Group #30686384 — subgroups

Subgroups — C₆₇×#7133 #30686384

Order 17152.

216 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 74 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×#1985	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1377	C₂	1	2	yes	maximal
3	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
4	C₂×M₃₂×C₆₇	—	2	2	no	—
5	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
6	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
7	#336×C₆₇	—	2	2	no	—
8	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₆₇	—	2	1	no	—
10	C₂²×C₈×C₆₇	D₄	1	2	yes	—
11	C₄×Q₈×C₆₇	—	4	1	no	—
12	C₄×Q₈×C₆₇	—	4	1	no	—
13	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
15	C₂×C₄²×C₆₇	—	2	1	no	—
16	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
17	M₃₂×C₆₇	—	2	4	no	—
18	M₃₂×C₆₇	—	4	2	no	—
19	C₂×C₁₆×C₆₇	—	4	2	no	—
20	C₄⋊C₈×C₆₇	—	4	1	no	—
21	#97×C₆₇	—	4	1	no	—
22	C₂×Q₈×C₆₇	—	4	2	no	—
23	C₂×Q₈×C₆₇	—	4	1	no	—
24	C₂²×C₄×C₆₇	—	2	1	no	—
25	C₂²×C₄×C₆₇	—	2	1	no	—
26	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	—	4	1	no	—
32	C₂×C₈×C₆₇	—	4	2	no	—
33	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
34	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
35	C₄⋊C₄×C₆₇	—	4	1	no	—
36	C₄⋊C₄×C₆₇	—	4	2	no	—
37	C₄²×C₆₇	—	2	1	no	—
38	C₄²×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	—	4	1	no	—
40	C₄²×C₆₇	—	4	1	no	—
41	C₁₆×C₆₇	—	4	4	no	—
42	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
43	Q₈×C₆₇	—	4	2	no	—
44	Q₈×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	1	no	—
47	C₂×C₄×C₆₇	—	2	1	no	—
48	C₂×C₄×C₆₇	—	4	1	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	—	4	1	no	—
51	C₂×C₄×C₆₇	—	4	1	no	—
52	C₂×C₄×C₆₇	—	4	2	no	—
53	C₂×C₄×C₆₇	—	4	1	no	—
54	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#100	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#103	1	2	yes	—
57	C₂×C₄×C₆₇	#104	1	2	yes	—
58	C₈×C₆₇	—	2	2	no	—
59	C₈×C₆₇	—	2	2	no	—
60	C₈×C₆₇	—	8	1	no	—
61	C₂²×C₆₇	—	2	1	no	—
62	C₂²×C₆₇	—	2	1	no	—
63	C₂²×C₆₇	#323	1	1	yes	characteristic
64	C₂²×C₆₇	Q₈⋊C₈	1	2	yes	—
65	C₄×C₆₇	—	4	2	no	—
66	C₄×C₆₇	—	4	1	no	—
67	C₄×C₆₇	—	4	1	no	—
68	C₄×C₆₇	—	8	1	no	—
69	C₄×C₆₇	#328	1	2	yes	—
70	C₄×C₆₇	#329	1	1	yes	characteristic
71	C₄×C₆₇	#333	1	1	yes	characteristic
72	#7133	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₆₇	—	2	2	no	—
74	C₂×C₆₇	#1272	1	1	yes	characteristic
75	C₂×C₆₇	#1320	1	1	yes	characteristic
76	C₂×C₆₇	#1321	1	1	yes	characteristic
77	#1985	C₂×C₆₇	1	1	yes	characteristic
78	#1377	C₂×C₆₇	1	2	yes	—
79	C₆₇	#7133	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
81	C₂×M₃₂	—	2	2	no	—
82	C₂×C₄⋊C₈	C₄×C₆₇	1	1	yes	characteristic
83	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
84	#336	—	2	2	no	—
85	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	D₄×C₆₇	1	2	yes	—
88	C₄×Q₈	—	4	1	no	—
89	C₄×Q₈	—	4	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	C₈×C₆₇	1	1	yes	characteristic
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
94	M₃₂	—	2	4	no	—
95	M₃₂	—	4	2	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	#97	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
104	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
105	C₂²×C₄	M₁₆×C₆₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	SD₁₆×C₆₇	1	2	yes	—
111	C₂×C₈	Q₁₆×C₆₇	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₁₆	—	4	4	no	—
119	C₂³	C₂²⋊C₈×C₆₇	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#99×C₆₇	1	1	yes	characteristic
132	C₂×C₄	#100×C₆₇	1	1	yes	characteristic
133	C₂×C₄	#103×C₆₇	1	2	yes	—
134	C₂×C₄	#104×C₆₇	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#323×C₆₇	1	1	yes	characteristic
141	C₂²	Q₈⋊C₈×C₆₇	1	2	yes	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	#328×C₆₇	1	2	yes	—
147	C₄	#329×C₆₇	1	1	yes	characteristic
148	C₄	#333×C₆₇	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₆₇×#1272	1	1	yes	characteristic
151	C₂	C₆₇×#1320	1	1	yes	characteristic
152	C₂	C₆₇×#1321	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#1985

