Group #30772191 — subgroups

Subgroups — #30772191

Order 17152.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22561965	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558290	C₂	1	4	yes	maximal
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₇	—	2	2	no	—
16	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
17	#14523983	—	2	4	no	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
20	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
21	C₈×Dic₆₇	—	2	4	no	—
22	C₈×Dic₆₇	—	2	2	no	—
23	C₈×Dic₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	4	no	—
28	C₂×C₁₆×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₄×C₈×C₆₇	Q₈	1	2	yes	—
31	C₂²×Dic₆₇	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
35	C₄×Dic₆₇	—	2	8	no	—
36	C₄×Dic₆₇	—	2	2	no	—
37	C₄×Dic₆₇	M₁₆	1	4	yes	—
38	C₆₇⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	#41	1	2	yes	—
46	C₂×C₈×C₆₇	#41	1	2	yes	—
47	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₁₆×C₆₇	—	4	2	no	—
51	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	4	no	—
53	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#117	1	2	yes	—
57	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₆₇	—	2	8	no	—
60	C₂×Dic₆₇	—	2	4	no	—
61	C₈×C₆₇	—	2	2	no	—
62	C₈×C₆₇	#124	1	1	yes	characteristic
63	C₈×C₆₇	#124	1	1	yes	characteristic
64	C₈×C₆₇	#125	1	2	yes	—
65	C₆₇⋊C₈	—	2	4	no	—
66	C₂²×C₆₇	—	2	2	no	—
67	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₆₇	#432	1	2	yes	—
69	C₄×C₆₇	—	2	2	no	—
70	C₄×C₆₇	#364	1	4	yes	—
71	C₄×C₆₇	#382	1	1	yes	characteristic
72	C₄×C₆₇	#423	1	1	yes	characteristic
73	C₄×C₆₇	#431	1	2	yes	—
74	Dic₆₇	—	2	8	no	—
75	#12883	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	—	2	2	no	—
77	C₂×C₆₇	#1831	1	1	yes	characteristic
78	C₂×C₆₇	C₂×#364	1	2	yes	—
79	#2150	—	67	1	no	—
80	#2150	D₆₇	1	1	yes	characteristic
81	C₄²×C₈	—	67	1	no	—
82	#1376	—	67	4	no	—
83	C₆₇	#12883	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₃₂	—	67	2	no	—
85	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
86	C₂×C₄×C₈	—	67	1	no	—
87	C₂×C₄×C₈	—	67	2	no	—
88	C₂×C₄×C₈	—	67	2	no	—
89	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
90	C₄³	—	67	1	no	—
91	C₄⋊C₁₆	—	2	2	no	—
92	C₄⋊C₁₆	—	134	2	no	—
93	C₂²×C₈	—	67	2	no	—
94	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
96	C₂×C₄²	—	67	2	no	—
97	C₂×C₄²	—	134	2	no	—
98	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
99	M₃₂	—	2	4	no	—
100	M₃₂	—	134	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	134	2	no	—
103	C₄×C₈	—	67	2	no	—
104	C₄×C₈	—	67	2	no	—
105	C₄×C₈	—	134	4	no	—
106	C₄×C₈	—	134	2	no	—
107	C₄×C₈	—	134	2	no	—
108	C₄×C₈	Dic₁₃₄	1	2	yes	—
109	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
110	C₂²×C₄	—	134	2	no	—
111	C₂²×C₄	#11761611	1	2	yes	—
112	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	67	4	no	—
115	C₂×C₈	—	134	2	no	—
116	C₂×C₈	—	134	2	no	—
117	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
119	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
120	C₂×C₈	#11761627	1	2	yes	—
121	C₂×C₈	#11761629	1	2	yes	—
122	C₄²	—	67	4	no	—
123	C₄²	—	134	8	no	—
124	C₄²	—	134	2	no	—
125	C₄²	Dic₆₇⋊C₄	1	2	yes	—
126	C₄²	C₈×D₆₇	1	1	yes	characteristic
127	C₄²	C₈×D₆₇	1	1	yes	characteristic
128	C₁₆	—	4	2	no	—
129	C₁₆	—	268	2	no	—
130	C₂³	C₂×#11761611	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	134	8	no	—
133	C₂×C₄	—	134	4	no	—
134	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
135	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
136	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
137	C₂×C₄	#14524001	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
140	C₈	—	2	2	no	—
141	C₈	—	134	4	no	—
142	C₈	#14524030	1	1	yes	characteristic
143	C₈	#14524032	1	1	yes	characteristic
144	C₈	#14524034	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
147	C₂²	#16863452	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	134	8	no	—
150	C₄	#16863216	1	4	yes	—
151	C₄	#16863261	1	1	yes	characteristic
152	C₄	#16863419	1	1	yes	characteristic
153	C₄	#16863448	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	#22560325	1	1	yes	characteristic
156	C₂	C₂×#16863216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561965

