Group #30746258 — subgroups

Subgroups — #30746258

Order 17152.

478 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 235 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	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₆₇⋊#1273	C₂	1	4	yes	maximal
4	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523954	C₄	1	2	yes	—
6	C₂×#14523954	C₄	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	#336×C₆₇	C₄	1	4	yes	—
13	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
15	#14523954	—	2	4	no	—
16	#14523954	C₈	1	8	yes	complemented
17	C₄×C₆₇⋊C₈	—	2	8	no	—
18	C₄×C₆₇⋊C₈	—	2	4	no	—
19	C₄×C₆₇⋊C₈	—	2	4	no	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₆₇	—	2	4	no	—
27	C₂×C₆₇⋊C₈	—	2	4	no	—
28	C₂×C₆₇⋊C₈	—	2	4	no	—
29	C₂×C₆₇⋊C₈	—	2	4	no	—
30	C₂×C₆₇⋊C₈	—	2	4	no	—
31	C₂×C₆₇⋊C₈	C₂×C₈	1	8	yes	—
32	C₂×C₆₇⋊C₈	M₁₆	1	8	yes	—
33	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
34	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	4	no	—
39	C₂×C₈×C₆₇	—	2	8	no	—
40	C₄²×C₆₇	—	2	4	no	—
41	C₄²×C₆₇	—	2	4	no	—
42	C₄²×C₆₇	C₂×C₈	1	4	yes	—
43	C₄²×C₆₇	M₁₆	1	4	yes	—
44	C₄²×C₆₇	#41	1	4	yes	—
45	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₆₇	#97	1	4	yes	—
52	C₂×C₄×C₆₇	#117	1	2	yes	—
53	C₂×C₄×C₆₇	#117	1	2	yes	—
54	C₂×C₄×C₆₇	#117	1	2	yes	—
55	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
57	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₆₇	#131	1	4	yes	—
59	C₈×C₆₇	—	4	4	no	—
60	C₆₇⋊C₈	—	2	8	no	—
61	C₆₇⋊C₈	—	2	8	no	—
62	C₂²×C₆₇	#376	1	1	yes	characteristic
63	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
65	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
66	C₂²×C₆₇	#405	1	2	yes	—
67	C₄×C₆₇	—	2	4	no	—
68	C₄×C₆₇	—	2	4	no	—
69	C₄×C₆₇	#431	1	4	yes	—
70	C₄×C₆₇	#432	1	4	yes	—
71	C₄×C₆₇	#433	1	4	yes	—
72	#8132	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	C₈×M₁₆	1	2	yes	—
74	C₂×C₆₇	#1451	1	2	yes	—
75	C₂×C₆₇	#1744	1	1	yes	characteristic
76	C₂×C₆₇	#1745	1	1	yes	characteristic
77	C₂×C₆₇	#1746	1	1	yes	characteristic
78	#1744	D₆₇	1	1	yes	characteristic
79	C₄×#97	—	67	1	no	—
80	C₄²×C₈	—	67	1	no	—
81	#1273	—	67	4	no	—
82	C₆₇	#8132	1	1	yes	characteristic, complemented, Sylow
83	C₂×#97	—	67	2	no	—
84	C₂×#97	—	67	2	no	—
85	C₂×C₄×C₈	—	67	4	no	—
86	C₂×C₄×C₈	—	67	2	no	—
87	C₂×C₄×C₈	—	67	2	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
89	C₄³	C₂×D₆₇	1	1	yes	characteristic
90	#336	Dic₆₇	1	4	yes	—
91	C₂²×C₈	—	67	4	no	—
92	C₂²×C₈	—	67	4	no	—
93	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
94	C₂×C₄²	C₄×D₆₇	1	2	yes	—
95	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
98	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
99	#97	—	67	8	no	—
100	#97	—	134	4	no	—
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	134	8	no	—
103	C₄×C₈	—	134	4	no	—
104	C₄×C₈	—	134	4	no	—
105	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
106	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	8	no	—
112	C₂×C₈	—	67	8	no	—
113	C₂×C₈	—	67	8	no	—
114	C₂×C₈	—	134	4	no	—
115	C₂×C₈	—	134	4	no	—
116	C₂×C₈	—	134	4	no	—
117	C₂×C₈	—	134	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₈×D₆₇	1	4	yes	—
121	C₄²	#11761611	1	4	yes	—
122	C₄²	#11761628	1	4	yes	—
123	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
129	C₂×C₄	#14523953	1	4	yes	—
130	C₂×C₄	#14524000	1	2	yes	—
131	C₂×C₄	#14524000	1	2	yes	—
132	C₂×C₄	#14524002	1	2	yes	—
133	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
134	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
135	C₂×C₄	C₂×#11761611	1	2	yes	—
136	C₂×C₄	#14524050	1	4	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	134	8	no	—
139	C₈	—	134	8	no	—
140	C₂²	#16863242	1	1	yes	characteristic
141	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
142	C₂²	C₂×#14523953	1	1	yes	characteristic
143	C₂²	M₁₆×Dic₆₇	1	2	yes	—
144	C₂²	#16863352	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#16863447	1	4	yes	—
148	C₄	#16863451	1	4	yes	—
149	C₄	#16863457	1	4	yes	—
150	C₂	#22558490	1	2	yes	—
151	C₂	#22558499	1	2	yes	—
152	C₂	#22559954	1	1	yes	characteristic
153	C₂	#22559957	1	1	yes	characteristic
154	C₂	#22559961	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523954

