Group #30712118 — subgroups

Subgroups — C₆₇×#32867 #30712118

Order 17152.

630 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 206 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	C₆₇×#2644	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2087	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2067	C₂	1	6	yes	maximal, complemented
4	C₆₇×#2054	C₂	1	3	yes	maximal
5	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×#107×C₆₇	C₂²	1	3	yes	—
8	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
10	C₂×#102×C₆₇	C₂²	1	6	yes	—
11	#397×C₆₇	—	2	6	no	—
12	#396×C₆₇	—	2	2	no	—
13	#390×C₆₇	C₂²	1	6	yes	—
14	#389×C₆₇	C₂²	1	6	yes	—
15	#376×C₆₇	C₂²	1	1	yes	characteristic
16	#340×C₆₇	C₂²	1	3	yes	—
17	#340×C₆₇	C₂²	1	3	yes	—
18	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
19	C₂×D₈×C₆₇	—	4	3	no	—
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
22	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
23	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
24	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
25	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
26	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
27	#107×C₆₇	—	4	3	no	—
28	C₄⋊C₈×C₆₇	—	4	3	no	—
29	C₄⋊C₈×C₆₇	—	4	3	no	—
30	#102×C₆₇	—	4	6	no	—
31	#95×C₆₇	—	2	6	no	—
32	#95×C₆₇	D₄	1	1	yes	characteristic
33	#95×C₆₇	D₄	1	3	yes	—
34	C₂⁴×C₆₇	—	2	2	no	—
35	C₂×D₄×C₆₇	—	2	6	no	—
36	C₂×D₄×C₆₇	—	2	6	no	—
37	C₂²×C₄×C₆₇	—	2	6	no	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
41	D₈×C₆₇	—	4	4	no	—
42	C₂×C₈×C₆₇	—	2	6	no	—
43	C₂×C₈×C₆₇	—	2	3	no	—
44	C₂×C₈×C₆₇	—	2	3	no	—
45	C₄⋊C₄×C₆₇	—	2	6	no	—
46	C₄⋊C₄×C₆₇	—	2	6	no	—
47	C₄⋊C₄×C₆₇	—	8	6	no	—
48	C₂²⋊C₄×C₆₇	—	8	6	no	—
49	C₂²⋊C₄×C₆₇	—	8	6	no	—
50	C₄²×C₆₇	—	2	3	no	—
51	C₄²×C₆₇	—	2	3	no	—
52	C₂³×C₆₇	—	4	6	no	—
53	C₂³×C₆₇	—	4	6	no	—
54	C₂³×C₆₇	—	4	2	no	—
55	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
56	D₄×C₆₇	—	4	2	no	—
57	D₄×C₆₇	—	4	6	no	—
58	C₂×C₄×C₆₇	—	2	3	no	—
59	C₂×C₄×C₆₇	—	2	3	no	—
60	C₂×C₄×C₆₇	—	2	3	no	—
61	C₂×C₄×C₆₇	—	2	6	no	—
62	C₂×C₄×C₆₇	—	2	3	no	—
63	C₂×C₄×C₆₇	—	4	6	no	—
64	C₂×C₄×C₆₇	—	4	6	no	—
65	C₂×C₄×C₆₇	—	4	6	no	—
66	C₂×C₄×C₆₇	H(5,𝔽₂)	1	3	yes	—
67	C₂×C₄×C₆₇	2₋¹⁺⁴	1	3	yes	—
68	C₈×C₆₇	—	4	3	no	—
69	C₈×C₆₇	—	4	1	no	—
70	C₂²×C₆₇	—	8	2	no	—
71	C₂²×C₆₇	—	8	6	no	—
72	C₂²×C₆₇	—	8	6	no	—
73	C₂²×C₆₇	#537	1	3	yes	—
74	C₂²×C₆₇	#576	1	3	yes	—
75	C₂²×C₆₇	#578	1	1	yes	characteristic
76	C₄×C₆₇	—	4	3	no	—
77	C₄×C₆₇	—	4	3	no	—
78	C₄×C₆₇	—	8	6	no	—
79	C₄×C₆₇	#560	1	1	yes	characteristic
80	C₄×C₆₇	#563	1	3	yes	—
81	#32867	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₆₇	—	16	2	no	—
83	C₂×C₆₇	#2823	1	1	yes	characteristic
