Group #22573195 — subgroups

Subgroups — #22573195

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	#16864116	C₂	1	2	yes	maximal, complemented
2	#16863779	C₂	1	2	yes	maximal
3	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863617	C₂	1	4	yes	maximal, complemented
5	#16863601	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863490	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₆₈	C₂²	1	2	yes	—
10	C₂×Dic₂₆₈	C₂²	1	2	yes	—
11	C₂×#11761616	C₂²	1	4	yes	—
12	#14524043	C₂²	1	2	yes	—
13	#14524036	—	2	4	no	—
14	#14524030	C₂²	1	4	yes	—
15	#14524022	—	2	2	no	—
16	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
17	#14524008	C₂²	1	4	yes	—
18	#14524002	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₆₇⋊C₈	C₂²	1	2	yes	—
21	#14523966	C₂²	1	4	yes	—
22	C₈×Dic₆₇	C₂²	1	2	yes	—
23	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	#107×C₆₇	C₂²	1	1	yes	characteristic
26	#102×C₆₇	C₂²	1	2	yes	—
27	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
29	C₂×Dic₁₃₄	C₂³	1	4	yes	—
30	Dic₂₆₈	—	2	4	no	—
31	Dic₂₆₈	—	4	2	no	—
32	#11761616	—	4	4	no	—
33	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₆₇	—	2	2	no	—
35	C₄⋊Dic₆₇	—	2	2	no	—
36	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
37	Dic₆₇⋊C₄	—	2	4	no	—
38	Dic₆₇⋊C₄	D₄	1	2	yes	—
39	C₂²⋊Dic₆₇	—	2	4	no	—
40	C₂²⋊Dic₆₇	—	2	4	no	—
41	C₄×Dic₆₇	—	2	4	no	—
42	C₄×Dic₆₇	C₂³	1	2	yes	—
43	C₂×D₄×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	D₈×C₆₇	—	2	2	no	—
46	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₆₇	—	2	2	no	—
52	D₄×C₆₇	—	2	2	no	—
53	D₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	4	4	no	—
56	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₆₇	—	2	4	no	—
60	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
61	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
62	Dic₁₃₄	—	2	2	no	—
63	Dic₁₃₄	—	2	4	no	—
64	Dic₁₃₄	—	2	4	no	—
65	C₈×C₆₇	—	2	1	no	—
66	C₈×C₆₇	#41	1	2	yes	—
67	C₆₇⋊C₈	—	2	2	no	—
68	C₂²×C₆₇	—	4	2	no	—
69	C₂²×C₆₇	—	4	2	no	—
70	C₂²×C₆₇	—	4	2	no	—
71	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	—	4	2	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	4	no	—
80	C₂×C₆₇	—	8	2	no	—
81	C₂×C₆₇	#550	1	1	yes	characteristic
82	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
83	C₂×C₆₇	#578	1	1	yes	characteristic
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	—	67	1	no	—
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	D₆₇	1	1	yes	characteristic
94	C₂×Q₁₆	—	67	2	no	—
95	C₂×Q₁₆	—	67	2	no	—
96	C₂×SD₁₆	—	67	4	no	—
97	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
98	C₄⋊Q₈	—	67	2	no	—
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	C₂×D₆₇	1	2	yes	—
105	#117	—	134	2	no	—
106	#107	C₂×D₆₇	1	1	yes	characteristic
107	C₄⋊C₈	—	67	2	no	—
108	C₄⋊C₈	—	67	1	no	—
109	#103	—	67	4	no	—
110	#102	C₂×D₆₇	1	2	yes	—
111	C₄×C₈	—	67	2	no	—
112	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
113	C₂×Q₈	—	67	1	no	—
114	C₂×Q₈	—	67	4	no	—
115	C₂×D₄	C₂²×D₆₇	1	2	yes	—
116	C₂²×C₄	—	2	2	no	—
117	Q₁₆	—	268	2	no	—
118	Q₁₆	—	134	4	no	—
119	SD₁₆	—	268	4	no	—
120	D₈	—	2	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	—	67	1	no	—
125	C₄⋊C₄	—	67	2	no	—
126	C₄⋊C₄	—	134	4	no	—
127	C₄⋊C₄	—	134	2	no	—
128	C₄⋊C₄	—	134	2	no	—
129	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	134	4	no	—
132	C₂²⋊C₄	—	134	4	no	—
133	C₄²	—	67	2	no	—
134	C₄²	—	134	4	no	—
135	C₄²	C₂²×D₆₇	1	1	yes	characteristic
136	C₂³	—	2	2	no	—
137	Q₈	—	134	4	no	—
138	Q₈	—	134	4	no	—
139	Q₈	—	134	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	67	2	no	—
145	C₂×C₄	—	67	2	no	—
146	C₂×C₄	—	134	4	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	#11761628	1	2	yes	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	268	4	no	—
160	C₄	—	134	2	no	—
161	C₄	—	134	2	no	—
162	C₄	#14524062	1	2	yes	—
163	C₄	C₂×#11761628	1	1	yes	characteristic
164	C₄	#14524089	1	1	yes	characteristic
165	C₂	—	8	2	no	—
166	C₂	#16864067	1	1	yes	characteristic
167	C₂	C₂×#14524062	1	1	yes	characteristic
168	C₂	#16864210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864116

