Subgroups — #27156876

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261640	C₂	1	2	yes	maximal
2	#20261101	C₂	1	1	yes	maximal, characteristic
3	#15690869	—	2	2	no	—
4	#15690807	C₂²	1	1	yes	characteristic
5	#15690730	—	2	2	no	—
6	#15690721	C₄	1	1	yes	characteristic, complemented
7	#125×C₉₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#87604	—	4	2	no	—
9	C₂×D₃₈₈	—	4	1	no	—
10	C₂×D₃₈₈	D₄	1	2	yes	—
11	C₂×#11715895	—	2	1	no	—
12	C₄⋊C₄×C₉₇	—	2	1	no	—
13	C₄⋊C₄×C₉₇	—	4	1	no	—
14	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×D₉₇	—	2	2	no	—
16	C₂×#87604	—	8	2	no	—
17	C₂×C₄×C₉₇	—	2	1	no	—
18	C₂×C₄×C₉₇	—	4	1	no	—
19	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	D₃₈₈	—	4	2	no	—
21	D₃₈₈	—	8	2	no	—
22	#11715895	—	4	1	no	—
23	C₂²×C₉₇	#99	1	1	yes	characteristic
24	C₂×D₉₇	—	4	2	no	—
25	C₂×D₉₇	—	8	2	no	—
26	#87604	—	16	2	no	—
27	C₄×C₉₇	—	2	1	no	—
28	C₄×C₉₇	—	4	1	no	—
29	C₄×C₉₇	—	8	1	no	—
30	C₂×C₉₇	—	2	1	no	—
31	C₂×C₉₇	#354	1	1	yes	characteristic
32	D₉₇	—	8	2	no	—
33	#1412	—	97	1	no	maximal, Sylow
34	C₉₇	#1412	1	1	yes	characteristic, complemented, Sylow
35	#490	—	97	1	no	—
36	#353	—	97	2	no	—
37	#127	—	97	1	no	—
38	#125	#87604	1	1	yes	characteristic, complemented
39	#102	—	194	2	no	—
40	#99	—	194	2	no	—
41	#97	—	97	1	no	—
42	C₂×D₄	—	97	2	no	—
43	C₂×D₄	—	388	1	no	—
44	C₂×C₈	—	194	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	388	2	no	—
48	C₄²	C₂×#87604	1	1	yes	characteristic
49	C₂³	—	194	2	no	—
50	D₄	—	776	2	no	—
51	D₄	—	388	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	776	2	no	—
55	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
56	C₈	—	388	1	no	—
57	C₂²	—	776	2	no	—
58	C₂²	—	388	2	no	—
59	C₂²	#15690868	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	1552	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	776	2	no	—
66	C₂	#20261641	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261641
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#125×C₉₇	C₄
socle	C₂×C₉₇	#354
prime radical, cosocle	#15690807	C₂²
hypercenter	#125	#87604
hypocenter	C₉₇	#1412
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
6. #15690721	C₄	semidirect
7. #125×C₉₇	C₄	semidirect
34. C₉₇	#1412	semidirect
38. #125	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261640	2
#20261101	2
#1412	97