Subgroups — #27141345

Order 12416.

168 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	#20261087	C₂	1	2	yes	maximal
3	#20261085	C₂	1	2	yes	maximal
4	C₈×#11715895	C₂	1	1	yes	maximal, characteristic
5	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690810	C₂²	1	2	yes	—
7	#15690744	—	2	2	no	—
8	#15690734	—	2	4	no	—
9	#15690733	—	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	C₂×#11715895	D₄	1	2	yes	—
17	C₂×#11715895	D₄	1	2	yes	—
18	C₄⋊Dic₉₇	—	2	2	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₂×Q₈×C₉₇	—	2	2	no	—
21	C₂×C₈×C₉₇	D₄	1	2	yes	—
22	C₄⋊C₄×C₉₇	—	2	2	no	—
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₄²×C₉₇	C₂³	1	1	yes	characteristic
25	Q₈×C₉₇	—	4	4	no	—
26	C₂×C₄×C₉₇	—	4	2	no	—
27	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₉₇	—	4	2	no	—
31	Dic₁₉₄	—	4	4	no	—
32	C₈×C₉₇	Q₁₆	1	4	yes	—
33	#11715895	—	2	2	no	—
34	#11715895	—	2	2	no	—
35	C₂²×C₉₇	#127	1	1	yes	characteristic
36	C₄×C₉₇	—	8	2	no	—
37	C₄×C₉₇	C₂×Q₁₆	1	2	yes	—
38	C₄×C₉₇	#135	1	2	yes	—
39	C₄×C₉₇	#135	1	2	yes	—
40	Dic₉₇	—	8	2	no	—
41	C₂×C₉₇	#494	1	1	yes	characteristic
42	C₂×C₉₇	#495	1	1	yes	characteristic
43	C₂×C₉₇	#495	1	1	yes	characteristic
44	#1710	—	97	1	no	maximal, Sylow
45	C₉₇	#1710	1	1	yes	characteristic, complemented, Sylow
46	#500	—	97	1	no	—
47	#500	D₉₇	1	1	yes	characteristic
48	#487	—	97	2	no	—
49	#487	—	97	2	no	—
50	C₈²	—	97	1	no	—
51	C₄⋊Q₈	—	97	2	no	—
52	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
53	#107	—	2	2	no	—
54	#107	—	194	2	no	—
55	#103	—	194	4	no	—
56	#103	—	194	4	no	—
57	C₄×C₈	—	97	1	no	—
58	C₄×C₈	—	97	1	no	—
59	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	194	2	no	—
62	C₂×C₈	—	97	2	no	—
63	C₂×C₈	—	97	2	no	—
64	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	194	2	no	—
68	C₄⋊C₄	—	194	2	no	—
69	C₄²	C₂²×D₉₇	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	388	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	388	2	no	—
74	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
75	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
77	C₈	—	194	2	no	—
78	C₈	—	194	2	no	—
79	C₈	#13095501	1	4	yes	—
80	C₂²	#15690806	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	776	2	no	—
83	C₄	C₂×#13095501	1	2	yes	—
84	C₄	#15690829	1	2	yes	—
85	C₄	#15690833	1	2	yes	—
86	C₂	#20261114	1	1	yes	characteristic
87	C₂	#20261118	1	1	yes	characteristic
88	C₂	#20261120	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	#1710
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
45. C₉₇	#1710	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20261087	2
#20261085	2
C₈×#11715895	2
#500×C₉₇	2
#1710	97