Group #22627174 — subgroups

Subgroups — #22627174

Order 9344.

396 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 191 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	Q₈×#10596842	C₂	1	8	yes	maximal, complemented
2	#17989700	C₂	1	6	yes	maximal
3	C₂×Q₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#10596842	C₂²	1	12	yes	—
5	C₂²⋊#10596842	C₂²	1	4	yes	—
6	C₄×#10596842	C₂²	1	12	yes	—
7	Q₈×#64408	C₄	1	4	yes	—
8	Q₈×#64408	C₂²	1	4	yes	—
9	C₂×C₄⋊#64408	C₄	1	3	yes	—
10	C₂×C₄×#64408	C₂²	1	3	yes	—
11	C₂×Q₈×D₇₃	C₄	1	1	yes	characteristic
12	C₂×#10596842	C₂³	1	8	yes	—
13	C₂²×#64408	C₂×C₄	1	3	yes	—
14	C₂²×#64408	C₂³	1	1	yes	characteristic
15	C₄⋊#64408	C₂×C₄	1	12	yes	—
16	C₄×#64408	C₂×C₄	1	6	yes	—
17	C₄×#64408	C₂³	1	6	yes	—
18	C₂×Dic₁₄₆	C₈	1	3	yes	complemented
19	Q₈×D₇₃	C₈	1	4	yes	complemented
20	Q₈×D₇₃	C₂×C₄	1	4	yes	—
21	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	—
22	C₂×Q₈×C₇₃	C₈	1	1	yes	characteristic, complemented
23	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
24	C₂×#64408	—	2	2	no	—
25	C₂×#64408	—	4	3	no	—
26	C₂×#64408	C₂²×C₄	1	1	yes	characteristic
27	C₂×#64408	C₂²×C₄	1	6	yes	—
28	C₂×#64408	C₂⁴	1	1	yes	characteristic
29	#10596842	—	2	8	no	—
30	Q₈×C₇₃	C₂×C₈	1	4	yes	—
31	C₂×C₄×C₇₃	C₂×C₈	1	3	yes	—
32	C₂×Dic₇₃	C₂×C₈	1	3	yes	—
33	C₄×D₇₃	C₂×C₈	1	6	yes	—
34	C₄×D₇₃	C₂²×C₄	1	6	yes	—
35	Dic₁₄₆	C₂×C₈	1	12	yes	—
36	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
37	C₂×D₇₃	—	2	1	no	—
38	C₂×D₇₃	—	2	1	no	—
39	C₂×D₇₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₇₃	C₂³×C₄	1	1	yes	characteristic
41	#64408	—	2	1	no	—
42	#64408	—	2	6	no	—
43	#64408	2₋¹⁺⁴	1	2	yes	—
44	C₄×C₇₃	C₂²×C₈	1	6	yes	—
45	Dic₇₃	C₂²×C₈	1	6	yes	—
46	C₂×C₇₃	—	2	1	no	—
47	C₂×C₇₃	C₂³×C₈	1	1	yes	characteristic
48	D₇₃	—	2	1	no	—
49	D₇₃	#519	1	1	yes	characteristic
50	D₇₃	#568	1	1	yes	characteristic
51	#2982	—	73	1	no	maximal, Sylow
52	C₇₃	#2982	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₄×Q₈	—	73	1	no	—
54	C₈×Q₈	—	73	8	no	—
55	#431	—	73	6	no	—
56	C₂²×Q₈	—	73	1	no	—
57	C₄×Q₈	—	73	4	no	—
58	C₄×Q₈	—	73	4	no	—
59	C₂×C₄⋊C₄	—	73	3	no	—
60	C₂×C₄²	—	73	3	no	—
61	C₄⋊C₈	—	73	12	no	—
62	C₂²⋊C₈	—	73	4	no	—
63	C₄×C₈	—	73	12	no	—
64	C₂×Q₈	—	73	4	no	—
65	C₂×Q₈	—	73	4	no	—
66	C₂×Q₈	—	73	3	no	—
67	C₂×Q₈	#10596842	1	1	yes	characteristic, complemented
68	C₂²×C₄	—	73	3	no	—
69	C₂²×C₄	—	73	3	no	—
70	C₂²×C₄	—	73	1	no	—
71	C₂×C₈	—	73	8	no	—
72	C₄⋊C₄	—	73	12	no	—
73	C₄²	—	73	6	no	—
74	C₄²	—	73	6	no	—
75	C₂³	—	73	1	no	—
76	Q₈	—	73	12	no	—
77	Q₈	C₂×#10596842	1	4	yes	—
78	C₂×C₄	—	292	3	no	—
79	C₂×C₄	—	73	1	no	—
80	C₂×C₄	—	73	3	no	—
81	C₂×C₄	—	73	6	no	—
82	C₂×C₄	—	73	6	no	—
83	C₂×C₄	—	73	1	no	—
84	C₂×C₄	—	73	6	no	—
85	C₂×C₄	—	146	2	no	—
86	C₂×C₄	C₂×#10596842	1	3	yes	—
87	C₈	—	146	8	no	—
88	C₂²	—	73	1	no	—
89	C₂²	—	73	1	no	—
90	C₂²	—	146	1	no	—
91	C₂²	—	146	1	no	—
92	C₂²	C₂²×#10596842	1	1	yes	characteristic
93	C₄	—	73	2	no	—
94	C₄	—	73	6	no	—
95	C₄	—	146	6	no	—
96	C₄	—	146	1	no	—
97	C₄	C₂²×#10596842	1	6	yes	—
98	C₂	—	2	1	no	—
99	C₂	—	73	1	no	—
100	C₂	—	73	1	no	—
101	C₂	—	146	1	no	—
102	C₂	C₂³×#10596842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#10596842

