Subgroups — #27158783

Order 12416.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261828	C₂	1	1	yes	maximal, characteristic
2	#20261825	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×#87604	C₂	1	1	yes	maximal, characteristic
4	#15690905	—	2	1	no	—
5	C₄⋊#11715906	—	2	1	no	—
6	Q₈×#87604	—	2	1	no	—
7	#15690879	C₄	1	1	yes	characteristic
8	#15690873	—	2	1	no	—
9	C₈×#87604	C₂²	1	1	yes	characteristic
10	Q₁₆×D₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715906	—	4	1	no	—
12	#13095523	—	2	2	no	—
13	#13095519	D₄	1	1	yes	characteristic
14	C₄⋊#87604	—	2	1	no	—
15	C₄⋊#87604	—	4	1	no	—
16	C₄×#87604	—	4	1	no	—
17	C₄×#87604	D₄	1	1	yes	characteristic
18	Q₈×D₉₇	—	2	1	no	—
19	Dic₃₈₈	C₈	1	1	yes	characteristic, complemented
20	#13095501	—	2	1	no	—
21	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
22	Q₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
23	C₂×#87604	—	4	1	no	—
24	C₂×#87604	D₈	1	1	yes	characteristic
25	C₂×#87604	SD₁₆	1	1	yes	characteristic
26	#11715906	—	8	1	no	—
27	Q₈×C₉₇	—	2	1	no	—
28	C₄×D₉₇	—	4	1	no	—
29	C₄×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
30	Dic₁₉₄	—	2	1	no	—
31	Dic₁₉₄	—	4	1	no	—
32	#11715897	—	2	1	no	—
33	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
34	#11715895	M₁₆	1	1	yes	characteristic
35	C₂×D₉₇	#102	1	1	yes	characteristic
36	#87604	—	2	1	no	—
37	#87604	—	8	1	no	—
38	#87604	SD₃₂	1	1	yes	characteristic
39	#87604	Q₃₂	1	1	yes	characteristic
40	C₄×C₉₇	—	4	1	no	—
41	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
42	Dic₉₇	—	4	1	no	—
43	Dic₉₇	#104	1	1	yes	characteristic
44	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
45	D₉₇	#358	1	1	yes	characteristic
46	D₉₇	#359	1	1	yes	characteristic
47	#1332	—	97	1	no	maximal, Sylow
48	C₉₇	#1332	1	1	yes	characteristic, complemented, Sylow
49	C₄×Q₁₆	—	97	1	no	—
50	C₄×C₁₆	—	97	1	no	—
51	#335	—	97	1	no	—
52	C₂×Q₁₆	—	97	1	no	—
53	C₄×Q₈	—	194	1	no	—
54	C₂×C₁₆	—	194	1	no	—
55	#107	—	97	1	no	—
56	C₄⋊C₈	—	194	1	no	—
57	#103	—	194	1	no	—
58	C₄×C₈	—	97	1	no	—
59	C₂×Q₈	—	194	1	no	—
60	Q₁₆	—	97	1	no	—
61	Q₁₆	—	194	1	no	—
62	Q₁₆	#11715906	1	1	yes	characteristic, complemented
63	C₂×C₈	—	97	1	no	—
64	C₂×C₈	—	97	1	no	—
65	C₂×C₈	—	388	1	no	—
66	C₄⋊C₄	—	388	1	no	—
67	C₄⋊C₄	—	194	1	no	—
68	C₄²	—	97	1	no	—
69	C₄²	—	388	1	no	—
70	C₁₆	—	194	2	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	388	1	no	—
73	Q₈	—	194	1	no	—
74	C₂×C₄	—	97	1	no	—
75	C₂×C₄	—	97	1	no	—
76	C₂×C₄	—	97	1	no	—
77	C₂×C₄	—	388	1	no	—
78	C₂×C₄	—	388	1	no	—
79	C₈	—	776	1	no	—
80	C₈	—	97	1	no	—
81	C₈	—	194	1	no	—
82	C₈	C₂×#11715906	1	1	yes	characteristic
83	C₂²	—	97	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	776	1	no	—
86	C₄	—	97	1	no	—
87	C₄	—	97	1	no	—
88	C₄	—	97	1	no	—
89	C₄	—	388	1	no	—
90	C₄	—	194	1	no	—
91	C₄	C₂²⋊#11715906	1	1	yes	characteristic
92	C₂	—	97	1	no	—
93	C₂	—	97	1	no	—
94	C₂	D₄⋊#11715906	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄⋊#11715906
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₈	C₂×#11715906
Fitting	Q₁₆×C₉₇	C₈
socle	C₂×C₉₇	D₄⋊C₈
prime radical	C₂×D₉₇	#102
cosocle	C₈×#87604	C₂²
hypercenter	Q₁₆	#11715906
hypocenter	C₉₇	#1332
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
19. Dic₃₈₈	C₈	semidirect
22. Q₁₆×C₉₇	C₈	semidirect
48. C₉₇	#1332	semidirect
62. Q₁₆	#11715906	semidirect

Maximal subgroups

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

subgroup	index
#20261828	2
#20261825	2
Q₁₆×#87604	2
#1332	97