Group #30770605 — subgroups

Subgroups — #30770605

Order 17152.

318 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22562465	C₂	1	2	yes	maximal, complemented
2	#22562433	C₂	1	2	yes	maximal, complemented
3	#22559824	C₂	1	1	yes	maximal, characteristic
4	#22559806	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523988	—	2	4	no	—
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863794	—	2	1	no	—
9	#16863776	C₂²	1	1	yes	characteristic
10	#16863773	—	2	1	no	—
11	#16863773	—	2	1	no	—
12	#16863773	C₂²	1	2	yes	—
13	#16863747	—	2	1	no	—
14	#16863198	—	2	4	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₈	—	2	2	no	—
18	C₂×D₅₃₆	—	2	2	no	—
19	C₂×D₅₃₆	—	2	2	no	—
20	C₂×D₅₃₆	—	4	2	no	—
21	C₂×D₅₃₆	—	4	2	no	—
22	#14524043	—	2	1	no	—
23	#14524043	—	2	1	no	—
24	#14524040	—	2	2	no	—
25	#14523988	—	4	8	no	—
26	#14523977	—	2	2	no	—
27	#14523975	—	4	2	no	—
28	#14523968	—	4	2	no	—
29	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₃₄	—	4	1	no	—
34	C₂×Dic₁₃₄	—	4	1	no	—
35	C₂×D₂₆₈	—	4	2	no	—
36	C₂×D₂₆₈	—	4	2	no	—
37	C₂×D₂₆₈	—	4	2	no	—
38	Dic₂₆₈	—	4	4	no	—
39	D₅₃₆	—	4	4	no	—
40	D₅₃₆	—	4	4	no	—
41	D₅₃₆	—	8	4	no	—
42	D₅₃₆	—	8	4	no	—
43	C₄⋊Dic₆₇	—	4	1	no	—
44	C₄⋊Dic₆₇	—	4	1	no	—
45	C₄⋊Dic₆₇	—	4	1	no	—
46	C₄⋊Dic₆₇	—	4	1	no	—
47	C₂×C₈×C₆₇	D₈	1	2	yes	—
48	C₂×C₈×C₆₇	D₈	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
50	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₁₆×C₆₇	—	2	4	no	—
52	C₂²×D₆₇	—	8	2	no	—
53	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₆₇	—	8	1	no	—
57	C₂×Dic₆₇	—	8	1	no	—
58	D₂₆₈	—	8	2	no	—
59	D₂₆₈	—	8	2	no	—
60	D₂₆₈	—	8	4	no	—
61	D₂₆₈	—	8	4	no	—
62	Dic₁₃₄	—	8	2	no	—
63	Dic₁₃₄	—	8	2	no	—
64	C₈×C₆₇	—	2	2	no	—
65	C₈×C₆₇	SD₃₂	1	4	yes	complemented
66	C₈×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
67	C₂²×C₆₇	#493	1	1	yes	characteristic
68	C₂×D₆₇	—	16	2	no	—
69	C₂×D₆₇	—	16	2	no	—
70	C₂×D₆₇	—	16	2	no	—
71	C₄×C₆₇	#496	1	2	yes	—
72	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
73	C₄×C₆₇	#509	1	2	yes	—
74	Dic₆₇	—	16	1	no	—
75	Dic₆₇	—	16	1	no	—
76	#12529	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	#1721	1	1	yes	characteristic
78	C₂×C₆₇	#2248	1	1	yes	characteristic
79	C₂×C₆₇	#2250	1	1	yes	characteristic
