Group #27154410 — subgroups

Subgroups — #27154410

Order 12416.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	#20261085	C₂	1	3	yes	maximal
3	#20261030	C₂	1	3	yes	maximal
4	#20260942	C₂	1	3	yes	maximal
5	Q₈×#11715895	C₂	1	1	yes	maximal, characteristic
6	C₄×#13095501	C₂	1	3	yes	maximal
7	#557×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095501	C₂²	1	3	yes	—
9	#15690810	C₂²	1	3	yes	—
10	#15690809	—	2	3	no	—
11	Q₈×Dic₉₇	—	2	1	no	—
12	C₄×Dic₁₉₄	C₂²	1	3	yes	—
13	#15690743	C₂²	1	3	yes	—
14	C₄⋊#11715895	C₂²	1	3	yes	—
15	#15690734	C₂²	1	6	yes	—
16	#15690734	C₂²	1	3	yes	—
17	#15690733	C₂²	1	1	yes	characteristic
18	C₄×#11715895	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₉₇	C₂²	1	3	yes	—
20	#125×C₉₇	—	2	3	no	—
21	C₄×Q₈×C₉₇	—	2	1	no	—
22	C₄×Q₈×C₉₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₉₄	—	2	3	no	—
25	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
26	#13095501	—	2	6	no	—
27	C₂×#11715895	C₂³	1	3	yes	—
28	C₂×#11715895	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₉₇	—	2	3	no	—
30	C₄⋊Dic₉₇	C₂³	1	3	yes	—
31	#13095490	—	2	3	no	—
32	C₄×Dic₉₇	—	2	3	no	—
33	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₉₇	C₂³	1	3	yes	—
35	C₄⋊C₄×C₉₇	—	2	3	no	—
36	C₄⋊C₄×C₉₇	—	2	3	no	—
37	C₄⋊C₄×C₉₇	—	2	3	no	—
38	C₄⋊C₄×C₉₇	D₄	1	3	yes	—
39	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
40	C₄²×C₉₇	—	2	3	no	—
41	C₄²×C₉₇	C₂³	1	3	yes	—
42	Q₈×C₉₇	—	2	3	no	—
43	Q₈×C₉₇	—	2	3	no	—
44	Q₈×C₉₇	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₉₇	—	2	1	no	—
46	C₂×C₄×C₉₇	—	2	3	no	—
47	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₉₇	—	2	1	no	—
51	C₂×Dic₉₇	—	2	3	no	—
52	Dic₁₉₄	—	2	1	no	—
53	Dic₁₉₄	—	8	3	no	—
54	Dic₁₉₄	#41	1	2	yes	—
55	#11715895	—	2	3	no	—
56	#11715895	—	2	1	no	—
57	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₉₇	—	2	3	no	—
59	C₄×C₉₇	—	4	3	no	—
60	C₄×C₉₇	—	4	1	no	—
61	C₄×C₉₇	C₂×Q₁₆	1	6	yes	—
62	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
63	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₉₇	—	2	2	no	—
65	Dic₉₇	—	4	3	no	—
66	C₂×C₉₇	#548	1	1	yes	characteristic
67	C₂×C₉₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₉₇	#577	1	1	yes	characteristic
69	#3363	—	97	1	no	maximal, Sylow
70	C₉₇	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	—	97	1	no	—
72	#557	D₉₇	1	1	yes	characteristic
73	#487	—	97	3	no	—
74	#477	—	97	3	no	—
75	#462	—	97	3	no	—
76	C₈×Q₈	—	97	1	no	—
77	C₄×Q₁₆	—	97	3	no	—
78	C₂×Q₁₆	—	97	3	no	—
79	C₄⋊Q₈	—	97	3	no	—
80	C₄⋊Q₈	—	194	3	no	—
81	C₄⋊Q₈	C₂×D₉₇	1	3	yes	—
82	#125	—	2	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	97	3	no	—
85	C₄×Q₈	—	194	1	no	—
86	C₄×Q₈	C₂×D₉₇	1	3	yes	—
87	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
88	#107	—	97	3	no	—
89	C₄⋊C₈	—	97	3	no	—
90	#103	—	97	3	no	—
91	#103	—	97	6	no	—
92	#103	—	97	1	no	—
93	C₄×C₈	—	97	3	no	—
94	C₂×Q₈	—	97	1	no	—
95	C₂×Q₈	—	194	3	no	—
96	C₂×Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₉₇	1	3	yes	—
98	Q₁₆	—	194	6	no	—
99	C₂×C₈	—	97	1	no	—
100	C₂×C₈	—	97	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	97	3	no	—
105	C₄⋊C₄	—	194	3	no	—
106	C₄⋊C₄	—	194	3	no	—
107	C₄⋊C₄	C₂²⋊D₉₇	1	3	yes	—
108	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	194	3	no	—
111	C₄²	C₂²×D₉₇	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	776	3	no	—
115	Q₈	—	97	2	no	—
116	Q₈	—	194	1	no	—
117	Q₈	#13095501	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	194	3	no	—
121	C₂×C₄	—	194	1	no	—
122	C₂×C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
124	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
125	C₈	—	194	3	no	—
126	C₈	—	194	1	no	—
127	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	388	3	no	—
132	C₄	—	194	2	no	—
133	C₄	C₂×#13095501	1	6	yes	—
134	C₄	#41×D₉₇	1	1	yes	characteristic
135	C₄	#15690857	1	1	yes	characteristic
136	C₂	#20261392	1	1	yes	characteristic
137	C₂	C₂²×#13095501	1	1	yes	characteristic
138	C₂	#20261543	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₉₄

