Group #22573214 — subgroups

Subgroups — #22573214

Order 8576.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16864124	C₂	1	2	yes	maximal
2	#16863776	C₂	1	1	yes	maximal, characteristic
3	#16863774	C₂	1	2	yes	maximal
4	#16863606	C₂	1	2	yes	maximal
5	#16863605	C₂	1	4	yes	maximal
6	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
7	#16863490	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₆₈	C₂²	1	2	yes	—
10	C₂×Dic₂₆₈	C₂²	1	2	yes	—
11	C₂×#11761618	C₂²	1	4	yes	—
12	#14524043	C₂²	1	2	yes	—
13	#14524041	C₂²	1	4	yes	—
14	#14524035	—	2	2	no	—
15	#14524033	—	2	4	no	—
16	C₄×Dic₁₃₄	—	2	2	no	—
17	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
18	#14524008	C₂²	1	4	yes	—
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×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	#107×C₆₇	C₂²	1	1	yes	characteristic
26	#103×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	#11761618	—	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	—	2	4	no	—
39	Dic₆₇⋊C₄	—	2	4	no	—
40	Dic₆₇⋊C₄	D₄	1	2	yes	—
41	C₄×Dic₆₇	—	2	4	no	—
42	C₄×Dic₆₇	C₂³	1	2	yes	—
43	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
44	Q₁₆×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	Q₈×C₆₇	—	2	2	no	—
52	Q₈×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	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	4	no	—
59	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
60	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
61	Dic₁₃₄	—	2	4	no	—
62	Dic₁₃₄	—	2	4	no	—
63	Dic₁₃₄	Q₁₆	1	4	yes	complemented
64	C₈×C₆₇	—	2	1	no	—
65	C₈×C₆₇	#41	1	2	yes	—
66	C₆₇⋊C₈	—	2	2	no	—
67	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₆₇	—	2	1	no	—
69	C₄×C₆₇	—	4	2	no	—
70	C₄×C₆₇	—	4	2	no	—
71	C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
72	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
73	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
74	Dic₆₇	—	2	2	no	—
75	Dic₆₇	—	4	4	no	—
76	Dic₆₇	C₂×Q₁₆	1	4	yes	—
77	C₂×C₆₇	#550	1	1	yes	characteristic
78	C₂×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
79	C₂×C₆₇	#578	1	1	yes	characteristic
80	#3395	—	67	1	no	maximal, Sylow
81	C₆₇	#3395	1	1	yes	characteristic, complemented, Sylow
82	#557	—	67	2	no	—
83	#494	—	67	2	no	—
84	#494	—	67	1	no	—
85	#462	—	67	2	no	—
86	#462	—	67	4	no	—
87	C₈×Q₈	—	67	1	no	—
88	C₄×Q₁₆	—	67	2	no	—
89	C₄×Q₁₆	D₆₇	1	1	yes	characteristic
90	C₂×Q₁₆	—	67	4	no	—
91	C₂×Q₁₆	—	67	2	no	—
92	C₂×Q₁₆	—	67	2	no	—
93	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
94	C₄⋊Q₈	—	67	4	no	—
95	C₄⋊Q₈	—	67	2	no	—
96	#125	—	134	2	no	—
97	#125	—	134	4	no	—
98	C₄×Q₈	—	67	1	no	—
99	C₄×Q₈	—	67	4	no	—
100	C₄×Q₈	—	134	2	no	—
101	C₄×Q₈	C₂×D₆₇	1	2	yes	—
102	#107	C₂×D₆₇	1	1	yes	characteristic
103	C₄⋊C₈	—	67	2	no	—
104	C₄⋊C₈	—	67	1	no	—
105	#103	—	67	4	no	—
106	#103	C₂×D₆₇	1	2	yes	—
107	C₄×C₈	—	67	2	no	—
108	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	C₂×Q₈	—	67	1	no	—
110	C₂×Q₈	—	67	4	no	—
111	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	268	2	no	—
114	Q₁₆	—	268	4	no	—
115	Q₁₆	—	134	4	no	—
116	C₂×C₈	—	67	2	no	—
117	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	67	2	no	—
121	C₄⋊C₄	—	67	1	no	—
122	C₄⋊C₄	—	134	4	no	—
123	C₄⋊C₄	—	134	4	no	—
124	C₄⋊C₄	—	134	4	no	—
125	C₄⋊C₄	—	134	2	no	—
126	C₄⋊C₄	—	134	2	no	—
127	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	—	67	2	no	—
130	C₄²	—	134	4	no	—
131	C₄²	C₂²×D₆₇	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	67	4	no	—
135	Q₈	—	134	4	no	—
136	Q₈	—	134	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	67	2	no	—
140	C₂×C₄	—	67	2	no	—
141	C₂×C₄	—	134	4	no	—
142	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	134	2	no	—
147	C₈	#11761628	1	2	yes	—
148	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	268	4	no	—
153	C₄	—	67	4	no	—
154	C₄	—	134	2	no	—
155	C₄	Q₁₆×D₆₇	1	2	yes	—
156	C₄	C₂×#11761628	1	1	yes	characteristic
157	C₄	#14524089	1	1	yes	characteristic
158	C₂	#16864067	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
160	C₂	#16864210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864124