C₂

yes

maximal, characteristic

C₆₇×#1377

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₄

yes

characteristic

C₂×M₃₂×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₄

yes

characteristic

C₂×#97×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×Q₈×C₆₇

C₈

yes

characteristic

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₈

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

M₁₆

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂²⋊C₈

yes

characteristic

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×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#104

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#323

yes

characteristic

C₂²×C₆₇

Q₈⋊C₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#328

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

characteristic

#1377

C₂×C₆₇

yes

—

C₆₇

#7133

yes

characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₄×C₆₇

yes

characteristic

C₂×M₃₂

—

C₂×C₄⋊C₈

C₄×C₆₇

yes

characteristic

C₂×#97

C₂²×C₆₇

yes

characteristic

#336

—

C₂²×Q₈

C₈×C₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

D₄×C₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₈×C₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₆₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

#97

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊C₄×C₆₇

yes

characteristic

104

C₂²×C₄

C₂×C₈×C₆₇

yes

characteristic

105

C₂²×C₄

M₁₆×C₆₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

SD₁₆×C₆₇

yes

—

111

C₂×C₈

Q₁₆×C₆₇

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

119

C₂³

C₂²⋊C₈×C₆₇

yes

characteristic

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#99×C₆₇

yes

characteristic

132

C₂×C₄

#100×C₆₇

yes

characteristic

133

C₂×C₄

#103×C₆₇

yes

—

134

C₂×C₄

#104×C₆₇

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

#323×C₆₇

yes

characteristic

141

C₂²

Q₈⋊C₈×C₆₇

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#328×C₆₇

yes

—

147

C₄

#329×C₆₇

yes

characteristic

148

C₄

#333×C₆₇

yes

characteristic

149

C₂

—

150

C₂

C₆₇×#1272

yes

characteristic

151

C₂

C₆₇×#1320

yes

characteristic

152

C₂

C₆₇×#1321

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#99
commutator	C₂²×C₄	C₂×C₈×C₆₇
Frattini, cosocle	C₂×#97	C₂²×C₆₇
socle	C₂²×C₆₇	#323
prime radical	C₂³×C₆₇	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

#99

commutator

C₂²×C₄

C₂×C₈×C₆₇

Frattini, cosocle

C₂×#97

C₂²×C₆₇

socle

C₂²×C₆₇

#323

prime radical

C₂³×C₆₇

C₂²⋊C₈

LCS

G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

N	H ≅ G/N	type
72. #7133	C₆₇	direct (N × H)
79. C₆₇	#7133	direct (N × H)

H ≅ G/N

type

72. #7133

C₆₇

direct (N × H)

79. C₆₇

#7133

direct (N × H)

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

subgroup	index
C₆₇×#1985	2
C₆₇×#1377	2
#7133	67

subgroup

index

C₆₇×#1985

C₆₇×#1377

#7133

Subgroups — C₆₇×#7133 #30686384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#7133 #30686384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7133 #30686384