C₂

yes

maximal, complemented

#17989700

C₂

yes

maximal

C₂×Q₈×#64408

C₂

yes

maximal, characteristic

C₄⋊#10596842

C₂²

yes

—

C₂²⋊#10596842

C₂²

yes

—

C₄×#10596842

C₂²

yes

—

Q₈×#64408

C₄

yes

—

Q₈×#64408

C₂²

yes

—

C₂×C₄⋊#64408

C₄

yes

—

C₂×C₄×#64408

C₂²

yes

—

C₂×Q₈×D₇₃

C₄

yes

characteristic

C₂×#10596842

C₂³

yes

—

C₂²×#64408

C₂×C₄

yes

—

C₂²×#64408

C₂³

yes

characteristic

C₄⋊#64408

C₂×C₄

yes

—

C₄×#64408

C₂×C₄

yes

—

C₄×#64408

C₂³

yes

—

C₂×Dic₁₄₆

C₈

yes

complemented

Q₈×D₇₃

C₈

yes

complemented

Q₈×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

C₈

yes

characteristic, complemented

C₂²×D₇₃

C₂²×C₄

yes

—

—

C₂²×C₄

yes

characteristic

C₂×#64408

C₂²×C₄

yes

—

C₂×#64408

C₂⁴

yes

characteristic

#10596842

—

Q₈×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂×C₈

yes

—

C₄×D₇₃

C₂×C₈

yes

—

C₄×D₇₃

C₂²×C₄

yes

—

Dic₁₄₆

C₂×C₈

yes

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

C₂³×C₄

yes

—

—

2₋¹⁺⁴

yes

—

C₄×C₇₃

C₂²×C₈

yes

—

Dic₇₃

C₂²×C₈

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₂³×C₈

yes

characteristic

D₇₃

—

D₇₃

yes

D₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₈×Q₈

—

#431

—

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#10596842

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

—

Q₈

—

Q₈

C₂×#10596842

yes

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

C₂×#10596842

yes

—

C₈

—

146

—

C₂²

—

C₂²

—

C₂²

—

146

—

C₂²

—

146

—

C₂²

C₂²×#10596842

yes

characteristic

C₄

—

C₄

—

C₄

—

146

—

C₄

—

146

—

C₄

C₂²×#10596842

yes

—

C₂

—

C₂

—

100

C₂

—

101

C₂

—

146

—

102

C₂

C₂³×#10596842

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#10596842
commutator, socle	C₂×C₇₃	C₂³×C₈
Fitting	C₂×Q₈×C₇₃	C₈
hypercenter	C₂×Q₈	#10596842
hypocenter	C₇₃	#2982
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂

C₂³×#10596842

commutator, socle

C₂×C₇₃

C₂³×C₈

Fitting

C₂×Q₈×C₇₃

C₈

C₂×Q₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂ ◁ C₂×Q₈

N	H ≅ G/N	type
1. Q₈×#10596842	C₂	semidirect
18. C₂×Dic₁₄₆	C₈	semidirect
19. Q₈×D₇₃	C₈	semidirect
22. C₂×Q₈×C₇₃	C₈	semidirect
52. C₇₃	#2982	semidirect
67. C₂×Q₈	#10596842	semidirect

H ≅ G/N

type

1. Q₈×#10596842

C₂

semidirect

18. C₂×Dic₁₄₆

C₈

semidirect

19. Q₈×D₇₃

C₈

semidirect

22. C₂×Q₈×C₇₃

C₈

semidirect

52. C₇₃

#2982

semidirect

67. C₂×Q₈

#10596842

semidirect

Subgroups — #22627174

Conjugacy classes of subgroups

Complements