Group #30775845 — subgroups

Subgroups — #30775845

Order 17152.

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

normal subgroups only

This group has 252 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	#22561964	C₂	1	1	yes	maximal, characteristic
2	#22560955	C₂	1	2	yes	maximal
3	#22558323	C₂	1	2	yes	maximal
4	#22558311	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2161	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	#16863434	C₂²	1	2	yes	—
8	#16863424	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
10	Dic₆₇⋊C₁₆	—	2	2	no	—
11	#16863157	—	2	2	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₆₇	C₂²	1	2	yes	—
14	#368×C₆₇	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
16	#14524002	—	2	2	no	—
17	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
18	#14523983	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	#14523977	—	2	2	no	—
21	#14523975	Q₈	1	4	yes	—
22	Dic₆₇⋊C₈	—	4	4	no	—
23	C₈×Dic₆₇	—	2	1	no	—
24	C₈×Dic₆₇	—	2	1	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
29	#108×C₆₇	—	2	2	no	—
30	#108×C₆₇	—	2	2	no	—
31	C₂²×Dic₆₇	—	2	1	no	—
32	C₂×C₆₇⋊C₈	—	2	1	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₄⋊Dic₆₇	—	2	4	no	—
35	C₂²⋊Dic₆₇	—	4	2	no	—
36	C₄×Dic₆₇	—	2	1	no	—
37	C₄×Dic₆₇	—	4	2	no	—
38	C₄×Dic₆₇	Q₁₆	1	1	yes	characteristic
39	C₄×Dic₆₇	Q₁₆	1	1	yes	characteristic
40	C₆₇⋊C₁₆	—	4	1	no	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	M₁₆×C₆₇	—	4	2	no	—
43	M₁₆×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₁₆×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	#135	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₂×Dic₆₇	—	2	1	no	—
56	C₂×Dic₆₇	—	2	1	no	—
57	C₂×Dic₆₇	—	4	2	no	—
58	C₂×Dic₆₇	—	4	4	no	—
59	C₈×C₆₇	—	8	2	no	—
60	C₈×C₆₇	#123	1	1	yes	characteristic
61	C₈×C₆₇	#125	1	2	yes	—
62	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₆₇⋊C₈	—	4	1	no	—
64	C₂²×C₆₇	—	4	1	no	—
65	C₂²×C₆₇	#467	1	1	yes	characteristic
66	C₂²×C₆₇	#478	1	2	yes	—
67	C₄×C₆₇	#398	1	1	yes	characteristic
68	C₄×C₆₇	#469	1	1	yes	characteristic
69	C₄×C₆₇	#477	1	2	yes	—
70	Dic₆₇	—	2	2	no	—
71	Dic₆₇	—	4	1	no	—
72	Dic₆₇	—	8	4	no	—
73	#13519	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	2	no	—
76	C₂×C₆₇	#2087	1	1	yes	characteristic
77	#2161	D₆₇	1	1	yes	characteristic
78	#2150	—	67	1	no	—
79	#1945	—	67	2	no	—
80	#1389	—	67	2	no	—
81	#1384	—	67	1	no	—
82	C₆₇	#13519	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	67	1	no	—
84	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
85	C₂×#108	C₂×D₆₇	1	2	yes	—
86	#427	—	67	2	no	—
87	#424	—	134	2	no	—
88	C₂×C₄×C₈	—	67	1	no	—
89	#368	—	2	2	no	—
90	C₄⋊C₁₆	—	134	2	no	—
91	#339	—	134	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	—	67	1	no	—
94	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
95	#117	—	134	2	no	—
96	C₂×C₄²	—	67	1	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	134	2	no	—
99	C₂×C₁₆	—	134	1	no	—
100	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	2	2	no	—
103	#107	—	134	2	no	—
104	#106	—	67	4	no	—
105	C₄⋊C₈	—	268	4	no	—
106	C₄×C₈	—	134	1	no	—
107	C₄×C₈	—	134	1	no	—
108	C₂²×C₄	—	134	1	no	—
109	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	134	1	no	—
114	C₂×C₈	—	134	2	no	—
115	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂×C₈	Q₈×D₆₇	1	2	yes	—
117	C₂×C₈	#11761627	1	1	yes	characteristic
118	C₂×C₈	#11761628	1	2	yes	—
119	C₄⋊C₄	—	134	4	no	—
120	C₂²⋊C₄	—	268	2	no	—
121	C₄²	—	268	2	no	—
122	C₄²	—	67	1	no	—
123	C₄²	—	67	1	no	—
124	C₄²	—	134	1	no	—
125	C₁₆	—	2	2	no	—
126	C₁₆	—	268	1	no	—
127	C₂³	#14524066	1	1	yes	characteristic
128	C₂×C₄	—	268	2	no	—
129	C₂×C₄	—	268	4	no	—
130	C₂×C₄	—	134	1	no	—
131	C₂×C₄	—	134	1	no	—
132	C₂×C₄	#14524014	1	1	yes	characteristic
133	C₂×C₄	#14524021	1	2	yes	—
134	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
135	C₂×C₄	#14524067	1	2	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	268	1	no	—
138	C₈	#14524026	1	1	yes	characteristic
139	C₈	#14524033	1	2	yes	—
140	C₈	#14524041	1	1	yes	characteristic
141	C₂²	—	4	1	no	—
142	C₂²	#16863634	1	1	yes	characteristic
143	C₂²	#16863699	1	2	yes	—
144	C₄	—	268	1	no	—
145	C₄	—	536	4	no	—
146	C₄	—	134	2	no	—
147	C₄	#16863327	1	1	yes	characteristic
148	C₄	#16863646	1	1	yes	characteristic
149	C₄	#16863690	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#22561703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561964

