Group #22621735 — subgroups

Subgroups — #22621735

Order 9344.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#17989400	C₂	1	4	yes	maximal, complemented
2	#17988859	C₂	1	4	yes	maximal, complemented
3	#17988769	C₂	1	4	yes	maximal, complemented
4	#17988705	C₂	1	2	yes	maximal, complemented
5	#494×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556639	C₂²	1	4	yes	—
7	#14556623	—	2	8	no	—
8	#14556622	C₂²	1	8	yes	—
9	#14556614	—	2	4	no	—
10	#14556613	C₂²	1	2	yes	—
11	Q₁₆×D₇₃	—	2	4	no	—
12	#14556597	C₂²	1	4	yes	—
13	#14556527	C₂²	1	2	yes	—
14	#14556519	C₂²	1	4	yes	—
15	#14556510	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₇₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
18	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#11924029	—	2	8	no	—
20	#11924028	—	4	4	no	—
21	#11924028	C₂³	1	4	yes	—
22	Q₈×D₇₃	—	2	8	no	—
23	Dic₂₉₂	D₄	1	2	yes	—
24	#11924018	—	2	8	no	—
25	#11924017	D₄	1	4	yes	—
26	#11924015	—	2	8	no	—
27	D₅₈₄	D₄	1	2	yes	—
28	#11924012	C₂³	1	4	yes	—
29	#11924011	—	2	4	no	—
30	C₈×D₇₃	—	2	4	no	—
31	C₂×Q₈×C₇₃	C₂³	1	4	yes	—
32	Q₁₆×C₇₃	—	2	4	no	—
33	Q₁₆×C₇₃	—	2	4	no	—
34	C₂×C₈×C₇₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₇₃	—	4	2	no	—
36	C₄²×C₇₃	C₂³	1	1	yes	characteristic
37	Q₈×C₇₃	—	2	8	no	—
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	—	2	4	no	—
40	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₇₃	—	2	4	no	—
42	D₂₉₂	—	4	4	no	—
43	D₂₉₂	C₂×D₄	1	4	yes	—
44	C₄×D₇₃	—	2	4	no	—
45	C₄×D₇₃	—	4	8	no	—
46	Dic₁₄₆	—	4	4	no	—
47	Dic₁₄₆	C₂×D₄	1	4	yes	—
48	C₈×C₇₃	C₂×D₄	1	2	yes	—
49	C₈×C₇₃	C₂×D₄	1	2	yes	—
50	#10596831	—	2	4	no	—
51	C₂²×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₇₃	—	2	4	no	—
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	—	4	4	no	—
55	C₄×C₇₃	C₂²×D₄	1	2	yes	—
56	Dic₇₃	—	2	4	no	—
57	C₂×C₇₃	—	2	1	no	—
58	C₂×C₇₃	D₄²	1	1	yes	characteristic
59	D₇₃	—	4	4	no	—
60	#3293	—	73	1	no	maximal, Sylow
61	C₇₃	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	73	4	no	—
63	#494	D₇₃	1	1	yes	characteristic, complemented
64	#473	—	73	4	no	—
65	#456	—	73	4	no	—
66	#443	—	73	2	no	—
67	2₋¹⁺⁴	—	73	4	no	—
68	#137	—	73	8	no	—
69	#137	—	146	8	no	—
70	#135	—	73	2	no	—
71	#135	—	146	4	no	—
72	C₂×Q₁₆	—	146	4	no	—
73	C₂×Q₁₆	C₂×D₇₃	1	4	yes	—
74	#131	—	73	4	no	—
75	C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
76	#108	—	73	2	no	—
77	#104	—	73	4	no	—
78	#101	—	73	4	no	—
79	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
80	#41	—	292	4	no	—
81	#41	—	73	4	no	—
82	#41	—	146	8	no	—
83	C₂×Q₈	—	146	8	no	—
84	C₂×Q₈	C₂²×D₇₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	73	2	no	—
88	Q₁₆	—	146	8	no	—
89	SD₁₆	—	73	4	no	—
90	SD₁₆	—	146	8	no	—
91	D₈	—	73	2	no	—
92	M₁₆	—	73	4	no	—
93	M₁₆	—	146	4	no	—
94	C₂×C₈	—	146	4	no	—
95	C₂×C₈	C₂²×D₇₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₇₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	292	4	no	—
100	Q₈	—	73	4	no	—
101	D₄	—	292	4	no	—
102	D₄	—	73	4	no	—
103	D₄	—	146	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	292	8	no	—
107	C₂×C₄	—	146	4	no	—
108	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
109	C₈	—	146	4	no	—
110	C₈	D₄×D₇₃	1	2	yes	—
111	C₈	D₄×D₇₃	1	2	yes	—
112	C₂²	—	146	4	no	—
113	C₂²	#14556636	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	146	4	no	—
117	C₄	C₂×D₄×D₇₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	292	4	no	—
120	C₂	#17989222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989400

