Group #22561119 — subgroups

Subgroups — #22561119

Order 8576.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863400	C₂	1	2	yes	maximal, complemented
4	C₂×#14523959	C₂	1	2	yes	maximal, complemented
5	#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	2	yes	—
7	C₂×D₅₃₆	—	2	2	no	—
8	C₂×D₅₃₆	—	2	1	no	—
9	C₂×D₅₃₆	—	2	1	no	—
10	C₂×D₅₃₆	C₄	1	4	yes	complemented
11	C₂×#11761612	C₂²	1	2	yes	—
12	#14523979	—	2	2	no	—
13	#14523963	—	2	4	no	—
14	#14523959	—	2	4	no	—
15	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#117×C₆₇	C₂²	1	1	yes	characteristic
18	#97×C₆₇	—	2	1	no	—
19	C₄×C₈×C₆₇	—	2	1	no	—
20	C₂³×D₆₇	—	2	2	no	—
21	C₂×D₂₆₈	—	2	2	no	—
22	C₂×D₂₆₈	—	2	2	no	—
23	C₂×D₂₆₈	—	2	2	no	—
24	C₂×D₂₆₈	—	2	2	no	—
25	C₂×D₂₆₈	—	2	2	no	—
26	C₂×D₂₆₈	C₂×C₄	1	4	yes	—
27	D₅₃₆	—	2	4	no	—
28	D₅₃₆	—	2	4	no	—
29	#11761612	—	2	2	no	—
30	#11761612	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₆₇	—	2	1	no	—
34	M₁₆×C₆₇	D₄	1	2	yes	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₆₇	D₄	1	2	yes	—
38	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
41	C₂²⋊C₄×C₆₇	—	2	1	no	—
42	C₄²×C₆₇	—	2	1	no	—
43	C₂³×C₆₇	#41	1	1	yes	characteristic
44	C₂²×D₆₇	—	2	4	no	—
45	C₂²×D₆₇	—	4	2	no	—
46	C₂²×D₆₇	—	4	2	no	—
47	C₂²×D₆₇	—	8	4	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
55	D₂₆₈	—	2	4	no	—
56	D₂₆₈	—	2	4	no	—
57	D₂₆₈	—	4	2	no	—
58	D₂₆₈	—	4	2	no	—
59	C₈×C₆₇	—	2	2	no	—
60	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₆₇⋊C₈	—	4	2	no	—
63	C₂²×C₆₇	—	2	2	no	—
64	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₆₇	#121	1	1	yes	characteristic
66	C₂²×C₆₇	#124	1	1	yes	characteristic
67	C₂×D₆₇	—	4	4	no	—
68	C₂×D₆₇	—	8	4	no	—
69	C₂×D₆₇	—	8	4	no	—
70	C₂×D₆₇	—	8	4	no	—
71	C₄×C₆₇	—	4	2	no	—
72	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
74	C₄×C₆₇	#121	1	1	yes	characteristic
75	C₄×C₆₇	#127	1	1	yes	characteristic
76	C₂×C₆₇	—	2	1	no	—
77	C₂×C₆₇	—	2	1	no	—
78	C₂×C₆₇	#390	1	1	yes	characteristic
79	C₂×C₆₇	#472	1	2	yes	—
80	D₆₇	—	8	4	no	—
81	#1978	—	67	1	no	maximal, Sylow
82	C₆₇	#1978	1	1	yes	characteristic, complemented, Sylow
83	C₂²×D₈	—	67	1	no	—
84	C₂×#108	—	67	1	no	—
85	#418	—	67	2	no	—
86	C₂×#100	—	67	2	no	—
87	#405	D₆₇	1	1	yes	characteristic, complemented
88	C₂²×D₄	—	67	2	no	—
89	C₂×D₈	—	67	4	no	—
90	C₂×D₈	—	134	2	no	—
91	C₂×D₈	—	134	1	no	—
92	C₂×D₈	—	134	1	no	—
93	C₂×M₁₆	—	67	2	no	—
94	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
96	#117	C₂×D₆₇	1	1	yes	characteristic
97	#108	—	134	2	no	—
98	#102	—	134	4	no	—
99	#100	—	134	4	no	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₂⁴	—	134	2	no	—
103	C₂×D₄	—	67	4	no	—
104	C₂×D₄	—	134	2	no	—
105	C₂×D₄	—	134	2	no	—
106	C₂×D₄	—	134	2	no	—
107	C₂×D₄	—	134	2	no	—
108	C₂×D₄	—	134	2	no	—
109	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
110	D₈	—	134	4	no	—
111	D₈	—	134	4	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	134	2	no	—
114	M₁₆	—	134	2	no	—
115	M₁₆	D₂₆₈	1	2	yes	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	134	2	no	—
118	C₂×C₈	C₄×D₆₇	1	2	yes	—
119	C₂×C₈	D₂₆₈	1	2	yes	—
120	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
122	C₄⋊C₄	D₂₆₈	1	2	yes	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₂³	—	268	2	no	—
126	C₂³	—	268	2	no	—
127	C₂³	—	536	4	no	—
128	C₂³	—	134	4	no	—
129	C₂³	#11761628	1	1	yes	characteristic
130	D₄	—	268	2	no	—
131	D₄	—	268	2	no	—
132	D₄	—	134	4	no	—
133	D₄	—	134	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
140	C₂×C₄	#11761628	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	268	2	no	—
143	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
144	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	268	4	no	—
147	C₂²	—	536	4	no	—
148	C₂²	—	536	4	no	—
149	C₂²	—	536	4	no	—
150	C₂²	C₄×D₂₆₈	1	1	yes	characteristic
151	C₂²	#14524017	1	1	yes	characteristic
152	C₂²	#14524031	1	1	yes	characteristic
153	C₄	—	4	2	no	—
154	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
155	C₄	C₄×D₂₆₈	1	1	yes	characteristic
156	C₄	#14524017	1	1	yes	characteristic
157	C₄	#14524040	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	536	4	no	—
161	C₂	#16863298	1	1	yes	characteristic
162	C₂	#16863666	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₃₆

