Group #30772197 — subgroups

Subgroups — #30772197

Order 17152.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22561975	C₂	1	1	yes	maximal, characteristic
2	#22559964	C₂	1	1	yes	maximal, characteristic
3	#22558297	C₂	1	2	yes	maximal
4	#22558296	C₂	1	2	yes	maximal
5	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	#16863351	C₂²	1	2	yes	—
8	#16863217	—	2	2	no	—
9	C₆₇⋊#338	C₄	1	1	yes	characteristic
10	#16863153	C₄	1	1	yes	characteristic
11	#16863152	C₄	1	2	yes	—
12	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#405×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₆₇	—	2	2	no	—
16	C₂×#11761612	C₂×C₄	1	2	yes	—
17	#14524002	C₂×C₄	1	2	yes	—
18	#14523983	—	2	2	no	—
19	#14523983	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	#14523953	—	2	2	no	—
22	C₈×Dic₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₆₇	—	2	2	no	—
27	C₂×C₁₆×C₆₇	—	2	1	no	—
28	C₂×C₁₆×C₆₇	—	2	1	no	—
29	C₂×C₁₆×C₆₇	—	2	2	no	—
30	#97×C₆₇	Q₈	1	2	yes	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	#11761612	—	2	2	no	—
33	#11761612	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₄⋊Dic₆₇	—	2	2	no	—
36	C₂²⋊Dic₆₇	—	2	2	no	—
37	C₄×Dic₆₇	—	2	2	no	—
38	C₆₇⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₆₇	#41	1	2	yes	—
45	C₂×C₈×C₆₇	#41	1	2	yes	—
46	C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₁₆×C₆₇	—	4	1	no	—
50	C₁₆×C₆₇	—	4	1	no	—
51	C₂³×C₆₇	#131	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#117	1	2	yes	—
55	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#131	1	2	yes	—
57	C₂×Dic₆₇	—	4	2	no	—
58	C₈×C₆₇	—	2	2	no	—
59	C₈×C₆₇	#124	1	1	yes	characteristic
60	C₈×C₆₇	#124	1	1	yes	characteristic
61	C₈×C₆₇	#125	1	2	yes	—
62	C₆₇⋊C₈	—	4	2	no	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	#424	1	1	yes	characteristic
65	C₂²×C₆₇	#433	1	2	yes	—
66	C₄×C₆₇	—	4	2	no	—
67	C₄×C₆₇	#382	1	1	yes	characteristic
68	C₄×C₆₇	#424	1	1	yes	characteristic
69	C₄×C₆₇	#433	1	2	yes	—
70	Dic₆₇	—	8	2	no	—
71	#12885	—	67	1	no	maximal, Sylow
72	C₂×C₆₇	—	2	2	no	—
73	C₂×C₆₇	—	2	1	no	—
74	C₂×C₆₇	#1832	1	1	yes	characteristic
75	#2153	—	67	1	no	—
76	#2151	D₆₇	1	1	yes	characteristic
77	#1747	—	67	1	no	—
78	#1379	—	67	2	no	—
79	#1378	—	67	2	no	—
80	C₆₇	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	67	2	no	—
82	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
84	#405	—	67	2	no	—
85	#405	C₂×D₆₇	1	1	yes	characteristic
86	#364	—	134	2	no	—
87	C₄⋊C₁₆	—	2	2	no	—
88	#338	—	67	1	no	—
89	H(3,ℤ/4)	—	67	1	no	—
90	H(3,ℤ/4)	—	67	2	no	—
91	C₂×M₁₆	—	67	2	no	—
92	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
94	#117	—	67	2	no	—
95	#117	C₄×D₆₇	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	134	2	no	—
98	M₃₂	—	134	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	134	2	no	—
103	#97	—	134	2	no	—
104	#97	Dic₁₃₄	1	2	yes	—
105	C₄×C₈	—	134	2	no	—
106	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	134	2	no	—
110	M₁₆	—	134	2	no	—
111	C₂×C₈	—	134	2	no	—
112	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
115	C₂×C₈	#11761627	1	2	yes	—
116	C₂×C₈	#11761629	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	134	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	134	2	no	—
121	C₄²	—	134	2	no	—
122	C₄²	Dic₆₇⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	268	2	no	—
126	C₂³	#14524049	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	268	2	no	—
129	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14524001	1	2	yes	—
131	C₂×C₄	#14524049	1	1	yes	characteristic
132	C₂×C₄	#14524051	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	268	2	no	—
135	C₈	#14524030	1	1	yes	characteristic
136	C₈	#14524032	1	1	yes	characteristic
137	C₈	#14524034	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#16863422	1	1	yes	characteristic
140	C₂²	#16863456	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	536	2	no	—
143	C₄	#16863261	1	1	yes	characteristic
144	C₄	#16863424	1	1	yes	characteristic
145	C₄	#16863456	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#22560328	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561975

