Group #30676472 — subgroups

Subgroups — #30676472

Order 16768.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#22553867	C₂	1	2	yes	maximal
2	#22553717	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553713	C₂	1	4	yes	maximal, complemented
4	#22553708	C₂	1	2	yes	maximal, complemented
5	#22553349	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₃₁	C₂²	1	2	yes	—
9	#16862246	C₂²	1	2	yes	—
10	#16862237	C₂²	1	4	yes	—
11	#16862226	—	2	2	no	—
12	#16862222	—	2	2	no	—
13	Q₈×Dic₁₃₁	—	2	2	no	—
14	#16862211	—	2	2	no	—
15	#16862204	C₂²	1	4	yes	—
16	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
17	#16862180	C₂²	1	4	yes	—
18	#16862177	C₂²	1	8	yes	—
19	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₃₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
22	#107×C₁₃₁	C₂²	1	4	yes	—
23	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₆₂	—	2	2	no	—
25	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
29	C₈×D₁₃₁	Q₈	1	8	yes	complemented
30	C₄⋊Dic₁₃₁	—	2	4	no	—
31	C₄⋊Dic₁₃₁	—	2	2	no	—
32	C₄⋊Dic₁₃₁	C₂³	1	4	yes	—
33	Dic₁₃₁⋊C₄	—	2	4	no	—
34	(C₂×D₁₃₁)⋊C₄	—	2	4	no	—
35	C₄×Dic₁₃₁	—	2	4	no	—
36	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
37	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
38	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₃₁	—	2	2	no	—
40	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
41	C₄⋊C₄×C₁₃₁	—	2	2	no	—
42	C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
43	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
45	Dic₂₆₂	—	4	4	no	—
46	C₂×Dic₁₃₁	—	2	4	no	—
47	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₃₁	—	2	2	no	—
51	C₄×D₁₃₁	—	2	2	no	—
52	C₄×D₁₃₁	C₂×Q₈	1	4	yes	—
53	C₁₃₁⋊C₈	C₂×Q₈	1	4	yes	—
54	Q₈×C₁₃₁	—	4	4	no	—
55	C₂×C₄×C₁₃₁	—	2	4	no	—
56	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
59	C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
60	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₁₃₁	—	2	1	no	—
62	C₂×D₁₃₁	—	2	1	no	—
63	C₂×D₁₃₁	C₄⋊Q₈	1	2	yes	—
64	C₄×C₁₃₁	—	4	4	no	—
65	C₄×C₁₃₁	#135	1	2	yes	—
66	C₄×C₁₃₁	#135	1	2	yes	—
67	C₄×C₁₃₁	C₂²×Q₈	1	2	yes	—
68	Dic₁₃₁	—	2	1	no	—
69	Dic₁₃₁	—	2	1	no	—
70	Dic₁₃₁	—	4	4	no	—
71	Dic₁₃₁	C₄⋊Q₈	1	2	yes	—
72	C₂×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
74	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
75	D₁₃₁	—	2	2	no	—
76	C₁₃₁	#3160	1	1	yes	characteristic, complemented, Sylow
77	#3160	—	131	1	no	maximal, Sylow
78	#533	—	131	2	no	—
79	#500	—	131	1	no	—
80	#500	D₁₃₁	1	1	yes	characteristic, complemented
81	#499	—	131	4	no	—
82	#498	—	131	2	no	—
83	#427	—	131	4	no	—
84	C₂×C₄×C₈	—	131	1	no	—
85	C₂²×C₈	—	131	2	no	—
86	C₄⋊Q₈	—	131	2	no	—
87	C₄⋊Q₈	C₂×D₁₃₁	1	2	yes	—
88	#125	—	131	4	no	—
89	C₂²⋊Q₈	—	262	2	no	—
90	C₂²⋊Q₈	—	262	2	no	—
91	C₄×Q₈	—	262	2	no	—
92	C₄×Q₈	—	262	2	no	—
93	#117	—	131	4	no	—
94	C₂×C₄²	—	131	1	no	—
95	#107	—	131	4	no	—
96	#107	C₂×D₁₃₁	1	4	yes	—
97	#106	—	131	8	no	—
98	C₄×C₈	—	131	2	no	—
99	C₄×C₈	—	131	1	no	—
100	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	262	2	no	—
103	C₂²×C₄	—	131	1	no	—
104	C₂²×C₄	—	131	1	no	—
105	C₂²×C₄	—	131	1	no	—
106	C₂×C₈	—	131	2	no	—
107	C₂×C₈	—	131	8	no	—
108	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	262	4	no	—
111	C₄⋊C₄	—	262	4	no	—
112	C₄⋊C₄	—	262	2	no	—
113	C₄⋊C₄	—	131	4	no	—
114	C₄⋊C₄	C₂²×D₁₃₁	1	4	yes	—
115	C₂²⋊C₄	—	262	4	no	—
116	C₄²	—	262	4	no	—
117	C₄²	—	131	1	no	—
118	C₄²	—	131	1	no	—
119	C₄²	—	131	1	no	—
120	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
121	C₂³	—	131	1	no	—
122	Q₈	—	4	4	no	—
123	Q₈	—	524	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	262	2	no	—
126	C₂×C₄	—	262	2	no	—
127	C₂×C₄	—	262	4	no	—
128	C₂×C₄	—	131	1	no	—
129	C₂×C₄	—	131	1	no	—
130	C₂×C₄	—	131	1	no	—
131	C₂×C₄	—	131	4	no	—
132	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
133	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
135	C₈	—	131	4	no	—
136	C₈	Q₈×D₁₃₁	1	4	yes	—
137	C₂²	—	262	1	no	—
138	C₂²	—	262	1	no	—
139	C₂²	—	131	2	no	—
140	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	262	1	no	—
143	C₄	—	262	1	no	—
144	C₄	—	524	4	no	—
145	C₄	—	131	2	no	—
146	C₄	#16862265	1	2	yes	—
147	C₄	#16862269	1	2	yes	—
148	C₄	C₂×Q₈×D₁₃₁	1	2	yes	—
149	C₂	—	262	2	no	—
150	C₂	C₄⋊Q₈×D₁₃₁	1	1	yes	characteristic
151	C₂	C₂×#16862265	1	1	yes	characteristic
152	C₂	C₂×#16862269	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553867

