Group #22573213 — subgroups

Subgroups — #22573213

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	#16864121	C₂	1	2	yes	maximal
2	#16863774	C₂	1	2	yes	maximal
3	#16863604	C₂	1	4	yes	maximal
4	#16863603	C₂	1	2	yes	maximal
5	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₆₇	C₂	1	2	yes	maximal
8	#494×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×#11761618	C₂²	1	4	yes	—
11	#14524041	C₂²	1	4	yes	—
12	#14524034	—	2	2	no	—
13	#14524033	—	2	4	no	—
14	Q₈×Dic₆₇	C₂²	1	4	yes	—
15	C₄×Dic₁₃₄	C₂²	1	2	yes	—
16	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
17	#14524008	—	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	#14523968	C₂²	1	2	yes	—
22	#14523967	C₂²	1	4	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×Dic₁₃₄	C₂³	1	2	yes	—
30	Dic₂₆₈	—	2	2	no	—
31	#11761618	—	4	4	no	—
32	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
33	C₄⋊Dic₆₇	—	2	4	no	—
34	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
35	C₄⋊Dic₆₇	C₂³	1	2	yes	—
36	Dic₆₇⋊C₄	—	2	2	no	—
37	Dic₆₇⋊C₄	—	2	4	no	—
38	Dic₆₇⋊C₄	—	2	4	no	—
39	Dic₆₇⋊C₄	D₄	1	2	yes	—
40	C₄×Dic₆₇	—	2	2	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	Q₈×C₆₇	—	2	4	no	—
52	Q₈×C₆₇	—	2	4	no	—
53	C₂×C₄×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₆₇	—	2	2	no	—
58	C₂×Dic₆₇	—	2	2	no	—
59	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
60	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
61	Dic₁₃₄	—	2	2	no	—
62	Dic₁₃₄	—	2	2	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	4	no	—
70	C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
71	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
72	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
73	Dic₆₇	—	2	2	no	—
74	Dic₆₇	—	4	2	no	—
75	Dic₆₇	—	4	2	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	D₆₇	1	1	yes	characteristic
85	#462	—	67	4	no	—
86	#462	—	67	2	no	—
87	C₈×Q₈	—	67	1	no	—
88	C₄×Q₁₆	—	67	2	no	—
89	C₄×Q₁₆	—	67	1	no	—
90	C₂×Q₁₆	—	67	4	no	—
91	C₂×Q₁₆	—	67	1	no	—
92	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
93	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
94	C₄⋊Q₈	—	67	4	no	—
95	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
96	#125	—	134	4	no	—
97	#125	—	134	2	no	—
98	C₄×Q₈	—	67	1	no	—
99	C₄×Q₈	—	67	4	no	—
100	C₄×Q₈	—	67	2	no	—
101	C₄×Q₈	—	134	2	no	—
102	#107	—	67	1	no	—
103	C₄⋊C₈	—	67	2	no	—
104	C₄⋊C₈	—	67	1	no	—
105	#103	—	67	4	no	—
106	#103	—	67	2	no	—
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	2	no	—
111	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
112	Q₁₆	—	2	4	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	268	4	no	—
115	Q₁₆	—	134	2	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	—	2	2	no	—
121	C₄⋊C₄	—	67	2	no	—
122	C₄⋊C₄	—	67	1	no	—
123	C₄⋊C₄	—	67	2	no	—
124	C₄⋊C₄	—	134	4	no	—
125	C₄⋊C₄	—	134	4	no	—
126	C₄⋊C₄	—	134	4	no	—
127	C₄⋊C₄	—	134	2	no	—
128	C₄²	—	67	2	no	—
129	C₄²	—	134	4	no	—
130	C₄²	—	134	2	no	—
131	C₄²	C₂²×D₆₇	1	1	yes	characteristic
132	Q₈	—	2	4	no	—
133	Q₈	—	2	4	no	—
134	Q₈	—	67	4	no	—
135	Q₈	—	134	2	no	—
136	Q₈	—	134	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	67	2	no	—
139	C₂×C₄	—	67	2	no	—
140	C₂×C₄	—	134	2	no	—
141	C₂×C₄	—	134	2	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	#11761627	1	2	yes	—
148	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	4	no	—
151	C₄	—	268	2	no	—
152	C₄	—	268	2	no	—
153	C₄	—	67	4	no	—
154	C₄	—	134	2	no	—
155	C₄	Q₁₆×D₆₇	1	2	yes	—
156	C₄	C₂×#11761627	1	1	yes	characteristic
157	C₄	#14524088	1	1	yes	characteristic
158	C₂	#16864066	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
160	C₂	#16864209	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864121

C₂

yes

maximal

#16863774

C₂

yes

maximal

#16863604

C₂

yes

maximal

#16863603

C₂

yes

maximal

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

C₄×Dic₂₆₈

C₂

yes

maximal, characteristic

Q₁₆×Dic₆₇

C₂

yes

maximal

#494×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×#11761618

C₂²

yes

—

#14524041

C₂²

yes

—

#14524034

—

#14524033

—

Q₈×Dic₆₇

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14524008

—

#14523977

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523968

C₂²

yes

—

#14523967

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×Dic₁₃₄

C₂³

yes

—

Dic₂₆₈

—

#11761618

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

D₄

yes

—

C₄×Dic₆₇

—

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

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×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

C₂×Q₁₆

yes

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

C₂×Q₁₆

yes

—

C₂×C₆₇

#550

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#557

—

#494

—

#494

D₆₇

yes

characteristic

#462

—

#462

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

—

#125

—

134

—

#125

—

134

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

134

—

102

#107

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

—

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

—

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

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

134

—

125

C₄⋊C₄

—

134

—

126

C₄⋊C₄

—

134

—

127

C₄⋊C₄

—

134

—

128

C₄²

—

129

C₄²

—

134

—

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

—

134

—

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

#11761627

yes

—

148

C₂²

C₂×D₄×D₆₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

268

—

152

C₄

—

268

—

153

C₄

—

154

C₄

—

134

—

155

C₄

Q₁₆×D₆₇

yes

—

156

C₄

C₂×#11761627

yes

characteristic

157

C₄

#14524088

yes

characteristic

158

C₂

#16864066

yes

characteristic

159

C₂

C₂×Q₁₆×D₆₇

yes

characteristic

160

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	#3395
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
63. Dic₁₃₄	Q₁₆	semidirect
81. C₆₇	#3395	semidirect

H ≅ G/N

type

63. Dic₁₃₄

Q₁₆

semidirect

81. C₆₇

#3395

semidirect

Subgroups — #22573213

Conjugacy classes of subgroups

Complements