Group #22626926 — subgroups

Subgroups — #22626926

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	#17989666	C₂	1	6	yes	maximal, complemented
2	#17989665	C₂	1	9	yes	maximal, complemented
3	D₇₃⋊#131	C₂	1	6	yes	maximal, complemented
4	C₂×#14556678	C₂	1	9	yes	maximal, complemented
5	#17989422	C₂	1	1	yes	maximal, characteristic
6	#14556679	C₂²	1	2	yes	complemented
7	#14556679	C₂²	1	6	yes	complemented
8	#14556678	C₂²	1	18	yes	complemented
9	#14556678	C₂²	1	36	yes	complemented
10	#14556678	C₂²	1	18	yes	complemented
11	C₂×#11924038	C₂²	1	9	yes	complemented
12	D₇₃⋊M₁₆	C₂²	1	18	yes	complemented
13	C₂×#11924037	C₂²	1	18	yes	complemented
14	C₂×#11924036	C₂²	1	9	yes	complemented
15	C₂²×#10596833	C₂²	1	6	yes	complemented
16	#14556638	C₄	1	6	yes	—
17	#14556636	C₄	1	9	yes	—
18	C₂×#11924027	C₂²	1	9	yes	—
19	#41×D₇₃	C₂²	1	6	yes	—
20	H(5,𝔽₂)×C₇₃	C₄	1	1	yes	characteristic
21	#11924038	C₂³	1	6	yes	—
22	#11924038	C₂³	1	18	yes	—
23	#11924037	C₂³	1	36	yes	—
24	#11924036	C₂³	1	6	yes	—
25	#11924036	C₂³	1	18	yes	—
26	C₂×#10596833	C₂³	1	9	yes	—
27	C₂×#10596833	C₂³	1	18	yes	—
28	C₂×#10596833	C₂³	1	9	yes	—
29	#11924029	C₂³	1	2	yes	—
30	#11924028	C₂×C₄	1	18	yes	—
31	#11924027	C₂×C₄	1	36	yes	—
32	#11924027	C₂³	1	18	yes	—
33	C₂×Dic₁₄₆	C₂×C₄	1	9	yes	—
34	Q₈×D₇₃	C₂×C₄	1	6	yes	—
35	C₂×C₂²⋊D₇₃	C₂×C₄	1	18	yes	—
36	D₄×D₇₃	C₂×C₄	1	18	yes	—
37	C₂²×Dic₇₃	C₂³	1	6	yes	—
38	C₂×C₄×D₇₃	C₂³	1	9	yes	—
39	#41×C₇₃	C₂×C₄	1	6	yes	—
40	C₂×D₄×C₇₃	C₂×C₄	1	9	yes	—
41	C₂³×C₇₃	C₂²×C₄	1	6	yes	—
42	C₂²×D₇₃	C₂²×C₄	1	9	yes	—
43	Q₈×C₇₃	C₂²×C₄	1	2	yes	—
44	D₄×C₇₃	C₂²×C₄	1	18	yes	—
45	C₂×C₄×C₇₃	C₂²×C₄	1	9	yes	—
46	C₂²⋊D₇₃	C₂²×C₄	1	36	yes	—
47	C₂×Dic₇₃	C₂²×C₄	1	18	yes	—
48	C₂×Dic₇₃	C₂⁴	1	9	yes	—
49	D₂₉₂	C₂²×C₄	1	6	yes	—
50	C₄×D₇₃	C₂²×C₄	1	18	yes	—
51	C₄×D₇₃	C₂⁴	1	6	yes	—
52	Dic₁₄₆	C₂²×C₄	1	18	yes	—
53	#10596833	C₂⁴	1	1	yes	characteristic
54	#10596833	C₂⁴	1	9	yes	—
55	#10596833	C₂⁴	1	6	yes	—
56	C₂²×C₇₃	—	4	6	no	—
57	C₂²×C₇₃	C₂³×C₄	1	9	yes	—
58	C₂×D₇₃	—	4	9	no	—
59	C₂×D₇₃	C₂³×C₄	1	6	yes	—
60	C₄×C₇₃	C₂³×C₄	1	6	yes	—
61	Dic₇₃	C₂³×C₄	1	9	yes	—
62	Dic₇₃	C₂⁵	1	1	yes	characteristic
63	C₂×C₇₃	—	2	9	no	—
64	C₂×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
65	D₇₃	—	2	6	no	—
66	#3573	—	73	1	no	maximal, Sylow
67	C₇₃	#3573	1	1	yes	characteristic, complemented, Sylow
68	#585	—	73	1	no	—
69	#568	—	73	9	no	—
70	#568	—	73	6	no	—
71	C₂×#131	—	73	9	no	—
72	C₂×#131	—	73	6	no	—
73	2₋¹⁺⁴	—	73	6	no	—
74	H(5,𝔽₂)	—	73	9	no	—
75	H(5,𝔽₂)	#64408	1	1	yes	characteristic
76	C₂×#41	—	73	6	no	—
77	C₂×#41	—	73	9	no	—
78	#131	—	73	18	no	—
79	#131	—	73	36	no	—
80	#131	—	73	18	no	—
81	#131	—	73	2	no	—
82	#131	—	73	6	no	—
83	C₂×M₁₆	—	73	18	no	—
84	C₂×M₁₆	—	73	18	no	—
85	C₂×M₁₆	—	73	9	no	—
86	C₂²×C₈	—	73	6	no	—
87	C₂²×C₈	—	73	9	no	—
88	#41	—	73	2	no	—
89	#41	—	73	36	no	—
90	#41	—	73	18	no	—
91	#41	—	73	18	no	—
92	#41	C₂×#64408	1	6	yes	—
93	C₂×Q₈	—	73	6	no	—
94	C₂×Q₈	—	73	9	no	—
95	C₂×D₄	—	73	18	no	—
96	C₂×D₄	—	73	18	no	—
97	C₂×D₄	C₂×#64408	1	9	yes	—
98	C₂²×C₄	—	73	6	no	—
99	C₂²×C₄	—	73	9	no	—
100	M₁₆	—	73	36	no	—
101	M₁₆	—	73	6	no	—
102	M₁₆	—	73	18	no	—
103	C₂×C₈	—	73	9	no	—
104	C₂×C₈	—	73	18	no	—
105	C₂×C₈	—	73	9	no	—
106	C₂×C₈	—	73	6	no	—
107	C₂×C₈	—	73	18	no	—
108	C₂³	—	73	9	no	—
109	C₂³	C₂²×#64408	1	6	yes	—
110	Q₈	—	73	18	no	—
111	Q₈	C₂²×#64408	1	2	yes	—
112	D₄	—	73	36	no	—
113	D₄	—	73	6	no	—
114	D₄	C₂²×#64408	1	18	yes	—
115	C₂×C₄	—	73	9	no	—
116	C₂×C₄	—	73	18	no	—
117	C₂×C₄	—	73	6	no	—
118	C₂×C₄	—	73	18	no	—
119	C₂×C₄	C₂²×#64408	1	9	yes	—
120	C₈	—	73	1	no	—
121	C₈	—	73	9	no	—
122	C₈	—	73	6	no	—
123	C₂²	—	4	6	no	—
124	C₂²	—	292	9	no	—
125	C₂²	—	73	6	no	—
126	C₂²	C₂³×#64408	1	9	yes	—
127	C₄	—	73	1	no	—
128	C₄	—	73	9	no	—
129	C₄	C₂³×#64408	1	6	yes	—
130	C₂	—	2	9	no	—
131	C₂	—	146	6	no	—
132	C₂	C₂⁴×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989666

