Group #22626900 — subgroups

Subgroups — #22626900

Order 9344.

360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 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	#17989646	C₂	1	8	yes	maximal, complemented
2	#17989643	C₂	1	4	yes	maximal, complemented
3	C₂×#14556665	C₂	1	2	yes	maximal, complemented
4	C₂×#14556599	C₂	1	1	yes	maximal, characteristic
5	#14556666	C₂²	1	8	yes	complemented
6	#14556665	C₂²	1	4	yes	—
7	#14556665	C₂²	1	4	yes	complemented
8	#14556664	C₂²	1	8	yes	complemented
9	C₂×#11924031	C₂²	1	4	yes	complemented
10	C₂×#11924028	C₄	1	1	yes	characteristic
11	#14556599	C₄	1	4	yes	—
12	#14556599	C₂²	1	4	yes	—
13	C₂×#11924011	C₄	1	2	yes	—
14	C₂²×#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₃	C₂²	1	2	yes	—
16	C₂×M₁₆×C₇₃	C₄	1	1	yes	characteristic
17	#11924031	C₂³	1	8	yes	—
18	#11924028	C₈	1	4	yes	—
19	#11924028	C₂×C₄	1	4	yes	—
20	C₂×Dic₁₄₆	C₈	1	1	yes	characteristic
21	C₂×D₂₉₂	C₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₇₃	C₈	1	2	yes	—
23	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
24	#11924011	C₂×C₄	1	8	yes	—
25	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂×#10596831	C₂×C₄	1	2	yes	—
27	C₂×#10596831	C₂³	1	1	yes	characteristic
28	C₂×#10596831	C₂³	1	2	yes	—
29	C₈×D₇₃	C₂×C₄	1	4	yes	—
30	C₈×D₇₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₇₃	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₇₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
40	C₂²⋊D₇₃	C₂×C₈	1	8	yes	—
41	C₂×Dic₇₃	C₂×C₈	1	2	yes	—
42	D₂₉₂	C₂×C₈	1	4	yes	—
43	C₄×D₇₃	C₂×C₈	1	4	yes	—
44	C₄×D₇₃	C₂²×C₄	1	4	yes	—
45	Dic₁₄₆	C₂×C₈	1	4	yes	—
46	C₈×C₇₃	C₂²×C₄	1	4	yes	—
47	#10596831	C₂²×C₄	1	1	yes	characteristic
48	#10596831	C₂²×C₄	1	2	yes	—
49	#10596831	C₂⁴	1	1	yes	characteristic
50	C₂²×C₇₃	—	4	1	no	—
51	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₇₃	C₂²×C₈	1	2	yes	—
53	C₂×D₇₃	—	4	2	no	—
54	C₂×D₇₃	C₂²×C₈	1	4	yes	—
55	C₄×C₇₃	C₂²×C₈	1	2	yes	—
56	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
58	Dic₇₃	C₂²×C₈	1	4	yes	—
59	C₂×C₇₃	—	2	1	no	—
60	C₂×C₇₃	—	2	2	no	—
61	C₂×C₇₃	C₂³×C₈	1	1	yes	characteristic
62	D₇₃	—	2	4	no	—
63	#3407	—	73	1	no	maximal, Sylow
64	C₇₃	#3407	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	—	73	1	no	—
66	#504	—	73	8	no	—
67	C₂×M₃₂	—	73	2	no	—
68	C₂×M₃₂	—	73	4	no	—
69	C₂×#41	—	73	1	no	—
70	#131	—	73	4	no	—
71	#131	—	73	4	no	—
72	C₂×M₁₆	—	73	2	no	—
73	C₂×M₁₆	#64408	1	1	yes	characteristic
74	C₂²×C₈	—	73	1	no	—
75	C₂²×C₈	—	73	2	no	—
76	M₃₂	—	73	4	no	—
77	M₃₂	—	73	4	no	—
78	M₃₂	—	73	8	no	—
79	C₂×C₁₆	—	73	4	no	—
80	C₂×C₁₆	—	73	8	no	—
81	#41	—	73	4	no	—
82	#41	—	73	4	no	—
83	C₂×Q₈	—	73	1	no	—
84	C₂×D₄	—	73	2	no	—
85	C₂×D₄	—	73	1	no	—
86	C₂²×C₄	—	73	2	no	—
87	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
88	M₁₆	—	73	8	no	—
89	M₁₆	C₂×#64408	1	4	yes	—
90	C₂×C₈	—	73	1	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	—	73	2	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	73	4	no	—
95	C₂×C₈	—	73	4	no	—
96	C₂×C₈	C₂×#64408	1	2	yes	—
97	C₁₆	—	73	8	no	—
98	C₂³	—	73	2	no	—
99	C₂³	#11924036	1	1	yes	characteristic
100	Q₈	—	73	4	no	—
101	D₄	—	73	8	no	—
102	D₄	—	73	4	no	—
103	C₂×C₄	—	73	2	no	—
104	C₂×C₄	—	73	4	no	—
105	C₂×C₄	—	73	4	no	—
106	C₂×C₄	#11924036	1	1	yes	characteristic
107	C₂×C₄	#11924036	1	2	yes	—
108	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
109	C₂×C₄	C₂²×#64408	1	2	yes	—
110	C₈	—	73	1	no	—
111	C₈	—	73	1	no	—
112	C₈	—	73	2	no	—
113	C₈	C₂²×#64408	1	4	yes	—
114	C₂²	—	4	1	no	—
115	C₂²	—	292	2	no	—
116	C₂²	—	73	4	no	—
117	C₂²	C₂×#11924036	1	1	yes	characteristic
118	C₂²	C₂×#11924036	1	2	yes	—
119	C₄	—	73	4	no	—
120	C₄	C₂×#11924036	1	1	yes	characteristic
121	C₄	C₂×#11924036	1	2	yes	—
122	C₄	C₂³×#64408	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	2	no	—
125	C₂	—	146	4	no	—
126	C₂	C₂²×#11924036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989646

