Group #30869537 — subgroups

Subgroups — #30869537

Order 17152.

594 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 203 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	#22570733	C₂	1	1	yes	maximal, characteristic
2	#22569146	C₂	1	2	yes	maximal
3	#22561185	C₂	1	2	yes	maximal
4	#22560807	C₂	1	8	yes	maximal
5	#22559957	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1838	C₂	1	1	yes	maximal, characteristic
7	#16863953	C₂²	1	2	yes	—
8	#16863805	C₄	1	8	yes	complemented
9	#16863525	—	2	8	no	—
10	#16863434	C₂²	1	4	yes	—
11	#16863424	C₂²	1	4	yes	—
12	#16863354	C₂²	1	4	yes	—
13	C₂×#14523954	C₂²	1	1	yes	characteristic
14	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
15	#16863157	C₂²	1	8	yes	—
16	#16863150	C₂²	1	4	yes	—
17	#423×C₆₇	C₂²	1	2	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	#341×C₆₇	C₂²	1	4	yes	—
20	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
21	#14524043	C₂×C₄	1	4	yes	—
22	#14524035	C₂×C₄	1	4	yes	—
23	#14524022	—	2	4	no	—
24	C₄×Dic₁₃₄	—	2	4	no	—
25	#14524002	C₂³	1	4	yes	—
26	C₂×C₄×Dic₆₇	—	2	2	no	—
27	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
28	#14523977	C₂×C₄	1	8	yes	—
29	#14523975	C₂×C₄	1	8	yes	—
30	Dic₆₇⋊C₈	—	2	8	no	—
31	#14523954	D₄	1	4	yes	—
32	#14523953	—	2	4	no	—
33	C₈×Dic₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	4	no	—
38	#97×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×Dic₁₃₄	—	2	4	no	—
40	C₂²×Dic₆₇	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
43	C₄⋊Dic₆₇	—	2	4	no	—
44	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
45	Dic₆₇⋊C₄	—	2	8	no	—
46	C₂²⋊Dic₆₇	—	2	4	no	—
47	C₄×Dic₆₇	—	2	4	no	—
48	C₄×Dic₆₇	—	2	4	no	—
49	C₄×Dic₆₇	—	4	4	no	—
50	C₂²×C₄×C₆₇	#41	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₆₇	—	2	8	no	—
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
56	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	4	2	no	—
60	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
61	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
64	C₂×Dic₆₇	—	2	4	no	—
65	C₂×Dic₆₇	—	2	8	no	—
66	C₂×Dic₆₇	—	4	2	no	—
67	C₂×Dic₆₇	—	4	2	no	—
68	Dic₁₃₄	—	8	4	no	—
69	C₈×C₆₇	—	2	4	no	—
70	C₈×C₆₇	—	4	4	no	—
71	C₆₇⋊C₈	—	2	4	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
74	C₂²×C₆₇	#541	1	2	yes	—
75	C₄×C₆₇	—	2	4	no	—
76	C₄×C₆₇	#519	1	2	yes	—
77	C₄×C₆₇	#536	1	2	yes	—
78	Dic₆₇	—	4	4	no	—
79	Dic₆₇	—	4	8	no	—
80	#22259	—	67	1	no	maximal, Sylow
81	C₂×C₆₇	—	2	2	no	—
82	C₂×C₆₇	#2327	1	1	yes	characteristic
83	C₂×C₆₇	#2956	1	2	yes	—
84	#3162	—	67	1	no	—
85	#2961	—	67	2	no	—
86	#1994	—	67	2	no	—
87	#1922	—	67	8	no	—
88	#1838	D₆₇	1	1	yes	characteristic
89	#1745	—	67	1	no	—
90	C₆₇	#22259	1	1	yes	characteristic, complemented, Sylow
91	#533	—	67	2	no	—
92	#501	—	67	8	no	—
93	#446	—	134	8	no	—
94	#427	—	67	4	no	—
95	#424	—	67	4	no	—
96	#423	C₂×D₆₇	1	2	yes	—
97	#405	—	67	4	no	—
98	C₂×#97	—	67	1	no	—
99	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
100	C₄³	—	67	1	no	—
101	#341	C₂×D₆₇	1	4	yes	—
102	#339	—	67	8	no	—
103	#336	—	67	4	no	—
104	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
105	C₂²×C₈	—	67	2	no	—
106	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
107	C₄⋊Q₈	—	67	4	no	—
108	#125	—	67	4	no	—
109	C₂²⋊Q₈	—	134	4	no	—
110	C₄×Q₈	—	134	4	no	—
111	#117	—	67	4	no	—
112	C₂×C₄²	—	67	2	no	—
113	C₂×C₄²	—	134	2	no	—
114	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	#107	—	67	8	no	—
116	#106	—	67	8	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	134	8	no	—
119	#97	—	67	4	no	—
120	#97	—	134	4	no	—
121	#97	C₄×D₆₇	1	4	yes	—
122	C₄×C₈	—	134	4	no	—
123	C₂×Q₈	—	134	4	no	—
124	C₂²×C₄	—	134	2	no	—
125	C₂²×C₄	#11761629	1	2	yes	—
126	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	67	4	no	—
131	C₂×C₈	—	134	4	no	—
132	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
133	C₄⋊C₄	—	67	8	no	—
134	C₄⋊C₄	—	134	8	no	—
135	C₄⋊C₄	—	134	4	no	—
136	C₂²⋊C₄	—	134	4	no	—
137	C₄²	—	268	4	no	—
138	C₄²	—	134	4	no	—
139	C₄²	—	134	4	no	—
140	C₄²	C₂×C₄×D₆₇	1	2	yes	—
141	C₄²	D₄×D₆₇	1	2	yes	—
142	C₂³	C₂×#11761629	1	1	yes	characteristic
143	Q₈	—	536	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	268	2	no	—
146	C₂×C₄	—	268	2	no	—
147	C₂×C₄	—	134	4	no	—
148	C₂×C₄	—	134	8	no	—
149	C₂×C₄	#14524005	1	4	yes	—
150	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
152	C₂×C₄	#14524090	1	4	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	4	4	no	—
155	C₈	—	134	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#14524005	1	1	yes	characteristic
158	C₂²	#16864009	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	268	4	no	—
161	C₄	—	268	8	no	—
162	C₄	#16863873	1	2	yes	—
163	C₄	#16863966	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#22562926	1	1	yes	characteristic
166	C₂	#22569118	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570733

