Group #30691313 — subgroups

Subgroups — C₆₇×#12062 #30691313

Order 17152.

252 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 130 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	D₄×C₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1367	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×D₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₆×C₆₇	—	2	2	no	—
7	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
8	C₈×D₄×C₆₇	C₂²	1	2	yes	—
9	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂²⋊C₁₆×C₆₇	—	2	2	no	—
11	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	#335×C₆₇	C₄	1	1	yes	characteristic
14	D₄⋊C₈×C₆₇	C₄	1	2	yes	—
15	C₈²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₆₇	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	—	2	2	no	—
18	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₆₇	—	2	2	no	—
20	C₂×C₁₆×C₆₇	—	2	1	no	—
21	C₂×C₁₆×C₆₇	—	4	4	no	—
22	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
23	#107×C₆₇	C₈	1	1	yes	characteristic
24	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
25	#102×C₆₇	C₈	1	2	yes	—
26	C₂²⋊C₈×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₆₇	—	2	2	no	—
32	D₈×C₆₇	C₁₆	1	4	yes	complemented
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	—	4	2	no	—
36	C₂×C₈×C₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
42	C₂²⋊C₄×C₆₇	—	2	2	no	—
43	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₁₆×C₆₇	—	2	2	no	—
45	C₁₆×C₆₇	—	4	1	no	—
46	C₁₆×C₆₇	—	4	2	no	—
47	C₂³×C₆₇	—	2	2	no	—
48	D₄×C₆₇	—	2	2	no	—
49	D₄×C₆₇	C₂×C₁₆	1	4	yes	complemented
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	4	2	no	—
52	C₂×C₄×C₆₇	—	4	2	no	—
53	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
56	C₈×C₆₇	—	2	1	no	—
57	C₈×C₆₇	—	2	1	no	—
58	C₈×C₆₇	—	4	1	no	—
59	C₈×C₆₇	—	4	2	no	—
60	C₈×C₆₇	C₂×C₁₆	1	2	yes	complemented
61	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₂²×C₆₇	—	2	4	no	—
63	C₂²×C₆₇	—	4	2	no	—
64	C₂²×C₆₇	—	4	2	no	—
65	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	4	2	no	—
68	C₄×C₆₇	#442	1	1	yes	characteristic
69	C₄×C₆₇	#444	1	1	yes	characteristic
70	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
71	C₄×C₆₇	#504	1	1	yes	characteristic
72	#12062	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₆₇	—	4	4	no	—
74	C₂×C₆₇	#1580	1	1	yes	characteristic
75	C₂×C₆₇	D₄×C₁₆	1	1	yes	characteristic
76	C₂×C₆₇	#2171	1	1	yes	characteristic
77	D₄×C₁₆	C₂×C₆₇	1	2	yes	complemented
78	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic
79	#1367	C₂×C₆₇	1	1	yes	characteristic, complemented
80	D₄⋊C₁₆	C₂×C₆₇	1	2	yes	complemented
81	#1312	C₂×C₆₇	1	1	yes	characteristic, complemented
82	C₆₇	#12062	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₁₆	—	2	2	no	—
84	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic
85	C₈×D₄	C₂²×C₆₇	1	2	yes	—
86	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
87	C₂²⋊C₁₆	—	2	2	no	—
88	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
89	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
90	#335	C₄×C₆₇	1	1	yes	characteristic
91	D₄⋊C₈	C₄×C₆₇	1	2	yes	—
92	C₈²	C₂²×C₆₇	1	1	yes	characteristic
93	C₂×D₈	C₈×C₆₇	1	1	yes	characteristic
94	C₂²×C₈	—	2	2	no	—
95	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	4	4	no	—
99	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
100	#107	C₈×C₆₇	1	1	yes	characteristic
101	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
102	#102	C₈×C₆₇	1	2	yes	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
105	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
106	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
107	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	D₈	C₁₆×C₆₇	1	4	yes	complemented
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
115	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
116	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
117	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
118	C₄⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	—	4	1	no	—
123	C₁₆	—	4	2	no	—
124	C₂³	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	C₂×C₁₆×C₆₇	1	4	yes	complemented
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
132	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	4	2	no	—
137	C₈	C₂×C₁₆×C₆₇	1	2	yes	complemented
138	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
139	C₂²	—	2	4	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	C₈×D₄×C₆₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	#442×C₆₇	1	1	yes	characteristic
146	C₄	#444×C₆₇	1	1	yes	characteristic
147	C₄	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
148	C₄	#504×C₆₇	1	1	yes	characteristic
149	C₂	—	4	4	no	—
150	C₂	C₆₇×#1580	1	1	yes	characteristic
151	C₂	D₄×C₁₆×C₆₇	1	1	yes	characteristic
152	C₂	C₆₇×#2171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×C₁₆×C₆₇

C₂

yes

maximal, complemented

C₈×D₈×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#1367

C₂

yes

maximal, characteristic, complemented

C₆₇×D₄⋊C₁₆

C₂

yes

maximal, complemented

C₆₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×C₆₇

—

C₄×D₈×C₆₇

C₄

yes

characteristic

C₈×D₄×C₆₇

C₂²

yes

—

C₄⋊C₁₆×C₆₇

C₂²

yes

—

C₂²⋊C₁₆×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#335×C₆₇

C₄

yes

characteristic

D₄⋊C₈×C₆₇

C₄

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₈

yes

characteristic

C₂²×C₈×C₆₇

—

C₄×D₄×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

C₈

yes

characteristic

C₄⋊C₈×C₆₇

C₂×C₄

yes

—

#102×C₆₇

C₈

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×D₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

