Group #30749504 — subgroups

Subgroups — #30749504

Order 17152.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22560326	C₂	1	1	yes	maximal, characteristic
2	#22559955	C₂	1	1	yes	maximal, characteristic
3	#22558043	C₂	1	4	yes	maximal
4	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
7	#16863151	—	2	2	no	—
8	#16863151	C₄	1	2	yes	—
9	#16863151	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	#336×C₆₇	—	2	2	no	—
13	C₂²×C₆₇⋊C₈	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
16	C₄×C₆₇⋊C₈	—	2	2	no	—
17	C₄×C₆₇⋊C₈	—	2	2	no	—
18	C₂²×C₈×C₆₇	—	2	2	no	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	D₄	1	2	yes	—
22	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₆₇	—	2	4	no	—
25	C₂×C₆₇⋊C₈	—	2	4	no	—
26	C₂×C₆₇⋊C₈	—	2	4	no	—
27	C₂×C₆₇⋊C₈	—	2	4	no	—
28	C₂×C₆₇⋊C₈	—	4	2	no	—
29	C₂×C₆₇⋊C₈	—	4	4	no	—
30	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	4	2	no	—
36	C₂×C₈×C₆₇	—	4	4	no	—
37	C₄²×C₆₇	—	2	2	no	—
38	C₄²×C₆₇	—	2	2	no	—
39	C₄²×C₆₇	—	2	4	no	—
40	C₄²×C₆₇	—	4	2	no	—
41	C₄²×C₆₇	#41	1	2	yes	—
42	C₄²×C₆₇	#41	1	2	yes	—
43	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
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	—	2	2	no	—
51	C₂×C₄×C₆₇	#117	1	2	yes	—
52	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#125	1	2	yes	—
55	C₂×C₄×C₆₇	#131	1	2	yes	—
56	C₂×C₄×C₆₇	#131	1	2	yes	—
57	C₂×C₄×C₆₇	#135	1	4	yes	—
58	C₈×C₆₇	—	8	2	no	—
59	C₆₇⋊C₈	—	4	4	no	—
60	C₆₇⋊C₈	—	8	2	no	—
61	C₂²×C₆₇	#382	1	1	yes	characteristic
62	C₂²×C₆₇	#424	1	1	yes	characteristic
63	C₂²×C₆₇	#424	1	1	yes	characteristic
64	C₂²×C₆₇	#427	1	2	yes	—
65	C₂²×C₆₇	#430	1	2	yes	—
66	C₄×C₆₇	—	2	2	no	—
67	C₄×C₆₇	—	4	2	no	—
68	C₄×C₆₇	—	4	2	no	—
69	C₄×C₆₇	#433	1	4	yes	—
70	C₄×C₆₇	#488	1	2	yes	—
71	C₄×C₆₇	#488	1	2	yes	—
72	#8788	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	#1572	1	2	yes	—
74	C₂×C₆₇	#1572	1	2	yes	—
75	C₂×C₆₇	#1832	1	1	yes	characteristic
76	C₂×C₆₇	#1835	1	1	yes	characteristic
77	C₂×C₆₇	#1838	1	1	yes	characteristic
78	#1831	—	67	1	no	—
79	#1831	D₆₇	1	1	yes	characteristic
80	#1744	—	67	1	no	—
81	#1277	—	67	4	no	—
82	C₆₇	#8788	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	67	2	no	—
84	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
85	C₂×C₄×C₈	—	67	2	no	—
86	C₄³	C₂×D₆₇	1	1	yes	characteristic
87	#336	—	2	2	no	—
88	#336	—	67	2	no	—
89	#336	—	67	2	no	—
90	#336	—	134	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	67	4	no	—
93	C₂²×C₈	—	134	2	no	—
94	C₂×C₄²	Dic₁₃₄	1	2	yes	—
95	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
98	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	134	4	no	—
101	C₄×C₈	—	134	2	no	—
102	C₄×C₈	—	134	2	no	—
103	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
104	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
105	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
107	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	268	2	no	—
111	C₂×C₈	—	268	4	no	—
112	C₂×C₈	—	134	4	no	—
113	C₂×C₈	—	134	4	no	—
114	C₂×C₈	—	134	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	4	2	no	—
119	C₄²	#11761627	1	2	yes	—
120	C₄²	#11761629	1	2	yes	—
121	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
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	—	2	2	no	—
129	C₂×C₄	#14524001	1	2	yes	—
130	C₂×C₄	#14524030	1	1	yes	characteristic
131	C₂×C₄	#14524032	1	1	yes	characteristic
132	C₂×C₄	#14524034	1	2	yes	—
133	C₂×C₄	#14524049	1	2	yes	—
134	C₂×C₄	#14524051	1	2	yes	—
135	C₂×C₄	#14524063	1	4	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	268	4	no	—
138	C₈	—	536	2	no	—
139	C₂²	#16863261	1	1	yes	characteristic
140	C₂²	#16863422	1	1	yes	characteristic
141	C₂²	#16863424	1	1	yes	characteristic
142	C₂²	#16863432	1	2	yes	—
143	C₂²	#16863444	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#16863456	1	4	yes	—
148	C₄	#16863750	1	2	yes	—
149	C₄	#16863753	1	2	yes	—
150	C₂	#22558987	1	2	yes	—
151	C₂	#22558990	1	2	yes	—
152	C₂	#22560328	1	1	yes	characteristic
153	C₂	#22560339	1	1	yes	characteristic
154	C₂	#22560349	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal, characteristic

