Group #27150819 — subgroups

Subgroups — #27150819

Order 12416.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	#20260865	C₂	1	4	yes	maximal, complemented
3	#20260862	C₂	1	4	yes	maximal, complemented
4	#20260789	C₂	1	4	yes	maximal
5	#20260788	C₂	1	1	yes	maximal, characteristic
6	#431×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₉₄	C₄	1	1	yes	characteristic
8	C₄×Dic₁₉₄	C₄	1	2	yes	—
9	C₄×Dic₁₉₄	C₄	1	2	yes	—
10	C₄×Dic₁₉₄	C₂²	1	2	yes	—
11	C₄×Dic₁₉₄	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
13	C₂×#13095490	C₄	1	2	yes	—
14	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
15	C₄⋊#11715895	C₂²	1	2	yes	—
16	#15690739	C₂²	1	8	yes	—
17	C₂²⋊#11715895	C₂²	1	2	yes	—
18	#15690720	C₂²	1	8	yes	—
19	C₄×#11715895	C₂²	1	2	yes	—
20	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
23	C₄×C₈×C₉₇	C₂²	1	2	yes	—
24	C₂×Dic₁₉₄	—	2	4	no	—
25	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
26	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
27	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
28	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
29	C₂×#11715895	C₂³	1	4	yes	—
30	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
32	#13095490	C₂×C₄	1	4	yes	—
33	#13095490	C₂×C₄	1	4	yes	—
34	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
35	C₄×Dic₉₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₉₇	C₂³	1	4	yes	—
40	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₉₇	C₂³	1	1	yes	characteristic
43	C₄²×C₉₇	C₂³	1	1	yes	characteristic
44	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	—	2	2	no	—
46	C₂×C₄×C₉₇	—	4	1	no	—
47	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₉₇	—	2	4	no	—
55	C₂×Dic₉₇	—	4	2	no	—
56	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
58	Dic₁₉₄	—	2	4	no	—
59	Dic₁₉₄	—	2	4	no	—
60	C₈×C₉₇	—	2	4	no	—
61	#11715895	—	2	4	no	—
62	C₂²×C₉₇	—	2	1	no	—
63	C₂²×C₉₇	—	2	1	no	—
64	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
65	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₉₇	—	2	1	no	—
67	C₄×C₉₇	—	2	1	no	—
68	C₄×C₉₇	—	2	1	no	—
69	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
70	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
71	C₄×C₉₇	2₋¹⁺⁴	1	2	yes	—
72	Dic₉₇	—	2	4	no	—
73	Dic₉₇	—	2	4	no	—
74	C₂×C₉₇	—	2	2	no	—
75	C₂×C₉₇	#519	1	1	yes	characteristic
76	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
77	C₂×C₉₇	#568	1	1	yes	characteristic
78	#2983	—	97	1	no	maximal, Sylow
79	C₉₇	#2983	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄×Q₈	—	97	1	no	—
81	#446	—	97	4	no	—
82	#446	—	97	4	no	—
83	#432	—	97	4	no	—
84	#431	—	97	1	no	—
85	#431	D₉₇	1	1	yes	characteristic
86	C₂²×Q₈	—	97	1	no	—
87	C₄×Q₈	—	97	2	no	—
88	C₄×Q₈	—	97	2	no	—
89	C₄×Q₈	—	97	2	no	—
90	C₄×Q₈	—	97	2	no	—
91	C₂×C₄⋊C₄	—	97	2	no	—
92	C₂×C₄⋊C₄	—	97	1	no	—
93	C₂×C₄²	—	97	2	no	—
94	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
95	C₄⋊C₈	—	97	8	no	—
96	C₄⋊C₈	—	97	2	no	—
97	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
98	C₂²⋊C₈	—	97	2	no	—
99	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
100	#97	—	97	8	no	—
101	C₄×C₈	—	97	2	no	—
102	C₄×C₈	C₂×D₉₇	1	2	yes	—
103	C₂×Q₈	—	97	2	no	—
104	C₂×Q₈	—	97	2	no	—
105	C₂×Q₈	—	194	4	no	—
106	C₂²×C₄	—	97	2	no	—
107	C₂²×C₄	—	97	2	no	—
108	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
111	C₂×C₈	—	97	4	no	—
112	C₂×C₈	C₂²×D₉₇	1	4	yes	—
113	C₄⋊C₄	—	97	4	no	—
114	C₄⋊C₄	—	97	4	no	—
115	C₄⋊C₄	—	97	2	no	—
116	C₄⋊C₄	—	97	2	no	—
117	C₄²	—	97	4	no	—
118	C₄²	—	97	4	no	—
119	C₄²	C₄×D₉₇	1	1	yes	characteristic
120	C₄²	C₄×D₉₇	1	1	yes	characteristic
121	C₄²	C₂²×D₉₇	1	1	yes	characteristic
122	C₄²	C₂²×D₉₇	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
124	Q₈	—	194	4	no	—
125	Q₈	—	194	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	97	4	no	—
129	C₂×C₄	—	97	4	no	—
130	C₂×C₄	—	388	2	no	—
131	C₂×C₄	—	194	4	no	—
132	C₂×C₄	#13095494	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	194	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#13095494	1	2	yes	—
144	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	194	4	no	—
149	C₄	—	194	4	no	—
150	C₄	C₂×#13095494	1	2	yes	—
151	C₄	C₂×#13095494	1	2	yes	—
152	C₄	#15690857	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#20261212	1	1	yes	characteristic
155	C₂	C₂²×#13095494	1	1	yes	characteristic
156	C₂	#20261501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₉₄