C₂

yes

maximal, characteristic

#22560955

C₂

yes

maximal

#22558323

C₂

yes

maximal

#22558311

C₂

yes

maximal, characteristic

C₆₇×#2161

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

#16863434

C₂²

yes

—

#16863424

—

C₂×C₈×Dic₆₇

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863157

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

#368×C₆₇

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

#14524002

—

C₂×C₄×Dic₆₇

D₄

yes

characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523977

—

#14523975

Q₈

yes

—

Dic₆₇⋊C₈

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

Q₁₆

yes

characteristic

C₄×Dic₆₇

Q₁₆

yes

characteristic

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#123

yes

characteristic

C₈×C₆₇

#125

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#467

yes

characteristic

C₂²×C₆₇

#478

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

#13519

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2087

yes

characteristic

#2161

D₆₇

yes

—

—

—

—

C₆₇

#13519

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂×#108

C₂×D₆₇

yes

—

#427

—

#424

—

134

—

C₂×C₄×C₈

—

#368

—

C₄⋊C₁₆

—

134

—

#339

—

134

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

—

134

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

134

—

100

C₂×C₁₆

C₂²⋊D₆₇

yes

—

101

#108

—

102

#108

—

103

#107

—

134

—

104

#106

—

105

C₄⋊C₈

—

268

—

106

C₄×C₈

—

134

—

107

C₄×C₈

—

134

—

108

C₂²×C₄

—

134

—

109

C₂²×C₄

D₄×D₆₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

134

—

114

C₂×C₈

—

134

—

115

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

116

C₂×C₈

Q₈×D₆₇

yes

—

117

C₂×C₈

#11761627

yes

characteristic

118

C₂×C₈

#11761628

yes

—

119

C₄⋊C₄

—

134

—

120

C₂²⋊C₄

—

268

—

121

C₄²

—

268

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

134

—

125

C₁₆

—

126

C₁₆

—

268

—

127

C₂³

#14524066

yes

characteristic

128

C₂×C₄

—

268

—

129

C₂×C₄

—

268

—

130

C₂×C₄

—

134

—

131

C₂×C₄

—

134

—

132

C₂×C₄

#14524014

yes

characteristic

133

C₂×C₄

#14524021

yes

—

134

C₂×C₄

Q₁₆×D₆₇

yes

characteristic

135

C₂×C₄

#14524067

yes

—

136

C₈

—

137

C₈

—

268

—

138

C₈

#14524026

yes

characteristic

139

C₈

#14524033

yes

—

140

C₈

#14524041

yes

characteristic

141

C₂²

—

142

C₂²

#16863634

yes

characteristic

143

C₂²

#16863699

yes

—

144

C₄

—

268

—

145

C₄

—

536

—

146

C₄

—

134

—

147

C₄

#16863327

yes

characteristic

148

C₄

#16863646

yes

characteristic

149

C₄

#16863690

yes

—

150

C₂

—

151

C₂

—

152

C₂

#22561703

yes

characteristic

	subgroup	G/H
center	C₄	#16863646
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2161	C₂
socle	C₂×C₆₇	#2087
prime radical	C₂³×C₆₇	#135
hypercenter	#2161	D₆₇
hypocenter	C₆₇	#13519
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#16863646

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2161

C₂

C₂×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
82. C₆₇	#13519	semidirect

H ≅ G/N

type

82. C₆₇

#13519

semidirect

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

subgroup	index
#22561964	2
#22560955	2
#22558323	2
#22558311	2
C₆₇×#2161	2
#13519	67

subgroup

index

C₆₇×#2161

Subgroups — #30775845

Conjugacy classes of subgroups

Complements

Maximal subgroups