C₂

yes

maximal, characteristic

#22559964

C₂

yes

maximal, characteristic

#22558297

C₂

yes

maximal

#22558296

C₂

yes

maximal

C₆₇×#2151

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

#16863351

C₂²

yes

—

#16863217

—

C₆₇⋊#338

C₄

yes

characteristic

#16863153

C₄

yes

characteristic

#16863152

C₄

yes

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#405×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂×#11761612

C₂×C₄

yes

—

#14524002

C₂×C₄

yes

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523953

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

#97×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×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

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#131

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#131

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#424

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12885

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

D₆₇

yes

—

—

—

C₆₇

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

#405

—

#405

C₂×D₆₇

yes

characteristic

#364

—

134

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

—

#117

C₄×D₆₇

yes

characteristic

M₃₂

—

M₃₂

—

134

—

M₃₂

—

134

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

134

—

103

#97

—

134

—

104

#97

Dic₁₃₄

yes

—

105

C₄×C₈

—

134

—

106

C₄×C₈

C₂²⋊D₆₇

yes

—

107

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

134

—

110

M₁₆

—

134

—

111

C₂×C₈

—

134

—

112

C₂×C₈

Dic₆₇⋊C₄

yes

—

113

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

114

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

115

C₂×C₈

#11761627

yes

—

116

C₂×C₈

#11761629

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

134

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

134

—

121

C₄²

—

134

—

122

C₄²

Dic₆₇⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

268

—

126

C₂³

#14524049

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

268

—

129

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

130

C₂×C₄

#14524001

yes

—

131

C₂×C₄

#14524049

yes

characteristic

132

C₂×C₄

#14524051

yes

—

133

C₈

—

134

C₈

—

268

—

135

C₈

#14524030

yes

characteristic

136

C₈

#14524032

yes

characteristic

137

C₈

#14524034

yes

—

138

C₂²

—

139

C₂²

#16863422

yes

characteristic

140

C₂²

#16863456

yes

—

141

C₄

—

142

C₄

—

536

—

143

C₄

#16863261

yes

characteristic

144

C₄

#16863424

yes

characteristic

145

C₄

#16863456

yes

—

146

C₂

—

147

C₂

—

148

C₂

#22560328

yes

characteristic

	subgroup	G/H
center	C₈	#14524032
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2151	C₂
socle	C₂×C₆₇	#1832
prime radical	C₂³×C₆₇	#131
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2151	D₆₇
hypocenter	C₆₇	#12885
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14524032

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2151

C₂

C₂×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
80. C₆₇	#12885	semidirect

H ≅ G/N

type

80. C₆₇

#12885

semidirect

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

subgroup	index
#22561975	2
#22559964	2
#22558297	2
#22558296	2
C₆₇×#2151	2
#12885	67

subgroup

index

C₆₇×#2151

Subgroups — #30772197

Conjugacy classes of subgroups

Complements

Maximal subgroups