Subgroups — #27144016

Order 12416.

128 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	#20260562	C₂	1	4	yes	maximal
3	C₄×#13095486	C₂	1	1	yes	maximal, characteristic
4	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690810	—	2	1	no	—
6	C₂×#13095486	C₂²	1	2	yes	—
7	#15690744	—	2	1	no	—
8	#15690744	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₉₇	—	2	1	no	—
10	#107×C₉₇	—	2	1	no	—
11	#107×C₉₇	C₂²	1	2	yes	—
12	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₄	—	4	1	no	—
14	C₄⋊Dic₉₇	—	2	2	no	—
15	C₄⋊Dic₉₇	—	4	1	no	—
16	#13095486	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₉₇	—	4	1	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	1	no	—
22	C₄²×C₉₇	D₄	1	1	yes	characteristic
23	Q₈×C₉₇	—	8	2	no	—
24	C₂×C₄×C₉₇	—	4	2	no	—
25	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	2	no	—
29	Dic₁₉₄	—	8	2	no	—
30	C₈×C₉₇	Q₁₆	1	2	yes	—
31	C₈×C₉₇	C₂×Q₈	1	2	yes	—
32	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₉₇	—	8	2	no	—
34	C₄×C₉₇	SD₃₂	1	2	yes	—
35	C₄×C₉₇	SD₃₂	1	2	yes	—
36	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₉₇	—	8	2	no	—
39	C₂×C₉₇	#500	1	1	yes	characteristic
40	C₂×C₉₇	C₂×SD₃₂	1	1	yes	characteristic
41	C₂×C₉₇	C₂×SD₃₂	1	1	yes	characteristic
42	#2254	—	97	1	no	maximal, Sylow
43	C₉₇	#2254	1	1	yes	characteristic, complemented, Sylow
44	#500	—	97	1	no	—
45	#500	D₉₇	1	1	yes	characteristic
46	#367	—	97	4	no	—
47	C₄×C₁₆	—	97	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	194	1	no	—
50	C₂×C₁₆	—	97	2	no	—
51	#107	—	2	1	no	—
52	#107	—	97	2	no	—
53	#107	—	194	1	no	—
54	#107	C₂×D₉₇	1	2	yes	—
55	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	388	1	no	—
58	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	388	1	no	—
63	C₄⋊C₄	—	194	2	no	—
64	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
65	C₁₆	—	97	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	776	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	388	2	no	—
70	C₂×C₄	#13095496	1	1	yes	characteristic
71	C₂×C₄	#13095496	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
73	C₈	#13095501	1	2	yes	—
74	C₈	Q₈×D₉₇	1	2	yes	—
75	C₂²	C₂×#13095496	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	776	2	no	—
78	C₄	#15690753	1	2	yes	—
79	C₄	#15690754	1	2	yes	—
80	C₄	#15690808	1	1	yes	characteristic
81	C₄	C₂×#13095501	1	1	yes	characteristic
82	C₂	#20261138	1	1	yes	characteristic
83	C₂	C₂×#15690753	1	1	yes	characteristic
84	C₂	C₂×#15690754	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13095496
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#500×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	#500	D₉₇
hypocenter	C₉₇	#2254
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. #13095486	Q₈	semidirect
43. C₉₇	#2254	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20260562	2
C₄×#13095486	2
#500×C₉₇	2
#2254	97