Group #27142696 — subgroups

Subgroups — #27142696

Order 12416.

300 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	—	2	1	no	—
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690810	—	2	2	no	—
8	#15690810	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₉₇	—	2	1	no	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
11	C₄×#11715895	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₉₄	—	2	4	no	—
17	C₂×Dic₁₉₄	—	4	2	no	—
18	C₂×Dic₁₉₄	—	4	2	no	—
19	C₂²×Dic₉₇	—	2	2	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	Q₈	1	4	yes	complemented
23	C₄⋊Dic₉₇	—	2	4	no	—
24	C₄⋊Dic₉₇	—	2	2	no	—
25	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₉₇	—	2	2	no	—
27	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₉₇	—	2	4	no	—
31	C₄⋊C₄×C₉₇	—	4	2	no	—
32	C₄⋊C₄×C₉₇	—	4	2	no	—
33	C₄²×C₉₇	C₂×C₄	1	2	yes	—
34	C₄²×C₉₇	D₄	1	2	yes	—
35	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	—	4	2	no	—
37	C₂×C₄×C₉₇	—	4	2	no	—
38	C₂×C₄×C₉₇	—	4	2	no	—
39	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₉₇	D₈	1	2	yes	—
42	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
45	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₉₇	#41	1	2	yes	—
49	C₂×Dic₉₇	—	2	4	no	—
50	C₂×Dic₉₇	—	4	2	no	—
51	C₂×Dic₉₇	—	4	2	no	—
52	Dic₁₉₄	—	4	8	no	—
53	#11715895	—	2	4	no	—
54	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₉₇	#124	1	1	yes	characteristic
56	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₉₇	—	8	2	no	—
62	C₄×C₉₇	#102	1	4	yes	—
63	C₄×C₉₇	#103	1	4	yes	—
64	C₄×C₉₇	C₄×Q₈	1	2	yes	—
65	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
66	Dic₉₇	—	4	4	no	—
67	C₂×C₉₇	#391	1	1	yes	characteristic
68	C₂×C₉₇	C₂×#102	1	1	yes	characteristic
69	C₂×C₉₇	C₂×#103	1	1	yes	characteristic
70	C₂×C₉₇	#486	1	1	yes	characteristic
71	C₂×C₉₇	#487	1	1	yes	characteristic
72	C₂×C₉₇	#498	1	1	yes	characteristic
73	C₂×C₉₇	#500	1	1	yes	characteristic
74	#1990	—	97	1	no	maximal, Sylow
75	C₉₇	#1990	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊Q₈	—	97	1	no	—
77	C₂×C₄×C₈	—	97	1	no	—
78	#384	D₉₇	1	1	yes	characteristic
79	#340	—	97	4	no	—
80	C₂²×Q₈	—	194	1	no	—
81	C₂²×C₈	—	97	2	no	—
82	C₄⋊Q₈	—	97	4	no	—
83	C₄⋊Q₈	—	194	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	97	2	no	—
87	C₂×C₄⋊C₄	—	194	1	no	—
88	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
89	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
90	C₄×C₈	—	194	2	no	—
91	C₂×Q₈	—	388	2	no	—
92	C₂×Q₈	—	388	2	no	—
93	C₂×Q₈	—	194	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	194	2	no	—
96	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	C₂×C₈	—	97	4	no	—
100	C₂×C₈	—	194	2	no	—
101	C₂×C₈	—	194	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	97	4	no	—
106	C₄⋊C₄	—	194	4	no	—
107	C₄⋊C₄	—	194	2	no	—
108	C₄²	C₄×D₉₇	1	2	yes	—
109	C₄²	D₃₈₈	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
111	Q₈	—	388	8	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	388	2	no	—
116	C₂×C₄	—	388	2	no	—
117	C₂×C₄	—	194	4	no	—
118	C₂×C₄	#13095488	1	2	yes	—
119	C₂×C₄	#13095488	1	2	yes	—
120	C₂×C₄	#13095496	1	2	yes	—
121	C₂×C₄	#13095498	1	2	yes	—
122	C₂×C₄	#13095499	1	2	yes	—
123	C₂×C₄	#13095501	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₉₇	1	2	yes	—
127	C₂×C₄	#13095510	1	2	yes	—
128	C₈	—	194	4	no	—
129	C₂²	C₂×#13095488	1	1	yes	characteristic
130	C₂²	#15690797	1	1	yes	characteristic
131	C₂²	#15690808	1	1	yes	characteristic
132	C₂²	C₂×#13095496	1	1	yes	characteristic
133	C₂²	C₂×#13095498	1	1	yes	characteristic
134	C₂²	C₂×#13095499	1	1	yes	characteristic
135	C₂²	C₂×#13095501	1	1	yes	characteristic
136	C₄	—	8	2	no	—
137	C₄	—	388	4	no	—
138	C₄	#15690730	1	4	yes	—
139	C₄	#15690733	1	4	yes	—
140	C₄	#15690775	1	2	yes	—
141	C₄	#15690790	1	2	yes	—
142	C₂	#20260640	1	1	yes	characteristic
143	C₂	C₂×#15690730	1	1	yes	characteristic
144	C₂	C₂×#15690733	1	1	yes	characteristic
145	C₂	#20261081	1	1	yes	characteristic
146	C₂	#20261085	1	1	yes	characteristic
147	C₂	#20261131	1	1	yes	characteristic
148	C₂	#20261138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

