Group #22620484 — subgroups

Subgroups — #22620484

Order 9344.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556583	C₂	1	1	yes	maximal, characteristic
2	#17988993	C₂	1	4	yes	maximal, complemented
3	#17988986	C₂	1	4	yes	maximal, complemented
4	#17988630	C₂	1	4	yes	maximal
5	C₄×#11924012	C₂	1	1	yes	maximal, characteristic
6	#533×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11924012	C₂²	1	2	yes	—
8	#14556583	—	2	2	no	—
9	#14556583	C₂²	1	2	yes	—
10	#14556583	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
12	C₂×#11924007	—	2	2	no	—
13	#14556524	C₂²	1	8	yes	—
14	#14556522	C₂²	1	8	yes	—
15	#14556502	C₂²	1	2	yes	—
16	C₄×#10596831	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
18	#125×C₇₃	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₇₃	—	2	2	no	—
20	C₄×Q₈×C₇₃	—	2	2	no	—
21	#117×C₇₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₇₃	—	2	2	no	—
24	#11924012	Q₈	1	8	yes	complemented
25	C₂×#10596831	C₂³	1	4	yes	—
26	C₄⋊Dic₇₃	—	2	2	no	—
27	C₄⋊Dic₇₃	C₂³	1	4	yes	—
28	#11924007	—	2	4	no	—
29	#11924007	—	2	4	no	—
30	C₂×Q₈×C₇₃	—	2	2	no	—
31	C₂²×C₄×C₇₃	D₄	1	2	yes	—
32	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	2	4	no	—
34	C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₇₃	—	2	2	no	—
37	C₄²×C₇₃	—	2	2	no	—
38	C₄²×C₇₃	D₄	1	2	yes	—
39	C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₇₃	—	4	4	no	—
43	C₂×C₄×C₇₃	—	2	4	no	—
44	C₂×C₄×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	4	no	—
51	C₂×Dic₇₃	—	4	4	no	—
52	#10596831	C₂×Q₈	1	8	yes	—
53	C₂²×C₇₃	—	2	2	no	—
54	C₂²×C₇₃	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	—	2	2	no	—
58	C₄×C₇₃	—	4	4	no	—
59	C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
60	C₄×C₇₃	C₂²×Q₈	1	2	yes	—
61	Dic₇₃	—	4	4	no	—
62	C₂×C₇₃	—	2	2	no	—
63	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₇₃	#575	1	2	yes	—
65	#3166	—	73	1	no	maximal, Sylow
66	C₇₃	#3166	1	1	yes	characteristic, complemented, Sylow
67	#533	D₇₃	1	1	yes	characteristic
68	C₂×#125	—	73	1	no	—
69	#501	—	73	4	no	—
70	#499	—	73	4	no	—
71	#428	—	73	4	no	—
72	C₄×M₁₆	—	73	1	no	—
73	C₂×M₁₆	—	73	2	no	—
74	C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
75	#125	—	73	2	no	—
76	#125	—	73	2	no	—
77	#125	—	146	2	no	—
78	#125	C₂×D₇₃	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	#117	C₂×D₇₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	73	2	no	—
83	C₂×C₄⋊C₄	—	146	2	no	—
84	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
85	#107	—	73	8	no	—
86	#106	—	73	8	no	—
87	#97	—	73	2	no	—
88	C₄×C₈	—	73	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	—	146	2	no	—
91	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
93	M₁₆	—	73	8	no	—
94	C₂×C₈	—	73	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	73	4	no	—
98	C₄⋊C₄	—	146	4	no	—
99	C₄⋊C₄	—	146	4	no	—
100	C₄⋊C₄	—	146	2	no	—
101	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₇₃	1	2	yes	—
105	C₄²	C₂²×D₇₃	1	1	yes	characteristic
106	C₄²	C₂²×D₇₃	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
108	Q₈	—	4	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	292	4	no	—
112	C₂×C₄	—	146	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₇₃	1	4	yes	—
117	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
118	C₈	—	73	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#14556589	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	292	4	no	—
126	C₄	#14556589	1	2	yes	—
127	C₄	C₂×Q₈×D₇₃	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#14556589	1	1	yes	characteristic
130	C₂	#17989385	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556583

C₂

yes

maximal, characteristic

#17988993

C₂

yes

maximal, complemented

#17988986

C₂

yes

maximal, complemented

#17988630

C₂

yes

maximal

C₄×#11924012

C₂

yes

maximal, characteristic

#533×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

—

#14556583

—

#14556583

C₂²

yes

—

#14556583

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924007

—

#14556524

C₂²

yes

—

#14556522

C₂²

yes

—

#14556502

C₂²

yes

—

C₄×#10596831

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

#125×C₇₃

C₂²

yes

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

#11924012

Q₈

yes

complemented

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

C₂×Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

—

C₄×C₇₃

C₂²×Q₈

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₃

#575

yes

—

#3166

—

maximal, Sylow

C₇₃

#3166

yes

characteristic, complemented, Sylow

#533

D₇₃

yes

—

—

—

—

C₄×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₇₃

yes

—

#125

—

#125

—

#125

—

146

—

#125

C₂×D₇₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₇₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²⋊D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

100

C₄⋊C₄

—

146

—

101

C₄⋊C₄

C₂²×D₇₃

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂²⋊D₇₃

yes

—

105

C₄²

C₂²×D₇₃

yes

characteristic

106

C₄²

C₂²×D₇₃

yes

characteristic

107

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

292

—

112

C₂×C₄

—

146

—

113

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

114

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

115

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

116

C₂×C₄

Q₈×D₇₃

yes

—

117

C₂×C₄

C₂³×D₇₃

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#14556589

yes

—

121

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

292

—

126

C₄

#14556589

yes

—

127

C₄

C₂×Q₈×D₇₃

yes

—

128

C₂

—

129

C₂

C₂×#14556589

yes

characteristic

130

C₂

#17989385

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#533×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#533	D₇₃
hypocenter	C₇₃	#3166
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#533×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #17988993	C₂	semidirect
3. #17988986	C₂	semidirect
24. #11924012	Q₈	semidirect
66. C₇₃	#3166	semidirect

H ≅ G/N

type

2. #17988993

C₂

semidirect

3. #17988986

C₂

semidirect

24. #11924012

Q₈

semidirect

66. C₇₃

#3166

semidirect

Subgroups — #22620484

Conjugacy classes of subgroups

Complements