Group #22626475 — subgroups

Subgroups — #22626475

Order 9344.

648 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 152 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	#17989535	C₂	1	12	yes	maximal, complemented
2	#17989532	C₂	1	2	yes	maximal, complemented
3	#127×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#64408	C₂²	1	4	yes	—
5	C₂×C₂²⋊#64408	C₂²	1	12	yes	—
6	(C₂×#64408)⋊C₄	C₂²	1	12	yes	—
7	C₂×D₄×D₇₃	C₂²	1	6	yes	—
8	#14556588	C₄	1	1	yes	characteristic, complemented
9	#14556587	C₄	1	6	yes	complemented
10	C₄²×D₇₃	C₂²	1	1	yes	characteristic
11	#127×C₇₃	C₄	1	1	yes	characteristic, complemented
12	C₂²×#64408	—	4	12	no	—
13	C₂²×#64408	C₂³	1	8	yes	—
14	C₂²⋊#64408	—	4	12	no	—
15	C₂³×D₇₃	C₂³	1	4	yes	—
16	C₂×D₂₉₂	C₂×C₄	1	6	yes	complemented
17	C₂×C₂²⋊D₇₃	C₂×C₄	1	12	yes	complemented
18	D₄×D₇₃	—	4	6	no	—
19	D₄×D₇₃	—	4	6	no	—
20	C₂×C₄×D₇₃	C₂³	1	3	yes	—
21	C₄×Dic₇₃	C₂×C₄	1	3	yes	complemented
22	C₂×D₄×C₇₃	C₂×C₄	1	6	yes	complemented
23	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
24	C₂³×C₇₃	C₂²×C₄	1	4	yes	—
25	C₂²×D₇₃	—	2	12	no	—
26	C₂²×D₇₃	—	2	12	no	—
27	C₂²×D₇₃	C₂²×C₄	1	4	yes	—
28	C₂²×D₇₃	C₂⁴	1	1	yes	characteristic
29	C₂×#64408	—	2	24	no	—
30	C₂×#64408	—	4	16	no	—
31	D₄×C₇₃	—	4	6	no	—
32	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
33	C₂²⋊D₇₃	—	4	12	no	—
34	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
35	D₂₉₂	—	4	6	no	—
36	C₄×D₇₃	—	2	6	no	—
37	C₂²×C₇₃	—	2	12	no	—
38	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
39	C₂×D₇₃	—	2	12	no	—
40	C₂×D₇₃	—	4	4	no	—
41	C₂×D₇₃	—	4	12	no	—
42	C₂×D₇₃	H(5,𝔽₂)	1	6	yes	—
43	#64408	—	4	8	no	—
44	C₄×C₇₃	—	2	3	no	—
45	Dic₇₃	—	2	3	no	—
46	C₂×C₇₃	—	4	4	no	—
47	C₂×C₇₃	#518	1	3	yes	—
48	D₇₃	—	4	4	no	—
49	D₇₃	#560	1	4	yes	—
50	#2380	—	73	1	no	maximal, Sylow
51	C₇₃	#2380	1	1	yes	characteristic, complemented, Sylow
52	C₂×#127	—	73	1	no	—
53	#386	—	73	12	no	—
54	#383	—	73	2	no	—
55	C₂²×D₄	—	73	6	no	—
56	C₂³×C₄	—	73	4	no	—
57	#127	—	73	6	no	—
58	#127	—	73	1	no	—
59	#127	#64408	1	1	yes	characteristic, complemented
60	C₂×C₂²⋊C₄	—	73	12	no	—
61	C₂×C₄²	—	73	1	no	—
62	#95	—	73	12	no	—
63	C₂⁴	—	73	4	no	—
64	C₂×D₄	—	292	6	no	—
65	C₂×D₄	—	292	6	no	—
66	C₂×D₄	—	73	12	no	—
67	C₂×D₄	—	73	6	no	—
68	C₂×D₄	C₂×#64408	1	6	yes	complemented
69	C₂²×C₄	—	292	12	no	—
70	C₂²×C₄	—	73	3	no	—
71	C₂²×C₄	—	73	8	no	—
72	C₂²⋊C₄	—	292	12	no	—
73	C₄²	—	73	3	no	—
74	C₄²	C₂×#64408	1	1	yes	characteristic, complemented
75	C₂³	—	73	1	no	—
76	C₂³	—	73	4	no	—
77	C₂³	—	146	12	no	—
78	C₂³	—	146	12	no	—
79	C₂³	C₂²×#64408	1	4	yes	—
80	D₄	—	4	6	no	—
81	D₄	—	292	6	no	—
82	D₄	—	292	12	no	—
83	C₂×C₄	—	292	16	no	—
84	C₂×C₄	—	73	3	no	—
85	C₂×C₄	—	146	6	no	—
86	C₂×C₄	—	146	24	no	—
87	C₂×C₄	C₂²×#64408	1	3	yes	—
88	C₂²	—	2	12	no	—
89	C₂²	—	292	4	no	—
90	C₂²	—	292	12	no	—
91	C₂²	—	73	6	no	—
92	C₂²	—	146	12	no	—
93	C₂²	C₂³×#64408	1	1	yes	characteristic
94	C₄	—	2	3	no	—
95	C₄	—	292	8	no	—
96	C₄	—	146	3	no	—
97	C₂	—	4	4	no	—
98	C₂	—	292	4	no	—
99	C₂	—	73	4	no	—
100	C₂	#17989654	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989535

