Group #22562523 — subgroups

Subgroups — #22562523

Order 8576.

112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 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	#16863829	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863233	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₁₄₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863231	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863229	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14524066	—	2	1	no	—
9	D₈×D₆₇	—	2	1	no	—
10	#14523986	—	2	1	no	—
11	D₁₀₇₂	C₂²	1	1	yes	characteristic
12	D₁₀₇₂	C₂²	1	1	yes	characteristic, complemented
13	#14523984	—	2	1	no	—
14	C₁₆×D₆₇	C₂²	1	1	yes	characteristic
15	C₆₇⋊C₃₂	C₂²	1	1	yes	characteristic, complemented
16	Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	D₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	#11761629	—	4	1	no	—
20	D₄×D₆₇	—	4	1	no	—
21	#11761615	—	4	1	no	—
22	D₅₃₆	—	2	1	no	—
23	D₅₃₆	—	2	1	no	—
24	#11761613	—	4	1	no	—
25	C₈×D₆₇	D₄	1	1	yes	characteristic
26	C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
27	Q₁₆×C₆₇	—	2	1	no	—
28	D₈×C₆₇	—	2	1	no	—
29	C₁₆×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×D₆₇	—	8	1	no	—
31	Q₈×C₆₇	—	4	1	no	—
32	D₄×C₆₇	—	4	1	no	—
33	C₂²⋊D₆₇	—	8	1	no	—
34	D₂₆₈	—	4	1	no	—
35	D₂₆₈	—	4	1	no	—
36	D₂₆₈	—	8	1	no	—
37	C₄×D₆₇	—	8	1	no	—
38	C₄×D₆₇	D₈	1	1	yes	characteristic
39	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₆₇⋊C₈	D₈	1	1	yes	characteristic
41	C₂²×C₆₇	—	8	1	no	—
42	C₂×D₆₇	—	8	1	no	—
43	C₂×D₆₇	—	8	1	no	—
44	C₂×D₆₇	—	16	2	no	—
45	C₂×D₆₇	D₁₆	1	1	yes	characteristic
46	C₄×C₆₇	—	8	1	no	—
47	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
48	Dic₆₇	D₁₆	1	1	yes	characteristic
49	C₂×C₆₇	—	16	1	no	—
50	C₂×C₆₇	C₂×D₁₆	1	1	yes	characteristic
51	D₆₇	—	2	1	no	—
52	D₆₇	—	16	1	no	—
53	D₆₇	—	16	1	no	—
54	#2263	—	67	1	no	maximal, Sylow
55	C₆₇	#2263	1	1	yes	characteristic, complemented, Sylow
56	#508	—	67	1	no	—
57	C₂×D₁₆	—	67	1	no	—
58	SD₆₄	—	67	1	no	—
59	SD₆₄	D₆₇	1	1	yes	characteristic, complemented
60	D₃₂	—	67	1	no	—
61	D₃₂	—	67	1	no	—
62	M₆₄	—	67	1	no	—
63	#135	—	134	1	no	—
64	C₂×D₈	—	134	1	no	—
65	Q₃₂	C₂×D₆₇	1	1	yes	characteristic, complemented
66	SD₃₂	—	134	1	no	—
67	D₁₆	—	67	1	no	—
68	D₁₆	—	67	1	no	—
69	D₁₆	—	134	1	no	—
70	D₁₆	C₂×D₆₇	1	1	yes	characteristic
71	C₂×C₁₆	—	67	1	no	—
72	C₃₂	—	67	1	no	—
73	C₃₂	C₂×D₆₇	1	1	yes	characteristic, complemented
74	#41	—	268	1	no	—
75	C₂×D₄	—	268	1	no	—
76	Q₁₆	—	2	1	no	—
77	SD₁₆	—	268	1	no	—
78	D₈	—	2	1	no	—
79	D₈	—	268	1	no	—
80	D₈	—	134	1	no	—
81	D₈	—	134	1	no	—
82	C₂×C₈	—	67	1	no	—
83	C₁₆	—	67	1	no	—
84	C₁₆	C₂²×D₆₇	1	1	yes	characteristic
85	C₂³	—	536	1	no	—
86	Q₈	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	268	1	no	—
89	D₄	—	268	1	no	—
90	D₄	—	536	1	no	—
91	D₄	—	536	1	no	—
92	C₂×C₄	—	536	1	no	—
93	C₂×C₄	—	67	1	no	—
94	C₈	—	67	1	no	—
95	C₈	D₄×D₆₇	1	1	yes	characteristic
96	C₂²	—	8	1	no	—
97	C₂²	—	536	1	no	—
98	C₂²	—	536	1	no	—
99	C₂²	—	1072	2	no	—
100	C₂²	—	67	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	67	1	no	—
103	C₄	D₈×D₆₇	1	1	yes	characteristic
104	C₂	—	16	1	no	—
105	C₂	—	1072	1	no	—
106	C₂	—	1072	1	no	—
107	C₂	—	134	1	no	—
108	C₂	D₁₆×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863829