84	C₂×C₆₇	#3259	1	3	yes	—
85	C₂×C₆₇	#3262	1	3	yes	—
86	#2644	C₂×C₆₇	1	2	yes	complemented
87	#2087	C₂×C₆₇	1	3	yes	complemented
88	#2067	C₂×C₆₇	1	6	yes	complemented
89	#2054	C₂×C₆₇	1	3	yes	—
90	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
91	C₆₇	#32867	1	1	yes	characteristic, direct product factor, Sylow
92	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
93	C₂×#107	C₂²×C₆₇	1	3	yes	—
94	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
95	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
96	C₂×#102	C₂²×C₆₇	1	6	yes	—
97	#397	—	2	6	no	—
98	#396	—	2	2	no	—
99	#390	C₂²×C₆₇	1	6	yes	—
100	#389	C₂²×C₆₇	1	6	yes	—
101	#376	C₂²×C₆₇	1	1	yes	characteristic
102	#340	C₂²×C₆₇	1	3	yes	—
103	#340	C₂²×C₆₇	1	3	yes	—
104	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
105	C₂×D₈	—	4	3	no	—
106	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
108	C₂×C₄⋊C₄	—	2	6	no	—
109	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
110	C₂×C₂²⋊C₄	—	2	6	no	—
111	C₂×C₂²⋊C₄	—	2	6	no	—
112	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
113	#107	—	4	3	no	—
114	C₄⋊C₈	—	4	3	no	—
115	C₄⋊C₈	—	4	3	no	—
116	#102	—	4	6	no	—
117	#95	—	2	6	no	—
118	#95	D₄×C₆₇	1	1	yes	characteristic
119	#95	D₄×C₆₇	1	3	yes	—
120	C₂⁴	—	2	2	no	—
121	C₂×D₄	—	2	6	no	—
122	C₂×D₄	—	2	6	no	—
123	C₂²×C₄	—	2	6	no	—
124	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
125	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
126	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
127	D₈	—	4	4	no	—
128	C₂×C₈	—	2	6	no	—
129	C₂×C₈	—	2	3	no	—
130	C₂×C₈	—	2	3	no	—
131	C₄⋊C₄	—	2	6	no	—
132	C₄⋊C₄	—	2	6	no	—
133	C₄⋊C₄	—	8	6	no	—
134	C₂²⋊C₄	—	8	6	no	—
135	C₂²⋊C₄	—	8	6	no	—
136	C₄²	—	2	3	no	—
137	C₄²	—	2	3	no	—
138	C₂³	—	4	6	no	—
139	C₂³	—	4	6	no	—
140	C₂³	—	4	2	no	—
141	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
142	D₄	—	4	2	no	—
143	D₄	—	4	6	no	—
144	C₂×C₄	—	2	3	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	2	3	no	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	2	3	no	—
149	C₂×C₄	—	4	6	no	—
150	C₂×C₄	—	4	6	no	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	H(5,𝔽₂)×C₆₇	1	3	yes	—
153	C₂×C₄	2₋¹⁺⁴×C₆₇	1	3	yes	—
154	C₈	—	4	1	no	—
155	C₈	—	4	3	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	6	no	—
158	C₂²	—	8	6	no	—
159	C₂²	#537×C₆₇	1	3	yes	—
160	C₂²	#576×C₆₇	1	3	yes	—
161	C₂²	#578×C₆₇	1	1	yes	characteristic
162	C₄	—	4	3	no	—
163	C₄	—	4	3	no	—
164	C₄	—	8	6	no	—
165	C₄	#560×C₆₇	1	1	yes	characteristic
166	C₄	#563×C₆₇	1	3	yes	—
167	C₂	—	16	2	no	—
168	C₂	C₆₇×#2823	1	1	yes	characteristic
169	C₂	C₆₇×#3259	1	3	yes	—
170	C₂	C₆₇×#3262	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2644

