Group #22596949 — subgroups

Subgroups — #22596949

Order 9088.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#17986766	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556135	C₂	1	1	yes	maximal, characteristic
3	#17986047	C₂	1	6	yes	maximal, complemented
4	C₂×#14556063	C₂	1	6	yes	maximal, complemented
5	#568×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#14556135	—	2	4	no	—
7	#14556135	C₄	1	8	yes	—
8	C₂×#11765849	C₂²	1	6	yes	—
9	C₂×#11765849	C₂²	1	3	yes	—
10	C₂²×C₇₁⋊C₈	C₂²	1	3	yes	—
11	#14556063	C₂²	1	4	yes	—
12	#14556063	C₂²	1	12	yes	—
13	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
14	#131×C₇₁	—	2	4	no	—
15	C₂×M₁₆×C₇₁	C₂²	1	6	yes	—
16	#11765849	—	2	6	no	—
17	#11765849	C₂×C₄	1	12	yes	—
18	#11765849	C₂³	1	4	yes	—
19	C₂×C₇₁⋊C₈	—	2	6	no	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
21	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
22	C₂×C₇₁⋊C₈	C₂³	1	3	yes	—
23	#41×C₇₁	C₂×C₄	1	4	yes	—
24	#41×C₇₁	D₄	1	4	yes	—
25	C₂×Q₈×C₇₁	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₇₁	Q₈	1	3	yes	—
27	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
28	M₁₆×C₇₁	—	2	6	no	—
29	M₁₆×C₇₁	C₂³	1	4	yes	—
30	C₂×C₈×C₇₁	—	2	6	no	—
31	C₂³×C₇₁	C₂×Q₈	1	3	yes	—
32	Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
33	D₄×C₇₁	C₄⋊C₄	1	12	yes	—
34	C₂×C₄×C₇₁	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₇₁	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₇₁	C₂×Q₈	1	3	yes	—
37	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
38	C₈×C₇₁	—	2	4	no	—
39	C₇₁⋊C₈	—	2	4	no	—
40	C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
41	C₇₁⋊C₈	C₂²×C₄	1	6	yes	—
42	C₂²×C₇₁	—	4	3	no	—
43	C₂²×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₇₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₇₁	—	2	1	no	—
49	C₂×C₇₁	—	2	6	no	—
50	C₂×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	#2916	—	71	1	no	maximal, Sylow
52	C₇₁	#2916	1	1	yes	characteristic, complemented, Sylow
53	#568	—	71	1	no	—
54	#568	D₇₁	1	1	yes	characteristic
55	C₂×#131	—	71	1	no	—
56	#430	—	71	6	no	—
57	C₂×#108	—	71	6	no	—
58	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	71	8	no	—
61	#131	—	142	4	no	—
62	C₂×M₁₆	—	71	3	no	—
63	C₂×M₁₆	—	71	6	no	—
64	C₂×M₁₆	C₂×D₇₁	1	6	yes	—
65	C₂²×C₈	—	71	3	no	—
66	#108	—	71	4	no	—
67	#108	—	71	12	no	—
68	#41	C₄×D₇₁	1	4	yes	—
69	#41	C₂²⋊D₇₁	1	4	yes	—
70	C₂×Q₈	Dic₁₄₂	1	1	yes	characteristic
71	C₂×D₄	Dic₁₄₂	1	3	yes	—
72	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	71	12	no	—
75	M₁₆	—	71	4	no	—
76	M₁₆	—	142	6	no	—
77	M₁₆	C₂²×D₇₁	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	71	12	no	—
80	C₂×C₈	—	71	1	no	—
81	C₂×C₈	—	71	3	no	—
82	C₂×C₈	—	142	6	no	—
83	C₂³	C₂×Dic₁₄₂	1	3	yes	—
84	Q₈	Dic₇₁⋊C₄	1	4	yes	—
85	D₄	Dic₇₁⋊C₄	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₇₁	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₇₁	1	6	yes	—
88	C₂×C₄	C₂×Dic₁₄₂	1	3	yes	—
89	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	71	2	no	—
92	C₈	—	71	6	no	—
93	C₈	—	142	4	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×Dic₇₁⋊C₄	1	6	yes	—
96	C₂²	C₂²×Dic₁₄₂	1	1	yes	characteristic
97	C₄	C₂×Dic₇₁⋊C₄	1	6	yes	—
98	C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×Dic₇₁⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986766

C₂

yes

maximal, characteristic

C₂×#14556135

C₂

yes

maximal, characteristic

#17986047

C₂

yes

maximal, complemented

C₂×#14556063

C₂

yes

maximal, complemented

#568×C₇₁

C₂

yes

maximal, characteristic

#14556135

—

#14556135

C₄

yes

—

C₂×#11765849

C₂²

yes

—

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

#14556063

C₂²

yes

—

#14556063

C₂²

yes

—

C₂×#41×C₇₁

C₂²

yes

characteristic

#131×C₇₁

—

C₂×M₁₆×C₇₁

C₂²

yes

—

#11765849

—

#11765849

C₂×C₄

yes

—

#11765849

C₂³

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

#41×C₇₁

C₂×C₄

yes

—

#41×C₇₁

D₄

yes

—

C₂×Q₈×C₇₁

Q₈

yes

characteristic

C₂×D₄×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

M₁₆×C₇₁

—

M₁₆×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

C₂×Q₈

yes

—

Q₈×C₇₁

C₄⋊C₄

yes

—

D₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²×C₄

yes

—

C₇₁⋊C₈

C₂²×C₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

C₂²×D₄

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂²×C₄⋊C₄

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#568

—

#568

D₇₁

yes

—

—

—

C₂×#41

C₂×D₇₁

yes

—

—

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂²×C₈

—

#108

—

#108

—

#41

C₄×D₇₁

yes

—

#41

C₂²⋊D₇₁

yes

—

C₂×Q₈

Dic₁₄₂

yes

characteristic

C₂×D₄

Dic₁₄₂

yes

—

C₂²×C₄

C₂²×D₇₁

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

C₂²×D₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂³

C₂×Dic₁₄₂

yes

—

Q₈

Dic₇₁⋊C₄

yes

—

D₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

C₂×C₄

C₂×Dic₁₄₂

yes

—

C₂×C₄

C₂³×D₇₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

142

—

C₂²

—

C₂²

C₂×Dic₇₁⋊C₄

yes

—

C₂²

C₂²×Dic₁₄₂

yes

characteristic

C₄

C₂×Dic₇₁⋊C₄

yes

—

C₄

C₂²×C₄×D₇₁

yes

characteristic

C₄

C₂²×C₂²⋊D₇₁

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×Dic₇₁⋊C₄

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#568×C₇₁	C₂
socle	C₂×C₇₁	C₂²×C₄⋊C₄
hypercenter	#568	D₇₁
hypocenter	C₇₁	#2916
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₇₁

commutator

C₄×C₇₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#568×C₇₁

C₂

socle

C₂×C₇₁

C₂²×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #17986047	C₂	semidirect
4. C₂×#14556063	C₂	semidirect
52. C₇₁	#2916	semidirect

H ≅ G/N

type

3. #17986047

C₂

semidirect

4. C₂×#14556063

C₂

semidirect

52. C₇₁

#2916

semidirect

Subgroups — #22596949

Conjugacy classes of subgroups

Complements