	1	yes	characteristic
35	#1328	—	97	1	no	maximal, Sylow
36	C₉₇	#1328	1	1	yes	characteristic, complemented, Sylow
37	#424	—	97	1	no	—
38	#349	—	97	2	no	—
39	C₂×M₁₆	—	97	1	no	—
40	C₂²×C₈	#87604	1	1	yes	characteristic
41	#117	—	97	1	no	—
42	M₃₂	—	194	2	no	—
43	C₄⋊C₈	—	194	2	no	—
44	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
45	M₁₆	—	388	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	97	2	no	—
49	C₄⋊C₄	—	194	1	no	—
50	C₂²⋊C₄	—	97	2	no	—
51	C₄²	—	194	1	no	—
52	C₁₆	—	388	2	no	—
53	C₂³	#13095519	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	194	2	no	—
57	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
58	C₂×C₄	#13095521	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₈	—	194	2	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#15690867	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	388	2	no	—
66	C₄	#15690868	1	1	yes	characteristic
67	C₄	#15690871	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#20261581	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690868
commutator, prime radical	C₂³×C₉₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#87604
Fitting	C₂²×C₈×C₉₇	C₄
socle	C₂×C₉₇	#324
cosocle	C₂×#13095495	C₂²
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	#1328
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
36. C₉₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#20261631	2
#20260761	2
#1328	97