Group #30760854 — subgroups

Subgroups — #30760854

Order 17152.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22560363	C₂	1	1	yes	maximal, characteristic
2	#22558985	C₂	1	2	yes	maximal
3	#22558612	C₂	1	2	yes	maximal
4	#22558038	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1995	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	#16863148	—	2	4	no	—
10	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
11	#533×C₆₇	C₂²	1	1	yes	characteristic
12	#423×C₆₇	C₂²	1	1	yes	characteristic
13	#376×C₆₇	C₂²	1	1	yes	characteristic
14	#339×C₆₇	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
22	C₂×M₁₆×C₆₇	—	2	1	no	—
23	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	—
24	#125×C₆₇	—	2	1	no	—
25	C₂²⋊Q₈×C₆₇	—	2	2	no	—
26	C₄×Q₈×C₆₇	—	2	2	no	—
27	#117×C₆₇	—	2	1	no	—
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	#95×C₆₇	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	4	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	4	no	—
35	C₂×Q₈×C₆₇	—	2	2	no	—
36	C₂²×C₄×C₆₇	—	2	2	no	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	#41	1	2	yes	—
39	M₁₆×C₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	—	4	1	no	—
41	C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₄⋊C₄×C₆₇	—	4	2	no	—
44	C₂²⋊C₄×C₆₇	—	4	1	no	—
45	C₄²×C₆₇	—	2	2	no	—
46	C₄²×C₆₇	—	4	1	no	—
47	C₄²×C₆₇	Q₁₆	1	4	yes	—
48	C₄²×C₆₇	C₂×D₄	1	2	yes	—
49	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₆₇	#41	1	1	yes	characteristic
51	C₂³×C₆₇	#123	1	1	yes	characteristic
52	Q₈×C₆₇	—	4	4	no	—
53	C₂×C₄×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	—	4	2	no	—
55	C₂×C₄×C₆₇	—	4	4	no	—
56	C₂×C₄×C₆₇	—	4	2	no	—
57	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#123	1	2	yes	—
60	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₈×C₆₇	—	8	1	no	—
64	C₆₇⋊C₈	—	4	4	no	—
65	C₆₇⋊C₈	—	8	2	no	—
66	C₂²×C₆₇	—	2	2	no	—
67	C₂²×C₆₇	#399	1	1	yes	characteristic
68	C₂²×C₆₇	#484	1	2	yes	—
69	C₄×C₆₇	—	2	2	no	—
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	4	4	no	—
72	C₄×C₆₇	—	8	2	no	—
73	C₄×C₆₇	#477	1	2	yes	—
74	C₄×C₆₇	#494	1	1	yes	characteristic
75	C₄×C₆₇	#496	1	1	yes	characteristic
76	#10668	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#2095	1	1	yes	characteristic
79	C₂×C₆₇	#2098	1	2	yes	—
80	#1995	D₆₇	1	1	yes	characteristic
81	#1843	—	67	1	no	—
82	#1571	—	67	2	no	—
83	#1481	—	67	2	no	—
84	#1274	—	67	1	no	—
85	C₆₇	#10668	1	1	yes	characteristic, complemented, Sylow
86	#533	C₂×D₆₇	1	1	yes	characteristic
87	#431	—	67	2	no	—
88	#423	C₂×D₆₇	1	1	yes	characteristic
89	C₂×C₄⋊C₈	—	67	2	no	—
90	C₂×C₄⋊C₈	—	134	2	no	—
91	#376	C₂×D₆₇	1	1	yes	characteristic
92	#339	—	2	2	no	—
93	#335	—	134	4	no	—
94	Q₈⋊C₈	—	134	4	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂²×C₈	—	134	2	no	—
97	C₄⋊Q₈	Dic₁₃₄	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
103	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	268	4	no	—
106	C₄⋊C₈	—	134	2	no	—
107	C₄⋊C₈	—	134	2	no	—
108	C₄⋊C₈	—	134	2	no	—
109	C₂²⋊C₈	—	134	2	no	—
110	C₄×C₈	—	67	4	no	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
115	C₂²×C₄	#11761627	1	2	yes	—
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	—	268	4	no	—
120	C₂×C₈	—	134	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#11761618	1	4	yes	—
128	C₄²	D₄×D₆₇	1	2	yes	—
129	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
130	C₄²	#11761628	1	1	yes	characteristic
131	C₂³	#14524024	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#14524020	1	2	yes	—
138	C₂×C₄	#14524022	1	1	yes	characteristic
139	C₂×C₄	#14524025	1	2	yes	—
140	C₂×C₄	#14524039	1	1	yes	characteristic
141	C₂×C₄	C₂×#11761618	1	2	yes	—
142	C₂×C₄	#14524068	1	2	yes	—
143	C₈	—	8	1	no	—
144	C₈	—	268	4	no	—
145	C₈	—	536	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#16863332	1	1	yes	characteristic
148	C₂²	#16863729	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	#16863691	1	2	yes	—
154	C₄	#16863775	1	1	yes	characteristic
155	C₄	#16863786	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#22561747	1	1	yes	characteristic
158	C₂	#22561764	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560363

