Group #22587969 — subgroups

Subgroups — #22587969

Order 9088.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986376	C₂	1	1	yes	maximal, characteristic
2	#17986373	C₂	1	2	yes	maximal, complemented
3	#17986347	C₂	1	2	yes	maximal, complemented
4	#17985720	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765852	—	2	4	no	—
7	C₂×D₅₆₈	—	2	2	no	—
8	#14556125	C₂²	1	1	yes	characteristic
9	#14556125	C₂²	1	1	yes	characteristic
10	#14556048	—	2	4	no	—
11	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
13	#106×C₇₁	—	2	2	no	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×D₂₈₄	—	2	2	no	—
16	C₂×D₂₈₄	—	2	1	no	—
17	C₂×D₂₈₄	—	2	1	no	—
18	C₂×D₂₈₄	—	2	2	no	—
19	#11765852	—	4	8	no	—
20	D₅₆₈	—	4	4	no	—
21	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
22	C₂×Q₈×C₇₁	—	2	2	no	—
23	C₂×C₈×C₇₁	D₄	1	2	yes	—
24	C₄⋊C₄×C₇₁	—	2	2	no	—
25	C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₄²×C₇₁	C₂³	1	1	yes	characteristic
27	C₂²×D₇₁	—	4	1	no	—
28	C₂²×D₇₁	—	4	1	no	—
29	Q₈×C₇₁	—	4	4	no	—
30	C₂×C₄×C₇₁	—	4	2	no	—
31	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	D₂₈₄	—	4	2	no	—
34	D₂₈₄	—	4	2	no	—
35	D₂₈₄	—	4	4	no	—
36	D₂₈₄	—	4	4	no	—
37	C₈×C₇₁	SD₁₆	1	4	yes	complemented
38	C₇₁⋊C₈	—	2	4	no	—
39	C₂²×C₇₁	#127	1	1	yes	characteristic
40	C₂×D₇₁	—	8	1	no	—
41	C₂×D₇₁	—	8	1	no	—
42	C₂×D₇₁	—	8	2	no	—
43	C₂×D₇₁	—	8	2	no	—
44	C₄×C₇₁	—	8	2	no	—
45	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
46	C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₇₁	#492	1	1	yes	characteristic
48	C₂×C₇₁	#496	1	2	yes	—
49	D₇₁	—	16	1	no	—
50	D₇₁	—	16	1	no	—
51	#1715	—	71	1	no	maximal, Sylow
52	C₇₁	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	D₇₁	1	1	yes	characteristic, complemented
54	#493	—	71	1	no	—
55	#492	—	71	2	no	—
56	#486	—	71	2	no	—
57	#322	—	71	1	no	—
58	C₂×SD₁₆	—	142	4	no	—
59	C₂×D₈	—	142	2	no	—
60	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
61	#127	—	71	1	no	—
62	#127	—	71	1	no	—
63	#106	—	2	2	no	—
64	#102	—	142	4	no	—
65	C₄×C₈	—	71	2	no	—
66	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	142	2	no	—
69	C₂×D₄	—	142	2	no	—
70	C₂×D₄	—	142	1	no	—
71	C₂×D₄	—	142	1	no	—
72	SD₁₆	—	284	8	no	—
73	D₈	—	284	4	no	—
74	C₂×C₈	—	71	4	no	—
75	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₇₁	1	1	yes	characteristic
79	C₂³	—	284	1	no	—
80	C₂³	—	284	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	284	2	no	—
83	D₄	—	284	2	no	—
84	D₄	—	284	4	no	—
85	D₄	—	284	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₇₁	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
89	C₈	—	142	4	no	—
90	C₈	#11765852	1	4	yes	complemented
91	C₂²	—	568	1	no	—
92	C₂²	—	568	1	no	—
93	C₂²	—	568	2	no	—
94	C₂²	—	568	2	no	—
95	C₂²	#14556124	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#11765852	1	2	yes	—
98	C₄	#14556155	1	4	yes	—
99	C₂	—	1136	1	no	—
100	C₂	—	1136	1	no	—
101	C₂	#17986373	1	1	yes	characteristic
102	C₂	#17986385	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986376

C₂

yes

maximal, characteristic

#17986373

C₂

yes

maximal, complemented

#17986347

C₂

yes

maximal, complemented

#17985720

C₂

yes

maximal, characteristic, complemented

#498×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765852

—

C₂×D₅₆₈

—

#14556125

C₂²

yes

characteristic

#14556125

C₂²

yes

characteristic

#14556048

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

#106×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765852

—

D₅₆₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

#127

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₇₁

#492

yes

characteristic

C₂×C₇₁

#496

yes

—

D₇₁

—

D₇₁

—

#1715

—

maximal, Sylow

C₇₁

#1715

yes

characteristic, complemented, Sylow

#498

D₇₁

yes

characteristic, complemented

—

—

—

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

142

—

C₄⋊Q₈

C₂×D₇₁

yes

—

—

—

—

—

142

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

SD₁₆

—

284

—

D₈

—

284

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

284

—

C₂³

—

284

—

Q₈

—

D₄

—

284

—

D₄

—

284

—

D₄

—

284

—

D₄

—

284

—

C₂×C₄

—

C₂×C₄

D₄×D₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₈

—

142

—

C₈

#11765852

yes

complemented

C₂²

—

568

—

C₂²

—

568

—

C₂²

—

568

—

C₂²

—

568

—

C₂²

#14556124

yes

characteristic

C₄

—

C₄

C₂×#11765852

yes

—

C₄

#14556155

yes

—

C₂

—

1136

—

100

C₂

—

1136

—

101

C₂

#17986373

yes

characteristic

102

C₂

#17986385

yes

—

	subgroup	G/H
center	C₂²	#14556124
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#498×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#498	D₇₁
hypocenter	C₇₁	#1715
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14556124

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#498×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #17986373	C₂	semidirect
3. #17986347	C₂	semidirect
4. #17985720	C₂	semidirect
5. #498×C₇₁	C₂	semidirect
37. C₈×C₇₁	SD₁₆	semidirect
52. C₇₁	#1715	semidirect
53. #498	D₇₁	semidirect
90. C₈	#11765852	semidirect

H ≅ G/N

type

2. #17986373

C₂

semidirect

3. #17986347

C₂

semidirect

4. #17985720

C₂

semidirect

5. #498×C₇₁

C₂

semidirect

37. C₈×C₇₁

SD₁₆

semidirect

52. C₇₁

#1715

semidirect

53. #498

D₇₁

semidirect

90. C₈

#11765852

semidirect

Subgroups — #22587969

Conjugacy classes of subgroups

Complements