Subgroups — #27157393

Order 12416.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690865	C₂	1	3	yes	maximal
2	C₂×C₄×#11715897	C₂	1	3	yes	maximal
3	#20260580	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715897	C₂²	1	1	yes	characteristic
5	C₂²×#11715897	C₂²	1	3	yes	—
6	#15690865	C₄	1	12	yes	complemented
7	C₄×#11715897	C₄	1	12	yes	complemented
8	C₂×C₄×Dic₉₇	C₂²	1	3	yes	—
9	#15690717	C₄	1	3	yes	—
10	#95×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	C₂×C₄	1	6	yes	—
12	C₂×#11715897	C₂×C₄	1	6	yes	—
13	C₂×#11715897	C₂×C₄	1	12	yes	—
14	C₂²×Dic₉₇	C₂×C₄	1	3	yes	—
15	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
16	C₄×Dic₉₇	C₂×C₄	1	6	yes	—
17	C₄×Dic₉₇	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
19	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₄×C₉₇	—	2	6	no	—
21	C₂×C₄×C₉₇	C₄²	1	6	yes	—
22	C₂×Dic₉₇	—	2	3	no	—
23	C₂×Dic₉₇	—	2	3	no	—
24	C₂×Dic₉₇	C₄²	1	6	yes	—
25	C₂×Dic₉₇	C₂²×C₄	1	3	yes	—
26	C₂×Dic₉₇	C₂²×C₄	1	3	yes	—
27	#11715897	C₄²	1	4	yes	—
28	#11715897	C₄²	1	12	yes	—
29	C₂²×C₉₇	C₂×C₄²	1	3	yes	—
30	C₂²×C₉₇	#131	1	3	yes	—
31	C₂²×C₉₇	#131	1	1	yes	characteristic
32	C₄×C₉₇	—	2	6	no	—
33	Dic₉₇	—	2	6	no	—
34	Dic₉₇	C₂×C₄²	1	1	yes	characteristic
35	Dic₉₇	C₂×C₄²	1	3	yes	—
36	C₂×C₉₇	C₄³	1	1	yes	characteristic
37	C₂×C₉₇	#405	1	3	yes	—
38	C₂×C₉₇	#405	1	3	yes	—
39	#1726	—	97	1	no	maximal, Sylow
40	C₉₇	#1726	1	1	yes	characteristic, complemented, Sylow
41	C₂×#97	—	97	3	no	—
42	C₂×C₄×C₈	—	97	3	no	—
43	#376	—	97	1	no	—
44	C₂²×C₈	—	97	1	no	—
45	C₂²×C₈	—	97	3	no	—
46	C₂×C₄²	—	97	3	no	—
47	#97	—	97	12	no	—
48	C₄×C₈	—	97	12	no	—
49	#95	—	97	3	no	—
50	#95	#87604	1	1	yes	characteristic
51	C₂²×C₄	—	97	1	no	—
52	C₂²×C₄	—	97	3	no	—
53	C₂²×C₄	C₂×#87604	1	3	yes	—
54	C₂×C₈	—	97	6	no	—
55	C₂×C₈	—	97	6	no	—
56	C₂×C₈	—	97	12	no	—
57	C₄²	—	97	6	no	—
58	C₄²	—	97	6	no	—
59	C₂³	C₂²×#87604	1	1	yes	characteristic
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	—	97	3	no	—
62	C₂×C₄	—	97	3	no	—
63	C₂×C₄	—	97	6	no	—
64	C₂×C₄	—	194	3	no	—
65	C₂×C₄	—	194	3	no	—
66	C₂×C₄	C₄×#87604	1	6	yes	—
67	C₈	—	97	4	no	—
68	C₈	—	97	12	no	—
69	C₂²	C₂×C₄×#87604	1	3	yes	—
70	C₂²	#15690897	1	3	yes	—
71	C₂²	#15690898	1	1	yes	characteristic
72	C₄	—	2	6	no	—
73	C₄	—	97	1	no	—
74	C₄	—	97	3	no	—
75	C₄	—	194	6	no	—
76	C₂	C₄²×#87604	1	1	yes	characteristic
77	C₂	#20261699	1	3	yes	—
78	C₂	#20261700	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator	C₂×C₉₇	C₄³
Fitting	#95×C₉₇	C₄
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	#95	#87604
hypocenter	C₉₇	#1726
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
6. #15690865	C₄	semidirect
7. C₄×#11715897	C₄	semidirect
40. C₉₇	#1726	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690865	2
C₂×C₄×#11715897	2
#20260580	2
#1726	97