Group #30676402 — subgroups

Subgroups — C₂×#22553707 #30676402

Order 16768.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14523503	C₂	1	2	yes	maximal, complemented
2	C₂²×#14523501	C₂	1	2	yes	maximal
3	C₂×#16862246	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553707	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16862157	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₂	C₂²	1	2	yes	—
8	C₂×#14523503	—	2	8	no	—
9	C₂×#14523503	C₂²	1	8	yes	complemented
10	C₂×#14523501	—	2	8	no	—
11	C₂×#14523501	C₂²	1	8	yes	—
12	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
13	#16862246	—	2	2	no	—
14	#16862246	C₂²	1	4	yes	complemented
15	C₂×C₄⋊Dic₁₃₁	—	2	2	no	—
16	#16862157	C₂²	1	4	yes	complemented
17	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
19	#124×C₁₃₁	—	2	2	no	—
20	#124×C₁₃₁	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
22	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₆₂	—	2	2	no	—
24	C₂×Dic₂₆₂	—	2	4	no	—
25	C₂×Dic₂₆₂	—	2	4	no	—
26	C₂×Dic₂₆₂	C₂³	1	4	yes	—
27	C₂²×Dic₁₃₁	—	2	2	no	—
28	#14523503	—	2	16	no	—
29	#14523501	—	2	16	no	—
30	C₂×C₁₃₁⋊C₈	D₄	1	8	yes	—
31	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₁₃₁	—	2	8	no	—
33	C₂⁴×C₁₃₁	—	2	1	no	—
34	C₂×Q₈×C₁₃₁	—	2	1	no	—
35	C₂×Q₈×C₁₃₁	—	2	4	no	—
36	C₂×Q₈×C₁₃₁	C₂³	1	2	yes	—
37	C₂×D₄×C₁₃₁	—	2	4	no	—
38	C₂×D₄×C₁₃₁	—	2	1	no	—
39	C₂×D₄×C₁₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃₁	—	2	1	no	—
41	C₂²×C₄×C₁₃₁	D₄	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
44	C₄²×C₁₃₁	D₄	1	2	yes	—
45	C₄²×C₁₃₁	C₂³	1	2	yes	—
46	Dic₂₆₂	—	2	4	no	—
47	Dic₂₆₂	—	2	4	no	—
48	Dic₂₆₂	—	4	4	no	—
49	C₂×Dic₁₃₁	—	2	4	no	—
50	C₂×Dic₁₃₁	—	4	4	no	—
51	C₁₃₁⋊C₈	C₂×D₄	1	8	yes	—
52	C₂³×C₁₃₁	—	2	2	no	—
53	C₂³×C₁₃₁	—	4	2	no	—
54	C₂³×C₁₃₁	—	4	4	no	—
55	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₁₃₁	—	2	4	no	—
57	Q₈×C₁₃₁	—	4	2	no	—
58	D₄×C₁₃₁	—	2	4	no	—
59	D₄×C₁₃₁	—	4	2	no	—
60	C₂×C₄×C₁₃₁	—	2	4	no	—
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	4	2	no	—
63	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
67	C₂²×C₁₃₁	—	4	2	no	—
68	C₂²×C₁₃₁	—	4	4	no	—
69	C₂²×C₁₃₁	—	8	4	no	—
70	C₂²×C₁₃₁	#127	1	2	yes	—
71	C₂²×C₁₃₁	#137	1	4	yes	—
72	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₃₁	—	2	4	no	—
74	C₄×C₁₃₁	—	4	2	no	—
75	C₄×C₁₃₁	#127	1	2	yes	—
