Group #22612497 — subgroups

Subgroups — #127×Dic₇₃ #22612497

Order 9344.

924 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 289 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	C₂×D₄×Dic₇₃	C₂	1	6	yes	maximal, complemented
2	#17988490	C₂	1	6	yes	maximal, complemented
3	#17988457	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₇₃	C₂²	1	4	yes	—
7	D₄×Dic₇₃	—	2	24	no	—
8	C₂×C₄⋊Dic₇₃	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₇₃	C₂²	1	12	yes	—
10	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
11	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
12	C₂²×D₄×C₇₃	C₂²	1	6	yes	—
13	#127×C₇₃	C₄	1	8	yes	complemented
14	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇₃	—	2	24	no	—
16	C₂²×Dic₇₃	C₂³	1	4	yes	—
17	C₂²×Dic₇₃	C₂³	1	3	yes	—
18	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
19	C₄⋊Dic₇₃	—	2	12	no	—
20	C₂²⋊Dic₇₃	—	2	24	no	—
21	C₄×Dic₇₃	—	2	6	no	—
22	C₄×Dic₇₃	D₄	1	12	yes	complemented
23	C₂⁴×C₇₃	C₂³	1	4	yes	—
24	C₂×D₄×C₇₃	—	2	12	no	—
25	C₂×D₄×C₇₃	—	2	12	no	—
26	C₂×D₄×C₇₃	C₂×C₄	1	24	yes	complemented
27	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
28	C₄²×C₇₃	C₂×C₄	1	4	yes	complemented
29	C₂³×C₇₃	—	2	12	no	—
30	C₂³×C₇₃	—	2	12	no	—
31	C₂³×C₇₃	C₂²×C₄	1	8	yes	—
32	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
33	D₄×C₇₃	—	2	48	no	—
34	C₂×C₄×C₇₃	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₇₃	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₇₃	#41	1	6	yes	—
37	C₂×Dic₇₃	—	2	6	no	—
38	C₂×Dic₇₃	—	2	12	no	—
39	C₂×Dic₇₃	—	4	16	no	—
40	C₂×Dic₇₃	C₂×D₄	1	6	yes	—
41	C₂×Dic₇₃	C₂×D₄	1	6	yes	—
42	C₂²×C₇₃	—	2	24	no	—
43	C₂²×C₇₃	—	4	4	no	—
44	C₂²×C₇₃	—	4	12	no	—
45	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₇₃	C₂²×D₄	1	3	yes	—
47	C₂²×C₇₃	C₂×#41	1	3	yes	—
48	C₄×C₇₃	C₄×D₄	1	12	yes	complemented
49	Dic₇₃	—	2	6	no	—
50	Dic₇₃	—	4	4	no	—
51	Dic₇₃	#127	1	4	yes	direct product factor
52	C₂×C₇₃	—	4	8	no	—
53	C₂×C₇₃	C₂×C₄×D₄	1	3	yes	—
54	C₂×C₇₃	C₂×#127	1	1	yes	characteristic
55	C₂×C₇₃	#532	1	3	yes	—
56	C₄×#127	—	73	1	no	maximal, Sylow
57	C₇₃	C₄×#127	1	1	yes	characteristic, complemented, Sylow
58	C₂×#127	D₇₃	1	1	yes	characteristic
59	C₂×C₄×D₄	—	73	6	no	—
60	#390	—	73	6	no	—
61	#384	—	73	1	no	—
62	C₄³	—	73	1	no	—
63	C₂²×D₄	C₂×D₇₃	1	6	yes	—
64	C₂³×C₄	—	73	4	no	—
65	#127	Dic₇₃	1	8	yes	direct product factor
66	C₄×D₄	—	146	24	no	—
67	C₂×C₄⋊C₄	—	73	6	no	—
68	C₂×C₂²⋊C₄	—	73	12	no	—
69	C₂×C₄²	—	73	3	no	—
70	C₂×C₄²	—	73	3	no	—
71	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
72	C₂⁴	C₂²×D₇₃	1	4	yes	—
73	C₂×D₄	—	2	12	no	—
74	C₂×D₄	—	2	12	no	—
75	C₂×D₄	C₂×Dic₇₃	1	24	yes	complemented
76	C₂²×C₄	—	73	1	no	—
77	C₂²×C₄	—	73	3	no	—
78	C₂²×C₄	—	73	4	no	—
79	C₂²×C₄	—	146	24	no	—
80	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
81	C₄⋊C₄	—	146	12	no	—
82	C₂²⋊C₄	—	146	24	no	—
83	C₄²	—	73	12	no	—
84	C₄²	—	146	6	no	—
85	C₄²	C₂×Dic₇₃	1	4	yes	complemented
86	C₂³	—	2	12	no	—
87	C₂³	—	2	12	no	—
88	C₂³	C₂²×Dic₇₃	1	8	yes	—
89	C₂³	C₂³×D₇₃	1	1	yes	characteristic
90	D₄	—	2	48	no	—
91	C₂×C₄	—	292	16	no	—
92	C₂×C₄	—	73	6	no	—
93	C₂×C₄	—	73	6	no	—
94	C₂×C₄	—	146	12	no	—
95	C₂×C₄	—	146	6	no	—
96	C₂×C₄	C₂²×Dic₇₃	1	6	yes	—
97	C₂×C₄	D₄×D₇₃	1	6	yes	—
98	C₂×C₄	#11924027	1	6	yes	—
99	C₂²	—	2	24	no	—
100	C₂²	—	4	4	no	—
101	C₂²	—	4	12	no	—
102	C₂²	C₂³×Dic₇₃	1	1	yes	characteristic
103	C₂²	C₂×D₄×D₇₃	1	3	yes	—
104	C₂²	C₂×#11924027	1	3	yes	—
105	C₄	—	292	4	no	—
106	C₄	—	73	4	no	—
107	C₄	—	146	6	no	—
108	C₄	D₄×Dic₇₃	1	12	yes	complemented
109	C₂	—	4	8	no	—
110	C₂	C₂×D₄×Dic₇₃	1	3	yes	—
111	C₂	#127×D₇₃	1	1	yes	characteristic
112	C₂	#17989135	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₇₃

