Group #30767465 — subgroups

Subgroups — #30767465

Order 17152.

232 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22561802	C₂	1	1	yes	maximal, characteristic
2	#22558963	C₂	1	2	yes	maximal
3	#22558275	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558274	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523982	C₄	1	1	yes	characteristic
7	C₂×#14523982	C₄	1	1	yes	characteristic
8	#16863418	—	2	2	no	—
9	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
10	#16863172	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₆₇	C₄	1	1	yes	characteristic
12	C₁₆×Dic₆₇	C₄	1	1	yes	characteristic
13	#16863147	C₂²	1	2	yes	—
14	#16863146	—	2	2	no	—
15	#346×C₆₇	C₂²	1	1	yes	characteristic
16	#334×C₆₇	C₂²	1	2	yes	—
17	C₂×#11761611	—	4	2	no	—
18	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
20	C₄²×D₆₇	D₄	1	1	yes	characteristic
21	#14523982	—	2	2	no	—
22	#14523982	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	Dic₆₇⋊C₈	—	4	2	no	—
27	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₆₇	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₆₇	SD₁₆	1	2	yes	—
36	#11761611	—	8	4	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₈×D₆₇	—	2	1	no	—
41	C₈×D₆₇	—	2	1	no	—
42	C₈×D₆₇	—	2	2	no	—
43	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₆₇	SD₁₆	1	2	yes	—
45	C₆₇⋊C₁₆	—	2	1	no	—
46	C₆₇⋊C₁₆	M₁₆	1	2	yes	complemented
47	C₂×C₈×C₆₇	—	4	2	no	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₁₆×C₆₇	—	2	1	no	—
52	C₁₆×C₆₇	M₁₆	1	2	yes	complemented
53	C₂²×D₆₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
56	C₂×Dic₆₇	#106	1	1	yes	characteristic
57	C₂×Dic₆₇	#106	1	2	yes	—
58	C₄×D₆₇	—	2	1	no	—
59	C₄×D₆₇	—	2	1	no	—
60	C₄×D₆₇	—	2	4	no	—
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	—	8	2	no	—
63	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
65	C₆₇⋊C₈	—	2	1	no	—
66	C₆₇⋊C₈	—	8	2	no	—
67	C₆₇⋊C₈	#108	1	2	yes	—
68	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
69	C₂×D₆₇	—	2	1	no	—
70	C₂×D₆₇	—	2	2	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	#423	1	1	yes	characteristic
73	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
74	Dic₆₇	—	2	1	no	—
75	Dic₆₇	#365	1	4	yes	—
76	#12020	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	#1566	1	1	yes	characteristic
78	C₂×C₆₇	C₂×#365	1	2	yes	—
79	D₆₇	—	4	1	no	—
80	C₄×M₃₂	—	67	1	no	—
81	#1566	—	67	2	no	—
82	#1368	—	67	1	no	—
83	#1368	—	67	2	no	—
84	#1368	D₆₇	1	1	yes	characteristic, complemented
85	C₆₇	#12020	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	67	1	no	—
87	C₂×M₃₂	—	67	1	no	—
88	#423	—	134	2	no	—
89	C₂×C₄×C₈	—	67	1	no	—
90	#346	—	67	1	no	—
91	#346	C₂×D₆₇	1	1	yes	characteristic
92	C₄×C₁₆	—	67	1	no	—
93	C₄×C₁₆	—	67	1	no	—
94	#334	—	67	2	no	—
95	#334	—	134	2	no	—
96	#334	C₂×D₆₇	1	2	yes	—
97	C₂×M₁₆	—	268	2	no	—
98	C₂²×C₈	—	67	1	no	—
99	C₂²×C₈	—	67	1	no	—
100	C₂×C₄²	—	67	1	no	—
101	M₃₂	—	134	2	no	—
102	M₃₂	—	134	2	no	—
103	C₂×C₁₆	—	67	1	no	—
104	C₂×C₁₆	—	67	1	no	—
105	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
106	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	268	2	no	—
109	C₄⋊C₈	—	134	2	no	—
110	C₄×C₈	—	67	1	no	—
111	C₄×C₈	—	67	1	no	—
112	C₄×C₈	—	67	1	no	—
113	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂²×C₄	—	67	1	no	—
115	C₂²×C₄	—	67	2	no	—
116	M₁₆	—	536	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	—	67	1	no	—
120	C₂×C₈	—	67	1	no	—
121	C₂×C₈	—	134	2	no	—
122	C₂×C₈	—	134	1	no	—
123	C₂×C₈	—	134	1	no	—
124	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
126	C₄²	—	67	1	no	—
127	C₄²	—	67	2	no	—
128	C₄²	D₄×D₆₇	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	67	2	no	—
131	C₁₆	—	134	1	no	—
132	C₁₆	#11761611	1	2	yes	complemented
133	C₂³	—	67	1	no	—
134	C₂×C₄	—	67	2	no	—
135	C₂×C₄	—	67	1	no	—
136	C₂×C₄	—	134	4	no	—
137	C₂×C₄	—	134	1	no	—
138	C₂×C₄	—	134	1	no	—
139	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
140	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	536	2	no	—
144	C₈	—	67	2	no	—
145	C₈	—	134	1	no	—
146	C₈	M₁₆×D₆₇	1	1	yes	characteristic
147	C₈	C₂×#11761611	1	1	yes	characteristic
148	C₂²	—	134	2	no	—
149	C₂²	—	134	1	no	—
150	C₂²	#106×D₆₇	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	67	4	no	—
153	C₄	—	134	1	no	—
154	C₄	#16863418	1	1	yes	characteristic
155	C₄	#108×D₆₇	1	1	yes	characteristic
156	C₂	—	268	1	no	—
157	C₂	#22558963	1	1	yes	characteristic
158	C₂	#365×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561802