C₂

yes

maximal, complemented

#17989665

C₂

yes

maximal, complemented

D₇₃⋊#131

C₂

yes

maximal, complemented

C₂×#14556678

C₂

yes

maximal, complemented

#17989422

C₂

yes

maximal, characteristic

#14556679

C₂²

yes

complemented

#14556679

C₂²

yes

complemented

#14556678

C₂²

yes

complemented

#14556678

C₂²

yes

complemented

#14556678

C₂²

yes

complemented

C₂×#11924038

C₂²

yes

complemented

D₇₃⋊M₁₆

C₂²

yes

complemented

C₂×#11924037

C₂²

yes

complemented

C₂×#11924036

C₂²

yes

complemented

C₂²×#10596833

C₂²

yes

complemented

#14556638

C₄

yes

—

#14556636

C₄

yes

—

C₂×#11924027

C₂²

yes

—

#41×D₇₃

C₂²

yes

—

H(5,𝔽₂)×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

—

C₂×#10596833

C₂³

yes

—

#11924029

C₂³

yes

—

#11924028

C₂×C₄

yes

—

#11924027

C₂×C₄

yes

—

#11924027

C₂³

yes

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

Q₈×D₇₃

C₂×C₄

yes

—

C₂×C₂²⋊D₇₃

C₂×C₄

yes

—

D₄×D₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

#41×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²×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

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

—

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

—

—

yes

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂³

—

109

C₂³

C₂²×#64408

yes

—

110

Q₈

—

111

Q₈

C₂²×#64408

yes

—

112

D₄

—

113

D₄

—

114

D₄

C₂²×#64408

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×#64408

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

292

—

125

C₂²

—

126

C₂²

C₂³×#64408

yes

—

127

C₄

—

128

C₄

—

129

C₄

C₂³×#64408

yes

—

130

C₂

—

131

C₂

—

146

—

132

C₂

C₂⁴×#64408

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#64408
commutator, socle	C₂×C₇₃	C₂⁴×C₄
Fitting	H(5,𝔽₂)×C₇₃	C₄
hypercenter	H(5,𝔽₂)	#64408
hypocenter	C₇₃	#3573
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

subgroup

G/H

center, Frattini

C₂

C₂⁴×#64408

commutator, socle

C₂×C₇₃

C₂⁴×C₄

Fitting

H(5,𝔽₂)×C₇₃

C₄

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #17989666	C₂	semidirect
2. #17989665	C₂	semidirect
3. D₇₃⋊#131	C₂	semidirect
4. C₂×#14556678	C₂	semidirect
6. #14556679	C₂²	semidirect
7. #14556679	C₂²	semidirect
8. #14556678	C₂²	semidirect
9. #14556678	C₂²	semidirect
10. #14556678	C₂²	semidirect
11. C₂×#11924038	C₂²	semidirect
12. D₇₃⋊M₁₆	C₂²	semidirect
13. C₂×#11924037	C₂²	semidirect
14. C₂×#11924036	C₂²	semidirect
15. C₂²×#10596833	C₂²	semidirect
67. C₇₃	#3573	semidirect

H ≅ G/N

type

1. #17989666

C₂

semidirect

2. #17989665

C₂

semidirect

3. D₇₃⋊#131

C₂

semidirect

4. C₂×#14556678

C₂

semidirect

6. #14556679

C₂²

semidirect

7. #14556679

C₂²

semidirect

8. #14556678

C₂²

semidirect

9. #14556678

C₂²

semidirect

10. #14556678

C₂²

semidirect

11. C₂×#11924038

C₂²

semidirect

12. D₇₃⋊M₁₆

C₂²

semidirect

13. C₂×#11924037

C₂²

semidirect

14. C₂×#11924036

C₂²

semidirect

15. C₂²×#10596833

C₂²

semidirect

67. C₇₃

#3573

semidirect

Subgroups — #22626926

Conjugacy classes of subgroups

Complements