C₂

yes

maximal, complemented

#17988859

C₂

yes

maximal, complemented

#17988769

C₂

yes

maximal, complemented

#17988705

C₂

yes

maximal, complemented

#494×C₇₃

C₂

yes

maximal, characteristic, complemented

#14556639

C₂²

yes

—

#14556623

—

#14556622

C₂²

yes

—

#14556614

—

#14556613

C₂²

yes

—

Q₁₆×D₇₃

—

#14556597

C₂²

yes

—

#14556527

C₂²

yes

—

#14556519

C₂²

yes

—

#14556510

C₂²

yes

—

C₂×Q₁₆×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₇₃

—

Dic₂₉₂

D₄

yes

—

#11924018

—

#11924017

D₄

yes

—

#11924015

—

D₅₈₄

D₄

yes

—

#11924012

C₂³

yes

—

#11924011

—

C₈×D₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

D₂₉₂

—

D₂₉₂

C₂×D₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

C₂×D₄

yes

—

C₈×C₇₃

C₂×D₄

yes

—

C₈×C₇₃

C₂×D₄

yes

—

#10596831

—

C₂²×C₇₃

H(5,𝔽₂)

yes

characteristic

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²×D₄

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

D₄²

yes

characteristic

D₇₃

—

#3293

—

maximal, Sylow

C₇₃

#3293

yes

characteristic, complemented, Sylow

#578

—

#494

D₇₃

yes

characteristic, complemented

#473

—

#456

—

#443

—

2₋¹⁺⁴

—

#137

—

#137

—

146

—

#135

—

#135

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

C₂×D₇₃

yes

—

#131

—

C₄⋊Q₈

C₂×D₇₃

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₇₃

yes

characteristic

#41

—

292

—

#41

—

#41

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

C₂²×D₇₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

146

—

SD₁₆

—

SD₁₆

—

146

—

D₈

—

M₁₆

—

M₁₆

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂²×D₇₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₇₃

yes

characteristic

Q₈

—

Q₈

—

292

—

100

Q₈

—

101

D₄

—

292

—

102

D₄

—

103

D₄

—

146

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

292

—

107

C₂×C₄

—

146

—

108

C₂×C₄

C₂³×D₇₃

yes

characteristic

109

C₈

—

146

—

110

C₈

D₄×D₇₃

yes

—

111

C₈

D₄×D₇₃

yes

—

112

C₂²

—

146

—

113

C₂²

#14556636

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

146

—

117

C₄

C₂×D₄×D₇₃

yes

—

118

C₂

—

119

C₂

—

292

—

120

C₂

#17989222

yes

characteristic

	subgroup	G/H
center	C₂	#17989222
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#494×C₇₃	C₂
socle	C₂×C₇₃	D₄²
hypercenter	#494	D₇₃
hypocenter	C₇₃	#3293
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#17989222

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#494×C₇₃

C₂

socle

C₂×C₇₃

D₄²

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #17989400	C₂	semidirect
2. #17988859	C₂	semidirect
3. #17988769	C₂	semidirect
4. #17988705	C₂	semidirect
5. #494×C₇₃	C₂	semidirect
61. C₇₃	#3293	semidirect
63. #494	D₇₃	semidirect

H ≅ G/N

type

1. #17989400

C₂

semidirect

2. #17988859

C₂

semidirect

3. #17988769

C₂

semidirect

4. #17988705

C₂

semidirect

5. #494×C₇₃

C₂

semidirect

61. C₇₃

#3293

semidirect

63. #494

D₇₃

semidirect

Subgroups — #22621735

Conjugacy classes of subgroups

Complements