C₂

yes

maximal, characteristic

#22558963

C₂

yes

maximal

#22558275

C₂

yes

maximal, characteristic, complemented

#22558274

C₂

yes

maximal, complemented

C₆₇×#1368

C₂

yes

maximal, characteristic, complemented

C₂×#14523982

C₄

yes

characteristic

C₂×#14523982

C₄

yes

characteristic

#16863418

—

C₄×C₈×D₆₇

C₂²

yes

characteristic

#16863172

C₂²

yes

characteristic

C₁₆×Dic₆₇

C₄

yes

characteristic

C₁₆×Dic₆₇

C₄

yes

characteristic

#16863147

C₂²

yes

—

#16863146

—

#346×C₆₇

C₂²

yes

characteristic

#334×C₆₇

C₂²

yes

—

C₂×#11761611

—

C₂×C₈×D₆₇

C₂×C₄

yes

characteristic

C₂×C₈×D₆₇

Q₈

yes

characteristic

C₄²×D₆₇

D₄

yes

characteristic

#14523982

—

#14523982

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₈×Dic₆₇

C₂×C₄

yes

characteristic

C₈×Dic₆₇

Q₈

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄×D₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₆₇

SD₁₆

yes

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

—

C₈×D₆₇

—

C₄×Dic₆₇

C₄⋊C₄

yes

characteristic

C₄×Dic₆₇

SD₁₆

yes

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

M₁₆

yes

complemented

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

M₁₆

yes

complemented

C₂²×D₆₇

#106

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×Dic₆₇

#106

yes

characteristic

C₂×Dic₆₇

#106

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×M₁₆

yes

characteristic

C₈×C₆₇

C₂×M₁₆

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#108

yes

—

C₂²×C₆₇

C₂×#106

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

#365

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

—

C₄×M₃₂

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#12020

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

#423

—

134

—

C₂×C₄×C₈

—

#346

—

#346

C₂×D₆₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#334

—

#334

—

134

—

#334

C₂×D₆₇

yes

—

C₂×M₁₆

—

268

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

M₃₂

—

134

—

102

M₃₂

—

134

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

C₄×D₆₇

yes

characteristic

106

C₂×C₁₆

C₄×D₆₇

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

268

—

109

C₄⋊C₈

—

134

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂²×D₆₇

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

M₁₆

—

536

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

134

—

122

C₂×C₈

—

134

—

123

C₂×C₈

—

134

—

124

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

125

C₂×C₈

Q₈×D₆₇

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

D₄×D₆₇

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

134

—

132

C₁₆

#11761611

yes

complemented

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

134

—

137

C₂×C₄

—

134

—

138

C₂×C₄

—

134

—

139

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

140

C₂×C₄

SD₁₆×D₆₇

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

536

—

144

C₈

—

145

C₈

—

134

—

146

C₈

M₁₆×D₆₇

yes

characteristic

147

C₈

C₂×#11761611

yes

characteristic

148

C₂²

—

134

—

149

C₂²

—

134

—

150

C₂²

#106×D₆₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

134

—

154

C₄

#16863418

yes

characteristic

155

C₄

#108×D₆₇

yes

characteristic

156

C₂

—

268

—

157

C₂

#22558963

yes

characteristic

158

C₂

#365×D₆₇

yes

—

	subgroup	G/H
center	C₂²	#106×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1368	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂²×D₆₇	#106
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1368	D₆₇
hypocenter	C₆₇	#12020
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₆₇

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1368

C₂

C₂²×C₆₇

C₂²×D₆₇

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #22558275	C₂	semidirect
4. #22558274	C₂	semidirect
5. C₆₇×#1368	C₂	semidirect
46. C₆₇⋊C₁₆	M₁₆	semidirect
52. C₁₆×C₆₇	M₁₆	semidirect
84. #1368	D₆₇	semidirect
85. C₆₇	#12020	semidirect
132. C₁₆	#11761611	semidirect

H ≅ G/N

type

3. #22558275

C₂

semidirect

4. #22558274

C₂

semidirect

5. C₆₇×#1368

C₂

semidirect

46. C₆₇⋊C₁₆

M₁₆

semidirect

52. C₁₆×C₆₇

M₁₆

semidirect

84. #1368

D₆₇

semidirect

85. C₆₇

#12020

semidirect

132. C₁₆

#11761611

semidirect

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

subgroup	index
#22561802	2
#22558963	2
#22558275	2
#22558274	2
C₆₇×#1368	2
#12020	67

subgroup

index

C₆₇×#1368

Subgroups — #30767465

Conjugacy classes of subgroups

Complements

Maximal subgroups