Group #22626927 — subgroups

Subgroups — #22626927

Order 9344.

1424 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 750 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	#17989667	C₂	1	5	yes	maximal, complemented
2	#17989666	C₂	1	10	yes	maximal, complemented
3	C₂×#14556679	C₂	1	5	yes	maximal, complemented
4	D₇₃⋊#131	C₂	1	10	yes	maximal, complemented
5	#17989424	C₂	1	1	yes	maximal, characteristic
6	#14556679	C₂²	1	10	yes	complemented
7	#14556679	C₂²	1	10	yes	complemented
8	#14556679	C₂²	1	20	yes	complemented
9	#14556678	C₂²	1	10	yes	complemented
10	#14556678	C₂²	1	30	yes	complemented
11	C₂×#11924038	C₂²	1	15	yes	complemented
12	D₇₃⋊M₁₆	C₂²	1	30	yes	complemented
13	C₂×#11924036	C₂²	1	15	yes	complemented
14	#14556639	C₄	1	5	yes	—
15	#14556637	C₄	1	10	yes	—
16	C₂×#11924029	C₂²	1	5	yes	—
17	#41×D₇₃	C₂²	1	10	yes	—
18	2₋¹⁺⁴×C₇₃	C₄	1	1	yes	characteristic
19	#11924038	C₂³	1	10	yes	—
20	#11924038	C₂³	1	30	yes	—
21	#11924037	C₂³	1	20	yes	—
22	#11924036	C₂³	1	10	yes	—
23	#11924036	C₂³	1	30	yes	—
24	C₂×#10596833	C₂³	1	5	yes	—
25	C₂×#10596833	C₂³	1	15	yes	—
26	#11924029	C₂×C₄	1	20	yes	—
27	#11924029	C₂³	1	10	yes	—
28	#11924028	C₂×C₄	1	30	yes	—
29	#11924027	C₂³	1	10	yes	—
30	Q₈×D₇₃	C₂×C₄	1	10	yes	—
31	C₂×D₂₉₂	C₂×C₄	1	15	yes	—
32	D₄×D₇₃	C₂×C₄	1	30	yes	—
33	C₂×C₄×D₇₃	C₂³	1	15	yes	—
34	#41×C₇₃	C₂×C₄	1	10	yes	—
35	C₂×Q₈×C₇₃	C₂×C₄	1	5	yes	—
36	C₂²×D₇₃	C₂²×C₄	1	15	yes	—
37	Q₈×C₇₃	C₂²×C₄	1	10	yes	—
38	D₄×C₇₃	C₂²×C₄	1	10	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	15	yes	—
40	C₂²⋊D₇₃	C₂²×C₄	1	20	yes	—
41	C₂×Dic₇₃	C₂⁴	1	5	yes	—
42	D₂₉₂	C₂²×C₄	1	30	yes	—
43	C₄×D₇₃	C₂²×C₄	1	30	yes	—
44	C₄×D₇₃	C₂⁴	1	10	yes	—
45	Dic₁₄₆	C₂²×C₄	1	10	yes	—
46	#10596833	C₂⁴	1	1	yes	characteristic
47	#10596833	C₂⁴	1	10	yes	—
48	#10596833	C₂⁴	1	5	yes	—
49	C₂²×C₇₃	C₂³×C₄	1	5	yes	—
50	C₂×D₇₃	—	4	15	no	—
51	C₂×D₇₃	C₂³×C₄	1	10	yes	—
52	C₄×C₇₃	C₂³×C₄	1	10	yes	—
53	Dic₇₃	C₂³×C₄	1	5	yes	—
54	Dic₇₃	C₂⁵	1	1	yes	characteristic
55	C₂×C₇₃	—	2	5	no	—
56	C₂×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
57	D₇₃	—	2	10	no	—
58	#3573	—	73	1	no	maximal, Sylow
59	C₇₃	#3573	1	1	yes	characteristic, complemented, Sylow
60	#585	—	73	1	no	—
61	#568	—	73	10	no	—
62	#568	—	73	5	no	—
63	C₂×#131	—	73	10	no	—
64	C₂×#131	—	73	5	no	—
65	2₋¹⁺⁴	—	73	5	no	—
66	2₋¹⁺⁴	#64408	1	1	yes	characteristic
67	H(5,𝔽₂)	—	73	10	no	—
68	C₂×#41	—	73	10	no	—
69	C₂×#41	—	73	5	no	—
70	#131	—	73	10	no	—
71	#131	—	73	30	no	—
72	#131	—	73	10	no	—
73	#131	—	73	10	no	—
74	#131	—	73	20	no	—
75	C₂×M₁₆	—	73	30	no	—
76	C₂×M₁₆	—	73	15	no	—
77	C₂²×C₈	—	73	15	no	—
78	#41	—	73	20	no	—
79	#41	—	73	10	no	—
80	#41	—	73	10	no	—
81	#41	—	73	30	no	—
82	#41	C₂×#64408	1	10	yes	—
83	C₂×Q₈	—	73	10	no	—
84	C₂×Q₈	C₂×#64408	1	5	yes	—
85	C₂×D₄	—	73	30	no	—
86	C₂×D₄	—	73	15	no	—
87	C₂²×C₄	—	73	15	no	—
88	M₁₆	—	73	20	no	—
89	M₁₆	—	73	10	no	—
90	M₁₆	—	73	30	no	—
91	C₂×C₈	—	73	5	no	—
92	C₂×C₈	—	73	15	no	—
93	C₂×C₈	—	73	10	no	—
94	C₂×C₈	—	73	30	no	—
95	C₂³	—	73	15	no	—
96	Q₈	—	73	10	no	—
97	Q₈	C₂²×#64408	1	10	yes	—
98	D₄	—	73	20	no	—
99	D₄	—	73	30	no	—
100	D₄	C₂²×#64408	1	10	yes	—
101	C₂×C₄	—	73	5	no	—
102	C₂×C₄	—	73	10	no	—
103	C₂×C₄	—	73	30	no	—
104	C₂×C₄	C₂²×#64408	1	15	yes	—
105	C₈	—	73	1	no	—
106	C₈	—	73	5	no	—
107	C₈	—	73	10	no	—
108	C₂²	—	292	15	no	—
109	C₂²	—	73	10	no	—
110	C₂²	C₂³×#64408	1	5	yes	—
111	C₄	—	73	1	no	—
112	C₄	—	73	5	no	—
113	C₄	C₂³×#64408	1	10	yes	—
114	C₂	—	2	5	no	—
115	C₂	—	146	10	no	—
116	C₂	C₂⁴×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989667