C₂

yes

maximal, characteristic

#22558985

C₂

yes

maximal

#22558612

C₂

yes

maximal

#22558038

C₂

yes

maximal, characteristic

C₆₇×#1995

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863148

—

Q₈⋊(C₆₇⋊C₈)

—

#533×C₆₇

C₂²

yes

characteristic

#423×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

Q₈

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#399

yes

characteristic

C₂²×C₆₇

#484

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#477

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#2095

yes

characteristic

C₂×C₆₇

#2098

yes

—

#1995

D₆₇

yes

—

—

—

—

C₆₇

#10668

yes

characteristic, complemented, Sylow

#533

C₂×D₆₇

yes

characteristic

#431

—

#423

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

134

—

#376

C₂×D₆₇

yes

characteristic

#339

—

#335

—

134

—

Q₈⋊C₈

—

134

—

C₂×M₁₆

—

C₂²×C₈

—

134

—

C₄⋊Q₈

Dic₁₃₄

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

C₂²⋊D₆₇

yes

—

103

C₂×C₄²

C₂²×D₆₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

268

—

106

C₄⋊C₈

—

134

—

107

C₄⋊C₈

—

134

—

108

C₄⋊C₈

—

134

—

109

C₂²⋊C₈

—

134

—

110

C₄×C₈

—

111

#95

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

115

C₂²×C₄

#11761627

yes

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

—

268

—

120

C₂×C₈

—

134

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11761618

yes

—

128

C₄²

D₄×D₆₇

yes

—

129

C₄²

C₂×Dic₁₃₄

yes

characteristic

130

C₄²

#11761628

yes

characteristic

131

C₂³

#14524024

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14524020

yes

—

138

C₂×C₄

#14524022

yes

characteristic

139

C₂×C₄

#14524025

yes

—

140

C₂×C₄

#14524039

yes

characteristic

141

C₂×C₄

C₂×#11761618

yes

—

142

C₂×C₄

#14524068

yes

—

143

C₈

—

144

C₈

—

268

—

145

C₈

—

536

—

146

C₂²

—

147

C₂²

#16863332

yes

characteristic

148

C₂²

#16863729

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16863691

yes

—

154

C₄

#16863775

yes

characteristic

155

C₄

#16863786

yes

characteristic

156

C₂

—

157

C₂

#22561747

yes

characteristic

158

C₂

#22561764

yes

—

	subgroup	G/H
center	C₂×C₄	#14524039
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1995	C₂
socle	C₂²×C₆₇	#399
prime radical	C₂³×C₆₇	#123
hypercenter	#1995	D₆₇
hypocenter	C₆₇	#10668
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#14524039

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1995

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
85. C₆₇	#10668	semidirect

H ≅ G/N

type

85. C₆₇

#10668

semidirect

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

subgroup	index
#22560363	2
#22558985	2
#22558612	2
#22558038	2
C₆₇×#1995	2
#10668	67

subgroup

index

C₆₇×#1995

Subgroups — #30760854

Conjugacy classes of subgroups

Complements

Maximal subgroups