C₂

yes

maximal, characteristic

#20260865

C₂

yes

maximal, complemented

#20260862

C₂

yes

maximal, complemented

#20260789

C₂

yes

maximal

#20260788

C₂

yes

maximal, characteristic

#431×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₄

yes

characteristic

C₄×Dic₁₉₄

C₄

yes

—

C₄×Dic₁₉₄

C₄

yes

—

C₄×Dic₁₉₄

C₂²

yes

—

C₄×Dic₁₉₄

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₄

yes

characteristic

C₂×#13095490

C₄

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₄⋊#11715895

C₂²

yes

—

#15690739

C₂²

yes

—

C₂²⋊#11715895

C₂²

yes

—

#15690720

C₂²

yes

—

C₄×#11715895

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₂²

yes

—

C₂²⋊C₈×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

C₂×C₄

yes

—

#13095490

C₂×C₄

yes

—

C₄×Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

C₂³

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂²×C₄

yes

—

C₂×Dic₉₇

C₂²×C₄

yes

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

2₋¹⁺⁴

yes

—

Dic₉₇

—

Dic₉₇

—

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

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₉₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₉₇

yes

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₉₇

yes

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

194

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₉₇

yes

characteristic

109

C₂²×C₄

C₄×D₉₇

yes

characteristic

110

C₂²×C₄

C₂²×D₉₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₉₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₉₇

yes

characteristic

120

C₄²

C₄×D₉₇

yes

characteristic

121

C₄²

C₂²×D₉₇

yes

characteristic

122

C₄²

C₂²×D₉₇

yes

characteristic

123

C₂³

C₂×C₄×D₉₇

yes

characteristic

124

Q₈

—

194

—

125

Q₈

—

194

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

388

—

131

C₂×C₄

—

194

—

132

C₂×C₄

#13095494

yes

—

133

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

134

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

135

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

136

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

137

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

138

C₂×C₄

C₂³×D₉₇

yes

characteristic

139

C₈

—

140

C₈

—

194

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#13095494

yes

—

144

C₂²

C₂²×C₄×D₉₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

194

—

149

C₄

—

194

—

150

C₄

C₂×#13095494

yes

—

151

C₄

C₂×#13095494

yes

—

152

C₄

#15690857

yes

—

153

C₂

—

154

C₂

#20261212

yes

characteristic

155

C₂

C₂²×#13095494

yes

characteristic

156

C₂

#20261501

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#431×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#431	D₉₇
hypocenter	C₉₇	#2983
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂³×C₄

Fitting

#431×C₉₇

C₂

prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂×C₄×C₉₇

C₂⁴

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #20260865	C₂	semidirect
3. #20260862	C₂	semidirect
79. C₉₇	#2983	semidirect

H ≅ G/N

type

2. #20260865

C₂

semidirect

3. #20260862

C₂

semidirect

79. C₉₇

#2983

semidirect

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

subgroup	index
C₂×C₄×Dic₁₉₄	2
#20260865	2
#20260862	2
#20260789	2
#20260788	2
#431×C₉₇	2
#2983	97

subgroup

index

C₂×C₄×Dic₁₉₄

#431×C₉₇

Subgroups — #27150819

Conjugacy classes of subgroups

Complements

Maximal subgroups