C₂

yes

maximal, characteristic, complemented

D₁₆×D₆₇

C₂

yes

maximal, characteristic, complemented

#16863233

C₂

yes

maximal, characteristic, complemented

D₂₁₄₄

C₂

yes

maximal, characteristic, complemented

#16863231

C₂

yes

maximal, characteristic, complemented

#16863229

C₂

yes

maximal, characteristic, complemented

SD₆₄×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524066

—

D₈×D₆₇

—

#14523986

—

D₁₀₇₂

C₂²

yes

characteristic

D₁₀₇₂

C₂²

yes

characteristic, complemented

#14523984

—

C₁₆×D₆₇

C₂²

yes

characteristic

C₆₇⋊C₃₂

C₂²

yes

characteristic, complemented

Q₃₂×C₆₇

C₂²

yes

characteristic, complemented

D₁₆×C₆₇

C₂²

yes

characteristic

C₃₂×C₆₇

C₂²

yes

characteristic, complemented

#11761629

—

D₄×D₆₇

—

#11761615

—

D₅₃₆

—

D₅₃₆

—

#11761613

—

C₈×D₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

D₄

yes

characteristic

Q₁₆×C₆₇

—

D₈×C₆₇

—

C₁₆×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

C₂²⋊D₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

D₈

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₆₇⋊C₈

D₈

yes

characteristic

C₂²×C₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

D₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

characteristic

Dic₆₇

D₁₆

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

C₂×D₁₆

yes

characteristic

D₆₇

—

D₆₇

—

D₆₇

—

#2263

—

maximal, Sylow

C₆₇

#2263

yes

characteristic, complemented, Sylow

#508

—

C₂×D₁₆

—

SD₆₄

—

SD₆₄

D₆₇

yes

characteristic, complemented

D₃₂

—

D₃₂

—

M₆₄

—

#135

—

134

—

C₂×D₈

—

134

—

Q₃₂

C₂×D₆₇

yes

characteristic, complemented

SD₃₂

—

134

—

D₁₆

—

D₁₆

—

D₁₆

—

134

—

D₁₆

C₂×D₆₇

yes

characteristic

C₂×C₁₆

—

C₃₂

—

C₃₂

C₂×D₆₇

yes

characteristic, complemented

#41

—

268

—

C₂×D₄

—

268

—

Q₁₆

—

SD₁₆

—

268

—

D₈

—

D₈

—

268

—

D₈

—

134

—

D₈

—

134

—

C₂×C₈

—

C₁₆

—

C₁₆

C₂²×D₆₇

yes

characteristic

C₂³

—

536

—

Q₈

—

D₄

—

D₄

—

268

—

D₄

—

268

—

D₄

—

536

—

D₄

—

536

—

C₂×C₄

—

536

—

C₂×C₄

—

C₈

—

C₈

D₄×D₆₇

yes

characteristic

C₂²

—

C₂²

—

536

—

C₂²

—

536

—

C₂²

—

1072

—

100

C₂²

—

101

C₄

—

102

C₄

—

103

C₄

D₈×D₆₇

yes

characteristic

104

C₂

—

105

C₂

—

1072

—

106

C₂

—

1072

—

107

C₂

—

134

—

108

C₂

D₁₆×D₆₇

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×D₆₇
commutator	C₁₆×C₆₇	C₂³
Frattini	C₁₆	C₂²×D₆₇
Fitting	SD₆₄×C₆₇	C₂
socle	C₂×C₆₇	C₂×D₁₆
hypercenter	SD₆₄	D₆₇
hypocenter	C₆₇	#2263
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

subgroup

G/H

center

C₂

D₁₆×D₆₇

commutator

C₁₆×C₆₇

C₂³

Frattini

C₁₆

C₂²×D₆₇

Fitting

SD₆₄×C₆₇

C₂

socle

C₂×C₆₇

C₂×D₁₆

hypercenter

SD₆₄

D₆₇

hypocenter

C₆₇

#2263

LCS

G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

N	H ≅ G/N	type
1. #16863829	C₂	semidirect
2. D₁₆×D₆₇	C₂	semidirect
3. #16863233	C₂	semidirect
4. D₂₁₄₄	C₂	semidirect
5. #16863231	C₂	semidirect
6. #16863229	C₂	semidirect
7. SD₆₄×C₆₇	C₂	semidirect
12. D₁₀₇₂	C₂²	semidirect
15. C₆₇⋊C₃₂	C₂²	semidirect
16. Q₃₂×C₆₇	C₂²	semidirect
18. C₃₂×C₆₇	C₂²	semidirect
55. C₆₇	#2263	semidirect
59. SD₆₄	D₆₇	semidirect
65. Q₃₂	C₂×D₆₇	semidirect
73. C₃₂	C₂×D₆₇	semidirect

H ≅ G/N

type

1. #16863829

C₂

semidirect

2. D₁₆×D₆₇

C₂

semidirect

3. #16863233

C₂

semidirect

4. D₂₁₄₄

C₂

semidirect

5. #16863231

C₂

semidirect

6. #16863229

C₂

semidirect

7. SD₆₄×C₆₇

C₂

semidirect

12. D₁₀₇₂

C₂²

semidirect

15. C₆₇⋊C₃₂

C₂²

semidirect

16. Q₃₂×C₆₇

C₂²

semidirect

18. C₃₂×C₆₇

C₂²

semidirect

55. C₆₇

#2263

semidirect

59. SD₆₄

D₆₇

semidirect

65. Q₃₂

C₂×D₆₇

semidirect

73. C₃₂

C₂×D₆₇

semidirect

Subgroups — #22562523

Conjugacy classes of subgroups

Complements