—

C₂²×#11715895

C₂²

yes

—

#15690810

—

#15690810

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₄×#11715895

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

Q₈

yes

complemented

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

D₈

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#124

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

#102

yes

—

C₄×C₉₇

#103

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×C₈

—

#384

D₉₇

yes

characteristic

#340

—

C₂²×Q₈

—

194

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄×C₈

—

194

—

C₂×Q₈

—

388

—

C₂×Q₈

—

388

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

194

—

101

C₂×C₈

—

194

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

194

—

107

C₄⋊C₄

—

194

—

108

C₄²

C₄×D₉₇

yes

—

109

C₄²

D₃₈₈

yes

—

110

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

111

Q₈

—

388

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

388

—

116

C₂×C₄

—

388

—

117

C₂×C₄

—

194

—

118

C₂×C₄

#13095488

yes

—

119

C₂×C₄

#13095488

yes

—

120

C₂×C₄

#13095496

yes

—

121

C₂×C₄

#13095498

yes

—

122

C₂×C₄

#13095499

yes

—

123

C₂×C₄

#13095501

yes

—

124

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

125

C₂×C₄

C₂×D₃₈₈

yes

characteristic

126

C₂×C₄

Q₈×D₉₇

yes

—

127

C₂×C₄

#13095510

yes

—

128

C₈

—

194

—

129

C₂²

C₂×#13095488

yes

characteristic

130

C₂²

#15690797

yes

characteristic

131

C₂²

#15690808

yes

characteristic

132

C₂²

C₂×#13095496

yes

characteristic

133

C₂²

C₂×#13095498

yes

characteristic

134

C₂²

C₂×#13095499

yes

characteristic

135

C₂²

C₂×#13095501

yes

characteristic

136

C₄

—

137

C₄

—

388

—

138

C₄

#15690730

yes

—

139

C₄

#15690733

yes

—

140

C₄

#15690775

yes

—

141

C₄

#15690790

yes

—

142

C₂

#20260640

yes

characteristic

143

C₂

C₂×#15690730

yes

characteristic

144

C₂

C₂×#15690733

yes

characteristic

145

C₂

#20261081

yes

characteristic

146

C₂

#20261085

yes

characteristic

147

C₂

#20261131

yes

characteristic

148

C₂

#20261138

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#384×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#384	D₉₇
hypocenter	C₉₇	#1990
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#384×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #15690810	C₄	semidirect
22. C₂×#11715895	Q₈	semidirect
75. C₉₇	#1990	semidirect

H ≅ G/N

type

8. #15690810

C₄

semidirect

22. C₂×#11715895

Q₈

semidirect

75. C₉₇

#1990

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×C₄×#11715895	2
#20260497	2
#384×C₉₇	2
#1990	97

subgroup

index

#384×C₉₇

Subgroups — #27142696

Conjugacy classes of subgroups

Complements

Maximal subgroups