Group #30689106 — subgroups

Subgroups — C₆₇×#9855 #30689106

Order 17152.

422 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 174 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×#3160	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1493	C₂	1	4	yes	maximal
4	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#533×C₆₇	C₂²	1	2	yes	—
6	#500×C₆₇	C₄	1	2	yes	—
7	#499×C₆₇	—	2	2	no	—
8	#498×C₆₇	C₄	1	2	yes	—
9	C₂×#108×C₆₇	—	2	2	no	—
10	#427×C₆₇	—	2	2	no	—
11	#423×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	Q₈⋊C₈×C₆₇	—	2	8	no	—
15	C₈²×C₆₇	—	2	2	no	—
16	C₂×M₁₆×C₆₇	—	2	2	no	—
17	C₂²×C₈×C₆₇	—	2	2	no	—
18	C₂²×C₈×C₆₇	D₄	1	2	yes	—
19	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
20	#125×C₆₇	—	2	2	no	—
21	C₂²⋊Q₈×C₆₇	—	2	4	no	—
22	C₄×Q₈×C₆₇	—	2	4	no	—
23	#117×C₆₇	—	2	2	no	—
24	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
25	#108×C₆₇	—	4	4	no	—
26	#107×C₆₇	—	2	4	no	—
27	#106×C₆₇	—	2	4	no	—
28	C₄⋊C₈×C₆₇	—	2	4	no	—
29	C₄×C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	D₄	1	4	yes	—
33	C₄×C₈×C₆₇	D₄	1	2	yes	—
34	C₂×Q₈×C₆₇	—	2	4	no	—
35	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	#41	1	2	yes	—
37	M₁₆×C₆₇	—	4	4	no	—
38	C₂×C₈×C₆₇	—	2	8	no	—
39	C₂×C₈×C₆₇	—	2	4	no	—
40	C₂×C₈×C₆₇	—	4	2	no	—
41	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
44	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
45	C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₂²⋊C₄×C₆₇	—	4	2	no	—
49	C₄²×C₆₇	—	4	2	no	—
50	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₂×D₄	1	2	yes	—
53	C₂³×C₆₇	#124	1	1	yes	characteristic
54	Q₈×C₆₇	—	4	8	no	—
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	4	4	no	—
57	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₆₇	#121	1	2	yes	—
60	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
63	C₈×C₆₇	—	2	8	no	—
64	C₈×C₆₇	—	8	2	no	—
65	C₈×C₆₇	#103	1	4	yes	—
66	C₈×C₆₇	#103	1	4	yes	—
67	C₂²×C₆₇	—	2	2	no	—
68	C₂²×C₆₇	#390	1	1	yes	characteristic
69	C₂²×C₆₇	#487	1	2	yes	—
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	8	4	no	—
72	C₄×C₆₇	C₂×#103	1	2	yes	—
73	C₄×C₆₇	#443	1	4	yes	—
74	C₄×C₆₇	#492	1	1	yes	characteristic
75	C₄×C₆₇	#494	1	1	yes	characteristic
76	#9855	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#1956	1	1	yes	characteristic
79	C₂×C₆₇	#1974	1	2	yes	—
80	#3160	C₂×C₆₇	1	1	yes	characteristic
81	#1848	C₂×C₆₇	1	1	yes	characteristic
82	#1493	C₂×C₆₇	1	4	yes	—
83	C₂×C₈²	C₂×C₆₇	1	1	yes	characteristic
84	C₆₇	#9855	1	1	yes	characteristic, direct product factor, Sylow
85	#533	C₂²×C₆₇	1	2	yes	—
86	#500	C₄×C₆₇	1	2	yes	—
87	#499	—	2	2	no	—
88	#498	C₄×C₆₇	1	2	yes	—
89	C₂×#108	—	2	2	no	—
90	#427	—	2	2	no	—
91	#423	C₂²×C₆₇	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
94	Q₈⋊C₈	—	2	8	no	—
95	C₈²	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	D₄×C₆₇	1	2	yes	—
99	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
100	#125	—	2	2	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	#117	—	2	2	no	—
104	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
105	#108	—	4	4	no	—
106	#107	—	2	4	no	—
107	#106	—	2	4	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
111	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
112	C₄×C₈	D₄×C₆₇	1	4	yes	—
113	C₄×C₈	D₄×C₆₇	1	2	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
116	C₂²×C₄	#41×C₆₇	1	2	yes	—
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	2	8	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
122	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
123	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
124	C₂×C₈	Q₁₆×C₆₇	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
131	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
132	C₄²	C₂×D₄×C₆₇	1	2	yes	—
133	C₂³	#124×C₆₇	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
138	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
139	C₂×C₄	#121×C₆₇	1	2	yes	—
140	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
141	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
142	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
143	C₈	—	2	8	no	—
144	C₈	—	8	2	no	—
145	C₈	#103×C₆₇	1	4	yes	—
146	C₈	#103×C₆₇	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#390×C₆₇	1	1	yes	characteristic
149	C₂²	#487×C₆₇	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#103×C₆₇	1	2	yes	—
153	C₄	#443×C₆₇	1	4	yes	—
154	C₄	#492×C₆₇	1	1	yes	characteristic
155	C₄	#494×C₆₇	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₆₇×#1956	1	1	yes	characteristic
158	C₂	C₆₇×#1974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#3160

