Group #22561624 — subgroups

Subgroups — #22561624

Order 8576.

258 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 79 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×#14523965	C₂	1	2	yes	maximal
2	#16863298	C₂	1	1	yes	maximal, characteristic
3	#16863262	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863159	C₂	1	2	yes	maximal, complemented
5	#336×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
10	#14523965	—	2	4	no	—
11	C₆₇⋊#95	—	2	2	no	—
12	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₆₇	—	2	1	no	—
15	C₂×D₂₆₈	—	2	2	no	—
16	C₂×D₂₆₈	—	2	2	no	—
17	C₂×D₂₆₈	—	2	1	no	—
18	C₂×D₂₆₈	D₄	1	2	yes	—
19	C₂²×Dic₆₇	—	2	2	no	—
20	C₂²×Dic₆₇	—	2	1	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	(C₂×D₆₇)⋊C₄	—	4	4	no	—
25	C₂²×C₄×C₆₇	D₄	1	2	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₆₇	—	2	2	no	—
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₄²×C₆₇	—	2	2	no	—
31	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₆₇	—	2	2	no	—
33	C₂²×D₆₇	—	4	1	no	—
34	C₂²×D₆₇	—	4	1	no	—
35	C₂²×D₆₇	—	4	2	no	—
36	C₂²×D₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#41	1	2	yes	—
42	C₂×C₄×C₆₇	#41	1	2	yes	—
43	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
44	C₂×Dic₆₇	—	4	2	no	—
45	C₂×Dic₆₇	—	4	1	no	—
46	C₂×Dic₆₇	—	4	2	no	—
47	C₂×Dic₆₇	—	4	2	no	—
48	C₂×Dic₆₇	—	4	2	no	—
49	D₂₆₈	—	2	4	no	—
50	D₂₆₈	—	4	2	no	—
51	C₈×C₆₇	—	4	2	no	—
52	C₂²×C₆₇	#121	1	1	yes	characteristic
53	C₂²×C₆₇	#123	1	1	yes	characteristic
54	C₂²×C₆₇	#124	1	1	yes	characteristic
55	C₂²×C₆₇	#135	1	2	yes	—
56	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₂×D₆₇	—	4	2	no	—
58	C₂×D₆₇	—	4	4	no	—
59	C₂×D₆₇	—	8	2	no	—
60	C₂×D₆₇	—	8	1	no	—
61	C₂×D₆₇	—	8	1	no	—
62	C₄×C₆₇	—	4	2	no	—
63	C₄×C₆₇	#123	1	2	yes	—
64	C₄×C₆₇	#126	1	2	yes	—
65	Dic₆₇	—	8	2	no	—
66	Dic₆₇	—	8	1	no	—
67	C₂×C₆₇	#397	1	1	yes	characteristic
68	C₂×C₆₇	#463	1	2	yes	—
69	C₂×C₆₇	#482	1	2	yes	—
70	C₂×C₆₇	#488	1	1	yes	characteristic
71	C₂×C₆₇	#490	1	1	yes	characteristic
72	D₆₇	—	8	2	no	—
73	#2073	—	67	1	no	maximal, Sylow
74	C₆₇	#2073	1	1	yes	characteristic, complemented, Sylow
75	C₂×#102	—	67	2	no	—
76	#390	—	67	1	no	—
77	#382	—	67	1	no	—
78	#340	—	67	2	no	—
79	#336	D₆₇	1	1	yes	characteristic, complemented
80	C₂²×D₄	—	67	1	no	—
81	C₂²×C₈	C₂×D₆₇	1	2	yes	—
82	C₂×C₄⋊C₄	—	67	1	no	—
83	C₂×C₄⋊C₄	—	67	2	no	—
84	C₂×C₂²⋊C₄	—	134	2	no	—
85	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
86	#102	—	134	4	no	—
87	#95	—	134	2	no	—
88	C₂⁴	—	134	1	no	—
89	C₂×D₄	—	67	2	no	—
90	C₂×D₄	—	134	2	no	—
91	C₂×D₄	—	134	2	no	—
92	C₂×D₄	—	134	1	no	—
93	C₂²×C₄	—	134	2	no	—
94	C₂²×C₄	—	134	1	no	—
95	C₂²×C₄	D₂₆₈	1	2	yes	—
96	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₄⋊C₄	—	134	2	no	—
101	C₄⋊C₄	—	134	2	no	—
102	C₄⋊C₄	—	134	2	no	—
103	C₂²⋊C₄	—	268	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	268	1	no	—
106	C₂³	—	268	1	no	—
107	C₂³	—	268	2	no	—
108	C₂³	—	268	2	no	—
109	C₂³	—	134	2	no	—
110	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
111	D₄	—	268	2	no	—
112	D₄	—	134	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	268	2	no	—
117	C₂×C₄	—	268	1	no	—
118	C₂×C₄	—	268	2	no	—
119	C₂×C₄	—	268	2	no	—
120	C₂×C₄	—	268	2	no	—
121	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
122	C₂×C₄	#11761627	1	2	yes	—
123	C₂×C₄	#11761628	1	2	yes	—
124	C₂×C₄	#11761629	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₂²	—	268	2	no	—
127	C₂²	—	268	4	no	—
128	C₂²	—	536	1	no	—
129	C₂²	—	536	1	no	—
130	C₂²	—	536	2	no	—
131	C₂²	#14524018	1	1	yes	characteristic
132	C₂²	#14524027	1	1	yes	characteristic
133	C₂²	#14524031	1	1	yes	characteristic
134	C₂²	#14524065	1	2	yes	—
135	C₂²	#14524071	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	536	1	no	—
138	C₄	—	536	2	no	—
139	C₄	#14524025	1	2	yes	—
140	C₄	#14524038	1	2	yes	—
141	C₂	—	536	2	no	—
142	C₂	#16863323	1	1	yes	characteristic
143	C₂	#16863612	1	2	yes	—
144	C₂	#16863721	1	2	yes	—
145	C₂	#16863751	1	1	yes	characteristic
146	C₂	#16863761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523965

