Subgroups — #27158787

Order 12416.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261840	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11715906	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261779	C₂	1	1	yes	maximal, characteristic
4	#15690907	—	2	1	no	—
5	C₄×#11715906	—	2	1	no	—
6	#15690898	—	2	1	no	—
7	#15690881	C₄	1	1	yes	characteristic
8	#15690875	—	2	1	no	—
9	C₈×#87604	C₂²	1	1	yes	characteristic
10	#15690833	C₄	1	1	yes	characteristic
11	C₂×#11715906	—	2	2	no	—
12	#13095523	—	4	1	no	—
13	#13095521	—	2	1	no	—
14	#13095521	—	4	1	no	—
15	#13095519	—	4	1	no	—
16	#13095519	D₄	1	1	yes	characteristic
17	C₄×#87604	D₄	1	1	yes	characteristic
18	#13095512	—	2	1	no	—
19	#13095498	—	2	1	no	—
20	D₇₇₆	C₈	1	1	yes	characteristic, complemented
21	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
22	Q₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
23	C₂×#87604	—	2	1	no	—
24	#11715906	—	2	4	no	—
25	Q₈×C₉₇	—	2	1	no	—
26	D₃₈₈	—	2	1	no	—
27	D₃₈₈	—	4	1	no	—
28	C₄×D₉₇	—	4	1	no	—
29	C₄×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
30	#11715897	—	4	1	no	—
31	#11715897	D₈	1	1	yes	characteristic
32	#11715897	SD₁₆	1	1	yes	characteristic
33	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
34	#11715895	M₁₆	1	1	yes	characteristic
35	C₂×D₉₇	—	4	1	no	—
36	C₂×D₉₇	#104	1	1	yes	characteristic
37	#87604	—	2	2	no	—
38	C₄×C₉₇	—	4	1	no	—
39	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
40	Dic₉₇	#102	1	1	yes	characteristic
41	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
42	D₉₇	—	2	1	no	—
43	D₉₇	—	8	1	no	—
44	#1335	—	97	1	no	maximal, Sylow
45	C₉₇	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	97	1	no	—
47	#364	—	97	1	no	—
48	C₈²	—	97	1	no	—
49	#135	—	97	1	no	—
50	#131	—	194	1	no	—
51	M₃₂	—	194	1	no	—
52	#108	—	97	1	no	—
53	#104	—	194	1	no	—
54	C₄×C₈	—	97	1	no	—
55	C₄×C₈	—	194	1	no	—
56	#41	—	194	1	no	—
57	Q₁₆	#11715906	1	1	yes	characteristic, complemented
58	SD₁₆	—	194	1	no	—
59	D₈	—	97	1	no	—
60	M₁₆	—	388	1	no	—
61	M₁₆	—	194	1	no	—
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	—	97	1	no	—
64	C₂×C₈	—	388	1	no	—
65	C₂×C₈	—	194	2	no	—
66	C₄²	—	97	1	no	—
67	C₁₆	—	388	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	388	1	no	—
70	D₄	—	194	1	no	—
71	C₂×C₄	—	97	1	no	—
72	C₂×C₄	—	388	1	no	—
73	C₂×C₄	—	194	1	no	—
74	C₈	—	97	1	no	—
75	C₈	—	97	1	no	—
76	C₈	—	97	1	no	—
77	C₈	—	388	1	no	—
78	C₈	—	194	4	no	—
79	C₈	C₂×#11715906	1	1	yes	characteristic
80	C₂²	—	97	1	no	—
81	C₂²	—	388	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	97	1	no	—
84	C₄	—	194	2	no	—
85	C₄	C₂²⋊#11715906	1	1	yes	characteristic
86	C₂	—	776	1	no	—
87	C₂	—	194	1	no	—
88	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	#15690833	C₄
cosocle	C₈×#87604	C₂²
hypercenter	Q₁₆	#11715906
hypocenter	C₉₇	#1335
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
1. #20261840	C₂	semidirect
2. C₈×#11715906	C₂	semidirect
20. D₇₇₆	C₈	semidirect
22. Q₁₆×C₉₇	C₈	semidirect
45. C₉₇	#1335	semidirect
57. Q₁₆	#11715906	semidirect

Maximal subgroups

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

subgroup	index
#20261840	2
C₈×#11715906	2
#20261779	2
#1335	97