C₂

yes

maximal, complemented

#16863779

C₂

yes

maximal

#16863776

C₂

yes

maximal, characteristic, complemented

#16863617

C₂

yes

maximal, complemented

#16863601

C₂

yes

maximal, complemented

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic, complemented

#16863490

C₂

yes

maximal, complemented

C₄×D₈×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761616

C₂²

yes

—

#14524043

C₂²

yes

—

#14524036

—

#14524030

C₂²

yes

—

#14524022

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14524008

C₂²

yes

—

#14524002

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523966

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

—

#107×C₆₇

C₂²

yes

characteristic

#102×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×Dic₁₃₄

C₂³

yes

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761616

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

D₄

yes

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×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

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

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

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×D₈

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

#126

—

134

—

100

#124

—

101

C₂²⋊Q₈

—

134

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

C₄×D₄

C₂×D₆₇

yes

—

105

#117

—

134

—

106

#107

C₂×D₆₇

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

110

#102

C₂×D₆₇

yes

—

111

C₄×C₈

—

112

C₄×C₈

C₂×D₆₇

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

C₂²×D₆₇

yes

—

116

C₂²×C₄

—

117

Q₁₆

—

268

—

118

Q₁₆

—

134

—

119

SD₁₆

—

268

—

120

D₈

—

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

—

134

—

127

C₄⋊C₄

—

134

—

128

C₄⋊C₄

—

134

—

129

C₄⋊C₄

C₂²×D₆₇

yes

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

134

—

132

C₂²⋊C₄

—

134

—

133

C₄²

—

134

C₄²

—

134

—

135

C₄²

C₂²×D₆₇

yes

characteristic

136

C₂³

—

137

Q₈

—

134

—

138

Q₈

—

134

—

139

Q₈

—

134

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

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

#11761628

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂×D₄×D₆₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

268

—

160

C₄

—

134

—

161

C₄

—

134

—

162

C₄

#14524062

yes

—

163

C₄

C₂×#11761628

yes

characteristic

164

C₄

#14524089

yes

characteristic

165

C₂

—

166

C₂

#16864067

yes

characteristic

167

C₂

C₂×#14524062

yes

characteristic

168

C₂

#16864210

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₄×D₈	D₆₇
hypocenter	C₆₇	#3393
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₄×D₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₄×D₈

D₆₇

hypocenter

C₆₇

#3393

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #16864116	C₂	semidirect
3. #16863776	C₂	semidirect
4. #16863617	C₂	semidirect
5. #16863601	C₂	semidirect
6. C₈×Dic₁₃₄	C₂	semidirect
7. #16863490	C₂	semidirect
85. C₆₇	#3393	semidirect

H ≅ G/N

type

1. #16864116

C₂

semidirect

3. #16863776

C₂

semidirect

4. #16863617

C₂

semidirect

5. #16863601

C₂

semidirect

6. C₈×Dic₁₃₄

C₂

semidirect

7. #16863490

C₂

semidirect

85. C₆₇

#3393

semidirect

Subgroups — #22573195

Conjugacy classes of subgroups

Complements