Group #30712052 — subgroups

Subgroups — C₆₇×#32801 #30712052

Order 17152.

762 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 238 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×#2868	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2863	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2043	C₂	1	6	yes	maximal
4	C₆₇×#2016	C₂	1	6	yes	maximal
5	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
8	C₂×#103×C₆₇	C₂²	1	12	yes	—
9	#391×C₆₇	C₂²	1	3	yes	—
10	#389×C₆₇	C₂²	1	3	yes	—
11	#387×C₆₇	—	2	3	no	—
12	#384×C₆₇	—	2	3	no	—
13	#384×C₆₇	—	2	1	no	—
14	#383×C₆₇	—	2	1	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	C₂²	1	6	yes	—
17	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
18	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
19	C₂×Q₁₆×C₆₇	—	4	12	no	—
20	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
21	C₄⋊Q₈×C₆₇	—	4	12	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
25	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
26	C₂×C₄²×C₆₇	D₄	1	3	yes	—
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
29	#103×C₆₇	—	4	12	no	—
30	#95×C₆₇	—	2	3	no	—
31	#95×C₆₇	—	2	3	no	—
32	#95×C₆₇	—	2	3	no	—
33	C₂×Q₈×C₆₇	—	2	12	no	—
34	C₂×Q₈×C₆₇	—	2	12	no	—
35	C₂×Q₈×C₆₇	—	2	6	no	—
36	C₂²×C₄×C₆₇	—	2	4	no	—
37	C₂²×C₄×C₆₇	—	2	3	no	—
38	C₂²×C₄×C₆₇	—	2	1	no	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
42	Q₁₆×C₆₇	—	4	16	no	—
43	C₂×C₈×C₆₇	—	2	12	no	—
44	C₄⋊C₄×C₆₇	—	2	6	no	—
45	C₄⋊C₄×C₆₇	—	4	12	no	—
46	C₄⋊C₄×C₆₇	—	4	12	no	—
47	C₄⋊C₄×C₆₇	—	4	6	no	—
48	C₄²×C₆₇	—	2	6	no	—
49	C₄²×C₆₇	—	2	6	no	—
50	C₄²×C₆₇	Q₁₆	1	4	yes	complemented
51	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
52	Q₈×C₆₇	—	4	4	no	—
53	Q₈×C₆₇	—	4	4	no	—
54	Q₈×C₆₇	—	4	12	no	—
55	C₂×C₄×C₆₇	—	2	3	no	—
56	C₂×C₄×C₆₇	—	2	3	no	—
57	C₂×C₄×C₆₇	—	2	3	no	—
58	C₂×C₄×C₆₇	—	2	3	no	—
59	C₂×C₄×C₆₇	—	4	3	no	—
60	C₂×C₄×C₆₇	—	4	12	no	—
61	C₂×C₄×C₆₇	—	4	3	no	—
62	C₂×C₄×C₆₇	—	4	6	no	—
63	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₆₇	#137	1	6	yes	—
65	C₂×C₄×C₆₇	H(5,𝔽₂)	1	6	yes	—
66	C₈×C₆₇	—	4	4	no	—
67	C₂²×C₆₇	#535	1	3	yes	—
68	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂²×C₆₇	C₂×#137	1	3	yes	—
70	C₄×C₆₇	—	2	6	no	—
71	C₄×C₆₇	—	2	6	no	—
72	C₄×C₆₇	—	8	4	no	—
73	C₄×C₆₇	—	8	1	no	—
74	C₄×C₆₇	—	8	3	no	—
75	C₄×C₆₇	#560	1	4	yes	—
76	#32801	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₆₇	#2818	1	1	yes	characteristic
78	C₂×C₆₇	#3209	1	3	yes	—
79	C₂×C₆₇	#3218	1	3	yes	—
80	#2868	C₂×C₆₇	1	1	yes	characteristic
81	#2863	C₂×C₆₇	1	1	yes	characteristic
82	#2043	C₂×C₆₇	1	6	yes	—
83	#2016	C₂×C₆₇	1	6	yes	—
84	#1839	C₂×C₆₇	1	1	yes	characteristic
85	C₆₇	#32801	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
87	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
88	C₂×#103	C₂²×C₆₇	1	12	yes	—
89	#391	C₂²×C₆₇	1	3	yes	—
90	#389	C₂²×C₆₇	1	3	yes	—
91	#387	—	2	3	no	—
92	#384	—	2	3	no	—
93	#384	—	2	1	no	—
94	#383	—	2	1	no	—
95	C₄³	C₂²×C₆₇	1	1	yes	characteristic
96	#336	C₂²×C₆₇	1	6	yes	—
97	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
98	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
99	C₂×Q₁₆	—	4	12	no	—
100	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
101	C₄⋊Q₈	—	4	12	no	—
102	C₂×C₄⋊C₄	—	2	6	no	—
103	C₂×C₄⋊C₄	—	2	6	no	—
104	C₂×C₄⋊C₄	—	2	3	no	—
105	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
106	C₂×C₄²	D₄×C₆₇	1	3	yes	—
107	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
108	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
109	#103	—	4	12	no	—
110	#95	—	2	3	no	—
111	#95	—	2	3	no	—
112	#95	—	2	3	no	—
113	C₂×Q₈	—	2	12	no	—
114	C₂×Q₈	—	2	12	no	—
115	C₂×Q₈	—	2	6	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	3	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
120	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
121	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
122	Q₁₆	—	4	16	no	—
123	C₂×C₈	—	2	12	no	—
124	C₄⋊C₄	—	2	6	no	—
125	C₄⋊C₄	—	4	12	no	—
126	C₄⋊C₄	—	4	12	no	—
127	C₄⋊C₄	—	4	6	no	—
128	C₄²	—	2	6	no	—
129	C₄²	—	2	6	no	—
130	C₄²	Q₁₆×C₆₇	1	4	yes	complemented
131	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	4	12	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	4	3	no	—
140	C₂×C₄	—	4	12	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	C₂×Q₁₆×C₆₇	1	6	yes	—
144	C₂×C₄	#137×C₆₇	1	6	yes	—
145	C₂×C₄	H(5,𝔽₂)×C₆₇	1	6	yes	—
146	C₈	—	4	4	no	—
147	C₂²	#535×C₆₇	1	3	yes	—
148	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
149	C₂²	C₂×#137×C₆₇	1	3	yes	—
150	C₄	—	2	6	no	—
151	C₄	—	2	6	no	—
152	C₄	—	8	4	no	—
153	C₄	—	8	1	no	—
154	C₄	—	8	3	no	—
155	C₄	#560×C₆₇	1	4	yes	—
156	C₂	C₆₇×#2818	1	1	yes	characteristic
157	C₂	C₆₇×#3209	1	3	yes	—
158	C₂	C₆₇×#3218	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2868

