Group #30770606 — subgroups

Subgroups — #30770606

Order 17152.

318 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22562464	C₂	1	2	yes	maximal, complemented
2	#22562432	C₂	1	2	yes	maximal, complemented
3	#22559806	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558127	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1712	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523987	—	2	4	no	—
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863794	—	2	1	no	—
9	#16863776	C₂²	1	1	yes	characteristic
10	#16863747	—	2	1	no	—
11	#16863197	—	2	4	no	—
12	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
13	#493×C₆₇	—	2	1	no	—
14	#493×C₆₇	—	2	1	no	—
15	#493×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₈	—	2	2	no	—
18	#14524043	—	2	1	no	—
19	#14524043	—	2	1	no	—
20	#14523987	—	4	8	no	—
21	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
22	#14523977	—	2	2	no	—
23	#14523975	—	4	2	no	—
24	#14523968	—	4	2	no	—
25	C₂×D₈×C₆₇	—	2	2	no	—
26	C₂×D₈×C₆₇	—	2	2	no	—
27	C₂×D₈×C₆₇	—	4	2	no	—
28	C₂×D₈×C₆₇	—	4	2	no	—
29	#127×C₆₇	—	2	2	no	—
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	Dic₂₆₈	—	4	4	no	—
36	C₄⋊Dic₆₇	—	4	1	no	—
37	C₄⋊Dic₆₇	—	4	1	no	—
38	C₄⋊Dic₆₇	—	4	1	no	—
39	C₄⋊Dic₆₇	—	4	1	no	—
40	C₆₇⋊C₁₆	—	2	4	no	—
41	C₂×D₄×C₆₇	—	4	2	no	—
42	C₂×D₄×C₆₇	—	4	2	no	—
43	C₂×D₄×C₆₇	—	4	2	no	—
44	D₈×C₆₇	—	4	4	no	—
45	D₈×C₆₇	—	4	4	no	—
46	D₈×C₆₇	—	8	4	no	—
47	D₈×C₆₇	—	8	4	no	—
48	C₂×C₈×C₆₇	D₈	1	2	yes	—
49	C₂×C₈×C₆₇	D₈	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₆₇	—	8	2	no	—
53	D₄×C₆₇	—	8	2	no	—
54	D₄×C₆₇	—	8	2	no	—
55	D₄×C₆₇	—	8	4	no	—
56	D₄×C₆₇	—	8	4	no	—
57	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₆₇	—	8	1	no	—
61	C₂×Dic₆₇	—	8	1	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	—	16	2	no	—
68	C₂²×C₆₇	—	16	2	no	—
69	C₂²×C₆₇	—	16	2	no	—
70	C₂²×C₆₇	#493	1	1	yes	characteristic
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	—	32	2	no	—
78	C₂×C₆₇	#1721	1	1	yes	characteristic
79	C₂×C₆₇	#2248	1	1	yes	characteristic
80	C₂×C₆₇	#2250	1	1	yes	characteristic
81	#2248	—	67	2	no	—
82	#2240	—	67	2	no	—
83	#1712	D₆₇	1	1	yes	characteristic
84	#1707	—	67	1	no	—
85	#1312	—	67	1	no	—
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	—	2	1	no	—
92	#493	—	2	1	no	—
93	#493	C₂×D₆₇	1	2	yes	—
94	#487	—	134	1	no	—
95	#357	—	134	4	no	—
96	C₄×C₁₆	—	67	2	no	—
97	C₈²	C₂×D₆₇	1	1	yes	characteristic
98	C₂×Q₁₆	—	134	2	no	—
99	C₂×D₈	—	2	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₂×D₈	—	4	2	no	—
102	C₂×D₈	—	4	2	no	—
103	C₄⋊Q₈	—	134	1	no	—
104	C₄⋊Q₈	—	134	1	no	—
105	#127	—	2	2	no	—
106	SD₃₂	—	268	8	no	—
107	C₂×C₁₆	—	67	4	no	—
108	#107	—	134	2	no	—
109	#106	—	268	2	no	—
110	#103	—	268	2	no	—
111	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₆₇	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	—	4	2	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	4	2	no	—
119	Q₁₆	—	268	4	no	—
120	D₈	—	4	4	no	—
121	D₈	—	4	4	no	—
122	D₈	—	8	4	no	—
123	D₈	—	8	4	no	—
124	C₂×C₈	#11761613	1	2	yes	—
125	C₂×C₈	#11761613	1	2	yes	—
126	C₂×C₈	D₄×D₆₇	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×C₂²⋊D₆₇	1	1	yes	characteristic
132	C₁₆	—	134	4	no	—
133	C₂³	—	8	2	no	—
134	Q₈	—	536	2	no	—
135	Q₈	—	536	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	4	no	—
139	D₄	—	8	4	no	—
140	C₂×C₄	—	536	1	no	—
141	C₂×C₄	—	536	1	no	—
142	C₂×C₄	#14524039	1	1	yes	characteristic
143	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
144	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#14523987	1	4	yes	complemented
147	C₈	#14524067	1	4	yes	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	2	no	—
150	C₂²	—	16	2	no	—
151	C₂²	#16863772	1	1	yes	characteristic
152	C₄	—	1072	1	no	—
153	C₄	—	1072	1	no	—
154	C₄	#16863784	1	2	yes	—
155	C₄	C₂×#14523987	1	2	yes	—
156	C₄	#16863832	1	2	yes	—
157	C₂	—	32	2	no	—
158	C₂	#22559876	1	1	yes	characteristic
159	C₂	#22562464	1	1	yes	characteristic
160	C₂	#22562473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562464