C₂

yes

maximal

#16863298

C₂

yes

maximal, characteristic

#16863262

C₂

yes

maximal, characteristic, complemented

#16863159

C₂

yes

maximal, complemented

#336×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×(C₂×D₆₇)⋊C₄

—

#14523965

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

D₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#123

yes

—

C₄×C₆₇

#126

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#397

yes

characteristic

C₂×C₆₇

#463

yes

—

C₂×C₆₇

#482

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#2073

—

maximal, Sylow

C₆₇

#2073

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂²×D₄

—

C₂²×C₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#102

—

134

—

#95

—

134

—

C₂⁴

—

134

—

C₂×D₄

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

134

—

101

C₄⋊C₄

—

134

—

102

C₄⋊C₄

—

134

—

103

C₂²⋊C₄

—

268

—

104

C₄²

—

105

C₂³

—

268

—

106

C₂³

—

268

—

107

C₂³

—

268

—

108

C₂³

—

268

—

109

C₂³

—

134

—

110

C₂³

C₂×D₂₆₈

yes

characteristic

111

D₄

—

268

—

112

D₄

—

134

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

268

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

268

—

119

C₂×C₄

—

268

—

120

C₂×C₄

—

268

—

121

C₂×C₄

D₄×D₆₇

yes

characteristic

122

C₂×C₄

#11761627

yes

—

123

C₂×C₄

#11761628

yes

—

124

C₂×C₄

#11761629

yes

characteristic

125

C₈

—

126

C₂²

—

268

—

127

C₂²

—

268

—

128

C₂²

—

536

—

129

C₂²

—

536

—

130

C₂²

—

536

—

131

C₂²

#14524018

yes

characteristic

132

C₂²

#14524027

yes

characteristic

133

C₂²

#14524031

yes

characteristic

134

C₂²

#14524065

yes

—

135

C₂²

#14524071

yes

—

136

C₄

—

137

C₄

—

536

—

138

C₄

—

536

—

139

C₄

#14524025

yes

—

140

C₄

#14524038

yes

—

141

C₂

—

536

—

142

C₂

#16863323

yes

characteristic

143

C₂

#16863612

yes

—

144

C₂

#16863721

yes

—

145

C₂

#16863751

yes

characteristic

146

C₂

#16863761

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#336×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#336	D₆₇
hypocenter	C₆₇	#2073
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#336×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
3. #16863262	C₂	semidirect
4. #16863159	C₂	semidirect
5. #336×C₆₇	C₂	semidirect
74. C₆₇	#2073	semidirect
79. #336	D₆₇	semidirect

H ≅ G/N

type

3. #16863262

C₂

semidirect

4. #16863159

C₂

semidirect

5. #336×C₆₇

C₂

semidirect

74. C₆₇

#2073

semidirect

79. #336

D₆₇

semidirect

Subgroups — #22561624

Conjugacy classes of subgroups

Complements