Group #30755597 — subgroups

Subgroups — #30755597

Order 17152.

558 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22561169	C₂	1	4	yes	maximal
3	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	2	no	—
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863776	—	2	2	no	—
8	#16863776	C₄	1	4	yes	complemented
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
10	#16863158	—	2	8	no	—
11	C₈×C₆₇⋊C₈	—	2	2	no	—
12	C₂×#107×C₆₇	—	2	2	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	C₂²	1	2	yes	—
15	C₂²×Dic₁₃₄	—	2	2	no	—
16	C₂×Dic₂₆₈	—	2	8	no	—
17	C₂×Dic₂₆₈	—	4	4	no	—
18	C₂×Dic₂₆₈	—	4	4	no	—
19	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
20	#14524043	—	2	2	no	—
21	#14524043	—	2	4	no	—
22	#14524043	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₆₇	—	2	4	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₆₇	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	#107×C₆₇	—	4	4	no	—
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₇	D₄	1	2	yes	—
33	C₂×Dic₁₃₄	—	2	4	no	—
34	C₂×Dic₁₃₄	—	4	4	no	—
35	C₂×Dic₁₃₄	—	4	4	no	—
36	C₂×Dic₁₃₄	—	4	2	no	—
37	C₂²×Dic₆₇	—	4	2	no	—
38	Dic₂₆₈	—	4	16	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₂×C₆₇⋊C₈	Q₁₆	1	8	yes	complemented
42	C₄⋊Dic₆₇	—	2	8	no	—
43	C₄⋊Dic₆₇	—	4	4	no	—
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₆₇	D₈	1	4	yes	—
49	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₆₇	—	4	4	no	—
51	C₄⋊C₄×C₆₇	—	4	8	no	—
52	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₂³×C₆₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	8	2	no	—
57	C₂×C₄×C₆₇	—	8	4	no	—
58	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₆₇	#121	1	2	yes	—
61	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
65	C₂×C₄×C₆₇	#135	1	4	yes	—
66	C₂×Dic₆₇	—	4	4	no	—
67	C₂×Dic₆₇	—	8	2	no	—
68	C₂×Dic₆₇	—	8	2	no	—
69	Dic₁₃₄	—	4	8	no	—
70	Dic₁₃₄	—	8	4	no	—
71	C₈×C₆₇	#102	1	8	yes	—
72	C₆₇⋊C₈	—	2	8	no	—
73	C₂²×C₆₇	#390	1	1	yes	characteristic
74	C₂²×C₆₇	#492	1	1	yes	characteristic
75	C₂²×C₆₇	#493	1	1	yes	characteristic
76	C₂²×C₆₇	#494	1	2	yes	—
77	C₂²×C₆₇	#495	1	2	yes	—
78	C₄×C₆₇	—	16	2	no	—
79	C₄×C₆₇	C₂×#102	1	2	yes	—
80	C₄×C₆₇	C₄×Q₁₆	1	4	yes	—
81	C₄×C₆₇	#467	1	4	yes	—
82	C₄×C₆₇	#486	1	2	yes	—
83	Dic₆₇	—	8	4	no	—
84	#9825	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	#1707	1	2	yes	—
86	C₂×C₆₇	#1709	1	1	yes	characteristic
87	C₂×C₆₇	#1713	1	1	yes	characteristic
88	C₂×C₆₇	#1957	1	1	yes	characteristic
89	C₂×C₆₇	#1969	1	2	yes	—
90	C₂×#494	—	67	1	no	—
91	#1990	—	67	4	no	—