C₂

yes

maximal, complemented

#22562432

C₂

yes

maximal, complemented

#22559806

C₂

yes

maximal, characteristic, complemented

#22558127

C₂

yes

maximal, characteristic, complemented

C₆₇×#1712

C₂

yes

maximal, characteristic

C₂×#14523987

—

#16863801

C₂²

yes

characteristic

#16863794

—

#16863776

C₂²

yes

characteristic

#16863747

—

#16863197

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#493×C₆₇

—

#493×C₆₇

—

#493×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

#14524043

—

#14524043

—

#14523987

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523975

—

#14523968

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

#127×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

SD₃₂

yes

complemented

C₈×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

#496

yes

—

C₄×C₆₇

C₂×SD₃₂

yes

—

C₄×C₆₇

#509

yes

—

Dic₆₇

—

Dic₆₇

—

#12529

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1707

—

#1312

—

C₆₇

#12529

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

134

—

#500

—

#498

—

134

—

—

—

—

C₂×D₆₇

yes

—

#487

—

134

—

#357

—

134

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×D₈

—

100

C₂×D₈

—

101

C₂×D₈

—

102

C₂×D₈

—

103

C₄⋊Q₈

—

134

—

104

C₄⋊Q₈

—

134

—

105

#127

—

106

SD₃₂

—

268

—

107

C₂×C₁₆

—

108

#107

—

134

—

109

#106

—

268

—

110

#103

—

268

—

111

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

112

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

113

C₄×C₈

C₂²×D₆₇

yes

characteristic

114

C₂×Q₈

—

268

—

115

C₂×Q₈

—

268

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

Q₁₆

—

268

—

120

D₈

—

121

D₈

—

122

D₈

—

123

D₈

—

124

C₂×C₈

#11761613

yes

—

125

C₂×C₈

#11761613

yes

—

126

C₂×C₈

D₄×D₆₇

yes

—

127

C₄⋊C₄

—

268

—

128

C₄⋊C₄

—

268

—

129

C₄⋊C₄

—

268

—

130

C₄⋊C₄

—

268

—

131

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

132

C₁₆

—

134

—

133

C₂³

—

134

Q₈

—

536

—

135

Q₈

—

536

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

536

—

141

C₂×C₄

—

536

—

142

C₂×C₄

#14524039

yes

characteristic

143

C₂×C₄

C₂×#11761613

yes

characteristic

144

C₂×C₄

C₂×#11761613

yes

characteristic

145

C₈

—

146

C₈

#14523987

yes

complemented

147

C₈

#14524067

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16863772

yes

characteristic

152

C₄

—

1072

—

153

C₄

—

1072

—

154

C₄

#16863784

yes

—

155

C₄

C₂×#14523987

yes

—

156

C₄

#16863832

yes

—

157

C₂

—

158

C₂

#22559876

yes

characteristic

159

C₂

#22562464

yes

characteristic

160

C₂

#22562473

yes

characteristic

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#1712	C₂
socle	C₂²×C₆₇	#493
hypercenter	#1712	D₆₇
hypocenter	C₆₇	#12529
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting, prime radical

C₆₇×#1712

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #22562464	C₂	semidirect
2. #22562432	C₂	semidirect
3. #22559806	C₂	semidirect
4. #22558127	C₂	semidirect
65. C₈×C₆₇	SD₃₂	semidirect
86. C₆₇	#12529	semidirect
146. C₈	#14523987	semidirect

H ≅ G/N

type

1. #22562464

C₂

semidirect

2. #22562432

C₂

semidirect

3. #22559806

C₂

semidirect

4. #22558127

C₂

semidirect

65. C₈×C₆₇

SD₃₂

semidirect

86. C₆₇

#12529

semidirect

146. C₈

#14523987

semidirect

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

subgroup	index
#22562464	2
#22562432	2
#22559806	2
#22558127	2
C₆₇×#1712	2
#12529	67

subgroup

index

C₆₇×#1712

Subgroups — #30770606

Conjugacy classes of subgroups

Complements

Maximal subgroups