Group #30767462 — subgroups

Subgroups — #30767462

Order 17152.

226 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
2	#22558972	C₂	1	1	yes	maximal, characteristic
3	#22558274	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
6	#16863453	—	2	1	no	—
7	#16863172	C₄	1	1	yes	characteristic
8	#16863172	C₄	1	1	yes	characteristic
9	#16863172	C₂²	1	2	yes	—
10	#16863149	—	2	1	no	—
11	#16863147	C₂²	1	2	yes	—
12	#423×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#334×C₆₇	—	2	1	no	—
15	#334×C₆₇	C₂²	1	2	yes	—
16	C₂×C₆₇⋊C₁₆	—	2	4	no	—
17	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
21	#14523954	—	4	1	no	—
22	C₂×M₁₆×C₆₇	—	4	1	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₇	—	2	2	no	—
27	C₄⋊C₈×C₆₇	—	4	1	no	—
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₆₇⋊C₈	—	4	2	no	—
33	C₆₇⋊C₁₆	—	2	2	no	—
34	C₆₇⋊C₁₆	M₁₆	1	4	yes	complemented
35	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
38	M₁₆×C₆₇	—	8	2	no	—
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	—	4	2	no	—
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
48	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂³×C₆₇	#106	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	—	4	1	no	—
55	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	—	2	1	no	—
63	C₈×C₆₇	—	8	2	no	—
64	C₈×C₆₇	#108	1	2	yes	—
65	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
66	C₆₇⋊C₈	—	8	2	no	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	—	2	1	no	—
69	C₂²×C₆₇	#365	1	2	yes	—
70	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	—	2	1	no	—
74	C₄×C₆₇	—	2	1	no	—
75	C₄×C₆₇	—	2	1	no	—
76	C₄×C₆₇	#423	1	1	yes	characteristic
77	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
78	#12019	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	—	2	2	no	—
80	C₂×C₆₇	#1566	1	1	yes	characteristic
81	C₂×C₆₇	C₂×#365	1	1	yes	characteristic
82	C₂×C₆₇	#2155	1	1	yes	characteristic
83	C₂×#346	—	67	1	no	—
84	#1568	—	67	1	no	—
85	#1566	D₆₇	1	1	yes	characteristic
86	#1368	—	67	4	no	—
87	C₆₇	#12019	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	67	2	no	—
89	#432	—	134	1	no	—
90	#423	—	2	1	no	—
91	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
92	#346	—	67	1	no	—
93	#346	—	67	1	no	—
94	#346	—	67	2	no	—
95	#335	—	134	1	no	—
96	#334	—	2	1	no	—
97	#334	—	67	2	no	—
98	#334	C₂×D₆₇	1	2	yes	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
101	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
103	C₂×C₁₆	—	67	2	no	—
104	C₂×C₁₆	—	67	2	no	—
105	C₂×C₁₆	—	134	4	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	134	2	no	—
109	C₂²⋊C₈	—	268	1	no	—
110	#97	—	268	1	no	—
111	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
112	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	#11761615	1	1	yes	characteristic
117	C₂²×C₄	#11761616	1	1	yes	characteristic
118	M₁₆	—	8	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	268	2	no	—
123	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
126	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
127	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
128	C₄²	#11761615	1	1	yes	characteristic
129	C₄²	#11761616	1	1	yes	characteristic
130	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	C₁₆	—	67	4	no	—
132	C₁₆	—	134	2	no	—
133	C₂³	#14523974	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	#14523974	1	1	yes	characteristic
139	C₂×C₄	#14523974	1	1	yes	characteristic
140	C₂×C₄	#14523974	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
143	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	536	2	no	—
148	C₈	#14523978	1	2	yes	—
149	C₈	M₁₆×D₆₇	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16863218	1	2	yes	—
153	C₂²	C₂×#14523974	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#16863419	1	1	yes	characteristic
160	C₄	C₂×#14523978	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#22558964	1	1	yes	characteristic
163	C₂	C₂×#16863218	1	1	yes	characteristic
164	C₂	#22561983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863172

C₂

yes

maximal, characteristic

#22558972

C₂

yes

maximal, characteristic

#22558274

C₂

yes

maximal, complemented

C₆₇×#1566

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₄

yes

—

#16863453

—

#16863172

C₄

yes

characteristic

#16863172

C₄

yes

characteristic

#16863172

C₂²

yes

—

#16863149

—

#16863147

C₂²

yes

—

#423×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#334×C₆₇

—

#334×C₆₇

C₂²

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

M₁₆

yes

complemented

C₂²×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#106

yes

characteristic

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

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#108

yes

—

C₈×C₆₇

C₂×M₁₆

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#365

yes

—

C₂²×C₆₇

C₂×#106

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1368

—

C₆₇

#12019

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#432

—

134

—

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

—

—

—

134

—

#334

—

#334

—

#334

C₂×D₆₇

yes

—

C₂×M₁₆

—

100

C₂²×C₈

Dic₁₃₄

yes

characteristic

101

C₂²×C₈

C₄×D₆₇

yes

characteristic

102

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

134

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

134

—

109

C₂²⋊C₈

—

268

—

110

#97

—

268

—

111

C₄×C₈

Dic₁₃₄

yes

characteristic

112

C₄×C₈

C₄×D₆₇

yes

characteristic

113

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

114

C₄×C₈

C₂²×D₆₇

yes

characteristic

115

C₂²×C₄

Dic₆₇⋊C₄

yes

characteristic

116

C₂²×C₄

#11761615

yes

characteristic

117

C₂²×C₄

#11761616

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

268

—

123

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

124

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

125

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

126

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

127

C₄²

Dic₆₇⋊C₄

yes

characteristic

128

C₄²

#11761615

yes

characteristic

129

C₄²

#11761616

yes

characteristic

130

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

131

C₁₆

—

132

C₁₆

—

134

—

133

C₂³

#14523974

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14523974

yes

characteristic

139

C₂×C₄

#14523974

yes

characteristic

140

C₂×C₄

#14523974

yes

characteristic

141

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

142

C₂×C₄

C₂×#11761615

yes

characteristic

143

C₂×C₄

C₂×#11761616

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

536

—

148

C₈

#14523978

yes

—

149

C₈

M₁₆×D₆₇

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

#16863218

yes

—

153

C₂²

C₂×#14523974

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#16863419

yes

characteristic

160

C₄

C₂×#14523978

yes

characteristic

161

C₂

—

162

C₂

#22558964

yes

characteristic

163

C₂

C₂×#16863218

yes

characteristic

164

C₂

#22561983

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14523974
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×C₆₇	#106
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#12019
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

subgroup

G/H

center

C₂²

C₂×#14523974

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1566

C₂

C₂²×C₆₇

C₂³×C₆₇

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

N	H ≅ G/N	type
3. #22558274	C₂	semidirect
34. C₆₇⋊C₁₆	M₁₆	semidirect
87. C₆₇	#12019	semidirect

H ≅ G/N

type

3. #22558274

C₂

semidirect

34. C₆₇⋊C₁₆

M₁₆

semidirect

87. C₆₇

#12019

semidirect

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

subgroup	index
C₂×#16863172	2
#22558972	2
#22558274	2
C₆₇×#1566	2
#12019	67

subgroup

index

C₆₇×#1566

Subgroups — #30767462

Conjugacy classes of subgroups

Complements

Maximal subgroups