C₂

yes

maximal, characteristic

C₂×#14523979

C₂

yes

maximal, characteristic, complemented

#16863400

C₂

yes

maximal, complemented

C₂×#14523959

C₂

yes

maximal, complemented

#405×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

C₄

yes

complemented

C₂×#11761612

C₂²

yes

—

#14523979

—

#14523963

—

#14523959

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂×C₄

yes

—

D₅₃₆

—

D₅₃₆

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#390

yes

characteristic

C₂×C₆₇

#472

yes

—

D₆₇

—

#1978

—

maximal, Sylow

C₆₇

#1978

yes

characteristic, complemented, Sylow

C₂²×D₈

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

C₂×D₆₇

yes

characteristic

#108

—

134

—

#102

—

134

—

#100

—

134

—

100

#97

—

101

C₄×C₈

—

102

C₂⁴

—

134

—

103

C₂×D₄

—

104

C₂×D₄

—

134

—

105

C₂×D₄

—

134

—

106

C₂×D₄

—

134

—

107

C₂×D₄

—

134

—

108

C₂×D₄

—

134

—

109

C₂²×C₄

C₂²×D₆₇

yes

characteristic

110

D₈

—

134

—

111

D₈

—

134

—

112

M₁₆

—

113

M₁₆

—

134

—

114

M₁₆

—

134

—

115

M₁₆

D₂₆₈

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

134

—

118

C₂×C₈

C₄×D₆₇

yes

—

119

C₂×C₈

D₂₆₈

yes

—

120

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

121

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

122

C₄⋊C₄

D₂₆₈

yes

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₂³

—

268

—

126

C₂³

—

268

—

127

C₂³

—

536

—

128

C₂³

—

134

—

129

C₂³

#11761628

yes

characteristic

130

D₄

—

268

—

131

D₄

—

268

—

132

D₄

—

134

—

133

D₄

—

134

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

137

C₂×C₄

C₂×D₂₆₈

yes

characteristic

138

C₂×C₄

C₂×D₂₆₈

yes

characteristic

139

C₂×C₄

C₂×D₂₆₈

yes

characteristic

140

C₂×C₄

#11761628

yes

characteristic

141

C₈

—

142

C₈

—

268

—

143

C₈

(C₂×D₆₇)⋊C₄

yes

—

144

C₈

(C₂×D₆₇)⋊C₄

yes

—

145

C₂²

—

146

C₂²

—

268

—

147

C₂²

—

536

—

148

C₂²

—

536

—

149

C₂²

—

536

—

150

C₂²

C₄×D₂₆₈

yes

characteristic

151

C₂²

#14524017

yes

characteristic

152

C₂²

#14524031

yes

characteristic

153

C₄

—

154

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

155

C₄

C₄×D₂₆₈

yes

characteristic

156

C₄

#14524017

yes

characteristic

157

C₄

#14524040

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

536

—

161

C₂

#16863298

yes

characteristic

162

C₂

#16863666

yes

—

	subgroup	G/H
center	C₂²	#14524017
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#405×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#405	D₆₇
hypocenter	C₆₇	#1978
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#14524017

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#405×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
2. C₂×#14523979	C₂	semidirect
3. #16863400	C₂	semidirect
4. C₂×#14523959	C₂	semidirect
5. #405×C₆₇	C₂	semidirect
10. C₂×D₅₃₆	C₄	semidirect
82. C₆₇	#1978	semidirect
87. #405	D₆₇	semidirect

H ≅ G/N

type

2. C₂×#14523979

C₂

semidirect

3. #16863400

C₂

semidirect

4. C₂×#14523959

C₂

semidirect

5. #405×C₆₇

C₂

semidirect

10. C₂×D₅₃₆

C₄

semidirect

82. C₆₇

#1978

semidirect

87. #405

D₆₇

semidirect

Subgroups — #22561119

Conjugacy classes of subgroups

Complements