Group #27142597 — subgroups

Subgroups — #27142597

Order 12416.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690744	C₂	1	1	yes	maximal, characteristic
3	C₂×#15690735	C₂	1	2	yes	maximal
4	#20260641	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
7	C₂×Dic₃₈₈	—	2	2	no	—
8	C₂×Dic₃₈₈	—	2	2	no	—
9	C₂×Dic₃₈₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
11	#15690744	—	2	2	no	—
12	#15690735	—	2	4	no	—
13	#15690717	—	2	4	no	—
14	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₉₇	—	2	2	no	—
18	C₂×Dic₁₉₄	—	2	2	no	—
19	C₂×Dic₁₉₄	—	2	2	no	—
20	C₂×Dic₁₉₄	—	2	2	no	—
21	C₂×Dic₁₉₄	—	2	2	no	—
22	C₂×Dic₁₉₄	—	2	2	no	—
23	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
24	C₂²×Dic₉₇	—	2	2	no	—
25	C₂²×Dic₉₇	—	2	2	no	—
26	Dic₃₈₈	—	2	8	no	—
27	C₄⋊Dic₉₇	—	2	2	no	—
28	C₄⋊Dic₉₇	—	2	2	no	—
29	C₂²×C₄×C₉₇	D₄	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₉₇	—	2	1	no	—
32	C₂×C₈×C₉₇	—	2	1	no	—
33	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₉₇	D₄	1	2	yes	—
35	C₂×C₈×C₉₇	D₄	1	2	yes	—
36	C₂×C₈×C₉₇	D₄	1	2	yes	—
37	C₄²×C₉₇	—	2	1	no	—
38	C₄²×C₉₇	—	2	1	no	—
39	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
49	C₂×Dic₉₇	—	2	4	no	—
50	C₂×Dic₉₇	—	4	2	no	—
51	C₂×Dic₉₇	—	4	2	no	—
52	C₂×Dic₉₇	—	4	2	no	—
53	C₂×Dic₉₇	—	4	4	no	—
54	Dic₁₉₄	—	2	4	no	—
55	Dic₁₉₄	—	2	4	no	—
56	Dic₁₉₄	—	4	2	no	—
57	Dic₁₉₄	—	4	2	no	—
58	C₈×C₉₇	—	2	2	no	—
59	C₈×C₉₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₉₇	#121	1	1	yes	characteristic
62	C₂²×C₉₇	#127	1	1	yes	characteristic
63	C₂²×C₉₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₉₇	#135	1	2	yes	—
65	C₄×C₉₇	—	2	4	no	—
66	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
68	C₄×C₉₇	#121	1	1	yes	characteristic
69	C₄×C₉₇	#124	1	1	yes	characteristic
70	Dic₉₇	—	4	4	no	—
71	Dic₉₇	—	8	2	no	—
72	C₂×C₉₇	#390	1	1	yes	characteristic
73	C₂×C₉₇	C₄×Q₁₆	1	2	yes	—
74	C₂×C₉₇	#467	1	2	yes	—
75	C₂×C₉₇	#494	1	1	yes	characteristic
76	C₂×C₉₇	#495	1	1	yes	characteristic
77	#1969	—	97	1	no	maximal, Sylow
78	C₉₇	#1969	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	97	1	no	—
80	C₂×#107	—	97	1	no	—
81	C₂×#103	—	97	2	no	—
82	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
83	#391	—	97	2	no	—
84	C₂²×Q₈	—	97	2	no	—
85	C₂×Q₁₆	—	97	4	no	—
86	C₂×Q₁₆	—	194	2	no	—
87	C₂×Q₁₆	—	194	2	no	—
88	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	97	2	no	—
91	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
92	#107	—	194	2	no	—
93	#103	—	194	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	194	4	no	—
96	C₂×Q₈	—	97	4	no	—
97	C₂×Q₈	—	194	2	no	—
98	C₂×Q₈	—	194	2	no	—
99	C₂×Q₈	—	194	2	no	—
100	C₂×Q₈	—	194	2	no	—
101	C₂×Q₈	—	194	2	no	—
102	C₂²×C₄	—	194	2	no	—
103	C₂²×C₄	—	194	2	no	—
104	C₂²×C₄	D₃₈₈	1	2	yes	—
105	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
106	Q₁₆	—	194	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₉₇	1	2	yes	—
110	C₂×C₈	D₃₈₈	1	2	yes	—
111	C₂×C₈	D₃₈₈	1	2	yes	—
112	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
113	C₄⋊C₄	—	194	2	no	—
114	C₄⋊C₄	—	194	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
118	Q₈	—	388	2	no	—
119	Q₈	—	388	2	no	—
120	Q₈	—	194	4	no	—
121	Q₈	—	194	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	388	2	no	—
125	C₂×C₄	—	388	4	no	—
126	C₂×C₄	—	388	2	no	—
127	C₂×C₄	—	388	2	no	—
128	C₂×C₄	—	194	4	no	—
129	C₂×C₄	Dic₃₈₈	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
132	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
134	C₂×C₄	#13095511	1	1	yes	characteristic
135	C₂×C₄	#13095511	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#13095488	1	4	yes	—
138	C₂²	C₄×D₃₈₈	1	1	yes	characteristic
139	C₂²	#15690784	1	1	yes	characteristic
140	C₂²	#15690807	1	1	yes	characteristic
141	C₂²	C₂×Dic₃₈₈	1	2	yes	—
142	C₂²	#15690832	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	776	2	no	—
145	C₄	—	388	4	no	—
146	C₄	C₂×#13095488	1	1	yes	characteristic
147	C₄	C₄×D₃₈₈	1	1	yes	characteristic
148	C₄	#15690784	1	1	yes	characteristic
149	C₄	#15690798	1	1	yes	characteristic
150	C₂	#20260637	1	1	yes	characteristic
151	C₂	C₄×Dic₃₈₈	1	2	yes	—
152	C₂	#20260976	1	2	yes	—
153	C₂	#20261115	1	1	yes	characteristic
154	C₂	#20261119	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₈₈

