Subgroups — #27143983

Order 12416.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261115	C₂	1	1	yes	maximal, characteristic
2	#20260538	C₂	1	4	yes	maximal
3	#20260511	C₂	1	1	yes	maximal, characteristic
4	#499×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₈	—	2	1	no	—
6	C₂×Dic₃₈₈	C₂²	1	2	yes	—
7	#15690810	—	2	1	no	—
8	C₂×#13095486	C₂²	1	2	yes	—
9	#125×C₉₇	—	2	1	no	—
10	#107×C₉₇	C₂²	1	2	yes	—
11	#106×C₉₇	—	2	1	no	—
12	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₄	—	2	2	no	—
14	Dic₃₈₈	—	2	4	no	—
15	Dic₃₈₈	—	4	2	no	—
16	C₄⋊Dic₉₇	—	4	2	no	—
17	#13095486	—	2	2	no	—
18	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₉₇	—	2	2	no	—
21	C₄⋊C₄×C₉₇	—	4	2	no	—
22	C₄²×C₉₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	4	2	no	—
24	C₂×C₄×C₉₇	D₈	1	2	yes	—
25	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
26	C₂×Dic₉₇	—	4	2	no	—
27	Dic₁₉₄	—	4	2	no	—
28	Dic₁₉₄	—	4	2	no	—
29	C₈×C₉₇	SD₁₆	1	2	yes	—
30	C₈×C₉₇	#41	1	2	yes	—
31	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₉₇	—	2	2	no	—
33	C₄×C₉₇	—	8	2	no	—
34	C₄×C₉₇	#124	1	1	yes	characteristic
35	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
36	Dic₉₇	—	8	2	no	—
37	C₂×C₉₇	#486	1	1	yes	characteristic
38	C₂×C₉₇	#510	1	2	yes	—
39	#2245	—	97	1	no	maximal, Sylow
40	C₉₇	#2245	1	1	yes	characteristic, complemented, Sylow
41	#499	D₉₇	1	1	yes	characteristic
42	#494	—	97	1	no	—
43	#358	—	97	4	no	—
44	#346	—	97	1	no	—
45	C₂×Q₁₆	—	97	2	no	—
46	C₂×Q₁₆	—	194	1	no	—
47	C₄⋊Q₈	—	194	1	no	—
48	#125	—	2	1	no	—
49	C₂×C₁₆	—	97	2	no	—
50	#107	C₂×D₉₇	1	2	yes	—
51	#106	—	2	1	no	—
52	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
53	C₂×Q₈	—	194	2	no	—
54	Q₁₆	—	388	2	no	—
55	Q₁₆	—	194	4	no	—
56	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	388	2	no	—
61	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
62	C₁₆	—	194	2	no	—
63	Q₈	—	388	2	no	—
64	Q₈	—	388	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	388	2	no	—
67	C₂×C₄	#13095496	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
69	C₈	#13095498	1	2	yes	—
70	C₈	#13095512	1	2	yes	—
71	C₂²	C₂×#13095496	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	776	2	no	—
75	C₄	#15690799	1	1	yes	characteristic
76	C₄	C₂×#13095498	1	1	yes	characteristic
77	C₂	#20261083	1	1	yes	characteristic
78	C₂	#20261175	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13095496
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#499×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	#499	D₉₇
hypocenter	C₉₇	#2245
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₉₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#20261115	2
#20260538	2
#20260511	2
#499×C₉₇	2
#2245	97