Group #30749644 — subgroups

Subgroups — #30749644

Order 17152.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22560380	C₂	1	2	yes	maximal
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558967	C₂	1	2	yes	maximal
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523978	—	2	4	no	—
7	#16863420	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863149	—	2	2	no	—
11	#16863148	—	2	4	no	—
12	#16863118	C₄	1	4	yes	—
13	#423×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#335×C₆₇	—	2	2	no	—
16	C₂×#11761612	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
18	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
19	#14523978	—	4	8	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
25	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₇	D₄	1	2	yes	—
33	#11761612	—	4	4	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
39	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₆₇	—	4	4	no	—
44	C₂×C₈×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	—	4	2	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
62	C₈×C₆₇	—	2	4	no	—
63	C₈×C₆₇	—	8	2	no	—
64	C₆₇⋊C₈	—	2	4	no	—
65	C₆₇⋊C₈	—	8	2	no	—
66	C₆₇⋊C₈	#107	1	8	yes	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	—	2	1	no	—
69	C₂²×C₆₇	#384	1	1	yes	characteristic
70	C₂²×C₆₇	#500	1	2	yes	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	C₂×#107	1	2	yes	—
74	C₄×C₆₇	#430	1	2	yes	—
75	C₄×C₆₇	#430	1	2	yes	—
76	C₄×C₆₇	#493	1	1	yes	characteristic
77	C₄×C₆₇	#494	1	1	yes	characteristic
78	#8826	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	—	2	2	no	—
80	C₂×C₆₇	#1845	1	1	yes	characteristic
81	C₂×C₆₇	#1849	1	1	yes	characteristic
82	C₂×C₆₇	#1849	1	1	yes	characteristic
83	#1848	—	67	2	no	—
84	#1567	—	67	2	no	—
85	#1567	—	67	1	no	—
86	#1567	D₆₇	1	1	yes	characteristic
87	C₂×#322	—	67	1	no	—
88	C₆₇	#8826	1	1	yes	characteristic, complemented, Sylow
89	C₂×#108	—	134	4	no	—
90	#423	—	67	2	no	—
91	#423	C₂×D₆₇	1	2	yes	—
92	C₂×C₄×C₈	—	67	1	no	—
93	C₂×C₄×C₈	—	67	1	no	—
94	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	#335	—	2	2	no	—
96	#335	—	134	4	no	—
97	#335	—	134	2	no	—
98	#322	—	67	4	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	134	2	no	—
101	C₂²×C₈	—	67	2	no	—
102	C₂²×C₈	—	67	2	no	—
103	C₂²×C₈	Dic₁₃₄	1	2	yes	—
104	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
105	#108	—	268	8	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	134	2	no	—
109	C₄⋊C₈	—	134	2	no	—
110	C₄×C₈	—	67	2	no	—
111	C₄×C₈	—	67	2	no	—
112	C₄×C₈	—	67	2	no	—
113	C₄×C₈	—	67	2	no	—
114	C₄×C₈	C₄×D₆₇	1	2	yes	—
115	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
116	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
117	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
118	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
119	M₁₆	—	4	4	no	—
120	M₁₆	—	268	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	268	2	no	—
124	C₂×C₈	—	268	2	no	—
125	C₂×C₈	—	67	4	no	—
126	C₂×C₈	—	67	4	no	—
127	C₂×C₈	—	67	4	no	—
128	C₂×C₈	—	67	4	no	—
129	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
130	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
131	C₄²	D₄×D₆₇	1	1	yes	characteristic
132	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₄²	Q₈×D₆₇	1	1	yes	characteristic
134	C₂³	#14524042	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
140	C₂×C₄	#14524039	1	1	yes	characteristic
141	C₂×C₄	#14524041	1	1	yes	characteristic
142	C₂×C₄	#14524042	1	1	yes	characteristic
143	C₂×C₄	D₈×D₆₇	1	2	yes	—
144	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
145	C₈	—	2	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	536	2	no	—
148	C₈	—	67	8	no	—
149	C₈	—	134	4	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16863265	1	1	yes	characteristic
153	C₂²	#16863800	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#107×D₆₇	1	2	yes	—
157	C₄	#16863443	1	2	yes	—
158	C₄	#16863444	1	2	yes	—
159	C₄	#16863771	1	1	yes	characteristic
160	C₄	#16863774	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#22560367	1	1	yes	characteristic
163	C₂	#22560382	1	1	yes	characteristic
164	C₂	#22560383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560380

C₂

yes

maximal

#22558968

C₂

yes

maximal, characteristic

#22558967

C₂

yes

maximal

C₂×#16863118

C₂

yes

maximal, characteristic

C₆₇×#1567

C₂

yes

maximal, characteristic

C₂×#14523978

—

#16863420

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

C₄

yes

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#335×C₆₇

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂²×C₆₇⋊C₈

Q₈

yes

—

#14523978

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₂×C₆₇⋊C₈

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#107

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#500

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#107

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

D₆₇

yes

characteristic

C₂×#322

—

C₆₇

#8826

yes

characteristic, complemented, Sylow

C₂×#108

—

134

—

#423

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#335

—

#335

—

134

—

#335

—

134

—

#322

—

C₂×M₁₆

—

100

C₂×M₁₆

—

134

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₁₃₄

yes

—

104

C₂×C₄²

C₂²×D₆₇

yes

characteristic

105

#108

—

268

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

134

—

109

C₄⋊C₈

—

134

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₄×D₆₇

yes

—

115

C₄×C₈

C₂²⋊D₆₇

yes

—

116

C₂²×C₄

D₄×D₆₇

yes

characteristic

117

C₂²×C₄

Q₈×D₆₇

yes

characteristic

118

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

119

M₁₆

—

120

M₁₆

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

268

—

124

C₂×C₈

—

268

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

Dic₆₇⋊C₄

yes

—

130

C₄²

C₂×C₄×D₆₇

yes

characteristic

131

C₄²

D₄×D₆₇

yes

characteristic

132

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

133

C₄²

Q₈×D₆₇

yes

characteristic

134

C₂³

#14524042

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

138

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

139

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

140

C₂×C₄

#14524039

yes

characteristic

141

C₂×C₄

#14524041

yes

characteristic

142

C₂×C₄

#14524042

yes

characteristic

143

C₂×C₄

D₈×D₆₇

yes

—

144

C₂×C₄

Q₁₆×D₆₇

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

536

—

148

C₈

—

149

C₈

—

134

—

150

C₂²

—

151

C₂²

—

152

C₂²

#16863265

yes

characteristic

153

C₂²

#16863800

yes

—

154

C₄

—

155

C₄

—

156

C₄

#107×D₆₇

yes

—

157

C₄

#16863443

yes

—

158

C₄

#16863444

yes

—

159

C₄

#16863771

yes

characteristic

160

C₄

#16863774

yes

characteristic

161

C₂

—

162

C₂

#22560367

yes

characteristic

163

C₂

#22560382

yes

characteristic

164

C₂

#22560383

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#8826
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1567

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
88. C₆₇	#8826	semidirect

H ≅ G/N

type

88. C₆₇

#8826

semidirect

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

subgroup	index
#22560380	2
#22558968	2
#22558967	2
C₂×#16863118	2
C₆₇×#1567	2
#8826	67

subgroup

index

C₆₇×#1567

Subgroups — #30749644

Conjugacy classes of subgroups

Complements

Maximal subgroups