C₂

yes

maximal, characteristic

C₂×#15690744

C₂

yes

maximal, characteristic

C₂×#15690735

C₂

yes

maximal

#20260641

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

C₂×Dic₃₈₈

—

C₂×Dic₃₈₈

—

C₂×Dic₃₈₈

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690744

—

#15690735

—

#15690717

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

Dic₃₈₈

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

Q₁₆

yes

complemented

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

C₂×C₄×C₉₇

yes

C₂×C₄×C₉₇

yes

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

C₂²⋊C₄

yes

—

C₂²×C₉₇

C₄×D₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×Q₁₆

yes

—

C₂²×C₉₇

#135

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

C₄×Q₁₆

yes

—

C₂×C₉₇

#467

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₉₇

yes

characteristic

#391

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

194

—

C₂×Q₁₆

—

194

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#107

—

194

—

#103

—

194

—

C₄×C₈

—

#95

—

194

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

100

C₂×Q₈

—

194

—

101

C₂×Q₈

—

194

—

102

C₂²×C₄

—

194

—

103

C₂²×C₄

—

194

—

104

C₂²×C₄

D₃₈₈

yes

—

105

C₂²×C₄

C₂²×D₉₇

yes

characteristic

106

Q₁₆

—

194

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₉₇

yes

—

110

C₂×C₈

D₃₈₈

yes

—

111

C₂×C₈

D₃₈₈

yes

—

112

C₂×C₈

C₂²⋊D₉₇

yes

—

113

C₄⋊C₄

—

194

—

114

C₄⋊C₄

—

194

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₃₈₈

yes

characteristic

118

Q₈

—

388

—

119

Q₈

—

388

—

120

Q₈

—

194

—

121

Q₈

—

194

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

388

—

125

C₂×C₄

—

388

—

126

C₂×C₄

—

388

—

127

C₂×C₄

—

388

—

128

C₂×C₄

—

194

—

129

C₂×C₄

Dic₃₈₈

yes

complemented

130

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

132

C₂×C₄

C₂×D₃₈₈

yes

characteristic

133

C₂×C₄

C₂×D₃₈₈

yes

characteristic

134

C₂×C₄

#13095511

yes

characteristic

135

C₂×C₄

#13095511

yes

characteristic

136

C₈

—

137

C₈

#13095488

yes

—

138

C₂²

C₄×D₃₈₈

yes

characteristic

139

C₂²

#15690784

yes

characteristic

140

C₂²

#15690807

yes

characteristic

141

C₂²

C₂×Dic₃₈₈

yes

—

142

C₂²

#15690832

yes

—

143

C₄

—

144

C₄

—

776

—

145

C₄

—

388

—

146

C₄

C₂×#13095488

yes

characteristic

147

C₄

C₄×D₃₈₈

yes

characteristic

148

C₄

#15690784

yes

characteristic

149

C₄

#15690798

yes

characteristic

150

C₂

#20260637

yes

characteristic

151

C₂

C₄×Dic₃₈₈

yes

—

152

C₂

#20260976

yes

—

153

C₂

#20261115

yes

characteristic

154

C₂

#20261119

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1969
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

#1969

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₃₈₈	C₄	semidirect
42. C₂×C₄×C₉₇	Q₁₆	semidirect
78. C₉₇	#1969	semidirect
129. C₂×C₄	Dic₃₈₈	semidirect

H ≅ G/N

type

9. C₂×Dic₃₈₈

C₄

semidirect

42. C₂×C₄×C₉₇

Q₁₆

semidirect

78. C₉₇

#1969

semidirect

129. C₂×C₄

Dic₃₈₈

semidirect

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

subgroup	index
C₂²×Dic₃₈₈	2
C₂×#15690744	2
C₂×#15690735	2
#20260641	2
C₂×C₄×C₈×C₉₇	2
#1969	97

subgroup

index

C₂²×Dic₃₈₈

C₂×C₄×C₈×C₉₇

Subgroups — #27142597

Conjugacy classes of subgroups

Complements

Maximal subgroups