Group #22559123 — subgroups

Subgroups — #22559123

Order 8576.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16863474	C₂	1	2	yes	maximal, complemented
2	#16863147	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#16863117	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	—	2	2	no	—
7	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
9	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₆₇	C₄	1	1	yes	characteristic
12	C₄×D₄×C₆₇	C₂²	1	2	yes	—
13	#107×C₆₇	C₄	1	1	yes	characteristic
14	#102×C₆₇	C₄	1	2	yes	—
15	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#11761612	—	4	4	no	—
17	C₂×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₆₇	—	2	2	no	—
22	D₈×C₆₇	—	2	2	no	—
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₆₇	—	2	2	no	—
27	C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₂³×C₆₇	—	2	2	no	—
29	D₄×C₆₇	—	2	2	no	—
30	D₄×C₆₇	—	2	2	no	—
31	C₂×C₄×C₆₇	—	2	2	no	—
32	C₂×C₄×C₆₇	—	4	2	no	—
33	C₂×C₄×C₆₇	—	4	2	no	—
34	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
37	C₈×C₆₇	—	2	1	no	—
38	C₈×C₆₇	M₁₆	1	2	yes	complemented
39	C₆₇⋊C₈	—	2	2	no	—
40	C₆₇⋊C₈	—	4	1	no	—
41	C₆₇⋊C₈	—	4	2	no	—
42	C₂²×C₆₇	—	4	2	no	—
43	C₂²×C₆₇	—	4	2	no	—
44	C₂²×C₆₇	—	4	2	no	—
45	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
46	C₄×C₆₇	—	2	1	no	—
47	C₄×C₆₇	—	4	2	no	—
48	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₆₇	#131	1	1	yes	characteristic
50	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₆₇	—	8	2	no	—
52	C₂×C₆₇	#436	1	1	yes	characteristic
53	C₂×C₆₇	#442	1	1	yes	characteristic
54	C₂×C₆₇	#444	1	1	yes	characteristic
55	#1594	—	67	1	no	maximal, Sylow
56	C₆₇	#1594	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	D₆₇	1	1	yes	characteristic
58	#436	—	67	2	no	—
59	#334	—	67	1	no	—
60	D₄⋊C₈	—	67	2	no	—
61	#322	—	67	1	no	—
62	C₂×D₈	Dic₆₇	1	1	yes	characteristic
63	C₂×M₁₆	—	134	2	no	—
64	C₄×D₄	C₂×D₆₇	1	2	yes	—
65	#107	Dic₆₇	1	1	yes	characteristic
66	C₄⋊C₈	—	67	2	no	—
67	#102	Dic₆₇	1	2	yes	—
68	C₂²⋊C₈	—	134	2	no	—
69	C₄×C₈	—	67	1	no	—
70	C₄×C₈	—	67	1	no	—
71	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	—	2	2	no	—
75	M₁₆	—	268	4	no	—
76	C₂×C₈	—	67	2	no	—
77	C₂×C₈	—	134	2	no	—
78	C₂×C₈	—	134	1	no	—
79	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₆₇	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
91	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
92	C₂×C₄	#11761627	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	268	1	no	—
95	C₈	—	268	2	no	—
96	C₈	—	134	2	no	—
97	C₈	#11761612	1	2	yes	complemented
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₆₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂×#11761612	1	1	yes	characteristic
105	C₄	#14524050	1	1	yes	characteristic
106	C₄	#14524067	1	2	yes	—
107	C₂	—	8	2	no	—
108	C₂	#16863474	1	1	yes	characteristic
109	C₂	#16863508	1	1	yes	characteristic
110	C₂	#16863518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863474

C₂

yes

maximal, complemented

#16863147

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, complemented

#16863117

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₄

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

—

#107×C₆₇

C₄

yes

characteristic

#102×C₆₇

C₄

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₄×D₈

D₆₇

yes

characteristic

#436

—

#334

—

D₄⋊C₈

—

#322

—

C₂×D₈

Dic₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₄×D₄

C₂×D₆₇

yes

—

#107

Dic₆₇

yes

characteristic

C₄⋊C₈

—

#102

Dic₆₇

yes

—

C₂²⋊C₈

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×D₄

C₂×Dic₆₇

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₄

D₄×D₆₇

yes

characteristic

C₂×C₄

#11761627

yes

characteristic

C₈

—

C₈

—

268

—

C₈

—

268

—

C₈

—

134

—

C₈

#11761612

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₆₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×#11761612

yes

characteristic

105

C₄

#14524050

yes

characteristic

106

C₄

#14524067

yes

—

107

C₂

—

108

C₂

#16863474

yes

characteristic

109

C₂

#16863508

yes

characteristic

110

C₂

#16863518

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	C₄×D₈	D₆₇
hypocenter	C₆₇	#1594
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

C₄×D₈×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

hypercenter

C₄×D₈

D₆₇

hypocenter

C₆₇

#1594

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #16863474	C₂	semidirect
2. #16863147	C₂	semidirect
3. D₄⋊(C₆₇⋊C₈)	C₂	semidirect
4. #16863117	C₂	semidirect
38. C₈×C₆₇	M₁₆	semidirect
56. C₆₇	#1594	semidirect
97. C₈	#11761612	semidirect

H ≅ G/N

type

1. #16863474

C₂

semidirect

2. #16863147

C₂

semidirect

3. D₄⋊(C₆₇⋊C₈)

C₂

semidirect

4. #16863117

C₂

semidirect

38. C₈×C₆₇

M₁₆

semidirect

56. C₆₇

#1594

semidirect

97. C₈

#11761612

semidirect

Subgroups — #22559123

Conjugacy classes of subgroups

Complements