76	C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
77	Dic₁₃₁	—	4	4	no	—
78	C₂×C₁₃₁	—	8	2	no	—
79	C₂×C₁₃₁	#497	1	4	yes	complemented
80	C₂×C₁₃₁	C₂×#127	1	1	yes	characteristic
81	C₂×C₁₃₁	C₂×#137	1	2	yes	—
82	C₁₃₁	C₂×#497	1	1	yes	characteristic, complemented, Sylow
83	C₂×#497	—	131	1	no	maximal, Sylow
84	C₂²×Q₁₆	—	131	2	no	—
85	C₂²×SD₁₆	—	131	2	no	—
86	C₂×C₄⋊Q₈	—	131	1	no	—
87	C₂×#124	D₁₃₁	1	1	yes	characteristic
88	#497	—	131	8	no	—
89	C₂×#97	—	131	1	no	—
90	C₂²×Q₈	—	131	2	no	—
91	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
92	C₂²×D₄	C₂×D₁₃₁	1	1	yes	characteristic
93	C₂×Q₁₆	—	262	8	no	—
94	C₂×Q₁₆	—	131	8	no	—
95	C₂×SD₁₆	—	262	8	no	—
96	C₂×SD₁₆	—	131	8	no	—
97	C₂²×C₈	—	131	2	no	—
98	C₄⋊Q₈	—	262	2	no	—
99	C₄⋊Q₈	—	131	4	no	—
100	#124	—	2	2	no	—
101	#124	C₂×D₁₃₁	1	4	yes	—
102	C₂×C₄⋊C₄	—	262	2	no	—
103	C₂×C₂²⋊C₄	—	2	2	no	—
104	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
105	#97	—	131	4	no	—
106	C₂⁴	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	262	2	no	—
110	C₂×Q₈	—	262	4	no	—
111	C₂×Q₈	—	262	4	no	—
112	C₂×Q₈	—	131	4	no	—
113	C₂×Q₈	C₂²×D₁₃₁	1	2	yes	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	C₂²×D₁₃₁	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	262	2	no	—
119	C₂²×C₄	C₂²⋊D₁₃₁	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
121	Q₁₆	—	262	16	no	—
122	SD₁₆	—	262	16	no	—
123	C₂×C₈	—	131	8	no	—
124	C₂×C₈	—	131	4	no	—
125	C₄⋊C₄	—	262	8	no	—
126	C₂²⋊C₄	—	2	8	no	—
127	C₄²	C₂²⋊D₁₃₁	1	2	yes	—
128	C₄²	C₂²×D₁₃₁	1	2	yes	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	4	no	—
132	C₂³	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	262	4	no	—
136	Q₈	—	262	4	no	—
137	Q₈	—	524	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	262	4	no	—
144	C₂×C₄	—	524	4	no	—
145	C₂×C₄	D₄×D₁₃₁	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
149	C₈	—	131	8	no	—
150	C₂²	—	4	4	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	4	no	—
153	C₂²	#16862242	1	2	yes	—
154	C₂²	#16862276	1	4	yes	—
155	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
156	C₄	—	2	4	no	—
157	C₄	—	4	2	no	—
158	C₄	—	524	4	no	—
159	C₄	#16862242	1	2	yes	—
160	C₄	C₂×D₄×D₁₃₁	1	2	yes	—
161	C₂	—	8	2	no	—
162	C₂	#22553707	1	4	yes	direct product factor
163	C₂	C₂×#16862242	1	1	yes	characteristic
164	C₂	C₂×#16862276	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14523503