80	D₆₇	—	32	2	no	—
81	#2248	—	67	2	no	—
82	#2240	—	67	2	no	—
83	#1712	—	67	1	no	—
84	#1707	—	67	1	no	—
85	#1312	D₆₇	1	1	yes	characteristic, complemented
86	C₆₇	#12529	1	1	yes	characteristic, complemented, Sylow
87	C₂×SD₃₂	—	134	4	no	—
88	#500	—	67	1	no	—
89	#498	—	134	1	no	—
90	#494	—	67	1	no	—
91	#493	—	67	2	no	—
92	#493	—	134	1	no	—
93	#493	—	134	1	no	—
94	#487	—	134	1	no	—
95	#357	—	134	4	no	—
96	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
97	C₈²	C₂×D₆₇	1	1	yes	characteristic
98	C₂×Q₁₆	—	134	2	no	—
99	C₂×D₈	—	268	2	no	—
100	C₂×D₈	—	268	2	no	—
101	C₂×D₈	—	134	2	no	—
102	C₂×D₈	—	134	2	no	—
103	C₄⋊Q₈	—	134	1	no	—
104	C₄⋊Q₈	—	134	1	no	—
105	#127	—	134	2	no	—
106	SD₃₂	—	268	8	no	—
107	C₂×C₁₆	D₂₆₈	1	4	yes	—
108	#107	—	134	2	no	—
109	#106	—	268	2	no	—
110	#103	—	268	2	no	—
111	C₄×C₈	D₂₆₈	1	1	yes	characteristic
112	C₄×C₈	D₂₆₈	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×Q₈	—	268	1	no	—
115	C₂×Q₈	—	268	1	no	—
116	C₂×D₄	—	268	2	no	—
117	C₂×D₄	—	268	2	no	—
118	C₂×D₄	—	268	2	no	—
119	Q₁₆	—	268	4	no	—
120	D₈	—	268	4	no	—
121	D₈	—	268	4	no	—
122	D₈	—	536	4	no	—
123	D₈	—	536	4	no	—
124	C₂×C₈	D₅₃₆	1	2	yes	—
125	C₂×C₈	D₅₃₆	1	2	yes	—
126	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
127	C₄⋊C₄	—	268	1	no	—
128	C₄⋊C₄	—	268	1	no	—
129	C₄⋊C₄	—	268	1	no	—
130	C₄⋊C₄	—	268	1	no	—
131	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₂³	—	536	2	no	—
134	Q₈	—	536	2	no	—
135	Q₈	—	536	2	no	—
136	D₄	—	536	2	no	—
137	D₄	—	536	2	no	—
138	D₄	—	536	4	no	—
139	D₄	—	536	4	no	—
140	C₂×C₄	—	536	1	no	—
141	C₂×C₄	—	536	1	no	—
142	C₂×C₄	#14524040	1	1	yes	characteristic
143	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
144	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#14523988	1	4	yes	complemented
147	C₈	#14524071	1	4	yes	—
148	C₂²	—	1072	2	no	—
149	C₂²	—	1072	2	no	—
150	C₂²	—	1072	2	no	—
151	C₂²	#16863773	1	1	yes	characteristic
152	C₄	—	1072	1	no	—
153	C₄	—	1072	1	no	—
154	C₄	#16863785	1	2	yes	—
155	C₄	C₂×#14523988	1	2	yes	—
156	C₄	#16863834	1	2	yes	—
157	C₂	—	2144	2	no	—
158	C₂	#22559875	1	1	yes	characteristic
159	C₂	#22562465	1	1	yes	characteristic
160	C₂	#22562475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562465

