Subgroups — #27157665

Order 12416.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261601	C₂	1	6	yes	maximal
2	#20260580	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715897	C₂²	1	4	yes	—
4	C₂×C₄×Dic₉₇	C₂²	1	3	yes	—
5	#15690717	C₄	1	3	yes	—
6	#95×C₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715897	—	2	12	no	—
8	C₂²×Dic₉₇	C₂×C₄	1	3	yes	—
9	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
10	C₄×Dic₉₇	—	2	6	no	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
12	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
13	C₂×C₄×C₉₇	—	2	3	no	—
14	C₂×C₄×C₉₇	—	2	6	no	—
15	C₂×Dic₉₇	—	2	3	no	—
16	C₂×Dic₉₇	—	2	3	no	—
17	C₂×Dic₉₇	—	2	3	no	—
18	C₂×Dic₉₇	#41	1	6	yes	—
19	#11715897	—	4	4	no	—
20	C₂²×C₉₇	#117	1	3	yes	—
21	C₂²×C₉₇	#131	1	3	yes	—
22	C₂²×C₉₇	#131	1	1	yes	characteristic
23	C₄×C₉₇	—	4	3	no	—
24	Dic₉₇	—	4	3	no	—
25	Dic₉₇	#126	1	4	yes	—
26	C₂×C₉₇	#383	1	1	yes	characteristic
27	C₂×C₉₇	#433	1	3	yes	—
28	C₂×C₉₇	#433	1	3	yes	—
29	#1840	—	97	1	no	maximal, Sylow
30	C₉₇	#1840	1	1	yes	characteristic, complemented, Sylow
31	#376	—	97	1	no	—
32	#336	—	97	6	no	—
33	C₂²×C₈	—	97	4	no	—
34	C₂×C₄²	—	97	3	no	—
35	#95	—	97	3	no	—
36	#95	#87604	1	1	yes	characteristic
37	C₂²×C₄	—	97	1	no	—
38	C₂²×C₄	—	97	3	no	—
39	C₂²×C₄	C₂×#87604	1	3	yes	—
40	C₂×C₈	—	194	12	no	—
41	C₄²	—	194	6	no	—
42	C₂³	C₂²×#87604	1	1	yes	characteristic
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	6	no	—
45	C₂×C₄	—	97	6	no	—
46	C₂×C₄	—	194	3	no	—
47	C₂×C₄	—	194	3	no	—
48	C₂×C₄	—	194	3	no	—
49	C₈	—	388	4	no	—
50	C₂²	#15690889	1	3	yes	—
51	C₂²	#15690897	1	3	yes	—
52	C₂²	#15690898	1	1	yes	characteristic
53	C₄	—	4	3	no	—
54	C₄	—	97	4	no	—
55	C₄	—	388	3	no	—
56	C₂	#20261680	1	1	yes	characteristic
57	C₂	#20261759	1	3	yes	—
58	C₂	#20261760	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	#95×C₉₇	C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	#95	#87604
hypocenter	C₉₇	#1840
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
30. C₉₇	#1840	semidirect

Maximal subgroups

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

subgroup	index
#20261601	2
#20260580	2
#1840	97