C₂

yes

maximal, characteristic

#22569146

C₂

yes

maximal

#22561185

C₂

yes

maximal

#22560807

C₂

yes

maximal

#22559957

C₂

yes

maximal, characteristic

C₆₇×#1838

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

#16863805

C₄

yes

complemented

#16863525

—

#16863434

C₂²

yes

—

#16863424

C₂²

yes

—

#16863354

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

C₄²×Dic₆₇

C₂²

yes

characteristic

#16863157

C₂²

yes

—

#16863150

C₂²

yes

—

#423×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

#341×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

#14524043

C₂×C₄

yes

—

#14524035

C₂×C₄

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂³

yes

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂³

yes

—

#14523977

C₂×C₄

yes

—

#14523975

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

#14523954

D₄

yes

—

#14523953

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

#97×C₆₇

C₂×C₄

yes

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂²×C₄

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

2₋¹⁺⁴

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#541

yes

—

C₄×C₆₇

—

C₄×C₆₇

#519

yes

—

C₄×C₆₇

#536

yes

—

Dic₆₇

—

Dic₆₇

—

#22259

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2327

yes

characteristic

C₂×C₆₇

#2956

yes

—

—

—

—

—

D₆₇

yes

characteristic

#1745

—

C₆₇

#22259

yes

characteristic, complemented, Sylow

#533

—

#501

—

#446

—

134

—

#427

—

#424

—

#423

C₂×D₆₇

yes

—

#405

—

C₂×#97

—

C₂×#97

C₂×D₆₇

yes

characteristic

100

C₄³

—

101

#341

C₂×D₆₇

yes

—

102

#339

—

103

#336

—

104

C₂×M₁₆

C₂²×D₆₇

yes

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₆₇

yes

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

134

—

110

C₄×Q₈

—

134

—

111

#117

—

112

C₂×C₄²

—

113

C₂×C₄²

—

134

—

114

C₂×C₄²

C₂²×D₆₇

yes

characteristic

115

#107

—

116

#106

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

134

—

119

#97

—

120

#97

—

134

—

121

#97

C₄×D₆₇

yes

—

122

C₄×C₈

—

134

—

123

C₂×Q₈

—

134

—

124

C₂²×C₄

—

134

—

125

C₂²×C₄

#11761629

yes

—

126

C₂²×C₄

C₂³×D₆₇

yes

characteristic

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

134

—

132

C₂×C₈

C₂×C₄×D₆₇

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

134

—

135

C₄⋊C₄

—

134

—

136

C₂²⋊C₄

—

134

—

137

C₄²

—

268

—

138

C₄²

—

134

—

139

C₄²

—

134

—

140

C₄²

C₂×C₄×D₆₇

yes

—

141

C₄²

D₄×D₆₇

yes

—

142

C₂³

C₂×#11761629

yes

characteristic

143

Q₈

—

536

—

144

C₂×C₄

—

145

C₂×C₄

—

268

—

146

C₂×C₄

—

268

—

147

C₂×C₄

—

134

—

148

C₂×C₄

—

134

—

149

C₂×C₄

#14524005

yes

—

150

C₂×C₄

C₂²×C₄×D₆₇

yes

characteristic

151

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

152

C₂×C₄

#14524090

yes

—

153

C₈

—

154

C₈

—

155

C₈

—

134

—

156

C₂²

—

157

C₂²

C₂×#14524005

yes

characteristic

158

C₂²

#16864009

yes

—

159

C₄

—

160

C₄

—

268

—

161

C₄

—

268

—

162

C₄

#16863873

yes

—

163

C₄

#16863966

yes

—

164

C₂

—

165

C₂

#22562926

yes

characteristic

166

C₂

#22569118

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1838	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1838	D₆₇
hypocenter	C₆₇	#22259
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#1838

C₂

socle

C₂²×C₆₇

C₂×C₄×D₄

prime radical

C₂³×C₆₇

C₂×#41

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
8. #16863805	C₄	semidirect
90. C₆₇	#22259	semidirect

H ≅ G/N

type

8. #16863805

C₄

semidirect

90. C₆₇

#22259

semidirect

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

subgroup	index
#22570733	2
#22569146	2
#22561185	2
#22560807	2
#22559957	2
C₆₇×#1838	2
#22259	67

subgroup

index

C₆₇×#1838

Subgroups — #30869537

Conjugacy classes of subgroups

Complements

Maximal subgroups