C₂

yes

maximal, complemented

C₆₇×#2087

C₂

yes

maximal, complemented

C₆₇×#2067

C₂

yes

maximal, complemented

C₆₇×#2054

C₂

yes

maximal

C₆₇×#1843

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₆₇

C₂²

yes

characteristic

C₂×#107×C₆₇

C₂²

yes

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×#102×C₆₇

C₂²

yes

—

#397×C₆₇

—

#396×C₆₇

—

#390×C₆₇

C₂²

yes

—

#389×C₆₇

C₂²

yes

—

#376×C₆₇

C₂²

yes

characteristic

#340×C₆₇

C₂²

yes

—

#340×C₆₇

C₂²

yes

—

C₂²×D₄×C₆₇

C₂³

yes

—

C₂×D₈×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

—

#107×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#102×C₆₇

—

#95×C₆₇

—

#95×C₆₇

D₄

yes

characteristic

#95×C₆₇

D₄

yes

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

D₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

H(5,𝔽₂)

yes

—

C₂×C₄×C₆₇

2₋¹⁺⁴

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#537

yes

—

C₂²×C₆₇

#576

yes

—

C₂²×C₆₇

#578

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#560

yes

characteristic

C₄×C₆₇

#563

yes

—

#32867

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2823

yes

characteristic

C₂×C₆₇

#3259

yes

—

C₂×C₆₇

#3262

yes

—

#2644

C₂×C₆₇

yes

complemented

#2087

C₂×C₆₇

yes

complemented

#2067

C₂×C₆₇

yes

complemented

#2054

C₂×C₆₇

yes

—

#1843

C₂×C₆₇

yes

characteristic, complemented

C₆₇

#32867

yes

characteristic, direct product factor, Sylow

C₂²×D₈

C₂²×C₆₇

yes

characteristic

C₂×#107

C₂²×C₆₇

yes

—

C₂×C₄⋊C₈

C₂²×C₆₇

yes

—

C₂×C₄⋊C₈

C₂²×C₆₇

yes

—

C₂×#102

C₂²×C₆₇

yes

—

#397

—

#396

—

#390

C₂²×C₆₇

yes

—

100

#389

C₂²×C₆₇

yes

—

101

#376

C₂²×C₆₇

yes

characteristic

102

#340

C₂²×C₆₇

yes

—

103

#340

C₂²×C₆₇

yes

—

104

C₂²×D₄

C₂³×C₆₇

yes

—

105

C₂×D₈

—

106

C₂²×C₈

C₂³×C₆₇

yes

characteristic

107

C₂²×C₈

C₂³×C₆₇

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

C₂³×C₆₇

yes

—

113

#107

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

#102

—

117

#95

—

118

#95

D₄×C₆₇

yes

characteristic

119

#95

D₄×C₆₇

yes

—

120

C₂⁴

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×D₄×C₆₇

yes

—

125

C₂²×C₄

C₂×D₄×C₆₇

yes

—

126

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

127

D₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

C₂²×D₄×C₆₇

yes

characteristic

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

H(5,𝔽₂)×C₆₇

yes

—

153

C₂×C₄

2₋¹⁺⁴×C₆₇

yes

—

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#537×C₆₇

yes

—

160

C₂²

#576×C₆₇

yes

—

161

C₂²

#578×C₆₇

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#560×C₆₇

yes

characteristic

166

C₄

#563×C₆₇

yes

—

167

C₂

—

168

C₂

C₆₇×#2823

yes

characteristic

169

C₂

C₆₇×#3259

yes

—

170

C₂

C₆₇×#3262

yes

—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

prime radical

C₂²×D₈×C₆₇

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

N	H ≅ G/N	type
1. C₆₇×#2644	C₂	semidirect
2. C₆₇×#2087	C₂	semidirect
3. C₆₇×#2067	C₂	semidirect
5. C₆₇×#1843	C₂	semidirect
81. #32867	C₆₇	direct (N × H)
86. #2644	C₂×C₆₇	semidirect
87. #2087	C₂×C₆₇	semidirect
88. #2067	C₂×C₆₇	semidirect
90. #1843	C₂×C₆₇	semidirect
91. C₆₇	#32867	direct (N × H)

H ≅ G/N

type

1. C₆₇×#2644

C₂

semidirect

2. C₆₇×#2087

C₂

semidirect

3. C₆₇×#2067

C₂

semidirect

5. C₆₇×#1843

C₂

semidirect

81. #32867

C₆₇

direct (N × H)

86. #2644

C₂×C₆₇

semidirect

87. #2087

C₂×C₆₇

semidirect

88. #2067

C₂×C₆₇

semidirect

90. #1843

C₂×C₆₇

semidirect

91. C₆₇

#32867

direct (N × H)

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

subgroup	index
C₆₇×#2644	2
C₆₇×#2087	2
C₆₇×#2067	2
C₆₇×#2054	2
C₆₇×#1843	2
#32867	67

subgroup

index

C₆₇×#2644

C₆₇×#2087

C₆₇×#2067

C₆₇×#2054

C₆₇×#1843

#32867

Subgroups — C₆₇×#32867 #30712118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#32867 #30712118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32867 #30712118