Group #30692136 — subgroups

Subgroups — C₆₇×#12885 #30692136

Order 17152.

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2153	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1747	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1379	C₂	1	2	yes	maximal
5	C₆₇×#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	#405×C₆₇	C₂²	1	2	yes	—
10	#405×C₆₇	C₂²	1	1	yes	characteristic
11	#364×C₆₇	—	2	2	no	—
12	C₄⋊C₁₆×C₆₇	—	2	2	no	—
13	#338×C₆₇	C₄	1	1	yes	characteristic
14	H(3,ℤ/4)×C₆₇	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)×C₆₇	C₄	1	2	yes	—
16	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
19	#117×C₆₇	C₂×C₄	1	2	yes	—
20	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₆₇	—	2	2	no	—
22	M₃₂×C₆₇	—	2	2	no	—
23	M₃₂×C₆₇	—	2	2	no	—
24	C₂×C₁₆×C₆₇	—	2	1	no	—
25	C₂×C₁₆×C₆₇	—	2	1	no	—
26	C₂×C₁₆×C₆₇	—	2	2	no	—
27	C₂×C₁₆×C₆₇	—	2	2	no	—
28	#97×C₆₇	—	2	2	no	—
29	#97×C₆₇	Q₈	1	2	yes	—
30	C₄×C₈×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₆₇	—	2	2	no	—
34	M₁₆×C₆₇	—	2	2	no	—
35	M₁₆×C₆₇	—	2	2	no	—
36	C₂×C₈×C₆₇	—	2	2	no	—
37	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₆₇	#41	1	2	yes	—
41	C₂×C₈×C₆₇	#41	1	2	yes	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂²⋊C₄×C₆₇	—	2	1	no	—
45	C₂²⋊C₄×C₆₇	—	2	2	no	—
46	C₄²×C₆₇	—	2	2	no	—
47	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₁₆×C₆₇	—	4	1	no	—
49	C₁₆×C₆₇	—	4	1	no	—
50	C₁₆×C₆₇	—	4	2	no	—
51	C₂³×C₆₇	#131	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	4	2	no	—
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#117	1	2	yes	—
56	C₂×C₄×C₆₇	#131	1	2	yes	—
57	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
58	C₈×C₆₇	—	2	2	no	—
59	C₈×C₆₇	—	4	2	no	—
60	C₈×C₆₇	#124	1	1	yes	characteristic
61	C₈×C₆₇	#124	1	1	yes	characteristic
62	C₈×C₆₇	#125	1	2	yes	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	#424	1	1	yes	characteristic
65	C₂²×C₆₇	#433	1	2	yes	—
66	C₄×C₆₇	—	4	2	no	—
67	C₄×C₆₇	—	8	2	no	—
68	C₄×C₆₇	#382	1	1	yes	characteristic
69	C₄×C₆₇	#424	1	1	yes	characteristic
70	C₄×C₆₇	#433	1	2	yes	—
71	#12885	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₆₇	—	2	1	no	—
73	C₂×C₆₇	—	2	2	no	—
74	C₂×C₆₇	#1832	1	1	yes	characteristic
75	#2153	C₂×C₆₇	1	1	yes	characteristic
76	#2151	C₂×C₆₇	1	1	yes	characteristic
77	#1747	C₂×C₆₇	1	1	yes	characteristic
78	#1379	C₂×C₆₇	1	2	yes	—
79	#1378	C₂×C₆₇	1	2	yes	—
80	C₆₇	#12885	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
82	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
83	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
84	#405	C₂²×C₆₇	1	2	yes	—
85	#405	C₂²×C₆₇	1	1	yes	characteristic
86	#364	—	2	2	no	—
87	C₄⋊C₁₆	—	2	2	no	—
88	#338	C₄×C₆₇	1	1	yes	characteristic
89	H(3,ℤ/4)	C₄×C₆₇	1	1	yes	characteristic
90	H(3,ℤ/4)	C₄×C₆₇	1	2	yes	—
91	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
92	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
93	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
94	#117	C₂×C₄×C₆₇	1	2	yes	—
95	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	2	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	#97	—	2	2	no	—
104	#97	Q₈×C₆₇	1	2	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	D₄×C₆₇	1	2	yes	—
107	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
113	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
115	C₂×C₈	#41×C₆₇	1	2	yes	—
116	C₂×C₈	#41×C₆₇	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	4	2	no	—
126	C₂³	#131×C₆₇	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
130	C₂×C₄	#117×C₆₇	1	2	yes	—
131	C₂×C₄	#131×C₆₇	1	2	yes	—
132	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	#124×C₆₇	1	1	yes	characteristic
136	C₈	#124×C₆₇	1	1	yes	characteristic
137	C₈	#125×C₆₇	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#424×C₆₇	1	1	yes	characteristic
140	C₂²	#433×C₆₇	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#382×C₆₇	1	1	yes	characteristic
144	C₄	#424×C₆₇	1	1	yes	characteristic
145	C₄	#433×C₆₇	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	C₆₇×#1832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2153