92	#1845	D₆₇	1	1	yes	characteristic
93	C₂×C₈²	—	67	1	no	—
94	C₆₇	#9825	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₁₆	—	134	2	no	—
96	C₂×C₄⋊Q₈	—	67	2	no	—
97	#494	—	67	4	no	—
98	#494	—	134	2	no	—
99	C₂×#107	—	2	2	no	—
100	C₂×C₄×C₈	—	67	2	no	—
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	#384	C₂×D₆₇	1	2	yes	—
103	#340	—	134	8	no	—
104	C₈²	—	134	2	no	—
105	C₂²×Q₈	—	134	2	no	—
106	C₂×Q₁₆	—	268	4	no	—
107	C₂×Q₁₆	—	268	4	no	—
108	C₂×Q₁₆	—	134	8	no	—
109	C₂²×C₈	—	67	4	no	—
110	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
111	C₄⋊Q₈	—	67	4	no	—
112	C₄⋊Q₈	—	134	2	no	—
113	C₄⋊Q₈	—	134	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	134	4	no	—
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	134	2	no	—
120	C₄×C₈	—	134	2	no	—
121	C₄×C₈	C₄×D₆₇	1	2	yes	—
122	C₄×C₈	D₂₆₈	1	2	yes	—
123	C₂×Q₈	—	268	2	no	—
124	C₂×Q₈	—	268	4	no	—
125	C₂×Q₈	—	268	4	no	—
126	C₂×Q₈	—	134	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	268	2	no	—
129	C₂²×C₄	D₄×D₆₇	1	2	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	Q₁₆	—	268	16	no	—
132	C₂×C₈	—	67	8	no	—
133	C₂×C₈	—	134	4	no	—
134	C₂×C₈	—	134	4	no	—
135	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
136	C₂×C₈	#11761613	1	4	yes	—
137	C₂×C₈	#11761615	1	4	yes	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	268	4	no	—
141	C₄⋊C₄	—	134	8	no	—
142	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
144	C₄²	#11761629	1	2	yes	—
145	C₂³	#14524039	1	1	yes	characteristic
146	Q₈	—	268	8	no	—
147	Q₈	—	536	4	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	268	4	no	—
151	C₂×C₄	—	536	2	no	—
152	C₂×C₄	—	536	2	no	—
153	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#14524005	1	2	yes	—
155	C₂×C₄	#14524018	1	2	yes	—
156	C₂×C₄	#14524032	1	1	yes	characteristic
157	C₂×C₄	C₂×#11761613	1	2	yes	—
158	C₂×C₄	C₂×#11761615	1	2	yes	—
159	C₂×C₄	Q₁₆×D₆₇	1	4	yes	—
160	C₂×C₄	#14524062	1	4	yes	—
161	C₈	—	134	8	no	—
162	C₈	#14523963	1	8	yes	—
163	C₂²	#16863297	1	1	yes	characteristic
164	C₂²	#16863770	1	1	yes	characteristic
165	C₂²	#16863772	1	1	yes	characteristic
166	C₂²	#16863774	1	2	yes	—
167	C₂²	#16863779	1	2	yes	—
168	C₄	—	16	2	no	—
169	C₄	—	536	4	no	—
170	C₄	C₂×#14523963	1	2	yes	—
171	C₄	#16863490	1	4	yes	—
172	C₄	#16863638	1	4	yes	—
173	C₄	#16863744	1	2	yes	—
174	C₂	#22559807	1	2	yes	—
175	C₂	#22559814	1	1	yes	characteristic
176	C₂	#22559825	1	1	yes	characteristic
177	C₂	#22561014	1	1	yes	characteristic
178	C₂	#22561068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