C₂

yes

maximal, characteristic

C₄×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#22558290

C₂

yes

maximal

C₆₇×#2150

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₆₇

C₄

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

—

C₄²×Dic₆₇

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₄×C₆₇⋊C₈

C₈

yes

—

C₄×C₆₇⋊C₈

C₈

yes

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

M₁₆

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

#364

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12883

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1831

yes

characteristic

C₂×C₆₇

C₂×#364

yes

—

#2150

—

#2150

D₆₇

yes

characteristic

C₄²×C₈

—

#1376

—

C₆₇

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

134

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

M₃₂

—

100

M₃₂

—

134

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

134

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

134

—

106

C₄×C₈

—

134

—

107

C₄×C₈

—

134

—

108

C₄×C₈

Dic₁₃₄

yes

—

109

C₄×C₈

C₂²⋊D₆₇

yes

—

110

C₂²×C₄

—

134

—

111

C₂²×C₄

#11761611

yes

—

112

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

134

—

116

C₂×C₈

—

134

—

117

C₂×C₈

Dic₆₇⋊C₄

yes

—

118

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

119

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

120

C₂×C₈

#11761627

yes

—

121

C₂×C₈

#11761629

yes

—

122

C₄²

—

123

C₄²

—

134

—

124

C₄²

—

134

—

125

C₄²

Dic₆₇⋊C₄

yes

—

126

C₄²

C₈×D₆₇

yes

characteristic

127

C₄²

C₈×D₆₇

yes

characteristic

128

C₁₆

—

129

C₁₆

—

268

—

130

C₂³

C₂×#11761611

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

134

—

133

C₂×C₄

—

134

—

134

C₂×C₄

Dic₆₇⋊C₈

yes

—

135

C₂×C₄

Dic₆₇⋊C₈

yes

—

136

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

137

C₂×C₄

#14524001

yes

—

138

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

139

C₂×C₄

M₁₆×D₆₇

yes

—

140

C₈

—

141

C₈

—

134

—

142

C₈

#14524030

yes

characteristic

143

C₈

#14524032

yes

characteristic

144

C₈

#14524034

yes

—

145

C₂²

—

146

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

147

C₂²

#16863452

yes

—

148

C₄

—

149

C₄

—

134

—

150

C₄

#16863216

yes

—

151

C₄

#16863261

yes

characteristic

152

C₄

#16863419

yes

characteristic

153

C₄

#16863448

yes

—

154

C₂

—

155

C₂

#22560325

yes

characteristic

156

C₂

C₂×#16863216

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#12883
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2150

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₈

prime radical

C₂³×C₆₇

C₂×M₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
83. C₆₇	#12883	semidirect

H ≅ G/N

type

83. C₆₇

#12883

semidirect

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

subgroup	index
#22561965	2
C₄×C₈×Dic₆₇	2
#22558290	2
C₆₇×#2150	2
#12883	67

subgroup

index

#22561965

C₄×C₈×Dic₆₇

#22558290

C₆₇×#2150

#12883

Subgroups — #30772191

Conjugacy classes of subgroups

Complements

Maximal subgroups