C₂

yes

maximal, complemented

#17989532

C₂

yes

maximal, complemented

#127×D₇₃

C₂

yes

maximal, characteristic

C₂³×#64408

C₂²

yes

—

C₂×C₂²⋊#64408

C₂²

yes

—

(C₂×#64408)⋊C₄

C₂²

yes

—

C₂×D₄×D₇₃

C₂²

yes

—

#14556588

C₄

yes

characteristic, complemented

#14556587

C₄

yes

complemented

C₄²×D₇₃

C₂²

yes

characteristic

#127×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

C₂²×#64408

C₂³

yes

—

C₂²⋊#64408

—

C₂³×D₇₃

C₂³

yes

—

C₂×D₂₉₂

C₂×C₄

yes

complemented

C₂×C₂²⋊D₇₃

C₂×C₄

yes

complemented

D₄×D₇₃

—

D₄×D₇₃

—

C₂×C₄×D₇₃

C₂³

yes

—

C₄×Dic₇₃

C₂×C₄

yes

complemented

C₂×D₄×C₇₃

C₂×C₄

yes

complemented

C₄²×C₇₃

C₂×C₄

yes

characteristic, complemented

C₂³×C₇₃

C₂²×C₄

yes

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂²×C₄

yes

—

C₂²×D₇₃

C₂⁴

yes

characteristic

C₂×#64408

—

C₂×#64408

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₂²×C₄

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

H(5,𝔽₂)

yes

—

#64408

—

C₄×C₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#518

yes

—

D₇₃

—

D₇₃

#560

yes

—

#2380

—

maximal, Sylow

C₇₃

#2380

yes

characteristic, complemented, Sylow

C₂×#127

—

#386

—

#383

—

C₂²×D₄

—

C₂³×C₄

—

—

—

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₂⁴

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#64408

yes

complemented

C₂²×C₄

—

292

—

C₂²×C₄

—

C₂²×C₄

—

C₂²⋊C₄

—

292

—

C₄²

—

C₄²

C₂×#64408

yes

characteristic, complemented

C₂³

—

C₂³

—

C₂³

—

146

—

C₂³

—

146

—

C₂³

C₂²×#64408

yes

—

D₄

—

D₄

—

292

—

D₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂²×#64408

yes

—

C₂²

—

C₂²

—

292

—

C₂²

—

292

—

C₂²

—

C₂²

—

146

—

C₂²

C₂³×#64408

yes

characteristic

C₄

—

C₄

—

292

—

C₄

—

146

—

C₂

—

C₂

—

292

—

C₂

—

100

C₂

#17989654

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	#127×C₇₃	C₄
hypercenter	#127	#64408
hypocenter	C₇₃	#2380
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator, socle

C₂²×C₇₃

C₂³×C₄

Fitting

#127×C₇₃

C₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. #17989535	C₂	semidirect
2. #17989532	C₂	semidirect
8. #14556588	C₄	semidirect
9. #14556587	C₄	semidirect
11. #127×C₇₃	C₄	semidirect
16. C₂×D₂₉₂	C₂×C₄	semidirect
17. C₂×C₂²⋊D₇₃	C₂×C₄	semidirect
21. C₄×Dic₇₃	C₂×C₄	semidirect
22. C₂×D₄×C₇₃	C₂×C₄	semidirect
23. C₄²×C₇₃	C₂×C₄	semidirect
51. C₇₃	#2380	semidirect
59. #127	#64408	semidirect
68. C₂×D₄	C₂×#64408	semidirect
74. C₄²	C₂×#64408	semidirect

H ≅ G/N

type

1. #17989535

C₂

semidirect

2. #17989532

C₂

semidirect

8. #14556588

C₄

semidirect

9. #14556587

C₄

semidirect

11. #127×C₇₃

C₄

semidirect

16. C₂×D₂₉₂

C₂×C₄

semidirect

17. C₂×C₂²⋊D₇₃

C₂×C₄

semidirect

21. C₄×Dic₇₃

C₂×C₄

semidirect

22. C₂×D₄×C₇₃

C₂×C₄

semidirect

23. C₄²×C₇₃

C₂×C₄

semidirect

51. C₇₃

#2380

semidirect

59. #127

#64408

semidirect

68. C₂×D₄

C₂×#64408

semidirect

74. C₄²

C₂×#64408

semidirect

Subgroups — #22626475

Conjugacy classes of subgroups

Complements