Subgroups — #27152260

Order 12416.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261292	C₂	1	2	yes	maximal
2	#20261144	C₂	1	8	yes	maximal, complemented
3	#20260773	C₂	1	4	yes	maximal
4	C₂×#97×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690810	C₂²	1	4	yes	—
6	#15690802	C₂²	1	4	yes	—
7	#15690789	—	2	4	no	—
8	C₄×Dic₁₉₄	—	2	4	no	—
9	#15690769	C₂²	1	4	yes	—
10	#15690744	C₂²	1	8	yes	—
11	#15690742	C₂²	1	8	yes	—
12	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#97×C₉₇	C₂²	1	4	yes	—
15	C₂×Dic₁₉₄	—	2	4	no	—
16	C₄⋊Dic₉₇	—	2	4	no	—
17	C₄⋊Dic₉₇	C₂³	1	8	yes	—
18	#13095490	—	2	8	no	—
19	C₂²⋊Dic₉₇	—	2	4	no	—
20	C₄×Dic₉₇	—	2	4	no	—
21	C₂²×C₄×C₉₇	D₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₉₇	Q₈	1	8	yes	complemented
24	C₂×C₈×C₉₇	C₂³	1	4	yes	—
25	C₄²×C₉₇	D₄	1	2	yes	—
26	C₄²×C₉₇	C₂³	1	2	yes	—
27	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	4	2	no	—
29	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
30	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
32	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
33	C₂×Dic₉₇	—	2	8	no	—
34	Dic₁₉₄	—	4	8	no	—
35	C₈×C₉₇	C₂×Q₈	1	8	yes	—
36	C₂²×C₉₇	—	2	2	no	—
37	C₂²×C₉₇	C₄⋊Q₈	1	2	yes	—
38	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₉₇	—	2	4	no	—
40	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
41	C₄×C₉₇	C₂²×Q₈	1	2	yes	—
42	Dic₉₇	—	4	8	no	—
43	C₂×C₉₇	—	2	2	no	—
44	C₂×C₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₉₇	#575	1	2	yes	—
46	#3162	—	97	1	no	maximal, Sylow
47	C₉₇	#3162	1	1	yes	characteristic, complemented, Sylow
48	#533	—	97	2	no	—
49	#501	—	97	8	no	—
50	#427	—	97	4	no	—
51	C₂×#97	D₉₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₉₇	1	2	yes	—
53	C₄⋊Q₈	—	97	4	no	—
54	#125	—	97	4	no	—
55	C₂²⋊Q₈	—	194	4	no	—
56	C₄×Q₈	—	194	4	no	—
57	#117	—	97	4	no	—
58	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
59	#107	—	97	8	no	—
60	#106	—	97	8	no	—
61	#97	C₂×D₉₇	1	4	yes	—
62	C₂×Q₈	—	194	4	no	—
63	C₂²×C₄	D₃₈₈	1	2	yes	—
64	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
65	C₂×C₈	Dic₁₉₄	1	8	yes	complemented
66	C₂×C₈	C₂²×D₉₇	1	4	yes	—
67	C₄⋊C₄	—	97	8	no	—
68	C₄⋊C₄	—	194	8	no	—
69	C₄⋊C₄	—	194	4	no	—
70	C₂²⋊C₄	—	194	4	no	—
71	C₄²	—	194	4	no	—
72	C₄²	D₃₈₈	1	2	yes	—
73	C₄²	C₂²×D₉₇	1	2	yes	—
74	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
75	Q₈	—	388	8	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	194	8	no	—
78	C₂×C₄	C₂×D₃₈₈	1	4	yes	—
79	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
81	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
82	C₈	C₂×Dic₁₉₄	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	#15690810	1	2	yes	—
85	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	388	8	no	—
88	C₄	#15690810	1	2	yes	—
89	C₄	C₂²×Dic₁₉₄	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×#15690810	1	1	yes	characteristic
92	C₂	#20261535	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#97×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×#97	D₉₇
hypocenter	C₉₇	#3162
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #20261144	C₂	semidirect
23. C₂×C₈×C₉₇	Q₈	semidirect
47. C₉₇	#3162	semidirect
65. C₂×C₈	Dic₁₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20261292	2
#20261144	2
#20260773	2
C₂×#97×C₉₇	2
#3162	97