Group #22559705 — subgroups

Subgroups — #22559705

Order 8576.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16863683	C₂	1	1	yes	maximal, characteristic
2	#16863598	C₂	1	1	yes	maximal, characteristic
3	#16863475	C₂	1	1	yes	maximal, characteristic
4	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863145	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863142	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761615	—	2	1	no	—
9	C₂×#11761611	—	2	1	no	—
10	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
11	#14523974	—	2	1	no	—
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#14523967	—	2	1	no	—
16	#14523963	—	2	1	no	—
17	(C₂×D₆₇)⋊C₈	—	2	1	no	—
18	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#106×C₆₇	—	2	1	no	—
21	#106×C₆₇	—	2	1	no	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×D₂₆₈	D₄	1	1	yes	characteristic
24	C₂×C₄×D₆₇	—	2	1	no	—
25	#11761615	—	4	2	no	—
26	#11761611	—	4	2	no	—
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	—	2	1	no	—
30	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
31	(C₂×D₆₇)⋊C₄	—	2	1	no	—
32	C₂×Q₈×C₆₇	—	2	1	no	—
33	C₂×Q₈×C₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₆₇	—	2	1	no	—
42	Q₈×C₆₇	—	4	1	no	—
43	Q₈×C₆₇	—	4	1	no	—
44	Q₈×C₆₇	—	8	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	—	4	1	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₂×Dic₆₇	—	2	1	no	—
51	D₂₆₈	—	2	1	no	—
52	D₂₆₈	—	2	1	no	—
53	C₄×D₆₇	—	4	1	no	—
54	C₄×D₆₇	—	4	1	no	—
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	SD₁₆	1	2	yes	complemented
57	C₆₇⋊C₈	—	4	1	no	—
58	C₆₇⋊C₈	—	4	1	no	—
59	C₆₇⋊C₈	—	4	1	no	—
60	C₂²×C₆₇	#121	1	1	yes	characteristic
61	C₂×D₆₇	—	4	1	no	—
62	C₂×D₆₇	—	4	1	no	—
63	C₂×D₆₇	—	4	1	no	—
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	—	8	1	no	—
66	C₄×C₆₇	—	8	1	no	—
67	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₆₇	#137	1	1	yes	characteristic
70	C₄×C₆₇	#137	1	1	yes	characteristic
71	Dic₆₇	—	4	1	no	—
72	C₂×C₆₇	#461	1	1	yes	characteristic
73	C₂×C₆₇	#468	1	1	yes	characteristic
74	C₂×C₆₇	#471	1	1	yes	characteristic
75	D₆₇	—	8	1	no	—
76	#1691	—	67	1	no	maximal, Sylow
77	C₆₇	#1691	1	1	yes	characteristic, complemented, Sylow
78	#498	D₆₇	1	1	yes	characteristic, complemented
79	#476	—	67	1	no	—
80	#461	—	67	1	no	—
81	#436	—	67	1	no	—
82	#335	—	67	1	no	—
83	#333	—	67	1	no	—
84	#332	—	67	1	no	—
85	C₂×SD₁₆	—	134	1	no	—
86	C₂×M₁₆	—	134	1	no	—
87	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
88	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
89	C₄×D₄	—	67	1	no	—
90	#106	—	2	1	no	—
91	#106	—	2	1	no	—
92	#106	—	134	1	no	—
93	C₄⋊C₈	—	67	1	no	—
94	C₄⋊C₈	—	67	1	no	—
95	C₄⋊C₈	—	67	1	no	—
96	#103	—	134	1	no	—
97	#102	—	134	1	no	—
98	C₂²⋊C₈	—	134	1	no	—
99	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	67	1	no	—
103	C₂²×C₄	—	134	1	no	—
104	SD₁₆	—	268	2	no	—
105	M₁₆	—	268	2	no	—
106	C₂×C₈	—	134	1	no	—
107	C₂×C₈	—	134	1	no	—
108	C₂×C₈	—	134	1	no	—
109	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	67	1	no	—
116	C₂²⋊C₄	—	134	1	no	—
117	C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₂³	—	134	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	8	1	no	—
122	D₄	—	134	1	no	—
123	D₄	—	134	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	268	1	no	—
127	C₂×C₄	—	268	1	no	—
128	C₂×C₄	—	134	1	no	—
129	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	C₂×C₄	#11761627	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	268	1	no	—
134	C₈	—	268	1	no	—
135	C₈	—	268	1	no	—
136	C₈	#11761615	1	2	yes	complemented
137	C₂²	—	268	1	no	—
138	C₂²	—	268	1	no	—
139	C₂²	—	268	1	no	—
140	C₂²	#14524014	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	268	1	no	—
145	C₄	C₂×#11761615	1	1	yes	characteristic
146	C₄	#14524070	1	1	yes	characteristic
147	C₄	#14524072	1	1	yes	characteristic
148	C₄	#14524074	1	1	yes	characteristic
149	C₂	—	536	1	no	—
150	C₂	#16863598	1	1	yes	characteristic
151	C₂	#16863639	1	1	yes	characteristic
152	C₂	#16863656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863683

