Group #22573199 — subgroups

Subgroups — #22573199

Order 8576.

360 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16864120	C₂	1	2	yes	maximal, complemented
2	#16863781	C₂	1	2	yes	maximal
3	#16863620	C₂	1	4	yes	maximal, complemented
4	#16863598	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
7	C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
8	#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11761615	C₂²	1	4	yes	—
10	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
11	#14524038	—	2	4	no	—
12	#14524032	C₂²	1	4	yes	—
13	#14524023	—	2	2	no	—
14	Q₈×Dic₆₇	C₂²	1	4	yes	—
15	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
16	C₄×D₂₆₈	C₂²	1	2	yes	—
17	#14524001	—	2	2	no	—
18	#14523977	C₂²	1	1	yes	characteristic
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₆₇⋊C₈	C₂²	1	2	yes	—
21	#14523967	C₂²	1	4	yes	—
22	#14523965	C₂²	1	2	yes	—
23	C₈×Dic₆₇	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
27	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
29	C₂×D₂₆₈	C₂³	1	2	yes	—
30	C₂×C₄×D₆₇	—	2	2	no	—
31	#11761615	—	4	4	no	—
32	D₅₃₆	—	2	2	no	—
33	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₆₇	—	2	4	no	—
35	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
36	C₄⋊Dic₆₇	C₂³	1	2	yes	—
37	Dic₆₇⋊C₄	D₄	1	2	yes	—
38	(C₂×D₆₇)⋊C₄	—	2	2	no	—
39	(C₂×D₆₇)⋊C₄	—	2	4	no	—
40	(C₂×D₆₇)⋊C₄	—	2	4	no	—
41	C₄×Dic₆₇	—	2	4	no	—
42	C₄×Dic₆₇	C₂³	1	2	yes	—
43	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
44	Q₁₆×C₆₇	—	2	4	no	—
45	Q₁₆×C₆₇	—	4	2	no	—
46	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×D₆₇	—	2	2	no	—
52	Q₈×C₆₇	—	2	4	no	—
53	Q₈×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	—	2	4	no	—
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₇	—	2	2	no	—
59	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
60	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
61	D₂₆₈	—	2	2	no	—
62	D₂₆₈	—	2	2	no	—
63	C₄×D₆₇	—	4	4	no	—
64	Dic₁₃₄	—	2	2	no	—
65	C₈×C₆₇	—	2	1	no	—
66	C₈×C₆₇	#41	1	2	yes	—
67	C₆₇⋊C₈	—	2	2	no	—
68	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₆₇	—	4	2	no	—
70	C₂×D₆₇	—	4	2	no	—
71	C₂×D₆₇	—	4	2	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	—	4	4	no	—
74	C₄×C₆₇	#135	1	2	yes	—
75	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
77	Dic₆₇	—	2	2	no	—
78	Dic₆₇	—	2	2	no	—
79	Dic₆₇	—	4	2	no	—
80	C₂×C₆₇	#550	1	1	yes	characteristic
81	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
82	C₂×C₆₇	#578	1	1	yes	characteristic
83	D₆₇	—	8	2	no	—
84	#3393	—	67	1	no	maximal, Sylow
85	C₆₇	#3393	1	1	yes	characteristic, complemented, Sylow
86	#556	—	67	2	no	—
87	#495	—	67	2	no	—
88	#494	D₆₇	1	1	yes	characteristic, complemented
89	#464	—	67	4	no	—
90	#461	—	67	2	no	—
91	C₈×Q₈	—	67	1	no	—
92	C₄×Q₁₆	—	67	2	no	—
93	C₄×D₈	—	67	1	no	—
94	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
95	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
96	C₂×SD₁₆	—	67	4	no	—
97	C₂×D₈	—	67	1	no	—
98	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
99	#126	—	134	4	no	—
100	#124	—	67	4	no	—
101	C₂²⋊Q₈	—	134	2	no	—
102	C₄×Q₈	—	67	1	no	—
103	C₄×Q₈	—	67	4	no	—
104	C₄×D₄	—	67	2	no	—
105	#117	—	134	2	no	—
106	#107	—	67	1	no	—
107	C₄⋊C₈	—	67	2	no	—
108	C₄⋊C₈	—	67	1	no	—
109	#103	—	67	4	no	—
110	#102	—	67	2	no	—
111	C₄×C₈	—	67	2	no	—
112	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
113	C₂×Q₈	—	67	1	no	—
114	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
115	C₂×D₄	—	67	2	no	—
116	C₂²×C₄	—	134	2	no	—
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	2	no	—
119	SD₁₆	—	268	4	no	—
120	D₈	—	134	2	no	—
121	C₂×C₈	—	67	2	no	—
122	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	67	1	no	—
127	C₄⋊C₄	—	67	2	no	—
128	C₄⋊C₄	—	67	2	no	—
129	C₄⋊C₄	—	134	4	no	—
130	C₂²⋊C₄	—	134	4	no	—
131	C₂²⋊C₄	—	134	4	no	—
132	C₂²⋊C₄	—	134	2	no	—
133	C₄²	—	67	2	no	—
134	C₄²	—	134	4	no	—
135	C₄²	C₂²×D₆₇	1	1	yes	characteristic
136	C₂³	—	134	2	no	—
137	Q₈	—	2	4	no	—
138	Q₈	—	2	4	no	—
139	Q₈	—	134	2	no	—
140	D₄	—	134	2	no	—
141	D₄	—	134	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	268	4	no	—
144	C₂×C₄	—	67	2	no	—
145	C₂×C₄	—	67	2	no	—
146	C₂×C₄	—	134	2	no	—
147	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
148	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	134	2	no	—
152	C₈	#11761627	1	2	yes	—
153	C₂²	—	268	2	no	—
154	C₂²	—	268	2	no	—
155	C₂²	—	268	2	no	—
156	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	4	no	—
159	C₄	—	268	2	no	—
160	C₄	—	134	2	no	—
161	C₄	—	134	2	no	—
162	C₄	#14524066	1	2	yes	—
163	C₄	C₂×#11761627	1	1	yes	characteristic
164	C₄	#14524088	1	1	yes	characteristic
165	C₂	—	536	2	no	—
166	C₂	#16864066	1	1	yes	characteristic
167	C₂	C₂×#14524066	1	1	yes	characteristic
168	C₂	#16864209	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864120