C₂

yes

maximal, characteristic

C₄²×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇⋊#1273

C₂

yes

maximal

C₆₇×#1744

C₂

yes

maximal, characteristic

C₂×#14523954

C₄

yes

—

C₂×#14523954

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

#14523954

—

#14523954

C₈

yes

complemented

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₈

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×C₈

yes

—

C₂×C₄×C₆₇

#97

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₇

C₂×#97

yes

characteristic

C₂²×C₆₇

C₄×M₁₆

yes

—

C₂²×C₆₇

#405

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#431

yes

—

C₄×C₆₇

#432

yes

—

C₄×C₆₇

#433

yes

—

#8132

—

maximal, Sylow

C₂×C₆₇

C₈×M₁₆

yes

—

C₂×C₆₇

#1451

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

C₄×#97

—

C₄²×C₈

—

#1273

—

C₆₇

#8132

yes

characteristic, complemented, Sylow

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

C₂×D₆₇

yes

characteristic

#336

Dic₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#97

—

100

#97

—

134

—

101

C₄×C₈

—

102

C₄×C₈

—

134

—

103

C₄×C₈

—

134

—

104

C₄×C₈

—

134

—

105

C₂²×C₄

C₄×Dic₆₇

yes

—

106

C₂²×C₄

C₄×Dic₆₇

yes

—

107

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

108

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

109

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

134

—

115

C₂×C₈

—

134

—

116

C₂×C₈

—

134

—

117

C₂×C₈

—

134

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₈×D₆₇

yes

—

121

C₄²

#11761611

yes

—

122

C₄²

#11761628

yes

—

123

C₂³

C₂×C₄×Dic₆₇

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×Dic₆₇

yes

—

129

C₂×C₄

#14523953

yes

—

130

C₂×C₄

#14524000

yes

—

131

C₂×C₄

#14524000

yes

—

132

C₂×C₄

#14524002

yes

—

133

C₂×C₄

C₂×C₈×D₆₇

yes

—

134

C₂×C₄

M₁₆×D₆₇

yes

—

135

C₂×C₄

C₂×#11761611

yes

—

136

C₂×C₄

#14524050

yes

—

137

C₈

—

138

C₈

—

134

—

139

C₈

—

134

—

140

C₂²

#16863242

yes

characteristic

141

C₂²

C₂×C₈×Dic₆₇

yes

characteristic

142

C₂²

C₂×#14523953

yes

characteristic

143

C₂²

M₁₆×Dic₆₇

yes

—

144

C₂²

#16863352

yes

—

145

C₄

—

146

C₄

—

147

C₄

#16863447

yes

—

148

C₄

#16863451

yes

—

149

C₄

#16863457

yes

—

150

C₂

#22558490

yes

—

151

C₂

#22558499

yes

—

152

C₂

#22559954

yes

characteristic

153

C₂

#22559957

yes

characteristic

154

C₂

#22559961

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8132
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₇

commutator

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₆₇×#1744

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
16. #14523954	C₈	semidirect
82. C₆₇	#8132	semidirect

H ≅ G/N

type

16. #14523954

C₈

semidirect

82. C₆₇

#8132

semidirect

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

subgroup	index
C₄×#14523954	2
C₄²×C₆₇⋊C₈	2
C₆₇⋊#1273	2
C₆₇×#1744	2
#8132	67

subgroup

index

C₄×#14523954

C₄²×C₆₇⋊C₈

C₆₇⋊#1273

C₆₇×#1744

#8132

Subgroups — #30746258

Conjugacy classes of subgroups

Complements

Maximal subgroups