Group #27150825 — subgroups

Subgroups — #27150825

Order 12416.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260866	C₂	1	6	yes	maximal, complemented
2	Q₈×#11715895	C₂	1	2	yes	maximal, complemented
3	#20260792	C₂	1	3	yes	maximal
4	#20260788	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11715895	C₂²	1	6	yes	—
7	C₄⋊#11715895	C₂²	1	6	yes	—
8	C₂²⋊#11715895	C₂²	1	1	yes	characteristic
9	C₂²⋊#11715895	C₂²	1	3	yes	—
10	#15690721	C₂²	1	6	yes	—
11	C₄×#11715895	C₂²	1	6	yes	—
12	C₂²×Q₈×C₉₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₉₇	C₄	1	3	yes	—
14	C₄×Q₈×C₉₇	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₉₇	C₂²	1	3	yes	—
16	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₉₇	C₄	1	3	yes	—
18	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
19	C₂×#11715895	C₂³	1	2	yes	—
20	C₂×#11715895	C₂³	1	6	yes	—
21	C₂×Q₈×C₉₇	—	2	4	no	—
22	C₂×Q₈×C₉₇	C₂×C₄	1	3	yes	—
23	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₉₇	C₂×C₄	1	6	yes	—
28	C₄⋊C₄×C₉₇	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₉₇	C₂×C₄	1	3	yes	—
30	C₄²×C₉₇	C₂×C₄	1	3	yes	—
31	C₄²×C₉₇	C₂×C₄	1	3	yes	—
32	C₄²×C₉₇	C₂³	1	3	yes	—
33	C₄²×C₉₇	C₂³	1	3	yes	—
34	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₉₇	—	2	6	no	—
36	Q₈×C₉₇	M₁₆	1	4	yes	—
37	C₂×C₄×C₉₇	—	2	6	no	—
38	C₂×C₄×C₉₇	—	2	6	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
46	#11715895	—	2	2	no	—
47	#11715895	—	2	6	no	—
48	C₂²×C₉₇	—	2	1	no	—
49	C₂²×C₉₇	—	2	1	no	—
50	C₂²×C₉₇	#131	1	2	yes	—
51	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₉₇	—	2	1	no	—
53	C₄×C₉₇	—	2	3	no	—
54	C₄×C₉₇	—	2	3	no	—
55	C₄×C₉₇	—	2	3	no	—
56	C₄×C₉₇	C₂×M₁₆	1	6	yes	—
57	C₄×C₉₇	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₉₇	—	2	2	no	—
59	C₂×C₉₇	#519	1	1	yes	characteristic
60	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
62	#2984	—	97	1	no	maximal, Sylow
63	C₉₇	#2984	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
65	#446	—	97	6	no	—
66	C₈×Q₈	—	97	2	no	—
67	#432	—	97	3	no	—
68	#431	—	97	3	no	—
69	C₂²×Q₈	Dic₉₇	1	1	yes	characteristic
70	C₄×Q₈	Dic₉₇	1	3	yes	—
71	C₄×Q₈	Dic₉₇	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₉₇	1	3	yes	—
74	C₂×C₄⋊C₄	Dic₉₇	1	3	yes	—
75	C₂×C₄²	C₂×D₉₇	1	3	yes	—
76	C₄⋊C₈	—	97	6	no	—
77	C₄⋊C₈	—	97	6	no	—
78	C₂²⋊C₈	—	97	1	no	—
79	C₂²⋊C₈	—	97	3	no	—
80	#97	—	97	6	no	—
81	C₄×C₈	—	97	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₉₇	1	3	yes	—
84	C₂×Q₈	C₂×Dic₉₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
86	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
87	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	C₂×C₈	—	97	2	no	—
89	C₂×C₈	—	97	6	no	—
90	C₄⋊C₄	C₂×Dic₉₇	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₉₇	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₉₇	1	3	yes	—
93	C₄²	C₂×Dic₉₇	1	3	yes	—
94	C₄²	C₂×Dic₉₇	1	3	yes	—
95	C₄²	C₂²×D₉₇	1	3	yes	—
96	C₄²	C₂²×D₉₇	1	3	yes	—
97	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#13095495	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₉₇	1	3	yes	—
105	C₂×C₄	C₂²×Dic₉₇	1	3	yes	—
106	C₂×C₄	C₂²×Dic₉₇	1	3	yes	—
107	C₂×C₄	C₂²×Dic₉₇	1	3	yes	—
108	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
109	C₈	—	194	2	no	—
110	C₈	—	194	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#15690817	1	2	yes	—
114	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#13095495	1	6	yes	—
120	C₄	#15690858	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#20261213	1	1	yes	characteristic
123	C₂	C₂²×#13095495	1	1	yes	characteristic
124	C₂	C₂×#15690817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260866

