Group #27142130 — subgroups

Subgroups — #27142130

Order 12416.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20261213	C₂	1	1	yes	maximal, characteristic
2	#20260744	C₂	1	2	yes	maximal
3	C₂×#15690729	C₂	1	1	yes	maximal, characteristic
4	#20260495	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	Q₈×Dic₉₇	—	2	4	no	—
8	#15690769	—	2	2	no	—
9	#15690769	C₂²	1	2	yes	—
10	#15690734	—	2	4	no	—
11	#15690729	—	2	2	no	—
12	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₉₇	—	2	2	no	—
15	C₄⋊Q₈×C₉₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	#13095495	—	2	2	no	—
19	#13095495	—	2	2	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₄⋊Dic₉₇	—	2	4	no	—
22	C₄⋊Dic₉₇	D₄	1	4	yes	—
23	C₂²⋊Dic₉₇	—	2	2	no	—
24	C₄×Dic₉₇	—	2	4	no	—
25	C₄×Dic₉₇	—	2	2	no	—
26	C₂×Q₈×C₉₇	—	2	2	no	—
27	C₂×Q₈×C₉₇	—	2	1	no	—
28	C₂×Q₈×C₉₇	—	2	2	no	—
29	C₂×Q₈×C₉₇	—	2	2	no	—
30	C₂×Q₈×C₉₇	—	2	2	no	—
31	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₉₇	D₄	1	2	yes	—
34	C₂²×C₄×C₉₇	—	2	1	no	—
35	C₂²×C₄×C₉₇	—	2	1	no	—
36	C₂²×C₄×C₉₇	D₄	1	2	yes	—
37	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₉₇	—	2	8	no	—
39	C₄²×C₉₇	—	2	1	no	—
40	C₄²×C₉₇	C₂×C₄	1	2	yes	—
41	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₉₇	—	2	4	no	—
43	Q₈×C₉₇	—	2	4	no	—
44	Q₈×C₉₇	—	2	4	no	—
45	Q₈×C₉₇	—	4	2	no	—
46	C₂×C₄×C₉₇	—	2	4	no	—
47	C₂×C₄×C₉₇	—	2	2	no	—
48	C₂×C₄×C₉₇	—	2	2	no	—
49	C₂×C₄×C₉₇	—	2	2	no	—
50	C₂×C₄×C₉₇	—	2	2	no	—
51	C₂×C₄×C₉₇	—	4	1	no	—
52	C₂×C₄×C₉₇	—	4	1	no	—
53	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	#41	1	2	yes	—
58	C₂×Dic₉₇	—	2	4	no	—
59	#11715895	—	4	2	no	—
60	C₂²×C₉₇	—	2	2	no	—
61	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₉₇	#120	1	1	yes	characteristic
63	C₂²×C₉₇	#123	1	1	yes	characteristic
64	C₄×C₉₇	—	2	4	no	—
65	C₄×C₉₇	—	2	4	no	—
66	C₄×C₉₇	—	4	2	no	—
67	C₄×C₉₇	C₄×D₄	1	2	yes	—
68	C₄×C₉₇	#121	1	2	yes	—
69	Dic₉₇	—	4	4	no	—
70	C₂×C₉₇	—	2	1	no	—
71	C₂×C₉₇	—	2	1	no	—
72	C₂×C₉₇	#386	1	1	yes	characteristic
73	C₂×C₉₇	#456	1	2	yes	—
74	#1886	—	97	1	no	maximal, Sylow
75	C₉₇	#1886	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊Q₈	D₉₇	1	1	yes	characteristic
77	#519	—	97	1	no	—
78	#419	—	97	2	no	—
79	C₂×#101	—	97	1	no	—
80	#339	—	97	2	no	—
81	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
83	C₂×M₁₆	—	97	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	Dic₉₇	1	4	yes	complemented
86	C₄×Q₈	—	194	4	no	—
87	#117	—	97	2	no	—
88	#117	—	194	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
91	#103	—	194	4	no	—
92	#101	—	194	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₉₇	1	2	yes	—
99	C₂×Q₈	C₂×Dic₉₇	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₉₇	1	2	yes	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
105	M₁₆	—	194	2	no	—
106	M₁₆	—	194	2	no	—
107	C₂×C₈	—	194	2	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	97	4	no	—
110	C₄⋊C₄	—	194	4	no	—
111	C₂²⋊C₄	—	194	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	194	4	no	—
114	C₄²	—	194	2	no	—
115	C₄²	C₂×Dic₉₇	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	194	4	no	—
129	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
130	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
131	C₂×C₄	D₄×D₉₇	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
133	C₂×C₄	#13095510	1	2	yes	—
134	C₈	—	388	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
137	C₂²	#15690778	1	1	yes	characteristic
138	C₂²	#15690791	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	388	4	no	—
143	C₄	D₄×Dic₉₇	1	2	yes	—
144	C₄	#15690782	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#20260615	1	1	yes	characteristic
148	C₂	#20260917	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261213