C₂

yes

maximal

#22553717

C₂

yes

maximal, characteristic, complemented

#22553713

C₂

yes

maximal, complemented

#22553708

C₂

yes

maximal, complemented

#22553349

C₂

yes

maximal

C₄×C₈×D₁₃₁

C₂

yes

maximal, characteristic

#500×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₁

C₂²

yes

—

#16862246

C₂²

yes

—

#16862237

C₂²

yes

—

#16862226

—

#16862222

—

Q₈×Dic₁₃₁

—

#16862211

—

#16862204

C₂²

yes

—

C₄²×D₁₃₁

C₂²

yes

characteristic

#16862180

C₂²

yes

—

#16862177

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₁

C₂²

yes

—

C₄⋊Q₈×C₁₃₁

C₂²

yes

—

#107×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

C₂×C₄×D₁₃₁

D₄

yes

characteristic

C₂×C₄×D₁₃₁

D₄

yes

characteristic

C₂×C₄×D₁₃₁

C₂³

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₈×D₁₃₁

Q₈

yes

complemented

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

C₂³

yes

—

Dic₁₃₁⋊C₄

—

(C₂×D₁₃₁)⋊C₄

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

D₄

yes

characteristic

C₄×Dic₁₃₁

D₄

yes

characteristic

C₄×Dic₁₃₁

C₂³

yes

characteristic

C₂×Q₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂³

yes

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

C₂³

yes

characteristic

C₂²×D₁₃₁

C₂×D₄

yes

characteristic

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

C₂×D₄

yes

characteristic

C₂×Dic₁₃₁

C₂×D₄

yes

characteristic

C₂×Dic₁₃₁

C₂×D₄

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

C₂×Q₈

yes

—

C₁₃₁⋊C₈

C₂×Q₈

yes

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₈×C₁₃₁

C₂×Q₈

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

C₄⋊Q₈

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#135

yes

—

C₄×C₁₃₁

#135

yes

—

C₄×C₁₃₁

C₂²×Q₈

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

C₄⋊Q₈

yes

—

C₂×C₁₃₁

C₂×C₄⋊Q₈

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

D₁₃₁

—

C₁₃₁

#3160

yes

characteristic, complemented, Sylow

#3160

—

131

maximal, Sylow

#533

—

131

—

#500

—

131

—

#500

D₁₃₁

yes

characteristic, complemented

#499

—

131

—

#498

—

131

—

#427

—

131

—

C₂×C₄×C₈

—

131

—

C₂²×C₈

—

131

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

—

#125

—

131

—

C₂²⋊Q₈

—

262

—

C₂²⋊Q₈

—

262

—

C₄×Q₈

—

262

—

C₄×Q₈

—

262

—

#117

—

131

—

C₂×C₄²

—

131

—

#107

—

131

—

#107

C₂×D₁₃₁

yes

—

#106

—

131

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

100

C₄×C₈

C₂×D₁₃₁

