Group #22561182 — subgroups

Subgroups — #22561182

Order 8576.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14524043	C₂	1	1	yes	maximal, characteristic
2	#16863262	C₂	1	1	yes	maximal, characteristic
3	#16863159	C₂	1	4	yes	maximal
4	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₃₄	—	2	1	no	—
6	#14524043	—	2	2	no	—
7	#14524043	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₆₇	—	2	1	no	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	C₆₇⋊#95	—	2	1	no	—
12	C₆₇⋊#95	—	2	1	no	—
13	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#97×C₆₇	—	2	2	no	—
16	C₂×Dic₁₃₄	—	2	4	no	—
17	C₂×Dic₁₃₄	—	4	4	no	—
18	C₂²×Dic₆₇	—	2	2	no	—
19	C₂²×Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	4	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	4	no	—
23	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
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	Q₈	1	4	yes	complemented
30	C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₄²×C₆₇	D₄	1	2	yes	—
32	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	—	2	2	no	—
34	C₂×C₄×C₆₇	—	2	2	no	—
35	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₆₇	#41	1	2	yes	—
41	C₂×Dic₆₇	—	2	4	no	—
42	C₂×Dic₆₇	—	4	4	no	—
43	C₂×Dic₆₇	—	4	2	no	—
44	C₂×Dic₆₇	—	4	2	no	—
45	C₂×Dic₆₇	—	4	2	no	—
46	Dic₁₃₄	—	4	8	no	—
47	C₈×C₆₇	—	2	4	no	—
48	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₆₇	#124	1	1	yes	characteristic
50	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₆₇	#137	1	2	yes	—
53	C₄×C₆₇	—	2	2	no	—
54	C₄×C₆₇	—	2	2	no	—
55	C₄×C₆₇	C₄×Q₈	1	2	yes	—
56	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
57	Dic₆₇	—	4	4	no	—
58	Dic₆₇	—	8	2	no	—
59	C₂×C₆₇	#391	1	1	yes	characteristic
60	C₂×C₆₇	#418	1	1	yes	characteristic
61	C₂×C₆₇	#419	1	1	yes	characteristic
62	C₂×C₆₇	#489	1	2	yes	—
63	C₂×C₆₇	#501	1	2	yes	—
64	#1993	—	67	1	no	maximal, Sylow
65	C₆₇	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	67	1	no	—
67	C₂×#97	D₆₇	1	1	yes	characteristic
68	#382	—	67	1	no	—
69	#340	—	67	4	no	—
70	C₂²×Q₈	—	134	1	no	—
71	C₂²×C₈	C₂×D₆₇	1	2	yes	—
72	C₄⋊Q₈	—	67	4	no	—
73	C₄⋊Q₈	—	134	2	no	—
74	C₂×C₄⋊C₄	—	67	2	no	—
75	C₂×C₄⋊C₄	—	67	2	no	—
76	C₂×C₄⋊C₄	—	134	1	no	—
77	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	134	1	no	—
80	#95	—	134	1	no	—
81	C₂×Q₈	—	268	4	no	—
82	C₂×Q₈	—	134	4	no	—
83	C₂²×C₄	—	134	2	no	—
84	C₂²×C₄	—	134	2	no	—
85	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
86	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₁₃₄	1	4	yes	complemented
91	C₄⋊C₄	—	67	4	no	—
92	C₄⋊C₄	—	134	4	no	—
93	C₄⋊C₄	—	134	4	no	—
94	C₄⋊C₄	—	134	2	no	—
95	C₄²	C₄×D₆₇	1	2	yes	—
96	C₄²	C₂²⋊D₆₇	1	2	yes	—
97	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
98	Q₈	—	268	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	268	4	no	—
102	C₂×C₄	—	268	2	no	—
103	C₂×C₄	—	268	2	no	—
104	C₂×C₄	—	268	2	no	—
105	C₂×C₄	—	134	4	no	—
106	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
111	C₂×C₄	#11761628	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
114	C₂²	#14524031	1	1	yes	characteristic
115	C₂²	#14524043	1	1	yes	characteristic
116	C₂²	#14524071	1	2	yes	—
117	C₂²	#14524076	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	268	4	no	—
121	C₄	—	536	2	no	—
122	C₄	C₄×Dic₁₃₄	1	2	yes	—
123	C₄	#14524022	1	2	yes	—
124	C₂	#16863302	1	1	yes	characteristic
125	C₂	#16863402	1	1	yes	characteristic
126	C₂	#16863406	1	1	yes	characteristic
127	C₂	#16863756	1	2	yes	—
128	C₂	#16863805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

#16863262

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal

C₂×#97×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

#14524043

—

#14524043

C₄

yes

complemented

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#97×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

Q₈

yes

complemented

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#124

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#501

yes

—

#1993

—

maximal, Sylow

C₆₇

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#97

D₆₇

yes

characteristic

#382

—

#340

—

C₂²×Q₈

—

134

—

C₂²×C₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#97

—

#95

—

134

—

#95

—

134

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₃₄

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

C₄×D₆₇

yes

—

C₄²

C₂²⋊D₆₇

yes

—

C₂³

C₂×D₂₆₈

yes

characteristic

Q₈

—

268

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

268

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

268

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

134

—

106

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

110

C₂×C₄

C₂×Dic₁₃₄

yes

—

111

C₂×C₄

#11761628

yes

—

112

C₈

—

113

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

114

C₂²

#14524031

yes

characteristic

115

C₂²

#14524043

yes

characteristic

116

C₂²

#14524071

yes

—

117

C₂²

#14524076

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

268

—

121

C₄

—

536

—

122

C₄

C₄×Dic₁₃₄

yes

—

123

C₄

#14524022

yes

—

124

C₂

#16863302

yes

characteristic

125

C₂

#16863402

yes

characteristic

126

C₂

#16863406

yes

characteristic

127

C₂

#16863756

yes

—

128

C₂

#16863805

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#97	D₆₇
hypocenter	C₆₇	#1993
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#97×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×#97

D₆₇

hypocenter

C₆₇

#1993

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #14524043	C₄	semidirect
29. C₂×C₈×C₆₇	Q₈	semidirect
65. C₆₇	#1993	semidirect
90. C₂×C₈	Dic₁₃₄	semidirect

H ≅ G/N

type

7. #14524043

C₄

semidirect

29. C₂×C₈×C₆₇

Q₈

semidirect

65. C₆₇

#1993

semidirect

90. C₂×C₈

Dic₁₃₄

semidirect

Subgroups — #22561182

Conjugacy classes of subgroups

Complements