C₂

yes

maximal

#16863776

C₂

yes

maximal, characteristic

#16863774

C₂

yes

maximal

#16863606

C₂

yes

maximal

#16863605

C₂

yes

maximal

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863490

C₂

yes

maximal

C₄×Q₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761618

C₂²

yes

—

#14524043

C₂²

yes

—

#14524041

C₂²

yes

—

#14524035

—

#14524033

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14524008

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523966

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

—

#107×C₆₇

C₂²

yes

characteristic

#103×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×Dic₁₃₄

C₂³

yes

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761618

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

D₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

C₂³

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×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

Q₁₆

yes

complemented

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

C₂×Q₁₆

yes

—

C₂×C₆₇

#550

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

134

—

#125

—

134

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

134

—

101

C₄×Q₈

C₂×D₆₇

yes

—

102

#107

C₂×D₆₇

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

C₂×D₆₇

yes

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₆₇

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂²×D₆₇

yes

—

112

Q₁₆

—

113

Q₁₆

—

268

—

114

Q₁₆

—

268

—

115

Q₁₆

—

134

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₆₇

yes

characteristic

118

C₂×C₈

C₂²×D₆₇

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

134

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

—

134

—

125

C₄⋊C₄

—

134

—

126

C₄⋊C₄

—

134

—

127

C₄⋊C₄

C₂²×D₆₇

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

134

—

131

C₄²

C₂²×D₆₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

134

—

136

Q₈

—

134

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

134

—

142

C₂×C₄

D₄×D₆₇

yes

characteristic

143

C₂×C₄

D₄×D₆₇

yes

characteristic

144

C₂×C₄

C₂³×D₆₇

yes

characteristic

145

C₈

—

146

C₈

—

134

—

147

C₈

#11761628

yes

—

148

C₂²

C₂×D₄×D₆₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

268

—

153

C₄

—

154

C₄

—

134

—

155

C₄

Q₁₆×D₆₇

yes

—

156

C₄

C₂×#11761628

yes

characteristic

157

C₄

#14524089

yes

characteristic

158

C₂

#16864067

yes

characteristic

159

C₂

C₂×Q₁₆×D₆₇

yes

characteristic

160

C₂

#16864210

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#3395
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₄×Q₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₄×Q₁₆

D₆₇

hypocenter

C₆₇

#3395

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
63. Dic₁₃₄	Q₁₆	semidirect
81. C₆₇	#3395	semidirect

H ≅ G/N

type

63. Dic₁₃₄

Q₁₆

semidirect

81. C₆₇

#3395

semidirect

Subgroups — #22573214

Conjugacy classes of subgroups

Complements