Group #30770094 — subgroups

Subgroups — #30770094

Order 17152.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22562462	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
3	#22559580	C₂	1	2	yes	maximal
4	#22558257	C₂	1	2	yes	maximal
5	C₆₇×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₆	—	2	1	no	—
7	C₂×Dic₅₃₆	—	2	1	no	—
8	#16863776	C₂²	1	2	yes	—
9	#16863682	—	2	2	no	—
10	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₆₇⋊C₁₆	—	2	1	no	—
13	C₁₆×Dic₆₇	—	2	1	no	—
14	#16863141	—	2	2	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#335×C₆₇	C₂²	1	2	yes	—
17	C₂×Dic₂₆₈	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	2	no	—
19	#14524043	—	2	2	no	—
20	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
21	Dic₅₃₆	—	4	2	no	—
22	Dic₅₃₆	—	4	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	1	no	—
24	#14523975	—	4	2	no	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
26	Dic₆₇⋊C₈	—	2	1	no	—
27	#14523968	—	4	2	no	—
28	C₈×Dic₆₇	—	2	1	no	—
29	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₃₄	—	4	2	no	—
34	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
35	Dic₂₆₈	—	4	2	no	—
36	Dic₂₆₈	—	4	2	no	—
37	Dic₂₆₈	—	8	2	no	—
38	C₂×C₆₇⋊C₈	—	2	1	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
42	Dic₆₇⋊C₄	—	2	1	no	—
43	C₄×Dic₆₇	—	2	1	no	—
44	C₆₇⋊C₁₆	—	4	1	no	—
45	C₂×C₈×C₆₇	—	4	2	no	—
46	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₆₇	—	2	1	no	—
50	C₁₆×C₆₇	SD₁₆	1	2	yes	—
51	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
54	C₂×Dic₆₇	—	2	1	no	—
55	C₂×Dic₆₇	—	2	1	no	—
56	C₂×Dic₆₇	—	8	2	no	—
57	Dic₁₃₄	—	2	1	no	—
58	Dic₁₃₄	—	8	2	no	—
59	Dic₁₃₄	—	8	2	no	—
60	Dic₁₃₄	Q₃₂	1	2	yes	complemented
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	—	8	2	no	—
63	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₆₇⋊C₈	—	4	1	no	—
66	C₂²×C₆₇	#466	1	1	yes	characteristic
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	#460	1	1	yes	characteristic
69	C₄×C₆₇	#472	1	1	yes	characteristic
70	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
71	C₄×C₆₇	#508	1	1	yes	characteristic
72	Dic₆₇	—	2	2	no	—
73	Dic₆₇	—	4	1	no	—
74	Dic₆₇	—	16	2	no	—
75	#12440	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	#1668	1	1	yes	characteristic
77	C₂×C₆₇	#2216	1	1	yes	characteristic
78	C₂×C₆₇	#2222	1	1	yes	characteristic
79	#2247	—	67	1	no	—
80	Q₈×C₁₆	—	67	1	no	—
81	#1670	—	67	2	no	—
82	#1371	D₆₇	1	1	yes	characteristic
83	#1362	—	67	2	no	—
84	C₆₇	#12440	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₃₂	—	134	1	no	—
86	C₂×Q₃₂	—	134	1	no	—
87	#494	—	67	2	no	—
88	#475	—	134	2	no	—
89	C₈×Q₈	—	67	1	no	—
90	C₄⋊C₁₆	—	67	1	no	—
91	C₄⋊C₁₆	—	134	1	no	—
92	C₄×C₁₆	—	134	1	no	—
93	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
94	#335	C₂×D₆₇	1	2	yes	—
95	#332	—	134	2	no	—
96	C₂×Q₁₆	—	134	2	no	—
97	C₂×Q₁₆	—	134	2	no	—
98	C₄⋊Q₈	—	134	2	no	—
99	C₄×Q₈	—	67	1	no	—
100	Q₃₂	—	268	2	no	—
101	Q₃₂	—	268	2	no	—
102	C₂×C₁₆	—	134	1	no	—
103	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
104	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
105	#106	—	268	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	67	1	no	—
108	C₄⋊C₈	—	134	1	no	—
109	#103	—	268	2	no	—
110	C₄×C₈	—	134	1	no	—
111	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
112	C₂×Q₈	—	268	2	no	—
113	C₂×Q₈	—	67	1	no	—
114	Q₁₆	—	268	2	no	—
115	Q₁₆	—	268	2	no	—
116	Q₁₆	—	536	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	134	1	no	—
119	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
120	C₂×C₈	#11761629	1	1	yes	characteristic
121	C₄⋊C₄	—	268	2	no	—
122	C₄⋊C₄	—	268	2	no	—
123	C₄⋊C₄	—	67	1	no	—
124	C₄⋊C₄	—	134	1	no	—
125	C₄²	—	134	1	no	—
126	C₄²	D₄×D₆₇	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	268	1	no	—
129	C₁₆	#11761617	1	2	yes	—
130	Q₈	—	536	2	no	—
131	Q₈	—	536	2	no	—
132	Q₈	—	67	2	no	—
133	Q₈	—	134	1	no	—
134	C₂×C₄	—	536	2	no	—
135	C₂×C₄	—	134	1	no	—
136	C₂×C₄	—	134	1	no	—
137	C₂×C₄	#14524018	1	1	yes	characteristic
138	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
139	C₂×C₄	#14524066	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	268	1	no	—
143	C₈	C₂×#11761617	1	1	yes	characteristic
144	C₈	#14524071	1	1	yes	characteristic
145	C₂²	#16863633	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	268	1	no	—
148	C₄	—	1072	2	no	—
149	C₄	—	134	2	no	—
150	C₄	#16863597	1	1	yes	characteristic
151	C₄	#16863667	1	1	yes	characteristic
152	C₄	Q₃₂×D₆₇	1	1	yes	characteristic
153	C₄	#16863825	1	1	yes	characteristic
154	C₂	#22559570	1	1	yes	characteristic
155	C₂	#22562301	1	1	yes	characteristic
156	C₂	#22562335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562462

