Group #30749648 — subgroups

Subgroups — #30749648

Order 17152.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22560346	C₂	1	2	yes	maximal
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558964	C₂	1	2	yes	maximal
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863160	—	2	4	no	—
10	#16863149	—	2	2	no	—
11	#16863146	—	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	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	#11761612	—	4	4	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₆₇	—	4	4	no	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#135	1	2	yes	—
60	C₂×C₄×C₆₇	#135	1	2	yes	—
61	C₈×C₆₇	—	2	4	no	—
62	C₈×C₆₇	—	8	2	no	—
63	C₆₇⋊C₈	—	2	4	no	—
64	C₆₇⋊C₈	—	2	4	no	—
65	C₆₇⋊C₈	—	8	2	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	#384	1	1	yes	characteristic
69	C₂²×C₆₇	#499	1	2	yes	—
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	#427	1	2	yes	—
73	C₄×C₆₇	#430	1	2	yes	—
74	C₄×C₆₇	#430	1	2	yes	—
75	C₄×C₆₇	#495	1	1	yes	characteristic
76	C₄×C₆₇	#495	1	1	yes	characteristic
77	#8827	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	—	2	2	no	—
79	C₂×C₆₇	#1846	1	1	yes	characteristic
80	C₂×C₆₇	#1849	1	1	yes	characteristic
81	C₂×C₆₇	#1849	1	1	yes	characteristic
82	#1837	—	67	2	no	—
83	#1567	—	67	1	no	—
84	#1567	D₆₇	1	1	yes	characteristic
85	#1566	—	67	2	no	—
86	C₂×#322	—	67	1	no	—
87	C₆₇	#8827	1	1	yes	characteristic, complemented, Sylow
88	#423	—	67	2	no	—
89	#423	C₂×D₆₇	1	2	yes	—
90	C₂×C₄×C₈	—	67	1	no	—
91	C₂×C₄×C₈	—	67	1	no	—
92	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	#341	—	134	4	no	—
94	#335	—	2	2	no	—
95	#335	—	134	2	no	—
96	#334	—	134	4	no	—
97	#322	—	67	4	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	134	2	no	—
100	C₂²×C₈	—	67	2	no	—
101	C₂²×C₈	—	67	2	no	—
102	C₂²×C₈	Dic₁₃₄	1	2	yes	—
103	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	134	2	no	—
107	C₄⋊C₈	—	134	2	no	—
108	C₄×C₈	—	67	2	no	—
109	C₄×C₈	—	67	2	no	—
110	C₄×C₈	—	67	2	no	—
111	C₄×C₈	—	67	2	no	—
112	C₄×C₈	C₄×D₆₇	1	2	yes	—
113	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
114	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	M₁₆	—	268	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	268	2	no	—
122	C₂×C₈	—	268	2	no	—
123	C₂×C₈	—	67	4	no	—
124	C₂×C₈	—	67	4	no	—
125	C₂×C₈	—	134	2	no	—
126	C₂×C₈	—	134	2	no	—
127	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
128	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
129	C₄²	D₄×D₆₇	1	1	yes	characteristic
130	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	C₄²	Q₈×D₆₇	1	1	yes	characteristic
132	C₂³	#14524042	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
136	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
138	C₂×C₄	#14524039	1	1	yes	characteristic
139	C₂×C₄	#14524041	1	1	yes	characteristic
140	C₂×C₄	#14524042	1	1	yes	characteristic
141	C₂×C₄	#14524062	1	2	yes	—
142	C₂×C₄	#14524066	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	536	2	no	—
146	C₈	—	134	4	no	—
147	C₈	—	134	4	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#16863265	1	1	yes	characteristic
151	C₂²	#16863797	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#16863433	1	2	yes	—
155	C₄	#16863443	1	2	yes	—
156	C₄	#16863444	1	2	yes	—
157	C₄	#16863779	1	1	yes	characteristic
158	C₄	#16863781	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#22560372	1	1	yes	characteristic
161	C₂	#22560382	1	1	yes	characteristic
162	C₂	#22560383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560346

C₂

yes

maximal

#22558968

C₂

yes

maximal, characteristic

#22558964

C₂

yes

maximal

C₂×#16863118

C₂

yes

maximal, characteristic

C₆₇×#1567

C₂

yes

maximal, characteristic

#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

—

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

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

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

#135

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#499

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#427

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

#1566

—

C₂×#322

—

C₆₇

#8827

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#341

—

134

—

#335

—

#335

—

134

—

#334

—

134

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₁₃₄

yes

—

103

C₂×C₄²

C₂²×D₆₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

134

—

107

C₄⋊C₈

—

134

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₄×D₆₇

yes

—

113

C₄×C₈

C₂²⋊D₆₇

yes

—

114

C₂²×C₄

D₄×D₆₇

yes

characteristic

115

C₂²×C₄

Q₈×D₆₇

yes

characteristic

116

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

117

M₁₆

—

118

M₁₆

—

268

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

268

—

122

C₂×C₈

—

268

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

134

—

126

C₂×C₈

—

134

—

127

C₂×C₈

Dic₆₇⋊C₄

yes

—

128

C₄²

C₂×C₄×D₆₇

yes

characteristic

129

C₄²

D₄×D₆₇

yes

characteristic

130

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

131

C₄²

Q₈×D₆₇

yes

characteristic

132

C₂³

#14524042

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

136

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

137

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

138

C₂×C₄

#14524039

yes

characteristic

139

C₂×C₄

#14524041

yes

characteristic

140

C₂×C₄

#14524042

yes

characteristic

141

C₂×C₄

#14524062

yes

—

142

C₂×C₄

#14524066

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

536

—

146

C₈

—

134

—

147

C₈

—

134

—

148

C₂²

—

149

C₂²

—

150

C₂²

#16863265

yes

characteristic

151

C₂²

#16863797

yes

—

152

C₄

—

153

C₄

—

154

C₄

#16863433

yes

—

155

C₄

#16863443

yes

—

156

C₄

#16863444

yes

—

157

C₄

#16863779

yes

characteristic

158

C₄

#16863781

yes

characteristic

159

C₂

—

160

C₂

#22560372

yes

characteristic

161

C₂

#22560382

yes

characteristic

162

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	#8827
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
87. C₆₇	#8827	semidirect

H ≅ G/N

type

87. C₆₇

#8827

semidirect

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

subgroup	index
#22560346	2
#22558968	2
#22558964	2
C₂×#16863118	2
C₆₇×#1567	2
#8827	67

subgroup

index

C₆₇×#1567

Subgroups — #30749648

Conjugacy classes of subgroups

Complements

Maximal subgroups