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

262

—

103

C₂²×C₄

—

131

—

104

C₂²×C₄

—

131

—

105

C₂²×C₄

—

131

—

106

C₂×C₈

—

131

—

107

C₂×C₈

—

131

—

108

C₂×C₈

C₂²×D₁₃₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

262

—

111

C₄⋊C₄

—

262

—

112

C₄⋊C₄

—

262

—

113

C₄⋊C₄

—

131

—

114

C₄⋊C₄

C₂²×D₁₃₁

yes

—

115

C₂²⋊C₄

—

262

—

116

C₄²

—

262

—

117

C₄²

—

131

—

118

C₄²

—

131

—

119

C₄²

—

131

—

120

C₄²

C₂²×D₁₃₁

yes

characteristic

121

C₂³

—

131

—

122

Q₈

—

123

Q₈

—

524

—

124

C₂×C₄

—

125

C₂×C₄

—

262

—

126

C₂×C₄

—

262

—

127

C₂×C₄

—

262

—

128

C₂×C₄

—

131

—

129

C₂×C₄

—

131

—

130

C₂×C₄

—

131

—

131

C₂×C₄

—

131

—

132

C₂×C₄

D₄×D₁₃₁

yes

characteristic

133

C₂×C₄

D₄×D₁₃₁

yes

characteristic

134

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

135

C₈

—

131

—

136

C₈

Q₈×D₁₃₁

yes

—

137

C₂²

—

262

—

138

C₂²

—

262

—

139

C₂²

—

131

—

140

C₂²

C₂×D₄×D₁₃₁

yes

characteristic

141

C₄

—

142

C₄

—

262

—

143

C₄

—

262

—

144

C₄

—

524

—

145

C₄

—

131

—

146

C₄

#16862265

yes

—

147

C₄

#16862269

yes

—

148

C₄

C₂×Q₈×D₁₃₁

yes

—

149

C₂

—

262

—

150

C₂

C₄⋊Q₈×D₁₃₁

yes

characteristic

151

C₂

C₂×#16862265

yes

characteristic

152

C₂

C₂×#16862269

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#500×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂²×D₁₃₁	C₂×D₄
hypercenter	#500	D₁₃₁
hypocenter	C₁₃₁	#3160
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₁

commutator, cosocle

C₂×C₄×C₁₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₁

Fitting

#500×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂²×D₄

prime radical

C₂²×D₁₃₁

C₂×D₄

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #22553717	C₂	semidirect
3. #22553713	C₂	semidirect
4. #22553708	C₂	semidirect
7. #500×C₁₃₁	C₂	semidirect
29. C₈×D₁₃₁	Q₈	semidirect
76. C₁₃₁	#3160	semidirect
80. #500	D₁₃₁	semidirect

H ≅ G/N

type

2. #22553717

C₂

semidirect

3. #22553713

C₂

semidirect

4. #22553708

C₂

semidirect

7. #500×C₁₃₁

C₂

semidirect

29. C₈×D₁₃₁

Q₈

semidirect

76. C₁₃₁

#3160

semidirect

80. #500

D₁₃₁

semidirect

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

subgroup	index
#22553867	2
#22553717	2
#22553713	2
#22553708	2
#22553349	2
C₄×C₈×D₁₃₁	2
#500×C₁₃₁	2
#3160	131

subgroup

index

C₄×C₈×D₁₃₁

#500×C₁₃₁

#3160

131

Subgroups — #30676472

Conjugacy classes of subgroups

Complements

Maximal subgroups