C₂

yes

maximal, complemented

#17988490

C₂

yes

maximal, complemented

#17988457

C₂

yes

maximal, characteristic, complemented

C₄²×Dic₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₇₃

C₂

yes

maximal, characteristic

C₂³×Dic₇₃

C₂²

yes

—

D₄×Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₂²⋊Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂²×D₄×C₇₃

C₂²

yes

—

#127×C₇₃

C₄

yes

complemented

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

complemented

C₂⁴×C₇₃

C₂³

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

complemented

C₂²×C₄×C₇₃

C₂³

yes

—

C₄²×C₇₃

C₂×C₄

yes

complemented

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂²×C₇₃

C₂²×D₄

yes

—

C₂²×C₇₃

C₂×#41

yes

—

C₄×C₇₃

C₄×D₄

yes

complemented

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#127

yes

direct product factor

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄×D₄

yes

—

C₂×C₇₃

C₂×#127

yes

characteristic

C₂×C₇₃

#532

yes

—

C₄×#127

—

maximal, Sylow

C₇₃

C₄×#127

yes

characteristic, complemented, Sylow

C₂×#127

D₇₃

yes

characteristic

C₂×C₄×D₄

—

#390

—

#384

—

C₄³

—

C₂²×D₄

C₂×D₇₃

yes

—

C₂³×C₄

—

#127

Dic₇₃

yes

direct product factor

C₄×D₄

—

146

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₂⁴

C₂²×D₇₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₇₃

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₄²

—

C₄²

—

146

—

C₄²

C₂×Dic₇₃

yes

complemented

C₂³

—

C₂³

—

C₂³

C₂²×Dic₇₃

yes

—

C₂³

C₂³×D₇₃

yes

characteristic

D₄

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂²×Dic₇₃

yes

—

C₂×C₄

D₄×D₇₃

yes

—

C₂×C₄

#11924027

yes

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂³×Dic₇₃

yes

characteristic

103

C₂²

C₂×D₄×D₇₃

yes

—

104

C₂²

C₂×#11924027

yes

—

105

C₄

—

292

—

106

C₄

—

107

C₄

—

146

—

108

C₄

D₄×Dic₇₃

yes

complemented

109

C₂

—

110

C₂

C₂×D₄×Dic₇₃

yes

—

111

C₂

#127×D₇₃

yes

characteristic

112

C₂

#17989135

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂³×C₄
Fitting	C₂×#127×C₇₃	C₂
socle	C₂³×C₇₃	C₂⁴
hypercenter	C₂×#127	D₇₃
hypocenter	C₇₃	C₄×#127
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator

C₂²×C₇₃

C₂³×C₄

Fitting

C₂×#127×C₇₃

C₂

socle

C₂³×C₇₃

C₂⁴

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×D₄×Dic₇₃	C₂	semidirect
2. #17988490	C₂	semidirect
3. #17988457	C₂	semidirect
4. C₄²×Dic₇₃	C₂	semidirect
13. #127×C₇₃	C₄	semidirect
22. C₄×Dic₇₃	D₄	semidirect
26. C₂×D₄×C₇₃	C₂×C₄	semidirect
28. C₄²×C₇₃	C₂×C₄	semidirect
48. C₄×C₇₃	C₄×D₄	semidirect
51. Dic₇₃	#127	direct (N × H)
57. C₇₃	C₄×#127	semidirect
65. #127	Dic₇₃	direct (N × H)
75. C₂×D₄	C₂×Dic₇₃	semidirect
85. C₄²	C₂×Dic₇₃	semidirect
108. C₄	D₄×Dic₇₃	semidirect

H ≅ G/N

type

1. C₂×D₄×Dic₇₃

C₂

semidirect

2. #17988490

C₂

semidirect

3. #17988457

C₂

semidirect

4. C₄²×Dic₇₃

C₂

semidirect

13. #127×C₇₃

C₄

semidirect

22. C₄×Dic₇₃

D₄

semidirect

26. C₂×D₄×C₇₃

C₂×C₄

semidirect

28. C₄²×C₇₃

C₂×C₄

semidirect

48. C₄×C₇₃

C₄×D₄

semidirect

51. Dic₇₃

#127

direct (N × H)

57. C₇₃

C₄×#127

semidirect

65. #127

Dic₇₃

direct (N × H)

75. C₂×D₄

C₂×Dic₇₃

semidirect

85. C₄²

C₂×Dic₇₃

semidirect

108. C₄

D₄×Dic₇₃

semidirect

Subgroups — #127×Dic₇₃ #22612497

Conjugacy classes of subgroups

Complements

Subgroups — #127×Dic73 #22612497

Conjugacy classes of subgroups

Complements

Subgroups — #127×Dic₇₃ #22612497