C₂

yes

maximal, characteristic

C₆₇×#1848

C₂

yes

maximal, characteristic

C₆₇×#1493

C₂

yes

maximal

C₂×C₈²×C₆₇

C₂

yes

maximal, characteristic

#533×C₆₇

C₂²

yes

—

#500×C₆₇

C₄

yes

—

#499×C₆₇

—

#498×C₆₇

C₄

yes

—

C₂×#108×C₆₇

—

#427×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

Q₈⋊C₈×C₆₇

—

C₈²×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

#107×C₆₇

—

#106×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

#124

yes

characteristic

Q₈×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

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

#103

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#487

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#103

yes

—

C₄×C₆₇

#443

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1956

yes

characteristic

C₂×C₆₇

#1974

yes

—

#3160

C₂×C₆₇

yes

characteristic

#1848

C₂×C₆₇

yes

characteristic

#1493

C₂×C₆₇

yes

—

C₂×C₈²

C₂×C₆₇

yes

characteristic

C₆₇

#9855

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₆₇

yes

—

#500

C₄×C₆₇

yes

—

#499

—

#498

C₄×C₆₇

yes

—

C₂×#108

—

#427

—

#423

C₂²×C₆₇

yes

—

C₂×C₄×C₈

C₂²×C₆₇

yes

—

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄×C₆₇

yes

—

C₄⋊Q₈

C₂×C₄×C₆₇

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂³×C₆₇

yes

characteristic

105

#108

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

111

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

112

C₄×C₈

D₄×C₆₇

yes

—

113

C₄×C₈

D₄×C₆₇

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

116

C₂²×C₄

#41×C₆₇

yes

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂²⋊C₄×C₆₇

yes

—

122

C₂×C₈

C₂²⋊C₄×C₆₇

yes

—

123

C₂×C₈

SD₁₆×C₆₇

yes

—

124

C₂×C₈

Q₁₆×C₆₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂²×C₄×C₆₇

yes

characteristic

131

C₄²

C₂×D₄×C₆₇

yes

characteristic

132

C₄²

C₂×D₄×C₆₇

yes

—

133

C₂³

#124×C₆₇

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊C₄×C₆₇

yes

characteristic

138

C₂×C₄

C₄×D₄×C₆₇

yes

—

139

C₂×C₄

#121×C₆₇

yes

—

140

C₂×C₄

#127×C₆₇

yes

characteristic

141

C₂×C₄

C₂×SD₁₆×C₆₇

yes

—

142

C₂×C₄

C₂×Q₁₆×C₆₇

yes

—

143

C₈

—

144

C₈

—

145

C₈

#103×C₆₇

yes

—

146

C₈

#103×C₆₇

yes

—

147

C₂²

—

148

C₂²

#390×C₆₇

yes

characteristic

149

C₂²

#487×C₆₇

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#103×C₆₇

yes

—

153

C₄

#443×C₆₇

yes

—

154

C₄

#492×C₆₇

yes

characteristic

155

C₄

#494×C₆₇

yes

characteristic

156

C₂

—

157

C₂

C₆₇×#1956

yes

characteristic

158

C₂

C₆₇×#1974

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

#127

commutator

C₄²

C₂²×C₄×C₆₇

Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

C₂²×C₆₇

C₂³×C₆₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
76. #9855	C₆₇	direct (N × H)
84. C₆₇	#9855	direct (N × H)

H ≅ G/N

type

76. #9855

C₆₇

direct (N × H)

84. C₆₇

#9855

direct (N × H)

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

subgroup	index
C₆₇×#3160	2
C₆₇×#1848	2
C₆₇×#1493	2
C₂×C₈²×C₆₇	2
#9855	67

subgroup

index

C₆₇×#3160

C₆₇×#1848

C₆₇×#1493

C₂×C₈²×C₆₇

#9855

Subgroups — C₆₇×#9855 #30689106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#9855 #30689106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9855 #30689106