Group #30755681 — subgroups

Subgroups — #30755681

Order 17152.

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

normal subgroups only

This group has 542 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	#22561171	C₂	1	4	yes	maximal
2	#22560366	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523975	—	2	2	no	—
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
7	#16863266	C₂²	1	2	yes	—
8	#16863158	—	2	8	no	—
9	#16863117	—	2	2	no	—
10	C₂²×Q₁₆×C₆₇	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
12	#494×C₆₇	—	2	2	no	—
13	#494×C₆₇	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₆₇	—	2	4	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	#14523975	—	4	4	no	—
19	C₄×C₆₇⋊C₈	—	2	2	no	—
20	C₄×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×Q₈×C₆₇	—	2	2	no	—
22	C₂×Q₁₆×C₆₇	—	2	8	no	—
23	C₂×Q₁₆×C₆₇	—	4	4	no	—
24	C₂×Q₁₆×C₆₇	—	4	4	no	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₄⋊Q₈×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	—	2	4	no	—
28	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₇	D₄	1	2	yes	—
33	C₂²×Dic₆₇	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	2	8	no	—
36	C₄⋊Dic₆₇	—	4	4	no	—
37	C₄⋊Dic₆₇	—	4	8	no	—
38	C₂×Q₈×C₆₇	—	2	4	no	—
39	C₂×Q₈×C₆₇	—	4	2	no	—
40	C₂×Q₈×C₆₇	—	4	4	no	—
41	C₂×Q₈×C₆₇	—	4	4	no	—
42	C₂²×C₄×C₆₇	—	4	2	no	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	Q₁₆×C₆₇	—	4	16	no	—
46	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₆₇	D₈	1	4	yes	—
48	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
49	C₄⋊C₄×C₆₇	—	2	8	no	—
50	C₄⋊C₄×C₆₇	—	4	4	no	—
51	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₆₇	#41	1	2	yes	—
54	C₂³×C₆₇	#127	1	1	yes	characteristic
55	Q₈×C₆₇	—	4	8	no	—
56	Q₈×C₆₇	—	8	4	no	—
57	C₂×C₄×C₆₇	—	4	4	no	—
58	C₂×C₄×C₆₇	—	8	2	no	—
59	C₂×C₄×C₆₇	—	8	2	no	—
60	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₆₇	#121	1	2	yes	—
63	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₆₇	#137	1	8	yes	—
67	C₂×Dic₆₇	—	8	2	no	—
68	C₂×Dic₆₇	—	8	4	no	—
69	C₈×C₆₇	#102	1	8	yes	—
70	C₆₇⋊C₈	—	4	4	no	—
71	C₂²×C₆₇	#390	1	1	yes	characteristic
72	C₂²×C₆₇	#492	1	1	yes	characteristic
73	C₂²×C₆₇	#493	1	1	yes	characteristic
74	C₂²×C₆₇	#497	1	4	yes	—
75	C₄×C₆₇	—	8	4	no	—
76	C₄×C₆₇	C₂×#102	1	2	yes	—
77	C₄×C₆₇	#441	1	4	yes	—
78	C₄×C₆₇	#470	1	4	yes	—
79	C₄×C₆₇	#486	1	2	yes	—
80	Dic₆₇	—	16	2	no	—
81	#9839	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	#1716	1	2	yes	—
83	C₂×C₆₇	#1722	1	2	yes	—
84	C₂×C₆₇	#1957	1	1	yes	characteristic
85	C₂×C₆₇	#1971	1	2	yes	—
86	C₂×#494	D₆₇	1	1	yes	characteristic
87	#1990	—	67	4	no	—
88	#1844	—	67	1	no	—
89	C₂×#322	—	67	1	no	—
90	C₆₇	#9839	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₁₆	—	2	2	no	—
92	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
93	#494	—	2	2	no	—
94	#494	Dic₆₇	1	4	yes	complemented
95	C₂×#106	—	134	2	no	—
96	C₂×C₄×C₈	—	67	2	no	—
97	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
98	#384	—	67	2	no	—
99	#340	—	134	8	no	—
100	#322	—	134	2	no	—
101	C₂²×Q₈	—	2	2	no	—
102	C₂×Q₁₆	—	2	8	no	—
103	C₂×Q₁₆	—	4	4	no	—
104	C₂×Q₁₆	—	4	4	no	—
105	C₂²×C₈	—	67	4	no	—
106	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	—	2	4	no	—
109	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	134	4	no	—
112	C₂×C₄⋊C₄	—	134	2	no	—
113	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
114	#106	—	268	4	no	—
115	C₄×C₈	—	134	2	no	—
116	C₄×C₈	—	134	2	no	—
117	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
118	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	4	4	no	—
122	C₂×Q₈	—	4	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	268	2	no	—
125	C₂²×C₄	D₄×D₆₇	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
127	Q₁₆	—	4	16	no	—
128	C₂×C₈	—	134	4	no	—
129	C₂×C₈	—	134	8	no	—
130	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
131	C₂×C₈	#11761613	1	4	yes	—
132	C₂×C₈	#11761616	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	268	4	no	—
136	C₄⋊C₄	—	268	8	no	—
137	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
138	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
139	C₄²	#11761627	1	2	yes	—
140	C₂³	#14524039	1	1	yes	characteristic
141	Q₈	—	4	8	no	—
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	536	2	no	—
147	C₂×C₄	—	536	4	no	—
148	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
149	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
150	C₂×C₄	#14524014	1	2	yes	—
151	C₂×C₄	#14524030	1	1	yes	characteristic
152	C₂×C₄	C₂×#11761613	1	2	yes	—
153	C₂×C₄	C₂×#11761616	1	2	yes	—
154	C₂×C₄	#14524075	1	8	yes	—
155	C₈	—	268	4	no	—
156	C₈	#14523964	1	8	yes	—
157	C₂²	#16863299	1	1	yes	characteristic
158	C₂²	#16863768	1	1	yes	characteristic
159	C₂²	#16863772	1	1	yes	characteristic
160	C₂²	#16863789	1	4	yes	—
161	C₄	—	8	4	no	—
162	C₄	—	1072	2	no	—
163	C₄	C₂×#14523964	1	2	yes	—
164	C₄	#16863503	1	4	yes	—
165	C₄	#16863651	1	4	yes	—
166	C₄	#16863742	1	2	yes	—
167	C₂	#22559840	1	2	yes	—
168	C₂	#22559881	1	2	yes	—
169	C₂	#22561016	1	1	yes	characteristic
170	C₂	#22561081	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561171

