Group #22587929 — subgroups

Subgroups — #22587929

Order 9088.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#17986404	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986371	C₂	1	2	yes	maximal, complemented
3	#17986345	C₂	1	2	yes	maximal, complemented
4	C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765853	—	2	4	no	—
7	#14556128	C₂²	1	2	yes	—
8	#14556062	—	2	2	no	—
9	#14556049	—	2	4	no	—
10	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₇₁	—	2	2	no	—
13	#127×C₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₂	—	2	2	no	—
16	#11765853	—	4	8	no	—
17	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
18	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×D₄×C₇₁	—	2	2	no	—
22	C₂×D₄×C₇₁	—	2	2	no	—
23	C₂×D₄×C₇₁	—	2	2	no	—
24	D₈×C₇₁	—	4	4	no	—
25	C₂×C₈×C₇₁	D₄	1	2	yes	—
26	C₄²×C₇₁	C₂³	1	1	yes	characteristic
27	C₂³×C₇₁	—	4	2	no	—
28	D₄×C₇₁	—	4	4	no	—
29	D₄×C₇₁	—	4	4	no	—
30	D₄×C₇₁	—	4	4	no	—
31	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₇₁	—	4	2	no	—
35	Dic₁₄₂	—	4	4	no	—
36	C₈×C₇₁	SD₁₆	1	4	yes	complemented
37	C₇₁⋊C₈	—	2	2	no	—
38	C₇₁⋊C₈	D₈	1	4	yes	complemented
39	C₂²×C₇₁	—	8	2	no	—
40	C₂²×C₇₁	—	8	2	no	—
41	C₂²×C₇₁	—	8	2	no	—
42	C₂²×C₇₁	#127	1	1	yes	characteristic
43	C₄×C₇₁	C₂×D₈	1	2	yes	—
44	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
45	C₄×C₇₁	#135	1	2	yes	—
46	Dic₇₁	—	8	2	no	—
47	C₂×C₇₁	—	16	2	no	—
48	C₂×C₇₁	#492	1	1	yes	characteristic
49	C₂×C₇₁	#493	1	1	yes	characteristic
50	C₂×C₇₁	#495	1	1	yes	characteristic
51	#1706	—	71	1	no	maximal, Sylow
52	C₇₁	#1706	1	1	yes	characteristic, complemented, Sylow
53	#500	—	71	1	no	—
54	#493	D₇₁	1	1	yes	characteristic
55	#492	—	71	2	no	—
56	#486	—	71	2	no	—
57	C₈²	—	71	1	no	—
58	C₂×SD₁₆	—	142	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	71	2	no	—
61	#127	C₂×D₇₁	1	2	yes	—
62	#107	—	142	2	no	—
63	#102	—	142	4	no	—
64	C₄×C₈	—	71	1	no	—
65	C₄×C₈	—	71	1	no	—
66	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
67	C₂×Q₈	—	142	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	284	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	—	71	2	no	—
74	C₂×C₈	—	71	2	no	—
75	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
76	C₄⋊C₄	—	142	2	no	—
77	C₄⋊C₄	—	142	2	no	—
78	C₄²	C₂²×D₇₁	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	284	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	284	2	no	—
85	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
86	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
88	C₈	—	71	4	no	—
89	C₈	—	142	2	no	—
90	C₈	#11765853	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#14556124	1	1	yes	characteristic
95	C₄	—	568	2	no	—
96	C₄	D₈×D₇₁	1	2	yes	—
97	C₄	C₂×#11765853	1	2	yes	—
98	C₄	#14556147	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#17986371	1	1	yes	characteristic
101	C₂	#17986374	1	1	yes	characteristic
102	C₂	#17986382	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986404

C₂

yes

maximal, characteristic, complemented

#17986371

C₂

yes

maximal, complemented

#17986345

C₂

yes

maximal, complemented

C₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#493×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

—

#14556128

C₂²

yes

—

#14556062

—

#14556049

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₇₁

—

#127×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

#11765853

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

D₈

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#127

yes

characteristic

C₄×C₇₁

C₂×D₈

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

#135

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

—

#493

D₇₁

yes

characteristic

#492

—

#486

—

C₈²

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₇₁

yes

—

#107

—

142

—

#102

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

284

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

Q₈

—

284

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

284

—

C₂×C₄

D₄×D₇₁

yes

characteristic

C₂×C₄

D₄×D₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₈

—

C₈

—

142

—

C₈

#11765853

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#14556124

yes

characteristic

C₄

—

568

—

C₄

D₈×D₇₁

yes

—

C₄

C₂×#11765853

yes

—

C₄

#14556147

yes

—

C₂

—

100

C₂

#17986371

yes

characteristic

101

C₂

#17986374

yes

characteristic

102

C₂

#17986382

yes

characteristic

	subgroup	G/H
center	C₂²	#14556124
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#493×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#493	D₇₁
hypocenter	C₇₁	#1706
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#14556124

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#493×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #17986404	C₂	semidirect
2. #17986371	C₂	semidirect
3. #17986345	C₂	semidirect
4. C₈×C₇₁⋊C₈	C₂	semidirect
36. C₈×C₇₁	SD₁₆	semidirect
38. C₇₁⋊C₈	D₈	semidirect
52. C₇₁	#1706	semidirect
90. C₈	#11765853	semidirect

H ≅ G/N

type

1. #17986404

C₂

semidirect

2. #17986371

C₂

semidirect

3. #17986345

C₂

semidirect

4. C₈×C₇₁⋊C₈

C₂

semidirect

36. C₈×C₇₁

SD₁₆

semidirect

38. C₇₁⋊C₈

D₈

semidirect

52. C₇₁

#1706

semidirect

90. C₈

#11765853

semidirect

Subgroups — #22587929

Conjugacy classes of subgroups

Complements