—

D₈×C₆₇

C₁₆

yes

complemented

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₈

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

C₂×C₁₆

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₁₆

yes

complemented

C₈×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂²×C₁₆

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1580

yes

characteristic

C₂×C₆₇

D₄×C₁₆

yes

characteristic

C₂×C₆₇

#2171

yes

characteristic

D₄×C₁₆

C₂×C₆₇

yes

complemented

C₈×D₈

C₂×C₆₇

yes

characteristic

#1367

C₂×C₆₇

yes

characteristic, complemented

D₄⋊C₁₆

C₂×C₆₇

yes

complemented

#1312

C₂×C₆₇

yes

characteristic, complemented

C₆₇

#12062

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

—

C₄×D₈

C₄×C₆₇

yes

characteristic

C₈×D₄

C₂²×C₆₇

yes

—

C₄⋊C₁₆

C₂²×C₆₇

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂²×C₆₇

yes

characteristic

C₄×C₁₆

C₂²×C₆₇

yes

characteristic

#335

C₄×C₆₇

yes

characteristic

D₄⋊C₈

C₄×C₆₇

yes

—

C₈²

C₂²×C₆₇

yes

characteristic

C₂×D₈

C₈×C₆₇

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂×C₄×C₆₇

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₆₇

yes

—

100

#107

C₈×C₆₇

yes

characteristic

101

C₄⋊C₈

C₂×C₄×C₆₇

yes

—

102

#102

C₈×C₆₇

yes

—

103

C₂²⋊C₈

—

104

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

105

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

106

C₄×C₈

C₂³×C₆₇

yes

characteristic

107

C₂×D₄

C₂×C₈×C₆₇

yes

—

108

C₂²×C₄

—

109

D₈

C₁₆×C₆₇

yes

complemented

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×C₈×C₆₇

yes

characteristic

115

C₂×C₈

C₂×C₈×C₆₇

yes

characteristic

116

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

117

C₂×C₈

#41×C₆₇

yes

characteristic

118

C₄⋊C₄

C₂×C₈×C₆₇

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×C₄×C₆₇

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

—

125

D₄

—

126

D₄

C₂×C₁₆×C₆₇

yes

complemented

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₄×D₄×C₆₇

yes

characteristic

131

C₂×C₄

C₂²×C₈×C₆₇

yes

characteristic

132

C₂×C₄

#131×C₆₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

C₂×C₁₆×C₆₇

yes

complemented

138

C₈

AGL(1,ℤ/8)×C₆₇

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₈×D₄×C₆₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#442×C₆₇

yes

characteristic

146

C₄

#444×C₆₇

yes

characteristic

147

C₄

C₂²×C₁₆×C₆₇

yes

characteristic

148

C₄

#504×C₆₇

yes

characteristic

149

C₂

—

150

C₂

C₆₇×#1580

yes

characteristic

151

C₂

D₄×C₁₆×C₆₇

yes

characteristic

152

C₂

C₆₇×#2171

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₆₇
Frattini, cosocle	C₄×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₈×D₄
prime radical	C₂×D₈×C₆₇	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₈×C₆₇

C₂×D₄

commutator

C₄

C₂²×C₁₆×C₆₇

Frattini, cosocle

C₄×C₈

C₂³×C₆₇

socle

C₂²×C₆₇

C₈×D₄

prime radical

C₂×D₈×C₆₇

C₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

N	H ≅ G/N	type
1. D₄×C₁₆×C₆₇	C₂	semidirect
3. C₆₇×#1367	C₂	semidirect
4. C₆₇×D₄⋊C₁₆	C₂	semidirect
5. C₆₇×#1312	C₂	semidirect
32. D₈×C₆₇	C₁₆	semidirect
49. D₄×C₆₇	C₂×C₁₆	semidirect
60. C₈×C₆₇	C₂×C₁₆	semidirect
72. #12062	C₆₇	direct (N × H)
77. D₄×C₁₆	C₂×C₆₇	semidirect
79. #1367	C₂×C₆₇	semidirect
80. D₄⋊C₁₆	C₂×C₆₇	semidirect
81. #1312	C₂×C₆₇	semidirect
82. C₆₇	#12062	direct (N × H)
109. D₈	C₁₆×C₆₇	semidirect
126. D₄	C₂×C₁₆×C₆₇	semidirect
137. C₈	C₂×C₁₆×C₆₇	semidirect

H ≅ G/N

type

1. D₄×C₁₆×C₆₇

C₂

semidirect

3. C₆₇×#1367

C₂

semidirect

4. C₆₇×D₄⋊C₁₆

C₂

semidirect

5. C₆₇×#1312

C₂

semidirect

32. D₈×C₆₇

C₁₆

semidirect

49. D₄×C₆₇

C₂×C₁₆

semidirect

60. C₈×C₆₇

C₂×C₁₆

semidirect

72. #12062

C₆₇

direct (N × H)

77. D₄×C₁₆

C₂×C₆₇

semidirect

79. #1367

C₂×C₆₇

semidirect

80. D₄⋊C₁₆

C₂×C₆₇

semidirect

81. #1312

C₂×C₆₇

semidirect

82. C₆₇

#12062

direct (N × H)

109. D₈

C₁₆×C₆₇

semidirect

126. D₄

C₂×C₁₆×C₆₇

semidirect

137. C₈

C₂×C₁₆×C₆₇

semidirect

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

subgroup	index
D₄×C₁₆×C₆₇	2
C₈×D₈×C₆₇	2
C₆₇×#1367	2
C₆₇×D₄⋊C₁₆	2
C₆₇×#1312	2
#12062	67

subgroup

index

D₄×C₁₆×C₆₇

C₈×D₈×C₆₇

C₆₇×#1367

C₆₇×D₄⋊C₁₆

C₆₇×#1312

#12062

Subgroups — C₆₇×#12062 #30691313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#12062 #30691313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#12062 #30691313