C₂

yes

maximal, complemented

#16863781

C₂

yes

maximal

#16863620

C₂

yes

maximal, complemented

#16863598

C₂

yes

maximal, complemented

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic, complemented

Q₁₆×Dic₆₇

C₂

yes

maximal, complemented

C₄×D₅₃₆

C₂

yes

maximal, characteristic

#494×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761615

C₂²

yes

—

C₂×D₅₃₆

C₂²

yes

characteristic

#14524038

—

#14524032

C₂²

yes

—

#14524023

—

Q₈×Dic₆₇

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

C₄×D₂₆₈

C₂²

yes

—

#14524001

—

#14523977

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523967

C₂²

yes

—

#14523965

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×D₂₆₈

C₂³

yes

—

C₂×C₄×D₆₇

—

#11761615

—

D₅₃₆

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

D₄

yes

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

C₂³

yes

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#3393

—

maximal, Sylow

C₆₇

#3393

yes

characteristic, complemented, Sylow

#556

—

#495

—

#494

D₆₇

yes

characteristic, complemented

#464

—

#461

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×D₈

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₆₇

yes

—

#126

—

134

—

100

#124

—

101

C₂²⋊Q₈

—

134

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

C₄×D₄

—

105

#117

—

134

—

106

#107

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

110

#102

—

111

C₄×C₈

—

112

C₄×C₈

C₂×D₆₇

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

C₂²×D₆₇

yes

—

115

C₂×D₄

—

116

C₂²×C₄

—

134

—

117

Q₁₆

—

118

Q₁₆

—

119

SD₁₆

—

268

—

120

D₈

—

134

—

121

C₂×C₈

—

122

C₂×C₈

C₂²×D₆₇

yes

characteristic

123

C₂×C₈

C₂²×D₆₇

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

134

—

130

C₂²⋊C₄

—

134

—

131

C₂²⋊C₄

—

134

—

132

C₂²⋊C₄

—

134

—

133

C₄²

—

134

C₄²

—

134

—

135

C₄²

C₂²×D₆₇

yes

characteristic

136

C₂³

—

134

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

134

—

140

D₄

—

134

—

141

D₄

—

134

—

142

C₂×C₄

—

143

C₂×C₄

—

268

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

134

—

147

C₂×C₄

D₄×D₆₇

yes

characteristic

148

C₂×C₄

D₄×D₆₇

yes

characteristic

149

C₂×C₄

C₂³×D₆₇

yes

characteristic

150

C₈

—

151

C₈

—

134

—

152

C₈

#11761627

yes

—

153

C₂²

—

268

—

154

C₂²

—

268

—

155

C₂²

—

268

—

156

C₂²

C₂×D₄×D₆₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

268

—

160

C₄

—

134

—

161

C₄

—

134

—

162

C₄

#14524066

yes

—

163

C₄

C₂×#11761627

yes

characteristic

164

C₄

#14524088

yes

characteristic

165

C₂

—

536

—

166

C₂

#16864066

yes

characteristic

167

C₂

C₂×#14524066

yes

characteristic

168

C₂

#16864209

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#494	D₆₇
hypocenter	C₆₇	#3393
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#494×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. #16864120	C₂	semidirect
3. #16863620	C₂	semidirect
4. #16863598	C₂	semidirect
5. C₈×Dic₁₃₄	C₂	semidirect
6. Q₁₆×Dic₆₇	C₂	semidirect
8. #494×C₆₇	C₂	semidirect
85. C₆₇	#3393	semidirect
88. #494	D₆₇	semidirect

H ≅ G/N

type

1. #16864120

C₂

semidirect

3. #16863620

C₂

semidirect

4. #16863598

C₂

semidirect

5. C₈×Dic₁₃₄

C₂

semidirect

6. Q₁₆×Dic₆₇

C₂

semidirect

8. #494×C₆₇

C₂

semidirect

85. C₆₇

#3393

semidirect

88. #494

D₆₇

semidirect

Subgroups — #22573199

Conjugacy classes of subgroups

Complements