C₂

yes

maximal, characteristic

C₆₇×#2863

C₂

yes

maximal, characteristic

C₆₇×#2043

C₂

yes

maximal

C₆₇×#2016

C₂

yes

maximal

C₆₇×#1839

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×#103×C₆₇

C₂²

yes

—

#391×C₆₇

C₂²

yes

—

#389×C₆₇

C₂²

yes

—

#387×C₆₇

—

#384×C₆₇

—

#384×C₆₇

—

#383×C₆₇

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂³

yes

—

C₂²×Q₈×C₆₇

C₂³

yes

characteristic

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

—

#103×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

complemented

C₂³×C₆₇

C₂²×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×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

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

H(5,𝔽₂)

yes

—

C₈×C₆₇

—

C₂²×C₆₇

#535

yes

—

C₂²×C₆₇

C₂²×Q₁₆

yes

characteristic

C₂²×C₆₇

C₂×#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#560

yes

—

#32801

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#2818

yes

characteristic

C₂×C₆₇

#3209

yes

—

C₂×C₆₇

#3218

yes

—

#2868

C₂×C₆₇

yes

characteristic

#2863

C₂×C₆₇

yes

characteristic

#2043

C₂×C₆₇

yes

—

#2016

C₂×C₆₇

yes

—

#1839

C₂×C₆₇

yes

characteristic

C₆₇

#32801

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂²×C₆₇

yes

—

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

C₂×#103

C₂²×C₆₇

yes

—

#391

C₂²×C₆₇

yes

—

#389

C₂²×C₆₇

yes

—

—

—

—

—

C₄³

C₂²×C₆₇

yes

characteristic

#336

C₂²×C₆₇

yes

—

C₂²×Q₈

C₂³×C₆₇

yes

—

C₂²×Q₈

C₂³×C₆₇

yes

characteristic

C₂×Q₁₆

—

100

C₂²×C₈

C₂³×C₆₇

yes

—

101

C₄⋊Q₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

106

C₂×C₄²

D₄×C₆₇

yes

—

107

C₂×C₄²

D₄×C₆₇

yes

characteristic

108

C₂×C₄²

C₂³×C₆₇

yes

—

109

#103

—

110

#95

—

111

#95

—

112

#95

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×D₄×C₆₇

yes

—

120

C₂²×C₄

C₂×D₄×C₆₇

yes

—

121

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

122

Q₁₆

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

Q₁₆×C₆₇

yes

complemented

131

C₂³

C₂²×D₄×C₆₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂×Q₁₆×C₆₇

yes

—

144

C₂×C₄

#137×C₆₇

yes

—

145

C₂×C₄

H(5,𝔽₂)×C₆₇

yes

—

146

C₈

—

147

C₂²

#535×C₆₇

yes

—

148

C₂²

C₂²×Q₁₆×C₆₇

yes

characteristic

149

C₂²

C₂×#137×C₆₇

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#560×C₆₇

yes

—

156

C₂

C₆₇×#2818

yes

characteristic

157

C₂

C₆₇×#3209

yes

—

158

C₂

C₆₇×#3218

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

N	H ≅ G/N	type
50. C₄²×C₆₇	Q₁₆	semidirect
76. #32801	C₆₇	direct (N × H)
85. C₆₇	#32801	direct (N × H)
130. C₄²	Q₁₆×C₆₇	semidirect

H ≅ G/N

type

50. C₄²×C₆₇

Q₁₆

semidirect

76. #32801

C₆₇

direct (N × H)

85. C₆₇

#32801

direct (N × H)

130. C₄²

Q₁₆×C₆₇

semidirect

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

subgroup	index
C₆₇×#2868	2
C₆₇×#2863	2
C₆₇×#2043	2
C₆₇×#2016	2
C₆₇×#1839	2
#32801	67

subgroup

index

C₆₇×#2868

C₆₇×#2863

C₆₇×#2043

C₆₇×#2016

C₆₇×#1839

#32801

Subgroups — C₆₇×#32801 #30712052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#32801 #30712052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32801 #30712052