C₂

yes

maximal, characteristic

#16863598

C₂

yes

maximal, characteristic

#16863475

C₂

yes

maximal, characteristic

#16863149

C₂

yes

maximal, characteristic, complemented

#16863145

C₂

yes

maximal, characteristic, complemented

#16863142

C₂

yes

maximal, characteristic, complemented

#498×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761615

—

C₂×#11761611

—

C₄×D₂₆₈

C₂²

yes

characteristic

#14523974

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523967

—

#14523963

—

(C₂×D₆₇)⋊C₈

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#106×C₆₇

—

#106×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₄×D₆₇

—

#11761615

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₄

yes

characteristic

(C₂×D₆₇)⋊C₄

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

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

#41

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1691

—

maximal, Sylow

C₆₇

#1691

yes

characteristic, complemented, Sylow

#498

D₆₇

yes

characteristic, complemented

—

—

—

—

—

—

C₂×SD₁₆

—

134

—

C₂×M₁₆

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

—

#106

—

#106

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

134

—

#102

—

134

—

C₂²⋊C₈

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

134

—

104

SD₁₆

—

268

—

105

M₁₆

—

268

—

106

C₂×C₈

—

134

—

107

C₂×C₈

—

134

—

108

C₂×C₈

—

134

—

109

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

110

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

134

—

117

C₄²

C₂²×D₆₇

yes

characteristic

118

C₂³

—

134

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

D₄

—

134

—

123

D₄

—

134

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

268

—

127

C₂×C₄

—

268

—

128

C₂×C₄

—

134

—

129

C₂×C₄

D₄×D₆₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

131

C₂×C₄

#11761627

yes

characteristic

132

C₈

—

133

C₈

—

268

—

134

C₈

—

268

—

135

C₈

—

268

—

136

C₈

#11761615

yes

complemented

137

C₂²

—

268

—

138

C₂²

—

268

—

139

C₂²

—

268

—

140

C₂²

#14524014

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

268

—

145

C₄

C₂×#11761615

yes

characteristic

146

C₄

#14524070

yes

characteristic

147

C₄

#14524072

yes

characteristic

148

C₄

#14524074

yes

characteristic

149

C₂

—

536

—

150

C₂

#16863598

yes

characteristic

151

C₂

#16863639

yes

characteristic

152

C₂

#16863656

yes

characteristic

	subgroup	G/H
center	C₂²	#14524014
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#498×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#498	D₆₇
hypocenter	C₆₇	#1691
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14524014

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#498×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
4. #16863149	C₂	semidirect
5. #16863145	C₂	semidirect
6. #16863142	C₂	semidirect
7. #498×C₆₇	C₂	semidirect
56. C₈×C₆₇	SD₁₆	semidirect
77. C₆₇	#1691	semidirect
78. #498	D₆₇	semidirect
136. C₈	#11761615	semidirect

H ≅ G/N

type

4. #16863149

C₂

semidirect

5. #16863145

C₂

semidirect

6. #16863142

C₂

semidirect

7. #498×C₆₇

C₂

semidirect

56. C₈×C₆₇

SD₁₆

semidirect

77. C₆₇

#1691

semidirect

78. #498

D₆₇

semidirect

136. C₈

#11761615

semidirect

Subgroups — #22559705

Conjugacy classes of subgroups

Complements