C₂

yes

maximal, characteristic

C₆₇×#2151

C₂

yes

maximal, characteristic

C₆₇×#1747

C₂

yes

maximal, characteristic

C₆₇×#1379

C₂

yes

maximal

C₆₇×#1378

C₂

yes

maximal

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#405×C₆₇

C₂²

yes

—

#405×C₆₇

C₂²

yes

characteristic

#364×C₆₇

—

C₄⋊C₁₆×C₆₇

—

#338×C₆₇

C₄

yes

characteristic

H(3,ℤ/4)×C₆₇

C₄

yes

characteristic

H(3,ℤ/4)×C₆₇

C₄

yes

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

C₂×C₄

yes

—

#117×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

#97×C₆₇

—

#97×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#131

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

#424

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#12885

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1832

yes

characteristic

#2153

C₂×C₆₇

yes

characteristic

#2151

C₂×C₆₇

yes

characteristic

#1747

C₂×C₆₇

yes

characteristic

#1379

C₂×C₆₇

yes

—

#1378

C₂×C₆₇

yes

—

C₆₇

#12885

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₆₇

yes

characteristic

C₂×M₃₂

C₂²×C₆₇

yes

—

C₂²×C₁₆

C₂²×C₆₇

yes

characteristic

#405

C₂²×C₆₇

yes

—

#405

C₂²×C₆₇

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

C₄×C₆₇

yes

characteristic

H(3,ℤ/4)

C₄×C₆₇

yes

characteristic

H(3,ℤ/4)

C₄×C₆₇

yes

—

C₂×M₁₆

C₂×C₄×C₆₇

yes

—

C₂×M₁₆

C₂×C₄×C₆₇

yes

characteristic

C₂²×C₈

C₂³×C₆₇

yes

characteristic

#117

C₂×C₄×C₆₇

yes

—

#117

C₂×C₄×C₆₇

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#97

—

104

#97

Q₈×C₆₇

yes

—

105

C₄×C₈

—

106

C₄×C₈

D₄×C₆₇

yes

—

107

C₂²×C₄

C₂²×C₄×C₆₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊C₄×C₆₇

yes

—

113

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

114

C₂×C₈

C₂×Q₈×C₆₇

yes

characteristic

115

C₂×C₈

#41×C₆₇

yes

—

116

C₂×C₈

#41×C₆₇

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₄⋊C₄×C₆₇

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

#131×C₆₇

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊C₄×C₆₇

yes

characteristic

130

C₂×C₄

#117×C₆₇

yes

—

131

C₂×C₄

#131×C₆₇

yes

—

132

C₂×C₄

#131×C₆₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

#124×C₆₇

yes

characteristic

136

C₈

#124×C₆₇

yes

characteristic

137

C₈

#125×C₆₇

yes

—

138

C₂²

—

139

C₂²

#424×C₆₇

yes

characteristic

140

C₂²

#433×C₆₇

yes

—

141

C₄

—

142

C₄

—

143

C₄

#382×C₆₇

yes

characteristic

144

C₄

#424×C₆₇

yes

characteristic

145

C₄

#433×C₆₇

yes

—

146

C₂

—

147

C₂

—

148

C₂

C₆₇×#1832

yes

characteristic

	subgroup	G/H
center	C₈×C₆₇	#124
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
socle	C₂×C₆₇	#1832
prime radical	C₂³×C₆₇	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ G

subgroup

G/H

center

C₈×C₆₇

#124

commutator

C₂²×C₄

C₂²×C₄×C₆₇

Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

C₂×C₆₇

C₂³×C₆₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
71. #12885	C₆₇	direct (N × H)
80. C₆₇	#12885	direct (N × H)

H ≅ G/N

type

71. #12885

C₆₇

direct (N × H)

80. C₆₇

#12885

direct (N × H)

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

subgroup	index
C₆₇×#2153	2
C₆₇×#2151	2
C₆₇×#1747	2
C₆₇×#1379	2
C₆₇×#1378	2
#12885	67

subgroup

index

C₆₇×#2153

C₆₇×#2151

C₆₇×#1747

C₆₇×#1379

C₆₇×#1378

#12885

Subgroups — C₆₇×#12885 #30692136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#12885 #30692136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#12885 #30692136