Subgroups — #27141353

Order 12416.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261140	C₂	1	1	yes	maximal, characteristic
2	#20261138	C₂	1	2	yes	maximal
3	#20261114	C₂	1	2	yes	maximal
4	C₈×#11715895	C₂	1	1	yes	maximal, characteristic
5	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095501	—	2	4	no	—
7	#15690810	C₂²	1	2	yes	—
8	#15690744	—	2	2	no	—
9	#15690743	—	2	4	no	—
10	C₄×#11715895	C₂²	1	1	yes	characteristic
11	C₄×#11715895	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
13	#107×C₉₇	—	2	2	no	—
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₄	—	2	2	no	—
16	#13095501	—	4	8	no	—
17	C₂×#11715895	D₄	1	2	yes	—
18	C₂×#11715895	D₄	1	2	yes	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	—	2	2	no	—
21	C₂×Q₈×C₉₇	—	2	2	no	—
22	C₂×C₈×C₉₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₄⋊C₄×C₉₇	—	2	2	no	—
25	C₄²×C₉₇	C₂³	1	1	yes	characteristic
26	Q₈×C₉₇	—	4	4	no	—
27	C₂×C₄×C₉₇	—	4	2	no	—
28	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₉₇	—	4	2	no	—
32	Dic₁₉₄	—	4	4	no	—
33	C₈×C₉₇	Q₁₆	1	4	yes	complemented
34	#11715895	D₈	1	4	yes	—
35	#11715895	Q₁₆	1	4	yes	complemented
36	C₂²×C₉₇	#127	1	1	yes	characteristic
37	C₄×C₉₇	—	8	2	no	—
38	C₄×C₉₇	C₂×D₈	1	2	yes	—
39	C₄×C₉₇	C₂×Q₁₆	1	2	yes	—
40	C₄×C₉₇	C₂×Q₁₆	1	2	yes	—
41	Dic₉₇	—	8	2	no	—
42	C₂×C₉₇	#493	1	1	yes	characteristic
43	C₂×C₉₇	#494	1	1	yes	characteristic
44	C₂×C₉₇	#494	1	1	yes	characteristic
45	#1713	—	97	1	no	maximal, Sylow
46	C₉₇	#1713	1	1	yes	characteristic, complemented, Sylow
47	#500	—	97	2	no	—
48	#500	—	97	1	no	—
49	#500	D₉₇	1	1	yes	characteristic
50	#494	—	97	2	no	—
51	C₈²	—	97	1	no	—
52	C₂×Q₁₆	—	194	4	no	—
53	C₄⋊Q₈	—	97	2	no	—
54	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
55	#107	—	2	2	no	—
56	#107	—	194	4	no	—
57	#107	—	194	2	no	—
58	C₄×C₈	—	97	1	no	—
59	C₄×C₈	—	97	1	no	—
60	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	194	2	no	—
63	Q₁₆	—	388	8	no	—
64	C₂×C₈	—	97	2	no	—
65	C₂×C₈	—	97	2	no	—
66	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	194	2	no	—
70	C₄⋊C₄	—	194	2	no	—
71	C₄²	C₂²×D₉₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	388	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	388	2	no	—
76	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
79	C₈	—	97	4	no	—
80	C₈	—	97	4	no	—
81	C₈	#13095501	1	4	yes	complemented
82	C₂²	#15690806	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	776	2	no	—
85	C₄	D₈×D₉₇	1	2	yes	—
86	C₄	Q₁₆×D₉₇	1	2	yes	—
87	C₄	C₂×#13095501	1	2	yes	—
88	C₂	#20261110	1	1	yes	characteristic
89	C₂	#20261113	1	1	yes	characteristic
90	C₂	#20261114	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#500×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#127
hypercenter	#500	D₉₇
hypocenter	C₉₇	#1713
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
33. C₈×C₉₇	Q₁₆	semidirect
35. #11715895	Q₁₆	semidirect
46. C₉₇	#1713	semidirect
81. C₈	#13095501	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20261138	2
#20261114	2
C₈×#11715895	2
#500×C₉₇	2
#1713	97