#22561169

C₂

yes

maximal

C₂×C₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#1845

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

—

C₂×#14524043

C₂²

yes

—

#16863776

—

#16863776

C₄

yes

complemented

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863158

—

C₈×C₆₇⋊C₈

—

C₂×#107×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂²×C₆₇⋊C₈

D₄

yes

—

#14524043

—

#14524043

—

#14524043

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₁₆

yes

complemented

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

#102

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#495

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#102

yes

—

C₄×C₆₇

C₄×Q₁₆

yes

—

C₄×C₆₇

#467

yes

—

C₄×C₆₇

#486

yes

—

Dic₆₇

—

#9825

—

maximal, Sylow

C₂×C₆₇

#1707

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×#494

—

#1990

—

#1845

D₆₇

yes

characteristic

C₂×C₈²

—

C₆₇

#9825

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

134

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

134

—

C₂×#107

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

102

#384

C₂×D₆₇

yes

—

103

#340

—

134

—

104

C₈²

—

134

—

105

C₂²×Q₈

—

134

—

106

C₂×Q₁₆

—

268

—

107

C₂×Q₁₆

—

268

—

108

C₂×Q₁₆

—

134

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂²⋊D₆₇

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

134

—

113

C₄⋊Q₈

—

134

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

134

—

117

C₂×C₄²

C₂²×D₆₇

yes

characteristic

118

#107

—

119

C₄×C₈

—

134

—

120

C₄×C₈

—

134

—

121

C₄×C₈

C₄×D₆₇

yes

—

122

C₄×C₈

D₂₆₈

yes

—

123

C₂×Q₈

—

268

—

124

C₂×Q₈

—

268

—

125

C₂×Q₈

—

268

—

126

C₂×Q₈

—

134

—

127

C₂²×C₄

—

128

C₂²×C₄

—

268

—

129

C₂²×C₄

D₄×D₆₇

yes

—

130

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

131

Q₁₆

—

268

—

132

C₂×C₈

—

133

C₂×C₈

—

134

—

134

C₂×C₈

—

134

—

135

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

136

C₂×C₈

#11761613

yes

—

137

C₂×C₈

#11761615

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

268

—

141

C₄⋊C₄

—

134

—

142

C₄²

C₂×C₄×D₆₇

yes

characteristic

143

C₄²

C₂×D₂₆₈

yes

characteristic

144

C₄²

#11761629

yes

—

145

C₂³

#14524039

yes

characteristic

146

Q₈

—

268

—

147

Q₈

—

536

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

536

—

152

C₂×C₄

—

536

—

153

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

154

C₂×C₄

#14524005

yes

—

155

C₂×C₄

#14524018

yes

—

156

C₂×C₄

#14524032

yes

characteristic

157

C₂×C₄

C₂×#11761613

yes

—

158

C₂×C₄

C₂×#11761615

yes

—

159

C₂×C₄

Q₁₆×D₆₇

yes

—

160

C₂×C₄

#14524062

yes

—

161

C₈

—

134

—

162

C₈

#14523963

yes

—

163

C₂²

#16863297

yes

characteristic

164

C₂²

#16863770

yes

characteristic

165

C₂²

#16863772

yes

characteristic

166

C₂²

#16863774

yes

—

167

C₂²

#16863779

yes

—

168

C₄

—

169

C₄

—

536

—

170

C₄

C₂×#14523963

yes

—

171

C₄

#16863490

yes

—

172

C₄

#16863638

yes

—

173

C₄

#16863744

yes

—

174

C₂

#22559807

yes

—

175

C₂

#22559814

yes

characteristic

176

C₂

#22559825

yes

characteristic

177

C₂

#22561014

yes

characteristic

178

C₂

#22561068

yes

—

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#9825
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1845

C₂

socle, prime radical

C₂³×C₆₇

#127

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
8. #16863776	C₄	semidirect
41. C₂×C₆₇⋊C₈	Q₁₆	semidirect
94. C₆₇	#9825	semidirect

H ≅ G/N

type

8. #16863776

C₄

semidirect

41. C₂×C₆₇⋊C₈

Q₁₆

semidirect

94. C₆₇

#9825

semidirect

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

subgroup	index
C₂×#16863776	2
#22561169	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1845	2
#9825	67

subgroup

index

C₂×#16863776

#22561169

C₂×C₈×C₆₇⋊C₈

C₆₇×#1845

#9825

Subgroups — #30755597

Conjugacy classes of subgroups

Complements

Maximal subgroups