C₂

yes

maximal, complemented

#17989643

C₂

yes

maximal, complemented

C₂×#14556665

C₂

yes

maximal, complemented

C₂×#14556599

C₂

yes

maximal, characteristic

#14556666

C₂²

yes

complemented

#14556665

C₂²

yes

—

#14556665

C₂²

yes

complemented

#14556664

C₂²

yes

complemented

C₂×#11924031

C₂²

yes

complemented

C₂×#11924028

C₄

yes

characteristic

#14556599

C₄

yes

—

#14556599

C₂²

yes

—

C₂×#11924011

C₄

yes

—

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

—

C₂×M₁₆×C₇₃

C₄

yes

characteristic

#11924031

C₂³

yes

—

#11924028

C₈

yes

—

#11924028

C₂×C₄

yes

—

C₂×Dic₁₄₆

C₈

yes

characteristic

C₂×D₂₉₂

C₈

yes

characteristic

C₂×C₂²⋊D₇₃

C₈

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

#11924011

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

C₂³

yes

characteristic

C₂×#10596831

C₂³

yes

—

C₈×D₇₃

C₂×C₄

yes

—

C₈×D₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂²⋊D₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂×C₈

yes

—

D₂₉₂

C₂×C₈

yes

—

C₄×D₇₃

C₂×C₈

yes

—

C₄×D₇₃

C₂²×C₄

yes

—

Dic₁₄₆

C₂×C₈

yes

—

C₈×C₇₃

C₂²×C₄

yes

—

#10596831

C₂²×C₄

yes

characteristic

#10596831

C₂²×C₄

yes

—

#10596831

C₂⁴

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂²×C₇₃

C₂²×C₈

yes

—

C₂×D₇₃

—

C₂×D₇₃

C₂²×C₈

yes

—

C₄×C₇₃

C₂²×C₈

yes

—

C₄×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

C₂³×C₄

yes

characteristic

Dic₇₃

C₂²×C₈

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂³×C₈

yes

characteristic

D₇₃

—

#3407

—

maximal, Sylow

C₇₃

#3407

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

characteristic

M₁₆

—

M₁₆

C₂×#64408

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#64408

yes

—

C₁₆

—

C₂³

—

C₂³

#11924036

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#11924036

yes

characteristic

107

C₂×C₄

#11924036

yes

—

108

C₂×C₄

C₂²×#64408

yes

characteristic

109

C₂×C₄

C₂²×#64408

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₂²×#64408

yes

—

114

C₂²

—

115

C₂²

—

292

—

116

C₂²

—

117

C₂²

C₂×#11924036

yes

characteristic

118

C₂²

C₂×#11924036

yes

—

119

C₄

—

120

C₄

C₂×#11924036

yes

characteristic

121

C₄

C₂×#11924036

yes

—

122

C₄

C₂³×#64408

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

146

—

126

C₂

C₂²×#11924036

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#64408
commutator, socle	C₂×C₇₃	C₂³×C₈
Fitting	C₂×M₁₆×C₇₃	C₄
hypercenter	C₂×M₁₆	#64408
hypocenter	C₇₃	#3407
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₄

C₂³×#64408

commutator, socle

C₂×C₇₃

C₂³×C₈

Fitting

C₂×M₁₆×C₇₃

C₄

C₂×M₁₆

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. #17989646	C₂	semidirect
2. #17989643	C₂	semidirect
3. C₂×#14556665	C₂	semidirect
5. #14556666	C₂²	semidirect
7. #14556665	C₂²	semidirect
8. #14556664	C₂²	semidirect
9. C₂×#11924031	C₂²	semidirect
64. C₇₃	#3407	semidirect

H ≅ G/N

type

1. #17989646

C₂

semidirect

2. #17989643

C₂

semidirect

3. C₂×#14556665

C₂

semidirect

5. #14556666

C₂²

semidirect

7. #14556665

C₂²

semidirect

8. #14556664

C₂²

semidirect

9. C₂×#11924031

C₂²

semidirect

64. C₇₃

#3407

semidirect

Subgroups — #22626900

Conjugacy classes of subgroups

Complements