Group #30772252 — subgroups

Subgroups — #30772252

Order 17152.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22561974	C₂	1	2	yes	maximal
3	#22561967	C₂	1	2	yes	maximal
4	#22559964	C₂	1	1	yes	maximal, characteristic
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	#16863216	—	2	4	no	—
10	Dic₆₇⋊C₁₆	—	2	4	no	—
11	C₆₇⋊#338	C₄	1	1	yes	characteristic
12	#16863153	C₄	1	1	yes	characteristic
13	#16863152	C₄	1	2	yes	—
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#405×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₆₇	—	2	2	no	—
18	C₂×#11761612	C₂×C₄	1	2	yes	—
19	#14524002	C₂×C₄	1	2	yes	—
20	#14523983	—	2	2	no	—
21	#14523983	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523953	Q₈	1	4	yes	—
24	C₈×Dic₆₇	D₄	1	4	yes	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₇	—	2	2	no	—
29	C₂×C₁₆×C₆₇	—	2	1	no	—
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	—	2	1	no	—
32	#97×C₆₇	Q₈	1	2	yes	—
33	C₄×C₈×C₆₇	D₄	1	2	yes	—
34	#11761612	—	2	2	no	—
35	#11761612	—	2	2	no	—
36	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₄⋊Dic₆₇	—	2	2	no	—
38	C₂²⋊Dic₆₇	—	2	2	no	—
39	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
40	C₆₇⋊C₁₆	—	4	2	no	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	M₁₆×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	C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₂²⋊C₄×C₆₇	—	2	1	no	—
50	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
51	C₁₆×C₆₇	—	4	1	no	—
52	C₁₆×C₆₇	—	4	1	no	—
53	C₂³×C₆₇	#131	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#131	1	2	yes	—
59	C₂×Dic₆₇	—	2	4	no	—
60	C₈×C₆₇	—	2	2	no	—
61	C₈×C₆₇	#127	1	1	yes	characteristic
62	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
64	C₆₇⋊C₈	—	2	4	no	—
65	C₂²×C₆₇	—	4	1	no	—
66	C₂²×C₆₇	#424	1	2	yes	—
67	C₂²×C₆₇	#424	1	1	yes	characteristic
68	C₄×C₆₇	—	4	2	no	—
69	C₄×C₆₇	#384	1	1	yes	characteristic
70	C₄×C₆₇	#424	1	2	yes	—
71	C₄×C₆₇	#424	1	1	yes	characteristic
72	Dic₆₇	—	4	4	no	—
73	#12902	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	2	2	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	#1843	1	1	yes	characteristic
77	#2153	—	67	2	no	—
78	#2153	—	67	1	no	—
79	#2151	—	67	2	no	—
80	#2151	D₆₇	1	1	yes	characteristic
81	#1747	—	67	1	no	—
82	C₆₇	#12902	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	67	2	no	—
84	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
86	#405	—	67	2	no	—
87	#405	C₂×D₆₇	1	1	yes	characteristic
88	#364	—	134	4	no	—
89	#364	—	134	2	no	—
90	C₄⋊C₁₆	—	2	2	no	—
91	C₄⋊C₁₆	—	134	4	no	—
92	#338	—	67	1	no	—
93	H(3,ℤ/4)	—	67	1	no	—
94	H(3,ℤ/4)	—	67	2	no	—
95	C₂×M₁₆	—	67	2	no	—
96	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	#117	—	67	2	no	—
99	#117	C₄×D₆₇	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	134	2	no	—
102	M₃₂	—	134	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	134	2	no	—
107	#97	—	67	4	no	—
108	#97	Dic₁₃₄	1	2	yes	—
109	C₄×C₈	—	67	4	no	—
110	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	134	2	no	—
114	M₁₆	—	134	2	no	—
115	C₂×C₈	—	67	4	no	—
116	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
117	C₂×C₈	D₄×D₆₇	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
119	C₂×C₈	Q₈×D₆₇	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	134	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	134	2	no	—
125	C₄²	—	67	4	no	—
126	C₄²	Dic₆₇⋊C₄	1	2	yes	—
127	C₁₆	—	4	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	—	268	2	no	—
130	C₂³	#14524049	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	134	4	no	—
133	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
134	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#14524049	1	1	yes	characteristic
136	C₂×C₄	#14524051	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	134	4	no	—
139	C₈	#14524039	1	1	yes	characteristic
140	C₈	#14524041	1	1	yes	characteristic
141	C₈	#14524042	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#16863421	1	2	yes	—
144	C₂²	#16863422	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	268	4	no	—
147	C₄	#16863265	1	1	yes	characteristic
148	C₄	#16863421	1	2	yes	—
149	C₄	#16863424	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#22560362	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561975

C₂

yes

maximal, characteristic

#22561974

C₂

yes

maximal

#22561967

C₂

yes

maximal

#22559964

C₂

yes

maximal, characteristic

C₆₇×#2151

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

#16863351

C₂²

yes

—

#16863217

—

#16863216

—

Dic₆₇⋊C₁₆

—

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

Q₈

yes

—

C₈×Dic₆₇

D₄

yes

—

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⋊C₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

C₄⋊C₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×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

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

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#127

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#424

yes

—

C₂²×C₆₇

#424

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#384

yes

characteristic

C₄×C₆₇

#424

yes

—

C₄×C₆₇

#424

yes

characteristic

Dic₆₇

—

#12902

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

—

—

D₆₇

yes

characteristic

#1747

—

C₆₇

#12902

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

—

#364

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

#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

100

M₃₂

—

101

M₃₂

—

134

—

102

M₃₂

—

134

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

134

—

107

#97

—

108

#97

Dic₁₃₄

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₆₇

yes

—

111

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

134

—

114

M₁₆

—

134

—

115

C₂×C₈

—

116

C₂×C₈

Dic₆₇⋊C₄

yes

—

117

C₂×C₈

D₄×D₆₇

yes

—

118

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

119

C₂×C₈

Q₈×D₆₇

yes

—

120

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

134

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

134

—

125

C₄²

—

126

C₄²

Dic₆₇⋊C₄

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

268

—

130

C₂³

#14524049

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

134

—

133

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

134

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

135

C₂×C₄

#14524049

yes

characteristic

136

C₂×C₄

#14524051

yes

—

137

C₈

—

138

C₈

—

134

—

139

C₈

#14524039

yes

characteristic

140

C₈

#14524041

yes

characteristic

141

C₈

#14524042

yes

—

142

C₂²

—

143

C₂²

#16863421

yes

—

144

C₂²

#16863422

yes

characteristic

145

C₄

—

146

C₄

—

268

—

147

C₄

#16863265

yes

characteristic

148

C₄

#16863421

yes

—

149

C₄

#16863424

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#22560362

yes

characteristic

	subgroup	G/H
center	C₈	#14524039
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2151	C₂
socle	C₂×C₆₇	#1843
prime radical	C₂³×C₆₇	#131
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2151	D₆₇
hypocenter	C₆₇	#12902
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14524039

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
82. C₆₇	#12902	semidirect

H ≅ G/N

type

82. C₆₇

#12902

semidirect

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

subgroup	index
#22561975	2
#22561974	2
#22561967	2
#22559964	2
C₆₇×#2151	2
#12902	67

subgroup

index

C₆₇×#2151

Subgroups — #30772252

Conjugacy classes of subgroups

Complements

Maximal subgroups