Group #30772247 — subgroups

Subgroups — #30772247

Order 17152.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	C₂×#16863216	C₂	1	4	yes	maximal
2	#22561965	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
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	#16863216	—	2	8	no	—
11	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	—	2	2	no	—
17	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
18	#14523983	—	2	4	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
21	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
22	C₈×Dic₆₇	—	2	4	no	—
23	C₈×Dic₆₇	D₄	1	4	yes	—
24	C₈×Dic₆₇	Q₈	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₇	—	2	4	no	—
29	C₂×C₁₆×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂²×Dic₆₇	—	2	2	no	—
33	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
34	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
35	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
36	C₄×Dic₆₇	—	2	8	no	—
37	C₄×Dic₆₇	—	2	2	no	—
38	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
39	C₆₇⋊C₁₆	—	4	2	no	—
40	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
51	C₁₆×C₆₇	—	4	2	no	—
52	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₆₇	—	2	8	no	—
61	C₂×Dic₆₇	—	2	4	no	—
62	C₈×C₆₇	—	2	2	no	—
63	C₈×C₆₇	#127	1	1	yes	characteristic
64	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
66	C₆₇⋊C₈	C₄⋊C₈	1	8	yes	—
67	C₂²×C₆₇	—	2	2	no	—
68	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₆₇	#423	1	2	yes	—
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	#364	1	4	yes	complemented
72	C₄×C₆₇	#384	1	1	yes	characteristic
73	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₆₇	#423	1	1	yes	characteristic
75	Dic₆₇	—	2	8	no	—
76	#12900	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#1842	1	1	yes	characteristic
79	C₂×C₆₇	C₂×#364	1	2	yes	—
80	C₂×#364	—	67	4	no	—
81	#2150	—	67	1	no	—
82	#2150	D₆₇	1	1	yes	characteristic
83	C₄²×C₈	—	67	1	no	—
84	C₆₇	#12900	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	67	2	no	—
86	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
87	C₂×C₄×C₈	—	67	2	no	—
88	C₂×C₄×C₈	—	67	1	no	—
89	C₂×C₄×C₈	—	67	2	no	—
90	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
91	C₄³	—	67	1	no	—
92	#364	—	134	8	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	134	2	no	—
95	C₂²×C₈	—	67	2	no	—
96	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	C₂×C₄²	—	67	2	no	—
99	C₂×C₄²	—	134	2	no	—
100	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	134	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	134	2	no	—
105	C₄×C₈	—	67	4	no	—
106	C₄×C₈	—	67	4	no	—
107	C₄×C₈	—	67	2	no	—
108	C₄×C₈	—	67	2	no	—
109	C₄×C₈	—	134	4	no	—
110	C₄×C₈	Dic₁₃₄	1	2	yes	—
111	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
112	C₂²×C₄	—	134	2	no	—
113	C₂²×C₄	#11761611	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	67	4	no	—
117	C₂×C₈	—	67	4	no	—
118	C₂×C₈	—	67	4	no	—
119	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
120	C₂×C₈	D₄×D₆₇	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₈	Q₈×D₆₇	1	2	yes	—
123	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
124	C₄²	—	67	4	no	—
125	C₄²	—	134	8	no	—
126	C₄²	—	134	2	no	—
127	C₄²	Dic₆₇⋊C₄	1	2	yes	—
128	C₄²	C₈×D₆₇	1	1	yes	characteristic
129	C₄²	C₈×D₆₇	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₁₆	—	268	2	no	—
132	C₂³	C₂×#11761611	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	134	8	no	—
135	C₂×C₄	—	134	4	no	—
136	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
137	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
139	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	67	8	no	—
144	C₈	#14524039	1	1	yes	characteristic
145	C₈	#14524041	1	1	yes	characteristic
146	C₈	#14524042	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
149	C₂²	#16863418	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	134	8	no	—
152	C₄	#16863216	1	4	yes	complemented
153	C₄	#16863265	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₆₇	1	2	yes	—
155	C₄	#16863419	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#22560360	1	1	yes	characteristic
158	C₂	C₂×#16863216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863216

C₂

yes

maximal

#22561965

C₂

yes

maximal, characteristic

C₄×C₈×Dic₆₇

C₂

yes

maximal, characteristic

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

#16863216

—

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

D₄

yes

—

C₈×Dic₆₇

Q₈

yes

—

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

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₄⋊C₄

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

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

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

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#127

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

—

C₆₇⋊C₈

C₄⋊C₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#423

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×C₄⋊C₈

yes

—

C₄×C₆₇

#423

yes

characteristic

Dic₆₇

—

#12900

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1842

yes

characteristic

C₂×C₆₇

C₂×#364

yes

—

C₂×#364

—

#2150

—

#2150

D₆₇

yes

characteristic

C₄²×C₈

—

C₆₇

#12900

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

—

#364

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

134

—

100

C₂×C₄²

C₄×D₆₇

yes

characteristic

101

M₃₂

—

102

M₃₂

—

134

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

134

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

134

—

110

C₄×C₈

Dic₁₃₄

yes

—

111

C₄×C₈

C₂²⋊D₆₇

yes

—

112

C₂²×C₄

—

134

—

113

C₂²×C₄

#11761611

yes

—

114

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

Dic₆₇⋊C₄

yes

—

120

C₂×C₈

D₄×D₆₇

yes

—

121

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₂×C₈

Q₈×D₆₇

yes

—

123

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

124

C₄²

—

125

C₄²

—

134

—

126

C₄²

—

134

—

127

C₄²

Dic₆₇⋊C₄

yes

—

128

C₄²

C₈×D₆₇

yes

characteristic

129

C₄²

C₈×D₆₇

yes

characteristic

130

C₁₆

—

131

C₁₆

—

268

—

132

C₂³

C₂×#11761611

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

134

—

135

C₂×C₄

—

134

—

136

C₂×C₄

Dic₆₇⋊C₈

yes

—

137

C₂×C₄

Dic₆₇⋊C₈

yes

—

138

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

139

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

140

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

141

C₂×C₄

M₁₆×D₆₇

yes

—

142

C₈

—

143

C₈

—

144

C₈

#14524039

yes

characteristic

145

C₈

#14524041

yes

characteristic

146

C₈

#14524042

yes

—

147

C₂²

—

148

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

149

C₂²

#16863418

yes

—

150

C₄

—

151

C₄

—

134

—

152

C₄

#16863216

yes

complemented

153

C₄

#16863265

yes

characteristic

154

C₄

C₄⋊C₈×D₆₇

yes

—

155

C₄

#16863419

yes

characteristic

156

C₂

—

157

C₂

#22560360

yes

characteristic

158

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	#12900
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
71. C₄×C₆₇	#364	semidirect
84. C₆₇	#12900	semidirect
152. C₄	#16863216	semidirect

H ≅ G/N

type

71. C₄×C₆₇

#364

semidirect

84. C₆₇

#12900

semidirect

152. C₄

#16863216

semidirect

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

subgroup	index
C₂×#16863216	2
#22561965	2
C₄×C₈×Dic₆₇	2
C₆₇×#2150	2
#12900	67

subgroup

index

C₂×#16863216

#22561965

C₄×C₈×Dic₆₇

C₆₇×#2150

#12900

Subgroups — #30772247

Conjugacy classes of subgroups

Complements

Maximal subgroups