C₂

yes

maximal, complemented

Q₈×#11715895

C₂

yes

maximal, complemented

#20260792

C₂

yes

maximal

#20260788

C₂

yes

maximal

C₂×C₄×Q₈×C₉₇

C₂

yes

maximal, characteristic

C₄⋊#11715895

C₂²

yes

—

C₄⋊#11715895

C₂²

yes

—

C₂²⋊#11715895

C₂²

yes

characteristic

C₂²⋊#11715895

C₂²

yes

—

#15690721

C₂²

yes

—

C₄×#11715895

C₂²

yes

—

C₂²×Q₈×C₉₇

C₄

yes

characteristic

C₄×Q₈×C₉₇

C₄

yes

—

C₄×Q₈×C₉₇

C₄

yes

characteristic

C₄×Q₈×C₉₇

C₂²

yes

—

C₄×Q₈×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×Q₈×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂³

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#131

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

2₋¹⁺⁴

yes

—

C₂×C₉₇

—

C₂×C₉₇

#519

yes

characteristic

C₂×C₉₇

C₂²×M₁₆

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₉₇

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂²×Q₈

Dic₉₇

yes

characteristic

C₄×Q₈

Dic₉₇

yes

—

C₄×Q₈

Dic₉₇

yes

characteristic

C₄×Q₈

C₂×D₉₇

yes

characteristic

C₄×Q₈

C₂×D₉₇

yes

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₉₇

yes

—

C₂×Q₈

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄²

C₂×Dic₉₇

yes

—

C₄²

C₂×Dic₉₇

yes

—

C₄²

C₂²×D₉₇

yes

—

C₄²

C₂²×D₉₇

yes

—

C₂³

C₂²×Dic₉₇

yes

characteristic

Q₈

—

Q₈

#13095495

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

104

C₂×C₄

C₂²×Dic₉₇

yes

—

105

C₂×C₄

C₂²×Dic₉₇

yes

—

106

C₂×C₄

C₂²×Dic₉₇

yes

—

107

C₂×C₄

C₂²×Dic₉₇

yes

—

108

C₂×C₄

C₂³×D₉₇

yes

characteristic

109

C₈

—

194

—

110

C₈

—

194

—

111

C₂²

—

112

C₂²

—

113

C₂²

#15690817

yes

—

114

C₂²

C₂³×Dic₉₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#13095495

yes

—

120

C₄

#15690858

yes

—

121

C₂

—

122

C₂

#20261213

yes

characteristic

123

C₂

C₂²×#13095495

yes

characteristic

124

C₂

C₂×#15690817

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	#2984
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₉₇

C₂

prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂×C₄×C₉₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #20260866	C₂	semidirect
2. Q₈×#11715895	C₂	semidirect
63. C₉₇	#2984	semidirect

H ≅ G/N

type

1. #20260866

C₂

semidirect

2. Q₈×#11715895

C₂

semidirect

63. C₉₇

#2984

semidirect

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

subgroup	index
#20260866	2
Q₈×#11715895	2
#20260792	2
#20260788	2
C₂×C₄×Q₈×C₉₇	2
#2984	97

subgroup

index

C₂×C₄×Q₈×C₉₇

Subgroups — #27150825

Conjugacy classes of subgroups

Complements

Maximal subgroups