Subgroups — #27144019

Order 12416.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261137	C₂	1	1	yes	maximal, characteristic
2	#20260562	C₂	1	2	yes	maximal
3	#20260560	C₂	1	2	yes	maximal
4	#20260511	C₂	1	1	yes	maximal, characteristic
5	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690802	—	2	1	no	—
7	C₂×#13095486	C₂²	1	2	yes	—
8	#15690744	C₂²	1	1	yes	characteristic
9	#15690744	C₂²	1	1	yes	characteristic
10	#15690742	—	2	1	no	—
11	C₄⋊Q₈×C₉₇	—	2	1	no	—
12	#107×C₉₇	—	2	1	no	—
13	#107×C₉₇	C₂²	1	2	yes	—
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₉₇	—	2	1	no	—
16	C₄⋊Dic₉₇	—	2	1	no	—
17	#13095490	—	4	1	no	—
18	#13095490	—	4	1	no	—
19	#13095486	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₉₇	—	4	1	no	—
21	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
22	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₄⋊C₄×C₉₇	—	4	1	no	—
25	C₄²×C₉₇	D₄	1	1	yes	characteristic
26	Q₈×C₉₇	—	8	2	no	—
27	C₂×C₄×C₉₇	—	4	2	no	—
28	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₉₇	—	4	1	no	—
32	C₂×Dic₉₇	—	4	1	no	—
33	C₈×C₉₇	Q₁₆	1	2	yes	—
34	C₈×C₉₇	C₂×Q₈	1	2	yes	—
35	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	—	2	1	no	—
38	C₄×C₉₇	—	8	2	no	—
39	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
40	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₉₇	—	8	1	no	—
42	Dic₉₇	—	8	1	no	—
43	C₂×C₉₇	#500	1	1	yes	characteristic
44	C₂×C₉₇	#509	1	1	yes	characteristic
45	C₂×C₉₇	#510	1	1	yes	characteristic
46	#2255	—	97	1	no	maximal, Sylow
47	C₉₇	#2255	1	1	yes	characteristic, complemented, Sylow
48	#500	D₉₇	1	1	yes	characteristic
49	#499	—	97	1	no	—
50	#367	—	97	2	no	—
51	#366	—	97	2	no	—
52	#346	—	97	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	194	1	no	—
55	C₂×C₁₆	—	97	2	no	—
56	#107	—	2	1	no	—
57	#107	—	97	1	no	—
58	#107	—	97	1	no	—
59	#107	C₂×D₉₇	1	2	yes	—
60	#106	—	194	1	no	—
61	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	388	1	no	—
68	C₄⋊C₄	—	388	1	no	—
69	C₄⋊C₄	—	194	1	no	—
70	C₄⋊C₄	—	194	1	no	—
71	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
72	C₁₆	—	97	4	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	388	1	no	—
76	C₂×C₄	—	388	1	no	—
77	C₂×C₄	#13095496	1	1	yes	characteristic
78	C₂×C₄	#13095496	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
80	C₈	#13095501	1	2	yes	—
81	C₈	Q₈×D₉₇	1	2	yes	—
82	C₂²	C₂×#13095496	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	2	no	—
86	C₄	—	776	1	no	—
87	C₄	—	776	1	no	—
88	C₄	#15690808	1	1	yes	characteristic
89	C₄	C₂×#13095501	1	1	yes	characteristic
90	C₂	#20261138	1	1	yes	characteristic
91	C₂	#20261171	1	1	yes	characteristic
92	C₂	#20261176	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	#2255
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
19. #13095486	Q₈	semidirect
47. C₉₇	#2255	semidirect

Maximal subgroups

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

subgroup	index
#20261137	2
#20260562	2
#20260560	2
#20260511	2
#500×C₉₇	2
#2255	97