Group #22560960 — subgroups

Subgroups — #22560960

Order 8576.

200 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16863434	C₂	1	1	yes	maximal, characteristic
2	#16863424	C₂	1	2	yes	maximal
3	C₂×#14523953	C₂	1	1	yes	maximal, characteristic
4	#16863157	C₂	1	2	yes	maximal
5	C₂×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	#14524002	C₂²	1	2	yes	—
8	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	#14523977	C₄	1	2	yes	complemented
10	#14523975	C₄	1	2	yes	complemented
11	Dic₆₇⋊C₈	—	2	4	no	—
12	#14523953	—	2	2	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#108×C₆₇	—	2	2	no	—
16	C₂²×Dic₆₇	—	2	1	no	—
17	C₂×C₆₇⋊C₈	—	2	1	no	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₆₇	—	2	2	no	—
22	C₄×Dic₆₇	—	2	1	no	—
23	C₄×Dic₆₇	—	2	2	no	—
24	C₄×Dic₆₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₆₇	—	2	2	no	—
27	M₁₆×C₆₇	—	2	2	no	—
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	D₄	1	2	yes	—
32	C₂×C₈×C₆₇	Q₈	1	2	yes	—
33	C₂³×C₆₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
40	C₂×Dic₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	2	2	no	—
42	C₂×Dic₆₇	—	4	2	no	—
43	C₈×C₆₇	—	4	2	no	—
44	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₆₇⋊C₈	—	2	2	no	—
47	C₂²×C₆₇	—	2	2	no	—
48	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₆₇	#125	1	1	yes	characteristic
51	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₆₇	#121	1	1	yes	characteristic
54	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₆₇	—	4	2	no	—
56	Dic₆₇	—	4	4	no	—
57	C₂×C₆₇	—	2	1	no	—
58	C₂×C₆₇	—	2	1	no	—
59	C₂×C₆₇	#389	1	1	yes	characteristic
60	C₂×C₆₇	#444	1	2	yes	—
61	#1946	—	67	1	no	maximal, Sylow
62	C₆₇	#1946	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	D₆₇	1	1	yes	characteristic
64	#427	—	67	1	no	—
65	#424	—	67	2	no	—
66	C₂×#97	—	67	1	no	—
67	#339	—	67	2	no	—
68	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
69	C₂²×C₈	—	67	1	no	—
70	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
71	#117	—	67	2	no	—
72	C₂×C₄²	—	67	1	no	—
73	#108	—	2	2	no	—
74	#107	—	67	2	no	—
75	#106	—	67	2	no	—
76	C₄⋊C₈	—	134	4	no	—
77	#97	—	134	2	no	—
78	C₂²×C₄	—	134	1	no	—
79	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	67	2	no	—
84	C₂×C₈	—	134	1	no	—
85	C₂×C₈	—	134	1	no	—
86	C₂×C₈	Dic₁₃₄	1	2	yes	—
87	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
88	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
89	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
90	C₄⋊C₄	—	67	4	no	—
91	C₂²⋊C₄	—	134	2	no	—
92	C₄²	—	67	2	no	—
93	C₄²	—	134	2	no	—
94	C₄²	—	134	1	no	—
95	C₂³	#11761629	1	1	yes	characteristic
96	C₂×C₄	—	268	2	no	—
97	C₂×C₄	—	134	4	no	—
98	C₂×C₄	—	134	2	no	—
99	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
104	C₂×C₄	#11761627	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	134	2	no	—
107	C₈	Dic₆₇⋊C₄	1	2	yes	—
108	C₈	Dic₆₇⋊C₄	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#14524005	1	1	yes	characteristic
111	C₂²	#14524019	1	1	yes	characteristic
112	C₂²	#14524034	1	1	yes	characteristic
113	C₄	—	268	2	no	—
114	C₄	—	268	4	no	—
115	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
116	C₄	#14524008	1	1	yes	characteristic
117	C₄	#14524014	1	1	yes	characteristic
118	C₄	#14524042	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#16863292	1	1	yes	characteristic
122	C₂	#16863517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863434

C₂

yes

maximal, characteristic

#16863424

C₂

yes

maximal

C₂×#14523953

C₂

yes

maximal, characteristic

#16863157

C₂

yes

maximal

C₂×#108×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523977

C₄

yes

complemented

#14523975

C₄

yes

complemented

Dic₆₇⋊C₈

—

#14523953

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#108×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

—

C₈×C₆₇

C₄⋊C₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#125

yes

characteristic

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#389

yes

characteristic

C₂×C₆₇

#444

yes

—

#1946

—

maximal, Sylow

C₆₇

#1946

yes

characteristic, complemented, Sylow

C₂×#108

D₆₇

yes

characteristic

#427

—

#424

—

C₂×#97

—

#339

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

C₂×C₄²

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

134

—

#97

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

134

—

C₄²

—

C₄²

—

134

—

C₄²

—

134

—

C₂³

#11761629

yes

characteristic

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

100

C₂×C₄

D₄×D₆₇

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

102

C₂×C₄

Q₈×D₆₇

yes

characteristic

103

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

104

C₂×C₄

#11761627

yes

characteristic

105

C₈

—

106

C₈

—

134

—

107

C₈

Dic₆₇⋊C₄

yes

—

108

C₈

Dic₆₇⋊C₄

yes

—

109

C₂²

—

110

C₂²

#14524005

yes

characteristic

111

C₂²

#14524019

yes

characteristic

112

C₂²

#14524034

yes

characteristic

113

C₄

—

268

—

114

C₄

—

268

—

115

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

116

C₄

#14524008

yes

characteristic

117

C₄

#14524014

yes

characteristic

118

C₄

#14524042

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#16863292

yes

characteristic

122

C₂

#16863517

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	C₂×#108	D₆₇
hypocenter	C₆₇	#1946
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#108×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #14523977	C₄	semidirect
10. #14523975	C₄	semidirect
62. C₆₇	#1946	semidirect

H ≅ G/N

type

9. #14523977

C₄

semidirect

10. #14523975

C₄

semidirect

62. C₆₇

#1946

semidirect

Subgroups — #22560960

Conjugacy classes of subgroups

Complements