C₂

yes

maximal, complemented

#22562433

C₂

yes

maximal, complemented

#22559824

C₂

yes

maximal, characteristic

#22559806

C₂

yes

maximal, characteristic, complemented

C₆₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#14523988

—

#16863801

C₂²

yes

characteristic

#16863794

—

#16863776

C₂²

yes

—

—

C₂²

yes

—

#16863747

—

#16863198

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

—

—

—

—

—

—

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

Dic₂₆₈

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

SD₃₂

yes

complemented

C₈×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#496

yes

—

C₄×C₆₇

C₂×SD₃₂

yes

—

C₄×C₆₇

#509

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#12529

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

134

—

#500

—

#498

—

134

—

#494

—

#493

—

#493

—

134

—

#493

—

134

—

#487

—

134

—

#357

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×D₈

—

268

—

100

C₂×D₈

—

268

—

101

C₂×D₈

—

134

—

102

C₂×D₈

—

134

—

103

C₄⋊Q₈

—

134

—

104

C₄⋊Q₈

—

134

—

105

#127

—

134

—

106

SD₃₂

—

268

—

107

C₂×C₁₆

D₂₆₈

yes

—

108

#107

—

134

—

109

#106

—

268

—

110

#103

—

268

—

111

C₄×C₈

D₂₆₈

yes

characteristic

112

C₄×C₈

D₂₆₈

yes

characteristic

113

C₄×C₈

C₂²×D₆₇

yes

characteristic

114

C₂×Q₈

—

268

—

115

C₂×Q₈

—

268

—

116

C₂×D₄

—

268

—

117

C₂×D₄

—

268

—

118

C₂×D₄

—

268

—

119

Q₁₆

—

268

—

120

D₈

—

268

—

121

D₈

—

268

—

122

D₈

—

536

—

123

D₈

—

536

—

124

C₂×C₈

D₅₃₆

yes

—

125

C₂×C₈

D₅₃₆

yes

—

126

C₂×C₈

C₂×D₂₆₈

yes

—

127

C₄⋊C₄

—

268

—

128

C₄⋊C₄

—

268

—

129

C₄⋊C₄

—

268

—

130

C₄⋊C₄

—

268

—

131

C₄²

C₂×D₂₆₈

yes

characteristic

132

C₁₆

—

133

C₂³

—

536

—

134

Q₈

—

536

—

135

Q₈

—

536

—

136

D₄

—

536

—

137

D₄

—

536

—

138

D₄

—

536

—

139

D₄

—

536

—

140

C₂×C₄

—

536

—

141

C₂×C₄

—

536

—

142

C₂×C₄

#14524040

yes

characteristic

143

C₂×C₄

C₂×D₅₃₆

yes

characteristic

144

C₂×C₄

C₂×D₅₃₆

yes

characteristic

145

C₈

—

146

C₈

#14523988

yes

complemented

147

C₈

#14524071

yes

—

148

C₂²

—

1072

—

149

C₂²

—

1072

—

150

C₂²

—

1072

—

151

C₂²

#16863773

yes

characteristic

152

C₄

—

1072

—

153

C₄

—

1072

—

154

C₄

#16863785

yes

—

155

C₄

C₂×#14523988

yes

—

156

C₄

#16863834

yes

—

157

C₂

—

2144

—

158

C₂

#22559875

yes

characteristic

159

C₂

#22562465

yes

characteristic

160

C₂

#22562475

yes

characteristic

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1312	C₂
socle	C₂²×C₆₇	#493
prime radical	#22559824	C₂
hypercenter	#1312	D₆₇
hypocenter	C₆₇	#12529
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#16863773

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1312

C₂

C₂²×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #22562465	C₂	semidirect
2. #22562433	C₂	semidirect
4. #22559806	C₂	semidirect
5. C₆₇×#1312	C₂	semidirect
65. C₈×C₆₇	SD₃₂	semidirect
85. #1312	D₆₇	semidirect
86. C₆₇	#12529	semidirect
146. C₈	#14523988	semidirect

H ≅ G/N

type

1. #22562465

C₂

semidirect

2. #22562433

C₂

semidirect

4. #22559806

C₂

semidirect

5. C₆₇×#1312

C₂

semidirect

65. C₈×C₆₇

SD₃₂

semidirect

85. #1312

D₆₇

semidirect

86. C₆₇

#12529

semidirect

146. C₈

#14523988

semidirect

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

subgroup	index
#22562465	2
#22562433	2
#22559824	2
#22559806	2
C₆₇×#1312	2
#12529	67

subgroup

index

C₆₇×#1312

Subgroups — #30770605

Conjugacy classes of subgroups

Complements

Maximal subgroups