Group #22561629 — subgroups

Subgroups — #22561629

Order 8576.

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

normal subgroups only

This group has 222 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×#14523968	C₂	1	2	yes	maximal
2	#16863302	C₂	1	1	yes	maximal, characteristic
3	#16863262	C₂	1	1	yes	maximal, characteristic
4	#16863159	C₂	1	2	yes	maximal
5	#336×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
9	#14523968	—	2	4	no	—
10	C₆₇⋊#95	—	2	2	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	C₂×Dic₁₃₄	—	2	2	no	—
15	C₂×Dic₁₃₄	—	2	2	no	—
16	C₂×Dic₁₃₄	—	2	1	no	—
17	C₂×Dic₁₃₄	D₄	1	2	yes	—
18	C₂²×Dic₆₇	—	2	1	no	—
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	C₂²×C₄×C₆₇	D₄	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₆₇	—	2	2	no	—
27	C₂×C₈×C₆₇	—	2	2	no	—
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₄²×C₆₇	—	2	2	no	—
30	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	—	2	2	no	—
32	C₂×C₄×C₆₇	—	2	2	no	—
33	C₂×C₄×C₆₇	—	2	2	no	—
34	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	#41	1	2	yes	—
36	C₂×C₄×C₆₇	#41	1	2	yes	—
37	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
38	C₂×Dic₆₇	—	2	2	no	—
39	C₂×Dic₆₇	—	4	1	no	—
40	C₂×Dic₆₇	—	4	1	no	—
41	C₂×Dic₆₇	—	4	2	no	—
42	C₂×Dic₆₇	—	4	1	no	—
43	C₂×Dic₆₇	—	4	2	no	—
44	C₂×Dic₆₇	—	4	2	no	—
45	C₂×Dic₆₇	—	4	2	no	—
46	Dic₁₃₄	—	2	4	no	—
47	Dic₁₃₄	—	4	2	no	—
48	C₈×C₆₇	—	4	2	no	—
49	C₂²×C₆₇	#121	1	1	yes	characteristic
50	C₂²×C₆₇	#123	1	1	yes	characteristic
51	C₂²×C₆₇	#124	1	1	yes	characteristic
52	C₂²×C₆₇	#135	1	2	yes	—
53	C₂²×C₆₇	#137	1	2	yes	—
54	C₄×C₆₇	—	4	2	no	—
55	C₄×C₆₇	#123	1	2	yes	—
56	C₄×C₆₇	#126	1	2	yes	—
57	Dic₆₇	—	4	2	no	—
58	Dic₆₇	—	8	2	no	—
59	Dic₆₇	—	8	1	no	—
60	C₂×C₆₇	#397	1	1	yes	characteristic
61	C₂×C₆₇	#464	1	2	yes	—
62	C₂×C₆₇	#485	1	2	yes	—
63	C₂×C₆₇	#488	1	1	yes	characteristic
64	C₂×C₆₇	#491	1	1	yes	characteristic
65	#2074	—	67	1	no	maximal, Sylow
66	C₆₇	#2074	1	1	yes	characteristic, complemented, Sylow
67	C₂×#103	—	67	2	no	—
68	#391	—	67	1	no	—
69	#382	—	67	1	no	—
70	#340	—	67	2	no	—
71	#336	D₆₇	1	1	yes	characteristic
72	C₂²×Q₈	—	67	1	no	—
73	C₂²×C₈	C₂×D₆₇	1	2	yes	—
74	C₂×C₄⋊C₄	—	67	1	no	—
75	C₂×C₄⋊C₄	—	67	2	no	—
76	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
77	#103	—	134	4	no	—
78	#95	—	134	2	no	—
79	#95	—	134	2	no	—
80	C₂×Q₈	—	67	2	no	—
81	C₂×Q₈	—	134	2	no	—
82	C₂×Q₈	—	134	2	no	—
83	C₂×Q₈	—	134	1	no	—
84	C₂²×C₄	—	134	2	no	—
85	C₂²×C₄	—	134	1	no	—
86	C₂²×C₄	—	134	1	no	—
87	C₂²×C₄	D₂₆₈	1	2	yes	—
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	134	2	no	—
93	C₄⋊C₄	—	134	2	no	—
94	C₄⋊C₄	—	134	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
97	Q₈	—	268	2	no	—
98	Q₈	—	134	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	268	1	no	—
103	C₂×C₄	—	268	1	no	—
104	C₂×C₄	—	268	2	no	—
105	C₂×C₄	—	268	1	no	—
106	C₂×C₄	—	268	2	no	—
107	C₂×C₄	—	268	2	no	—
108	C₂×C₄	—	268	2	no	—
109	C₂×C₄	—	134	2	no	—
110	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
111	C₂×C₄	#11761627	1	2	yes	—
112	C₂×C₄	#11761628	1	2	yes	—
113	C₂×C₄	#11761629	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#14524018	1	1	yes	characteristic
116	C₂²	#14524027	1	1	yes	characteristic
117	C₂²	#14524031	1	1	yes	characteristic
118	C₂²	#14524065	1	2	yes	—
119	C₂²	#14524076	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	268	2	no	—
122	C₄	—	536	1	no	—
123	C₄	—	536	2	no	—
124	C₄	#14524025	1	2	yes	—
125	C₄	#14524038	1	2	yes	—
126	C₂	#16863323	1	1	yes	characteristic
127	C₂	#16863619	1	2	yes	—
128	C₂	#16863738	1	2	yes	—
129	C₂	#16863751	1	1	yes	characteristic
130	C₂	#16863765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523968

C₂

yes

maximal

#16863302

C₂

yes

maximal, characteristic

#16863262

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal

#336×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

#14523968

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

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

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

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

#123

yes

—

C₄×C₆₇

#126

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#397

yes

characteristic

C₂×C₆₇

#464

yes

—

C₂×C₆₇

#485

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#103

—

134

—

#95

—

134

—

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

C₂³

C₂×D₂₆₈

yes

characteristic

Q₈

—

268

—

Q₈

—

134

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

268

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

268

—

106

C₂×C₄

—

268

—

107

C₂×C₄

—

268

—

108

C₂×C₄

—

268

—

109

C₂×C₄

—

134

—

110

C₂×C₄

D₄×D₆₇

yes

characteristic

111

C₂×C₄

#11761627

yes

—

112

C₂×C₄

#11761628

yes

—

113

C₂×C₄

#11761629

yes

characteristic

114

C₈

—

115

C₂²

#14524018

yes

characteristic

116

C₂²

#14524027

yes

characteristic

117

C₂²

#14524031

yes

characteristic

118

C₂²

#14524065

yes

—

119

C₂²

#14524076

yes

—

120

C₄

—

121

C₄

—

268

—

122

C₄

—

536

—

123

C₄

—

536

—

124

C₄

#14524025

yes

—

125

C₄

#14524038

yes

—

126

C₂

#16863323

yes

characteristic

127

C₂

#16863619

yes

—

128

C₂

#16863738

yes

—

129

C₂

#16863751

yes

characteristic

130

C₂

#16863765

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	#2074
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
66. C₆₇	#2074	semidirect

H ≅ G/N

type

66. C₆₇

#2074

semidirect

Subgroups — #22561629

Conjugacy classes of subgroups

Complements