C₂

yes

maximal

#22560366

C₂

yes

maximal, characteristic

C₂×#16863117

C₂

yes

maximal, characteristic

C₂×#494×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523975

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863266

C₂²

yes

—

#16863158

—

#16863117

—

C₂²×Q₁₆×C₆₇

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#494×C₆₇

—

#494×C₆₇

C₄

yes

complemented

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523975

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

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

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

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

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#102

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#102

yes

—

C₄×C₆₇

#441

yes

—

C₄×C₆₇

#470

yes

—

C₄×C₆₇

#486

yes

—

Dic₆₇

—

#9839

—

maximal, Sylow

C₂×C₆₇

#1716

yes

—

C₂×C₆₇

#1722

yes

—

C₂×C₆₇

#1957

yes

characteristic

C₂×C₆₇

#1971

yes

—

C₂×#494

D₆₇

yes

—

—

—

C₆₇

#9839

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

#494

—

#494

Dic₆₇

yes

complemented

C₂×#106

—

134

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#384

—

#340

—

134

—

100

#322

—

134

—

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²⋊D₆₇

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

C₂×Dic₆₇

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

134

—

112

C₂×C₄⋊C₄

—

134

—

113

C₂×C₄²

C₂²×D₆₇

yes

characteristic

114

#106

—

268

—

115

C₄×C₈

—

134

—

116

C₄×C₈

—

134

—

117

C₄×C₈

C₂×Dic₆₇

yes

—

118

C₄×C₈

C₂²⋊D₆₇

yes

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

268

—

125

C₂²×C₄

D₄×D₆₇

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

134

—

129

C₂×C₈

—

134

—

130

C₂×C₈

C₂²⋊Dic₆₇

yes

—

131

C₂×C₈

#11761613

yes

—

132

C₂×C₈

#11761616

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

268

—

136

C₄⋊C₄

—

268

—

137

C₄²

C₂²×Dic₆₇

yes

characteristic

138

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

139

C₄²

#11761627

yes

—

140

C₂³

#14524039

yes

characteristic

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

536

—

147

C₂×C₄

—

536

—

148

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

149

C₂×C₄

D₄×Dic₆₇

yes

—

150

C₂×C₄

#14524014

yes

—

151

C₂×C₄

#14524030

yes

characteristic

152

C₂×C₄

C₂×#11761613

yes

—

153

C₂×C₄

C₂×#11761616

yes

—

154

C₂×C₄

#14524075

yes

—

155

C₈

—

268

—

156

C₈

#14523964

yes

—

157

C₂²

#16863299

yes

characteristic

158

C₂²

#16863768

yes

characteristic

159

C₂²

#16863772

yes

characteristic

160

C₂²

#16863789

yes

—

161

C₄

—

162

C₄

—

1072

—

163

C₄

C₂×#14523964

yes

—

164

C₄

#16863503

yes

—

165

C₄

#16863651

yes

—

166

C₄

#16863742

yes

—

167

C₂

#22559840

yes

—

168

C₂

#22559881

yes

—

169

C₂

#22561016

yes

characteristic

170

C₂

#22561081

yes

—

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#9839
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#494×C₆₇

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 ◁ C₂×#494

N	H ≅ G/N	type
13. #494×C₆₇	C₄	semidirect
90. C₆₇	#9839	semidirect
94. #494	Dic₆₇	semidirect

H ≅ G/N

type

13. #494×C₆₇

C₄

semidirect

90. C₆₇

#9839

semidirect

94. #494

Dic₆₇

semidirect

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

subgroup	index
#22561171	2
#22560366	2
C₂×#16863117	2
C₂×#494×C₆₇	2
#9839	67

subgroup

index

C₂×#494×C₆₇

Subgroups — #30755681

Conjugacy classes of subgroups

Complements

Maximal subgroups