Group #22586324 — subgroups

Subgroups — #22586324

Order 9088.

114 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 31 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	#17986252	C₂	1	1	yes	maximal, characteristic
2	#17985781	C₂	1	1	yes	maximal, characteristic, complemented
3	H(3,ℤ/4)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₆₈	C₄	1	1	yes	characteristic, complemented
5	#14556102	—	2	1	no	—
6	#14556068	—	2	1	no	—
7	#14556060	C₄	1	1	yes	characteristic, complemented
8	#14556050	—	2	1	no	—
9	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	#117×C₇₁	—	2	1	no	—
11	C₂×D₂₈₄	—	2	1	no	—
12	C₂×C₂²⋊D₇₁	—	4	1	no	—
13	D₅₆₈	—	4	1	no	—
14	C₄⋊Dic₇₁	—	2	1	no	—
15	(C₂×D₇₁)⋊C₄	—	4	1	no	—
16	C₇₁⋊C₁₆	—	4	1	no	—
17	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
18	M₁₆×C₇₁	—	4	1	no	—
19	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₇₁	—	4	1	no	—
22	C₂²⋊C₄×C₇₁	—	4	1	no	—
23	C₄²×C₇₁	—	2	2	no	—
24	C₂³×C₇₁	D₈	1	1	yes	characteristic
25	C₂²×D₇₁	—	4	1	no	—
26	C₂×C₄×C₇₁	—	2	1	no	—
27	C₂×C₄×C₇₁	—	2	1	no	—
28	C₂×C₄×C₇₁	—	4	2	no	—
29	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
31	C₂²⋊D₇₁	—	8	2	no	—
32	C₂×Dic₇₁	—	4	1	no	—
33	D₂₈₄	—	4	1	no	—
34	D₂₈₄	—	4	1	no	—
35	C₈×C₇₁	—	2	1	no	—
36	C₈×C₇₁	—	2	1	no	—
37	C₂²×C₇₁	—	2	1	no	—
38	C₂²×C₇₁	—	4	1	no	—
39	C₂²×C₇₁	#102	1	1	yes	characteristic
40	C₂×D₇₁	—	8	1	no	—
41	C₂×D₇₁	—	8	1	no	—
42	C₂×D₇₁	—	8	1	no	—
43	C₄×C₇₁	—	2	1	no	—
44	C₄×C₇₁	—	8	2	no	—
45	C₄×C₇₁	#99	1	1	yes	characteristic
46	C₄×C₇₁	#104	1	1	yes	characteristic
47	Dic₇₁	—	8	1	no	—
48	C₂×C₇₁	—	2	1	no	—
49	C₂×C₇₁	—	4	1	no	—
50	C₂×C₇₁	#327	1	1	yes	characteristic
51	D₇₁	—	16	1	no	—
52	#1339	—	71	1	no	maximal, Sylow
53	C₇₁	#1339	1	1	yes	characteristic, complemented, Sylow
54	#469	—	71	1	no	—
55	#349	—	71	1	no	—
56	H(3,ℤ/4)	D₇₁	1	1	yes	characteristic, complemented
57	C₂×D₈	—	71	1	no	—
58	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
59	#121	—	142	1	no	—
60	#117	—	2	1	no	—
61	M₃₂	—	142	1	no	—
62	#106	—	71	1	no	—
63	#102	—	142	1	no	—
64	C₂×D₄	—	284	1	no	—
65	C₂×D₄	—	142	1	no	—
66	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
67	D₈	—	284	1	no	—
68	M₁₆	—	4	1	no	—
69	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
70	C₂×C₈	D₂₈₄	1	1	yes	characteristic
71	C₄⋊C₄	—	4	1	no	—
72	C₄⋊C₄	—	142	1	no	—
73	C₂²⋊C₄	—	4	1	no	—
74	C₂²⋊C₄	—	284	1	no	—
75	C₄²	—	2	2	no	—
76	C₁₆	—	284	1	no	—
77	C₂³	—	284	1	no	—
78	C₂³	#11765850	1	1	yes	characteristic
79	D₄	—	284	1	no	—
80	D₄	—	284	1	no	—
81	D₄	—	568	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	284	1	no	—
86	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	#11765852	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	568	1	no	—
93	C₂²	—	568	1	no	—
94	C₂²	—	568	1	no	—
95	C₂²	#14556048	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	8	2	no	—
98	C₄	—	568	1	no	—
99	C₄	#14556042	1	1	yes	characteristic
100	C₄	#14556056	1	1	yes	characteristic
101	C₂	—	2	1	no	—
102	C₂	—	4	1	no	—
103	C₂	—	1136	1	no	—
104	C₂	#17985732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986252

C₂

yes

maximal, characteristic

#17985781

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₄

yes

characteristic, complemented

#14556102

—

#14556068

—

#14556060

C₄

yes

characteristic, complemented

#14556050

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

—

C₂×D₂₈₄

—

C₂×C₂²⋊D₇₁

—

D₅₆₈

—

C₄⋊Dic₇₁

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

D₈

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#102

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#327

yes

characteristic

D₇₁

—

#1339

—

maximal, Sylow

C₇₁

#1339

yes

characteristic, complemented, Sylow

#469

—

#349

—

H(3,ℤ/4)

D₇₁

yes

characteristic, complemented

C₂×D₈

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#121

—

142

—

#117

—

M₃₂

—

142

—

#106

—

#102

—

142

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

D₈

—

284

—

M₁₆

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

C₂²⋊C₄

—

284

—

C₄²

—

C₁₆

—

284

—

C₂³

—

284

—

C₂³

#11765850

yes

characteristic

D₄

—

284

—

D₄

—

284

—

D₄

—

568

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

#11765852

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

568

—

C₂²

—

568

—

C₂²

—

568

—

C₂²

#14556048

yes

characteristic

C₄

—

C₄

—

C₄

—

568

—

C₄

#14556042

yes

characteristic

100

C₄

#14556056

yes

characteristic

101

C₂

—

102

C₂

—

103

C₂

—

1136

—

104

C₂

#17985732

yes

characteristic

	subgroup	G/H
center	C₂	#17985732
commutator	C₂×C₈×C₇₁	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₇₁
Fitting	H(3,ℤ/4)×C₇₁	C₂
socle	C₂×C₇₁	#327
hypercenter	H(3,ℤ/4)	D₇₁
hypocenter	C₇₁	#1339
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₂

#17985732

commutator

C₂×C₈×C₇₁

C₂×C₄

Frattini

C₂×M₁₆

C₂×D₇₁

Fitting

H(3,ℤ/4)×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #17985781	C₂	semidirect
3. H(3,ℤ/4)×C₇₁	C₂	semidirect
4. C₂×D₅₆₈	C₄	semidirect
7. #14556060	C₄	semidirect
53. C₇₁	#1339	semidirect
56. H(3,ℤ/4)	D₇₁	semidirect

H ≅ G/N

type

2. #17985781

C₂

semidirect

3. H(3,ℤ/4)×C₇₁

C₂

semidirect

4. C₂×D₅₆₈

C₄

semidirect

7. #14556060

C₄

semidirect

53. C₇₁

#1339

semidirect

56. H(3,ℤ/4)

D₇₁

semidirect

Subgroups — #22586324

Conjugacy classes of subgroups

Complements