#22559955

C₂

yes

maximal, characteristic

#22558043

C₂

yes

maximal

C₆₇×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863151

—

#16863151

C₄

yes

—

#16863151

C₄

yes

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

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

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#430

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#488

yes

—

C₄×C₆₇

#488

yes

—

#8788

—

maximal, Sylow

C₂×C₆₇

#1572

yes

—

C₂×C₆₇

#1572

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

#1744

—

#1277

—

C₆₇

#8788

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₆₇

yes

—

—

—

—

134

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

134

—

C₂×C₄²

Dic₁₃₄

yes

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

134

—

101

C₄×C₈

—

134

—

102

C₄×C₈

—

134

—

103

C₂²×C₄

Dic₆₇⋊C₄

yes

—

104

C₂²×C₄

Dic₆₇⋊C₄

yes

—

105

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

106

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

107

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

268

—

111

C₂×C₈

—

268

—

112

C₂×C₈

—

134

—

113

C₂×C₈

—

134

—

114

C₂×C₈

—

134

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

#11761627

yes

—

120

C₄²

#11761629

yes

—

121

C₂³

C₂×Dic₆₇⋊C₄

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#14524001

yes

—

130

C₂×C₄

#14524030

yes

characteristic

131

C₂×C₄

#14524032

yes

characteristic

132

C₂×C₄

#14524034

yes

—

133

C₂×C₄

#14524049

yes

—

134

C₂×C₄

#14524051

yes

—

135

C₂×C₄

#14524063

yes

—

136

C₈

—

137

C₈

—

268

—

138

C₈

—

536

—

139

C₂²

#16863261

yes

characteristic

140

C₂²

#16863422

yes

characteristic

141

C₂²

#16863424

yes

characteristic

142

C₂²

#16863432

yes

—

143

C₂²

#16863444

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#16863456

yes

—

148

C₄

#16863750

yes

—

149

C₄

#16863753

yes

—

150

C₂

#22558987

yes

—

151

C₂

#22558990

yes

—

152

C₂

#22560328

yes

characteristic

153

C₂

#22560339

yes

characteristic

154

C₂

#22560349

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#8788
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1831

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
82. C₆₇	#8788	semidirect

H ≅ G/N

type

82. C₆₇

#8788

semidirect

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

subgroup	index
#22560326	2
#22559955	2
#22558043	2
C₆₇×#1831	2
#8788	67

subgroup

index

C₆₇×#1831

Subgroups — #30749504

Conjugacy classes of subgroups

Complements

Maximal subgroups