C₂

yes

maximal, complemented

C₂²×#14523501

C₂

yes

maximal

C₂×#16862246

C₂

yes

maximal, characteristic, complemented

#22553707

C₂

yes

maximal, direct product factor

C₂×#16862157

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₂

C₂²

yes

—

C₂×#14523503

—

C₂×#14523503

C₂²

yes

complemented

C₂×#14523501

—

C₂×#14523501

C₂²

yes

—

C₂²×C₁₃₁⋊C₈

C₂²

yes

—

#16862246

—

#16862246

C₂²

yes

complemented

C₂×C₄⋊Dic₁₃₁

—

#16862157

C₂²

yes

complemented

C₂²×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂²×D₄×C₁₃₁

C₂²

yes

characteristic

#124×C₁₃₁

—

#124×C₁₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

C₂³

yes

—

C₂²×Dic₁₃₁

—

#14523503

—

#14523501

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₁

—

C₂⁴×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

C₂³

yes

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₄

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

D₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

C₂×D₄

yes

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂×D₄

yes

characteristic

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#127

yes

—

C₂²×C₁₃₁

#137

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#127

yes

—

C₄×C₁₃₁

C₂²×D₄

yes

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

—

C₁₃₁

C₂×#497

yes

characteristic, complemented, Sylow

C₂×#497

—

131

maximal, Sylow

C₂²×Q₁₆

—

131

—

C₂²×SD₁₆

—

131

—

C₂×C₄⋊Q₈

—

131

—

C₂×#124

D₁₃₁

yes

characteristic

#497

—

131

—

C₂×#97

—

131

—

C₂²×Q₈

—

131

—

C₂²×Q₈

C₂×D₁₃₁

yes

characteristic

C₂²×D₄

C₂×D₁₃₁

yes

characteristic

C₂×Q₁₆

—

262

—

C₂×Q₁₆

—

131

—

C₂×SD₁₆

—

262

—

C₂×SD₁₆

—

131

—

C₂²×C₈

—

131

—

C₄⋊Q₈

—

262

—

C₄⋊Q₈

—

131

—

100

#124

—

101

#124

C₂×D₁₃₁

yes

—

102

C₂×C₄⋊C₄

—

262

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

105

#97

—

131

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

262

—

110

C₂×Q₈

—

262

—

111

C₂×Q₈

—

262

—

112

C₂×Q₈

—

131

—

113

C₂×Q₈

C₂²×D₁₃₁

yes

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

C₂²×D₁₃₁

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

262

—

119

C₂²×C₄

C₂²⋊D₁₃₁

yes

—

120

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

121

Q₁₆

—

262

—

122

SD₁₆

—

262

—

123

C₂×C₈

—

131

—

124

C₂×C₈

—

131

—

125

C₄⋊C₄

—

262

—

126

C₂²⋊C₄

—

127

C₄²

C₂²⋊D₁₃₁

yes

—

128

C₄²

C₂²×D₁₃₁

yes

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂×C₂²⋊D₁₃₁

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

262

—

136

Q₈

—

262

—

137

Q₈

—

524

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

262

—

144

C₂×C₄

—

524

—

145

C₂×C₄

D₄×D₁₃₁

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

148

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

149

C₈

—

131

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#16862242

yes

—

154

C₂²

#16862276

yes

—

155

C₂²

C₂²×C₂²⋊D₁₃₁

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

524

—

159

C₄

#16862242

yes

—

160

C₄

C₂×D₄×D₁₃₁

yes

—

161

C₂

—

162

C₂

#22553707

yes

direct product factor

163

C₂

C₂×#16862242

yes

characteristic

164

C₂

C₂×#16862276

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₂×#124×C₁₃₁	C₂
socle	C₂³×C₁₃₁	C₂×D₄
prime radical	C₂²×D₄×C₁₃₁	C₂²
hypercenter	C₂×#124	D₁₃₁
hypocenter	C₁₃₁	C₂×#497
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₁

commutator, cosocle

C₂×C₄×C₁₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₁

Fitting

C₂×#124×C₁₃₁

C₂

socle

C₂³×C₁₃₁

C₂×D₄

prime radical

C₂²×D₄×C₁₃₁

C₂²

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂²×#14523503	C₂	semidirect
3. C₂×#16862246	C₂	semidirect
4. #22553707	C₂	direct (N × H)
5. C₂×#16862157	C₂	semidirect
9. C₂×#14523503	C₂²	semidirect
14. #16862246	C₂²	semidirect
16. #16862157	C₂²	semidirect
79. C₂×C₁₃₁	#497	semidirect
82. C₁₃₁	C₂×#497	semidirect
162. C₂	#22553707	direct (N × H)

H ≅ G/N

type

1. C₂²×#14523503

C₂

semidirect

3. C₂×#16862246

C₂

semidirect

4. #22553707

C₂

direct (N × H)

5. C₂×#16862157

C₂

semidirect

9. C₂×#14523503

C₂²

semidirect

14. #16862246

C₂²

semidirect

16. #16862157

C₂²

semidirect

79. C₂×C₁₃₁

#497

semidirect

82. C₁₃₁

C₂×#497

semidirect

162. C₂

#22553707

direct (N × H)

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

subgroup	index
C₂²×#14523503	2
C₂²×#14523501	2
C₂×#16862246	2
#22553707	2
C₂×#16862157	2
C₂×#124×C₁₃₁	2
C₂×#497	131

subgroup

index

C₂×#124×C₁₃₁

131

Subgroups — C₂×#22553707 #30676402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22553707 #30676402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553707 #30676402