Group #30770093 — subgroups

Subgroups — #30770093

Order 17152.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22559579	C₂	1	2	yes	maximal
3	#22558281	C₂	1	1	yes	maximal, characteristic
4	#22558259	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₆₇	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	#16863681	—	2	2	no	—
10	#16863149	C₂²	1	2	yes	—
11	#16863143	—	2	2	no	—
12	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	1	no	—
14	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₇	—	2	1	no	—
16	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₈	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	2	no	—
19	#14524043	—	2	2	no	—
20	Dic₅₃₆	—	4	2	no	—
21	Dic₅₃₆	—	4	2	no	—
22	#14523974	—	4	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	#14523967	—	4	2	no	—
25	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	—	2	1	no	—
27	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	1	no	—
30	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	—	2	1	no	—
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₃₄	—	4	2	no	—
34	Dic₂₆₈	—	4	2	no	—
35	Dic₂₆₈	—	4	2	no	—
36	Dic₂₆₈	—	8	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
46	C₄²×C₆₇	—	2	1	no	—
47	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₆₇	—	2	1	no	—
49	C₁₆×C₆₇	—	4	1	no	—
50	C₁₆×C₆₇	SD₁₆	1	2	yes	—
51	Q₈×C₆₇	—	2	1	no	—
52	Q₈×C₆₇	Q₃₂	1	2	yes	complemented
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	—	2	1	no	—
55	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
58	C₂×Dic₆₇	—	8	2	no	—
59	Dic₁₃₄	—	8	2	no	—
60	Dic₁₃₄	—	8	2	no	—
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	—	4	1	no	—
63	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₆₇⋊C₈	—	8	2	no	—
66	C₂²×C₆₇	#466	1	1	yes	characteristic
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	2	1	no	—
69	C₄×C₆₇	—	4	1	no	—
70	C₄×C₆₇	#460	1	1	yes	characteristic
71	C₄×C₆₇	#472	1	1	yes	characteristic
72	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
73	C₄×C₆₇	#508	1	1	yes	characteristic
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	D₆₇	1	1	yes	characteristic
81	#1670	—	67	2	no	—
82	#1371	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
90	C₄⋊C₁₆	—	2	1	no	—
91	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
92	C₄×C₁₆	—	2	1	no	—
93	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
94	#335	—	67	2	no	—
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	D₂₆₈	1	1	yes	characteristic
100	Q₃₂	—	268	2	no	—
101	Q₃₂	—	268	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
105	#106	—	268	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	134	2	no	—
108	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
109	#103	—	268	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
112	C₂×Q₈	—	268	2	no	—
113	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
114	Q₁₆	—	268	2	no	—
115	Q₁₆	—	268	2	no	—
116	Q₁₆	—	536	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
120	C₂×C₈	#11761628	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	268	2	no	—
123	C₄⋊C₄	—	268	2	no	—
124	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#11761615	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	536	2	no	—
132	Q₈	—	536	2	no	—
133	Q₈	Dic₅₃₆	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	536	2	no	—
137	C₂×C₄	#14524017	1	1	yes	characteristic
138	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
139	C₂×C₄	#14524065	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	536	2	no	—
143	C₈	C₂×#11761615	1	1	yes	characteristic
144	C₈	#14524068	1	1	yes	characteristic
145	C₂²	#16863632	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	1072	2	no	—
150	C₄	#16863596	1	1	yes	characteristic
151	C₄	#16863666	1	1	yes	characteristic
152	C₄	C₂×Dic₅₃₆	1	1	yes	characteristic
153	C₄	#16863828	1	1	yes	characteristic
154	C₂	#22559569	1	1	yes	characteristic
155	C₂	#22562300	1	1	yes	characteristic
156	C₂	#22562334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562462

C₂

yes

maximal, characteristic

#22559579

C₂

yes

maximal

#22558281

C₂

yes

maximal, characteristic

#22558259

C₂

yes

maximal

Q₈×C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

—

C₂×Dic₅₃₆

—

#16863776

C₂²

yes

—

#16863681

—

#16863149

C₂²

yes

—

#16863143

—

C₈×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₄⋊C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

Dic₅₃₆

—

Dic₅₃₆

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

—

#14523967

—

C₄×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

SD₁₆

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

Q₃₂

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

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

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×Q₃₂

yes

characteristic

C₄×C₆₇

#508

yes

characteristic

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

Q₈×C₁₆

D₆₇

yes

—

—

—

C₆₇

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

134

—

C₂×Q₃₂

—

134

—

#494

—

#475

—

134

—

C₈×Q₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₆₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

—

#332

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

134

—

C₄×Q₈

D₂₆₈

yes

characteristic

100

Q₃₂

—

268

—

101

Q₃₂

—

268

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

105

#106

—

268

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

134

—

108

C₄⋊C₈

D₂₆₈

yes

characteristic

109

#103

—

268

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₆₇

yes

characteristic

112

C₂×Q₈

—

268

—

113

C₂×Q₈

D₅₃₆

yes

characteristic

114

Q₁₆

—

268

—

115

Q₁₆

—

268

—

116

Q₁₆

—

536

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

120

C₂×C₈

#11761628

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

268

—

123

C₄⋊C₄

—

268

—

124

C₄⋊C₄

D₅₃₆

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₂₆₈

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#11761615

yes

—

130

Q₈

—

131

Q₈

—

536

—

132

Q₈

—

536

—

133

Q₈

Dic₅₃₆

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

536

—

137

C₂×C₄

#14524017

yes

characteristic

138

C₂×C₄

C₂×D₅₃₆

yes

characteristic

139

C₂×C₄

#14524065

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

536

—

143

C₈

C₂×#11761615

yes

characteristic

144

C₈

#14524068

yes

characteristic

145

C₂²

#16863632

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

1072

—

150

C₄

#16863596

yes

characteristic

151

C₄

#16863666

yes

characteristic

152

C₄

C₂×Dic₅₃₆

yes

characteristic

153

C₄

#16863828

yes

characteristic

154

C₂

#22559569

yes

characteristic

155

C₂

#22562300

yes

characteristic

156

C₂

#22562334

yes

characteristic

	subgroup	G/H
center	C₂²	#16863632
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	Q₈×C₁₆×C₆₇	C₂
socle, prime radical	C₂²×C₆₇	#466
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	#12440
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#16863632

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

Q₈×C₁₆×C₆₇

C₂

socle, prime radical

C₂²×C₆₇

#466

hypercenter

Q₈×C₁₆

D₆₇

hypocenter

C₆₇

#12440

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
52. Q₈×C₆₇	Q₃₂	semidirect
84. C₆₇	#12440	semidirect
133. Q₈	Dic₅₃₆	semidirect

H ≅ G/N

type

52. Q₈×C₆₇

Q₃₂

semidirect

84. C₆₇

#12440

semidirect

133. Q₈

Dic₅₃₆

semidirect

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

subgroup	index
#22562462	2
#22559579	2
#22558281	2
#22558259	2
Q₈×C₁₆×C₆₇	2
#12440	67

subgroup

index

Q₈×C₁₆×C₆₇

Subgroups — #30770093

Conjugacy classes of subgroups

Complements

Maximal subgroups