C₂

yes

maximal, characteristic

#20261085

C₂

yes

maximal

#20261030

C₂

yes

maximal

#20260942

C₂

yes

maximal

Q₈×#11715895

C₂

yes

maximal, characteristic

C₄×#13095501

C₂

yes

maximal

#557×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095501

C₂²

yes

—

#15690810

C₂²

yes

—

#15690809

—

Q₈×Dic₉₇

—

C₄×Dic₁₉₄

C₂²

yes

—

#15690743

C₂²

yes

—

C₄⋊#11715895

C₂²

yes

—

#15690734

C₂²

yes

—

#15690734

C₂²

yes

—

#15690733

C₂²

yes

characteristic

C₄×#11715895

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

#125×C₉₇

—

C₄×Q₈×C₉₇

—

C₄×Q₈×C₉₇

C₂²

yes

—

C₄×Q₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂³

yes

characteristic

#13095501

—

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

characteristic

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

D₄

yes

characteristic

C₂×Q₈×C₉₇

C₂³

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

—

C₄²×C₉₇

C₂³

yes

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

Q₁₆

yes

complemented

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

#41

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×Q₁₆

yes

—

C₄×C₉₇

C₂×#41

yes

characteristic

C₄×C₉₇

2₋¹⁺⁴

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#548

yes

characteristic

C₂×C₉₇

C₂²×Q₁₆

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

Q₈²

—

#557

D₉₇

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

194

—

C₄×Q₈

C₂×D₉₇

yes

—

C₄×Q₈

C₂×D₉₇

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂²⋊D₉₇

yes

characteristic

C₂×Q₈

C₂²×D₉₇

yes

—

Q₁₆

—

194

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

194

—

106

C₄⋊C₄

—

194

—

107

C₄⋊C₄

C₂²⋊D₉₇

yes

—

108

C₄⋊C₄

C₂²×D₉₇

yes

characteristic

109

C₄²

—

110

C₄²

—

194

—

111

C₄²

C₂²×D₉₇

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

776

—

115

Q₈

—

116

Q₈

—

194

—

117

Q₈

#13095501

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

194

—

121

C₂×C₄

—

194

—

122

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

124

C₂×C₄

C₂³×D₉₇

yes

characteristic

125

C₈

—

194

—

126

C₈

—

194

—

127

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

388

—

132

C₄

—

194

—

133

C₄

C₂×#13095501

yes

—

134

C₄

#41×D₉₇

yes

characteristic

135

C₄

#15690857

yes

characteristic

136

C₂

#20261392

yes

characteristic

137

C₂

C₂²×#13095501

yes

characteristic

138

C₂

#20261543

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#557×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂²×D₄
hypercenter	#557	D₉₇
hypocenter	C₉₇	#3363
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

#557×C₉₇

C₂

socle, prime radical

C₂²×C₉₇

C₂²×D₄

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
44. Q₈×C₉₇	Q₁₆	semidirect
70. C₉₇	#3363	semidirect
117. Q₈	#13095501	semidirect

H ≅ G/N

type

44. Q₈×C₉₇

Q₁₆

semidirect

70. C₉₇

#3363

semidirect

117. Q₈

#13095501

semidirect

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

subgroup	index
Q₈×Dic₁₉₄	2
#20261085	2
#20261030	2
#20260942	2
Q₈×#11715895	2
C₄×#13095501	2
#557×C₉₇	2
#3363	97

subgroup

index

Q₈×Dic₁₉₄

#557×C₉₇

Subgroups — #27154410

Conjugacy classes of subgroups

Complements

Maximal subgroups