C₂

yes

maximal, characteristic

C₁₆×Dic₁₃₄

C₂

yes

maximal, characteristic

#22559580

C₂

yes

maximal

#22558257

C₂

yes

maximal

C₆₇×#1371

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

—

C₂×Dic₅₃₆

—

#16863776

C₂²

yes

—

#16863682

—

C₈×Dic₁₃₄

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

C₁₆×Dic₆₇

—

#16863141

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#335×C₆₇

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

C₄×Dic₁₃₄

D₄

yes

characteristic

Dic₅₃₆

—

Dic₅₃₆

—

C₂×C₆₇⋊C₁₆

—

#14523975

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

characteristic

Dic₆₇⋊C₈

—

#14523968

—

C₈×Dic₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₈

yes

characteristic

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₈

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

Q₃₂

yes

complemented

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×SD₁₆

yes

characteristic

C₈×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×Q₃₂

yes

characteristic

C₄×C₆₇

#508

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

Q₈×C₁₆

—

#1670

—

#1371

D₆₇

yes

characteristic

#1362

—

C₆₇

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

134

—

C₂×Q₃₂

—

134

—

#494

—

#475

—

134

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₄×C₁₆

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

C₂×D₆₇

yes

—

#332

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

134

—

C₄×Q₈

—

100

Q₃₂

—

268

—

101

Q₃₂

—

268

—

102

C₂×C₁₆

—

134

—

103

C₂×C₁₆

D₂₆₈

yes

characteristic

104

C₂×C₁₆

D₂₆₈

yes

characteristic

105

#106

—

268

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

134

—

109

#103

—

268

—

110

C₄×C₈

—

134

—

111

C₄×C₈

C₂²×D₆₇

yes

characteristic

112

C₂×Q₈

—

268

—

113

C₂×Q₈

—

114

Q₁₆

—

268

—

115

Q₁₆

—

268

—

116

Q₁₆

—

536

—

117

C₂×C₈

—

118

C₂×C₈

—

134

—

119

C₂×C₈

C₂×D₂₆₈

yes

characteristic

120

C₂×C₈

#11761629

yes

characteristic

121

C₄⋊C₄

—

268

—

122

C₄⋊C₄

—

268

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

134

—

125

C₄²

—

134

—

126

C₄²

D₄×D₆₇

yes

characteristic

127

C₁₆

—

128

C₁₆

—

268

—

129

C₁₆

#11761617

yes

—

130

Q₈

—

536

—

131

Q₈

—

536

—

132

Q₈

—

133

Q₈

—

134

—

134

C₂×C₄

—

536

—

135

C₂×C₄

—

134

—

136

C₂×C₄

—

134

—

137

C₂×C₄

#14524018

yes

characteristic

138

C₂×C₄

D₈×D₆₇

yes

characteristic

139

C₂×C₄

#14524066

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

268

—

143

C₈

C₂×#11761617

yes

characteristic

144

C₈

#14524071

yes

characteristic

145

C₂²

#16863633

yes

characteristic

146

C₄

—

147

C₄

—

268

—

148

C₄

—

1072

—

149

C₄

—

134

—

150

C₄

#16863597

yes

characteristic

151

C₄

#16863667

yes

characteristic

152

C₄

Q₃₂×D₆₇

yes

characteristic

153

C₄

#16863825

yes

characteristic

154

C₂

#22559570

yes

characteristic

155

C₂

#22562301

yes

characteristic

156

C₂

#22562335

yes

characteristic

	subgroup	G/H
center	C₂²	#16863633
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1371	C₂
socle, prime radical	C₂²×C₆₇	#466
hypercenter	#1371	D₆₇
hypocenter	C₆₇	#12440
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#16863633

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1371

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
60. Dic₁₃₄	Q₃₂	semidirect
84. C₆₇	#12440	semidirect

H ≅ G/N

type

60. Dic₁₃₄

Q₃₂

semidirect

84. C₆₇

#12440

semidirect

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

subgroup	index
#22562462	2
C₁₆×Dic₁₃₄	2
#22559580	2
#22558257	2
C₆₇×#1371	2
#12440	67

subgroup

index

C₁₆×Dic₁₃₄

C₆₇×#1371

Subgroups — #30770094

Conjugacy classes of subgroups

Complements

Maximal subgroups