Group #22559627 — subgroups

Subgroups — #22559627

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	#16863744	C₂	1	1	yes	maximal, characteristic
2	#16863596	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863146	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863145	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863139	C₂	1	1	yes	maximal, characteristic
7	#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761615	—	2	1	no	—
9	#14524040	C₂²	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#14523963	—	2	1	no	—
14	#14523963	—	2	1	no	—
15	#14523963	—	2	1	no	—
16	C₄×C₆₇⋊C₈	—	2	1	no	—
17	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#107×C₆₇	—	2	1	no	—
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#103×C₆₇	—	2	1	no	—
23	C₂×D₂₆₈	—	2	1	no	—
24	C₂×D₂₆₈	—	2	1	no	—
25	C₂×D₂₆₈	—	2	1	no	—
26	#11761615	—	4	2	no	—
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
31	C₂×Q₈×C₆₇	—	2	1	no	—
32	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	—	2	1	no	—
34	C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₄²×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×D₆₇	—	4	1	no	—
41	Q₈×C₆₇	—	2	1	no	—
42	Q₈×C₆₇	—	8	1	no	—
43	Q₈×C₆₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	2	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	D₂₆₈	—	4	1	no	—
51	D₂₆₈	—	4	1	no	—
52	D₂₆₈	—	4	1	no	—
53	D₂₆₈	—	4	1	no	—
54	D₂₆₈	—	8	1	no	—
55	C₈×C₆₇	—	4	1	no	—
56	C₆₇⋊C₈	—	2	1	no	—
57	C₆₇⋊C₈	—	2	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	—	8	1	no	—
62	C₂×D₆₇	—	8	1	no	—
63	C₂×D₆₇	—	8	1	no	—
64	C₄×C₆₇	—	2	2	no	—
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	4	1	no	—
67	C₄×C₆₇	—	8	1	no	—
68	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₆₇	#135	1	1	yes	characteristic
70	C₄×C₆₇	#135	1	1	yes	characteristic
71	C₄×C₆₇	#137	1	1	yes	characteristic
72	C₂×C₆₇	#464	1	1	yes	characteristic
73	C₂×C₆₇	#465	1	1	yes	characteristic
74	C₂×C₆₇	#472	1	1	yes	characteristic
75	D₆₇	—	16	1	no	—
76	#1678	—	67	1	no	maximal, Sylow
77	C₆₇	#1678	1	1	yes	characteristic, complemented, Sylow
78	#486	—	67	1	no	—
79	#477	D₆₇	1	1	yes	characteristic, complemented
80	#460	—	67	1	no	—
81	C₈×Q₈	—	67	1	no	—
82	#334	—	67	1	no	—
83	#333	—	67	1	no	—
84	#331	—	67	1	no	—
85	C₂×SD₁₆	—	134	1	no	—
86	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
87	#127	—	67	1	no	—
88	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	C₄⋊C₈	—	67	1	no	—
91	C₄⋊C₈	—	67	1	no	—
92	C₄⋊C₈	—	134	1	no	—
93	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
94	#103	—	2	1	no	—
95	#102	—	134	1	no	—
96	#102	—	134	1	no	—
97	#102	—	134	1	no	—
98	C₄×C₈	—	67	1	no	—
99	C₄×C₈	—	134	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
102	C₂×D₄	—	134	1	no	—
103	C₂×D₄	—	134	1	no	—
104	C₂×D₄	—	134	1	no	—
105	SD₁₆	—	268	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	67	1	no	—
108	C₂×C₈	—	67	1	no	—
109	C₂×C₈	—	134	1	no	—
110	C₂×C₈	—	134	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂²×D₆₇	1	1	yes	characteristic
117	C₂³	—	268	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	8	1	no	—
120	Q₈	#11761615	1	2	yes	—
121	D₄	—	268	1	no	—
122	D₄	—	268	1	no	—
123	D₄	—	268	1	no	—
124	D₄	—	268	1	no	—
125	D₄	—	536	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	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	—	4	1	no	—
133	C₈	—	268	1	no	—
134	C₈	—	268	1	no	—
135	C₈	—	134	1	no	—
136	C₈	—	134	1	no	—
137	C₂²	—	536	1	no	—
138	C₂²	—	536	1	no	—
139	C₂²	—	536	1	no	—
140	C₂²	#14524015	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	8	1	no	—
145	C₄	C₂×#11761615	1	1	yes	characteristic
146	C₄	#14524063	1	1	yes	characteristic
147	C₄	#14524066	1	1	yes	characteristic
148	C₄	#14524075	1	1	yes	characteristic
149	C₂	—	1072	1	no	—
150	C₂	#16863620	1	1	yes	characteristic
151	C₂	#16863627	1	1	yes	characteristic
152	C₂	#16863664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863744

C₂

yes

maximal, characteristic

#16863596

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16863146

C₂

yes

maximal, characteristic, complemented

#16863145

C₂

yes

maximal, characteristic, complemented

#16863139

C₂

yes

maximal, characteristic

#477×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761615

—

#14524040

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

#107×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

#11761615

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

SD₁₆

yes

—

C₂×C₄×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

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₆₇⋊C₈

—

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

—

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1678

—

maximal, Sylow

C₆₇

#1678

yes

characteristic, complemented, Sylow

#486

—

#477

D₆₇

yes

characteristic, complemented

#460

—

C₈×Q₈

—

#334

—

#333

—

#331

—

C₂×SD₁₆

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#127

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

#102

—

134

—

#102

—

134

—

#102

—

134

—

C₄×C₈

—

C₄×C₈

—

134

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

102

C₂×D₄

—

134

—

103

C₂×D₄

—

134

—

104

C₂×D₄

—

134

—

105

SD₁₆

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

134

—

110

C₂×C₈

—

134

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

115

C₄²

—

116

C₄²

C₂²×D₆₇

yes

characteristic

117

C₂³

—

268

—

118

Q₈

—

119

Q₈

—

120

Q₈

#11761615

yes

—

121

D₄

—

268

—

122

D₄

—

268

—

123

D₄

—

268

—

124

D₄

—

268

—

125

D₄

—

536

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

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

—

134

—

136

C₈

—

134

—

137

C₂²

—

536

—

138

C₂²

—

536

—

139

C₂²

—

536

—

140

C₂²

#14524015

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

C₂×#11761615

yes

characteristic

146

C₄

#14524063

yes

characteristic

147

C₄

#14524066

yes

characteristic

148

C₄

#14524075

yes

characteristic

149

C₂

—

1072

—

150

C₂

#16863620

yes

characteristic

151

C₂

#16863627

yes

characteristic

152

C₂

#16863664

yes

characteristic

	subgroup	G/H
center	C₂²	#14524015
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#477×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#477	D₆₇
hypocenter	C₆₇	#1678
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14524015

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#477×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
3. Q₈×C₆₇⋊C₈	C₂	semidirect
4. #16863146	C₂	semidirect
5. #16863145	C₂	semidirect
7. #477×C₆₇	C₂	semidirect
77. C₆₇	#1678	semidirect
79. #477	D₆₇	semidirect

H ≅ G/N

type

3. Q₈×C₆₇⋊C₈

C₂

semidirect

4. #16863146

C₂

semidirect

5. #16863145

C₂

semidirect

7. #477×C₆₇

C₂

semidirect

77. C₆₇

#1678

semidirect

79. #477

D₆₇

semidirect

Subgroups — #22559627

Conjugacy classes of subgroups

Complements