C₂

yes

maximal, characteristic

#20260744

C₂

yes

maximal

C₂×#15690729

C₂

yes

maximal, characteristic

#20260495

C₂

yes

maximal

C₂×C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

Q₈×Dic₉₇

—

#15690769

—

#15690769

C₂²

yes

—

#15690734

—

#15690729

—

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

D₄

yes

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

#121

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#386

yes

characteristic

C₂×C₉₇

#456

yes

—

#1886

—

maximal, Sylow

C₉₇

#1886

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₉₇

yes

—

—

—

—

C₂²×Q₈

C₂×D₉₇

yes

characteristic

C₂²×Q₈

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₉₇

yes

complemented

C₄×Q₈

—

194

—

#117

—

#117

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

194

—

#101

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₉₇

yes

—

C₂×Q₈

C₂×Dic₉₇

yes

—

100

C₂×Q₈

C₂²⋊D₉₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₉₇

yes

—

104

C₂²×C₄

C₂²×D₉₇

yes

characteristic

105

M₁₆

—

194

—

106

M₁₆

—

194

—

107

C₂×C₈

—

194

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

194

—

111

C₂²⋊C₄

—

194

—

112

C₄²

—

113

C₄²

—

194

—

114

C₄²

—

194

—

115

C₄²

C₂×Dic₉₇

yes

—

116

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

194

—

129

C₂×C₄

C₂²⋊Dic₉₇

yes

—

130

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

131

C₂×C₄

D₄×D₉₇

yes

—

132

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

133

C₂×C₄

#13095510

yes

—

134

C₈

—

388

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

137

C₂²

#15690778

yes

characteristic

138

C₂²

#15690791

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

388

—

143

C₄

D₄×Dic₉₇

yes

—

144

C₄

#15690782

yes

—

145

C₂

—

146

C₂

—

147

C₂

#20260615

yes

characteristic

148

C₂

#20260917

yes

—

	subgroup	G/H
center	C₂²	#15690778
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄⋊Q₈×C₉₇	C₂
socle	C₂²×C₉₇	#120
prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#1886
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#15690778

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄⋊Q₈×C₉₇

C₂

socle

C₂²×C₉₇

#120

prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₉₇

hypocenter

C₉₇

#1886

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
15. C₄⋊Q₈×C₉₇	C₄	semidirect
75. C₉₇	#1886	semidirect
85. C₄⋊Q₈	Dic₉₇	semidirect

H ≅ G/N

type

15. C₄⋊Q₈×C₉₇

C₄

semidirect

75. C₉₇

#1886

semidirect

85. C₄⋊Q₈

Dic₉₇

semidirect

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

subgroup	index
#20261213	2
#20260744	2
C₂×#15690729	2
#20260495	2
C₂×C₄⋊Q₈×C₉₇	2
#1886	97

subgroup

index

C₂×C₄⋊Q₈×C₉₇

Subgroups — #27142130

Conjugacy classes of subgroups

Complements

Maximal subgroups