C₂

yes

maximal, complemented

#17989666

C₂

yes

maximal, complemented

C₂×#14556679

C₂

yes

maximal, complemented

D₇₃⋊#131

C₂

yes

maximal, complemented

#17989424

C₂

yes

maximal, characteristic

#14556679

C₂²

yes

complemented

#14556679

C₂²

yes

complemented

#14556679

C₂²

yes

complemented

#14556678

C₂²

yes

complemented

#14556678

C₂²

yes

complemented

C₂×#11924038

C₂²

yes

complemented

D₇₃⋊M₁₆

C₂²

yes

complemented

C₂×#11924036

C₂²

yes

complemented

#14556639

C₄

yes

—

#14556637

C₄

yes

—

C₂×#11924029

C₂²

yes

—

#41×D₇₃

C₂²

yes

—

2₋¹⁺⁴×C₇₃

C₄

yes

characteristic

#11924038

C₂³

yes

—

#11924038

C₂³

yes

—

#11924037

C₂³

yes

—

#11924036

C₂³

yes

—

#11924036

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

#11924029

C₂×C₄

yes

—

#11924029

C₂³

yes

—

#11924028

C₂×C₄

yes

—

#11924027

C₂³

yes

—

Q₈×D₇₃

C₂×C₄

yes

—

C₂×D₂₉₂

C₂×C₄

yes

—

D₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

#41×C₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×D₇₃

C₂²×C₄

yes

—

Q₈×C₇₃

C₂²×C₄

yes

—

D₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂²⋊D₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂⁴

yes

—

D₂₉₂

C₂²×C₄

yes

—

C₄×D₇₃

C₂²×C₄

yes

—

C₄×D₇₃

C₂⁴

yes

—

Dic₁₄₆

C₂²×C₄

yes

—

#10596833

C₂⁴

yes

characteristic

#10596833

C₂⁴

yes

—

#10596833

C₂⁴

yes

—

C₂²×C₇₃

C₂³×C₄

yes

—

C₂×D₇₃

—

C₂×D₇₃

C₂³×C₄

yes

—

C₄×C₇₃

C₂³×C₄

yes

—

Dic₇₃

C₂³×C₄

yes

—

Dic₇₃

C₂⁵

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂⁴×C₄

yes

characteristic

D₇₃

—

#3573

—

maximal, Sylow

C₇₃

#3573

yes

characteristic, complemented, Sylow

—

—

—

—

—

2₋¹⁺⁴

—

2₋¹⁺⁴

#64408

yes

characteristic

H(5,𝔽₂)

—

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

Q₈

—

Q₈

C₂²×#64408

yes

—

D₄

—

D₄

—

100

D₄

C₂²×#64408

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²×#64408

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

292

—

109

C₂²

—

110

C₂²

C₂³×#64408

yes

—

111

C₄

—

112

C₄

—

113

C₄

C₂³×#64408

yes

—

114

C₂

—

115

C₂

—

146

—

116

C₂

C₂⁴×#64408

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#64408
commutator, socle	C₂×C₇₃	C₂⁴×C₄
Fitting	2₋¹⁺⁴×C₇₃	C₄
hypercenter	2₋¹⁺⁴	#64408
hypocenter	C₇₃	#3573
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂

C₂⁴×#64408

commutator, socle

C₂×C₇₃

C₂⁴×C₄

Fitting

2₋¹⁺⁴×C₇₃

C₄

2₋¹⁺⁴

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂ ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
1. #17989667	C₂	semidirect
2. #17989666	C₂	semidirect
3. C₂×#14556679	C₂	semidirect
4. D₇₃⋊#131	C₂	semidirect
6. #14556679	C₂²	semidirect
7. #14556679	C₂²	semidirect
8. #14556679	C₂²	semidirect
9. #14556678	C₂²	semidirect
10. #14556678	C₂²	semidirect
11. C₂×#11924038	C₂²	semidirect
12. D₇₃⋊M₁₆	C₂²	semidirect
13. C₂×#11924036	C₂²	semidirect
59. C₇₃	#3573	semidirect

H ≅ G/N

type

1. #17989667

C₂

semidirect

2. #17989666

C₂

semidirect

3. C₂×#14556679

C₂

semidirect

4. D₇₃⋊#131

C₂

semidirect

6. #14556679

C₂²

semidirect

7. #14556679

C₂²

semidirect

8. #14556679

C₂²

semidirect

9. #14556678

C₂²

semidirect

10. #14556678

C₂²

semidirect

11. C₂×#11924038

C₂²

semidirect

12. D₇₃⋊M₁₆

C₂²

semidirect

13. C₂×#11924036

C₂²

semidirect

59. C₇₃

#3573

semidirect

Subgroups — #22626927

Conjugacy classes of subgroups

Complements