Group #22559886 — subgroups

Subgroups — #22559886

Order 8576.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16863801	C₂	1	1	yes	maximal, characteristic
2	#16863801	C₂	1	1	yes	maximal, characteristic
3	#16863794	C₂	1	1	yes	maximal, characteristic
4	#16863776	C₂	1	1	yes	maximal, characteristic
5	#16863747	C₂	1	1	yes	maximal, characteristic
6	#16863747	C₂	1	1	yes	maximal, characteristic
7	#322×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₈	—	2	2	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14524043	C₂²	1	1	yes	characteristic
11	#14524043	C₂²	1	1	yes	characteristic
12	#14524043	C₂²	1	1	yes	characteristic
13	#14523977	—	2	2	no	—
14	#14523977	—	2	2	no	—
15	#14523975	—	2	2	no	—
16	#14523968	—	2	2	no	—
17	#14523968	—	2	2	no	—
18	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂×Dic₁₃₄	—	2	1	no	—
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	—	2	1	no	—
25	Dic₂₆₈	—	4	4	no	—
26	C₄⋊Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	D₄	1	2	yes	—
35	C₂×C₈×C₆₇	D₄	1	2	yes	—
36	C₂×C₈×C₆₇	D₄	1	2	yes	—
37	C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₆₇	—	4	1	no	—
42	C₂×Dic₆₇	—	4	1	no	—
43	C₂×Dic₆₇	—	4	1	no	—
44	C₂×Dic₆₇	—	4	1	no	—
45	Dic₁₃₄	—	4	2	no	—
46	Dic₁₃₄	—	4	2	no	—
47	Dic₁₃₄	—	4	2	no	—
48	Dic₁₃₄	—	4	2	no	—
49	C₈×C₆₇	—	2	2	no	—
50	C₈×C₆₇	—	2	2	no	—
51	C₈×C₆₇	Q₁₆	1	4	yes	complemented
52	C₂²×C₆₇	#127	1	1	yes	characteristic
53	C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
54	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₆₇	#137	1	2	yes	—
56	Dic₆₇	—	8	1	no	—
57	Dic₆₇	—	8	1	no	—
58	Dic₆₇	—	8	1	no	—
59	Dic₆₇	—	8	1	no	—
60	C₂×C₆₇	#494	1	1	yes	characteristic
61	C₂×C₆₇	#496	1	1	yes	characteristic
62	C₂×C₆₇	#497	1	1	yes	characteristic
63	#1723	—	67	1	no	maximal, Sylow
64	C₆₇	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	67	1	no	—
66	#500	—	67	1	no	—
67	#498	—	67	1	no	—
68	#494	—	67	1	no	—
69	#487	—	67	1	no	—
70	#487	—	67	1	no	—
71	#322	D₆₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	134	2	no	—
73	C₄⋊Q₈	—	67	1	no	—
74	C₄⋊Q₈	—	67	1	no	—
75	C₄⋊Q₈	—	67	1	no	—
76	C₄⋊Q₈	—	67	1	no	—
77	#107	—	134	2	no	—
78	#107	—	134	2	no	—
79	#106	—	134	2	no	—
80	#103	—	134	2	no	—
81	#103	—	134	2	no	—
82	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
83	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
84	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
85	C₂×Q₈	—	134	1	no	—
86	C₂×Q₈	—	134	1	no	—
87	C₂×Q₈	—	134	1	no	—
88	C₂×Q₈	—	134	1	no	—
89	Q₁₆	—	268	4	no	—
90	C₂×C₈	D₂₆₈	1	2	yes	—
91	C₂×C₈	D₂₆₈	1	2	yes	—
92	C₂×C₈	D₂₆₈	1	2	yes	—
93	C₄⋊C₄	—	134	1	no	—
94	C₄⋊C₄	—	134	1	no	—
95	C₄⋊C₄	—	134	1	no	—
96	C₄⋊C₄	—	134	1	no	—
97	C₄⋊C₄	—	134	1	no	—
98	C₄⋊C₄	—	134	1	no	—
99	C₄⋊C₄	—	134	1	no	—
100	C₄⋊C₄	—	134	1	no	—
101	C₄²	C₂²×D₆₇	1	1	yes	characteristic
102	Q₈	—	268	2	no	—
103	Q₈	—	268	2	no	—
104	Q₈	—	268	2	no	—
105	Q₈	—	268	2	no	—
106	C₂×C₄	—	268	1	no	—
107	C₂×C₄	—	268	1	no	—
108	C₂×C₄	—	268	1	no	—
109	C₂×C₄	—	268	1	no	—
110	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
111	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
112	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₂₆₈	1	4	yes	complemented
116	C₂²	#14524040	1	1	yes	characteristic
117	C₄	—	536	1	no	—
118	C₄	—	536	1	no	—
119	C₄	—	536	1	no	—
120	C₄	—	536	1	no	—
121	C₄	C₂×Dic₂₆₈	1	2	yes	—
122	C₄	#14524071	1	2	yes	—
123	C₄	#14524076	1	2	yes	—
124	C₂	#16863776	1	1	yes	characteristic
125	C₂	#16863785	1	1	yes	characteristic
126	C₂	#16863790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863801

C₂

yes

maximal, characteristic

#16863801

C₂

yes

maximal, characteristic

#16863794

C₂

yes

maximal, characteristic

#16863776

C₂

yes

maximal, characteristic

#16863747

C₂

yes

maximal, characteristic

#16863747

C₂

yes

maximal, characteristic

#322×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

—

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

—

—

—

—

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

Q₁₆

yes

complemented

C₂²×C₆₇

#127

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

#137

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#107

—

134

—

#107

—

134

—

#106

—

134

—

#103

—

134

—

#103

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

Q₁₆

—

268

—

C₂×C₈

D₂₆₈

yes

—

C₂×C₈

D₂₆₈

yes

—

C₂×C₈

D₂₆₈

yes

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

100

C₄⋊C₄

—

134

—

101

C₄²

C₂²×D₆₇

yes

characteristic

102

Q₈

—

268

—

103

Q₈

—

268

—

104

Q₈

—

268

—

105

Q₈

—

268

—

106

C₂×C₄

—

268

—

107

C₂×C₄

—

268

—

108

C₂×C₄

—

268

—

109

C₂×C₄

—

268

—

110

C₂×C₄

C₂×D₂₆₈

yes

characteristic

111

C₂×C₄

C₂×D₂₆₈

yes

characteristic

112

C₂×C₄

C₂×D₂₆₈

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₂₆₈

yes

complemented

116

C₂²

#14524040

yes

characteristic

117

C₄

—

536

—

118

C₄

—

536

—

119

C₄

—

536

—

120

C₄

—

536

—

121

C₄

C₂×Dic₂₆₈

yes

—

122

C₄

#14524071

yes

—

123

C₄

#14524076

yes

—

124

C₂

#16863776

yes

characteristic

125

C₂

#16863785

yes

characteristic

126

C₂

#16863790

yes

characteristic

	subgroup	G/H
center	C₂²	#14524040
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#322×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#322	D₆₇
hypocenter	C₆₇	#1723
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#14524040

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#322×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₆₇	Q₁₆	semidirect
64. C₆₇	#1723	semidirect
115. C₈	Dic₂₆₈	semidirect

H ≅ G/N

type

51. C₈×C₆₇

Q₁₆

semidirect

64. C₆₇

#1723

semidirect

115. C₈

Dic₂₆₈

semidirect

Subgroups — #22559886

Conjugacy classes of subgroups

Complements