Group #22561355 — subgroups

Subgroups — #22561355

Order 8576.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#14524035	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523964	C₂	1	2	yes	maximal
3	#16863158	C₂	1	2	yes	maximal, complemented
4	#16863151	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
7	#14524035	—	2	4	no	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×Dic₆₇⋊C₄	—	2	2	no	—
10	#14523964	—	2	4	no	—
11	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₆₇	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	D₄	1	4	yes	—
21	Dic₆₇⋊C₄	—	2	8	no	—
22	C₂⁴×C₆₇	—	2	1	no	—
23	C₂×D₄×C₆₇	—	2	2	no	—
24	C₂×D₄×C₆₇	—	2	2	no	—
25	C₂×D₄×C₆₇	—	2	1	no	—
26	C₂×D₄×C₆₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄×C₆₇	D₄	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂²⋊C₄×C₆₇	—	4	4	no	—
32	C₄²×C₆₇	—	2	2	no	—
33	C₂³×C₆₇	—	2	2	no	—
34	C₂³×C₆₇	—	4	1	no	—
35	C₂³×C₆₇	—	4	1	no	—
36	C₂³×C₆₇	—	4	2	no	—
37	C₂³×C₆₇	—	4	2	no	—
38	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
39	D₄×C₆₇	—	2	4	no	—
40	D₄×C₆₇	—	4	2	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#41	1	2	yes	—
49	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	4	2	no	—
52	C₂×Dic₆₇	—	4	2	no	—
53	C₆₇⋊C₈	—	4	2	no	—
54	C₂²×C₆₇	—	4	2	no	—
55	C₂²×C₆₇	—	4	4	no	—
56	C₂²×C₆₇	—	8	2	no	—
57	C₂²×C₆₇	—	8	1	no	—
58	C₂²×C₆₇	—	8	1	no	—
59	C₂²×C₆₇	#120	1	1	yes	characteristic
60	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₆₇	#124	1	1	yes	characteristic
62	C₂²×C₆₇	#135	1	2	yes	—
63	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₆₇	—	4	2	no	—
65	C₄×C₆₇	—	8	1	no	—
66	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₆₇	#124	1	2	yes	—
68	Dic₆₇	—	4	4	no	—
69	C₂×C₆₇	—	8	2	no	—
70	C₂×C₆₇	#393	1	1	yes	characteristic
71	C₂×C₆₇	#449	1	2	yes	—
72	C₂×C₆₇	#478	1	2	yes	—
73	C₂×C₆₇	#488	1	1	yes	characteristic
74	C₂×C₆₇	#490	1	1	yes	characteristic
75	#2025	—	67	1	no	maximal, Sylow
76	C₆₇	#2025	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	—	67	1	no	—
78	C₂×#102	—	67	2	no	—
79	#390	D₆₇	1	1	yes	characteristic
80	#340	—	67	2	no	—
81	#336	—	67	1	no	—
82	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
83	C₂²×C₈	—	67	2	no	—
84	#125	—	134	4	no	—
85	C₂×C₄⋊C₄	—	67	2	no	—
86	C₂×C₄⋊C₄	—	134	2	no	—
87	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
90	#102	—	134	4	no	—
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	Dic₁₃₄	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	134	2	no	—
98	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	C₂×C₈	—	134	2	no	—
101	C₂×C₈	—	134	2	no	—
102	C₂×C₈	—	134	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	67	4	no	—
105	C₄⋊C₄	—	134	8	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	268	2	no	—
123	C₂×C₄	—	268	2	no	—
124	C₂×C₄	—	134	4	no	—
125	C₂×C₄	D₄×D₆₇	1	2	yes	—
126	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
127	C₂×C₄	#11761627	1	2	yes	—
128	C₂×C₄	#11761628	1	1	yes	characteristic
129	C₈	—	268	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#14524011	1	1	yes	characteristic
136	C₂²	#14524022	1	1	yes	characteristic
137	C₂²	#14524030	1	1	yes	characteristic
138	C₂²	#14524063	1	2	yes	—
139	C₂²	#14524069	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	268	4	no	—
143	C₄	#14524020	1	2	yes	—
144	C₄	#14524029	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#16863308	1	1	yes	characteristic
147	C₂	#16863539	1	2	yes	—
148	C₂	#16863696	1	2	yes	—
149	C₂	#16863750	1	1	yes	characteristic
150	C₂	#16863760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic, complemented

C₂×#14523964

C₂

yes

maximal

#16863158

C₂

yes

maximal, complemented

#16863151

C₂

yes

maximal, characteristic, complemented

#390×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

#14524035

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

#14523964

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

Dic₆₇⋊C₄

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#124

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#393

yes

characteristic

C₂×C₆₇

#449

yes

—

C₂×C₆₇

#478

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×#102

—

#390

D₆₇

yes

characteristic

#340

—

#336

—

C₂²×D₄

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#102

—

134

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₃₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

100

C₂×C₈

—

134

—

101

C₂×C₈

—

134

—

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

268

—

123

C₂×C₄

—

268

—

124

C₂×C₄

—

134

—

125

C₂×C₄

D₄×D₆₇

yes

—

126

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

127

C₂×C₄

#11761627

yes

—

128

C₂×C₄

#11761628

yes

characteristic

129

C₈

—

268

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#14524011

yes

characteristic

136

C₂²

#14524022

yes

characteristic

137

C₂²

#14524030

yes

characteristic

138

C₂²

#14524063

yes

—

139

C₂²

#14524069

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

268

—

143

C₄

#14524020

yes

—

144

C₄

#14524029

yes

—

145

C₂

—

146

C₂

#16863308

yes

characteristic

147

C₂

#16863539

yes

—

148

C₂

#16863696

yes

—

149

C₂

#16863750

yes

characteristic

150

C₂

#16863760

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#390×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#390	D₆₇
hypocenter	C₆₇	#2025
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#390×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#14524035	C₂	semidirect
3. #16863158	C₂	semidirect
4. #16863151	C₂	semidirect
76. C₆₇	#2025	semidirect

H ≅ G/N

type

1. C₂×#14524035

C₂

semidirect

3. #16863158

C₂

semidirect

4. #16863151

C₂

semidirect

76. C₆₇

#2025

semidirect

Subgroups — #22561355

Conjugacy classes of subgroups

Complements