Group #22561539 — subgroups

Subgroups — #22561539

Order 8576.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14523974	C₂	1	2	yes	maximal
2	#16863294	C₂	1	1	yes	maximal, characteristic
3	#16863158	C₂	1	2	yes	maximal
4	#16863151	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₇⋊C₄	—	2	1	no	—
10	#14523974	—	2	4	no	—
11	C₆₇⋊#95	—	2	1	no	—
12	C₂²×Q₈×C₆₇	—	2	1	no	—
13	C₄⋊Q₈×C₆₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₆₇	—	2	1	no	—
18	C₂²×Dic₆₇	—	2	1	no	—
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₂×C₆₇⋊C₈	—	2	2	no	—
21	C₂×C₆₇⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Dic₆₇	—	2	1	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₄⋊Dic₆₇	Q₈	1	2	yes	complemented
25	Dic₆₇⋊C₄	—	4	2	no	—
26	C₂×Q₈×C₆₇	—	2	4	no	—
27	C₂×Q₈×C₆₇	—	4	2	no	—
28	C₂×Q₈×C₆₇	—	4	2	no	—
29	C₂²×C₄×C₆₇	—	2	2	no	—
30	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₄⋊C₄×C₆₇	D₄	1	4	yes	—
36	C₄²×C₆₇	—	2	2	no	—
37	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₆₇	—	4	8	no	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	2	1	no	—
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	4	no	—
44	C₂×C₄×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
47	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×Dic₆₇	—	2	2	no	—
53	C₂×Dic₆₇	—	4	1	no	—
54	C₂×Dic₆₇	—	4	1	no	—
55	C₂×Dic₆₇	—	4	1	no	—
56	C₂×Dic₆₇	—	4	1	no	—
57	C₂×Dic₆₇	—	4	1	no	—
58	C₆₇⋊C₈	—	2	4	no	—
59	C₂²×C₆₇	#120	1	1	yes	characteristic
60	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₆₇	#137	1	1	yes	characteristic
66	C₄×C₆₇	—	2	2	no	—
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	4	4	no	—
69	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
71	Dic₆₇	—	4	2	no	—
72	Dic₆₇	—	8	1	no	—
73	C₂×C₆₇	#395	1	1	yes	characteristic
74	C₂×C₆₇	#448	1	1	yes	characteristic
75	C₂×C₆₇	#450	1	1	yes	characteristic
76	C₂×C₆₇	#475	1	1	yes	characteristic
77	C₂×C₆₇	#476	1	1	yes	characteristic
78	C₂×C₆₇	#498	1	1	yes	characteristic
79	C₂×C₆₇	#501	1	1	yes	characteristic
80	#2058	—	67	1	no	maximal, Sylow
81	C₆₇	#2058	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
83	C₂×#106	—	67	2	no	—
84	#389	—	67	1	no	—
85	#340	—	67	2	no	—
86	#336	—	67	1	no	—
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	67	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	67	1	no	—
92	C₂×C₄⋊C₄	—	67	1	no	—
93	C₂×C₄⋊C₄	—	134	1	no	—
94	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
95	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
96	#106	—	134	4	no	—
97	#95	—	134	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	134	1	no	—
103	C₂²×C₄	—	134	1	no	—
104	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
107	C₂×C₈	—	67	4	no	—
108	C₂×C₈	—	134	2	no	—
109	C₂×C₈	—	134	2	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	268	2	no	—
113	C₄⋊C₄	—	67	2	no	—
114	C₄⋊C₄	—	134	2	no	—
115	C₄⋊C₄	—	134	1	no	—
116	C₄⋊C₄	C₂²⋊D₆₇	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	268	1	no	—
128	C₂×C₄	—	268	1	no	—
129	C₂×C₄	—	268	1	no	—
130	C₂×C₄	—	268	1	no	—
131	C₂×C₄	—	268	1	no	—
132	C₂×C₄	—	134	2	no	—
133	C₂×C₄	#11761615	1	2	yes	—
134	C₂×C₄	#11761616	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
136	C₂×C₄	Q₈×D₆₇	1	2	yes	—
137	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
138	C₂×C₄	#11761629	1	1	yes	characteristic
139	C₈	—	134	4	no	—
140	C₂²	#14524012	1	1	yes	characteristic
141	C₂²	#14524019	1	1	yes	characteristic
142	C₂²	#14524041	1	1	yes	characteristic
143	C₂²	C₂×#11761615	1	1	yes	characteristic
144	C₂²	C₂×#11761616	1	1	yes	characteristic
145	C₂²	#14524069	1	1	yes	characteristic
146	C₂²	#14524074	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	268	2	no	—
151	C₄	—	536	1	no	—
152	C₄	#14524019	1	2	yes	—
153	C₄	#14524041	1	2	yes	—
154	C₂	#16863315	1	1	yes	characteristic
155	C₂	#16863535	1	1	yes	characteristic
156	C₂	#16863547	1	1	yes	characteristic
157	C₂	#16863678	1	1	yes	characteristic
158	C₂	#16863683	1	1	yes	characteristic
159	C₂	#16863792	1	1	yes	characteristic
160	C₂	#16863802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523974

C₂

yes

maximal

#16863294

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal

#16863151

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

#14523974

—

C₆₇⋊#95

—

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

complemented

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

Q₈

yes

complemented

Dic₆₇⋊C₄

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₆₇

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#106

—

134

—

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

134

—

103

C₂²×C₄

—

134

—

104

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

105

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

106

C₂²×C₄

C₂²×D₆₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

134

—

109

C₂×C₈

—

134

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

268

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

134

—

115

C₄⋊C₄

—

134

—

116

C₄⋊C₄

C₂²⋊D₆₇

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

268

—

128

C₂×C₄

—

268

—

129

C₂×C₄

—

268

—

130

C₂×C₄

—

268

—

131

C₂×C₄

—

268

—

132

C₂×C₄

—

134

—

133

C₂×C₄

#11761615

yes

—

134

C₂×C₄

#11761616

yes

—

135

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

136

C₂×C₄

Q₈×D₆₇

yes

—

137

C₂×C₄

Q₈×D₆₇

yes

characteristic

138

C₂×C₄

#11761629

yes

characteristic

139

C₈

—

134

—

140

C₂²

#14524012

yes

characteristic

141

C₂²

#14524019

yes

characteristic

142

C₂²

#14524041

yes

characteristic

143

C₂²

C₂×#11761615

yes

characteristic

144

C₂²

C₂×#11761616

yes

characteristic

145

C₂²

#14524069

yes

characteristic

146

C₂²

#14524074

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

268

—

151

C₄

—

536

—

152

C₄

#14524019

yes

—

153

C₄

#14524041

yes

—

154

C₂

#16863315

yes

characteristic

155

C₂

#16863535

yes

characteristic

156

C₂

#16863547

yes

characteristic

157

C₂

#16863678

yes

characteristic

158

C₂

#16863683

yes

characteristic

159

C₂

#16863792

yes

characteristic

160

C₂

#16863802

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#2058
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₇

hypocenter

C₆₇

#2058

LCS

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. C₂×C₆₇⋊C₈	Q₈	semidirect
24. C₄⋊Dic₆₇	Q₈	semidirect
81. C₆₇	#2058	semidirect

H ≅ G/N

type

21. C₂×C₆₇⋊C₈

Q₈

semidirect

24. C₄⋊Dic₆₇

Q₈

semidirect

81. C₆₇

#2058

semidirect

Subgroups — #22561539

Conjugacy classes of subgroups

Complements