Group #22609587 — subgroups

Subgroups — #22609587

Order 9344.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988961	C₂	1	1	yes	maximal, characteristic
2	#17988939	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988935	C₂	1	1	yes	maximal, characteristic
5	#17988934	C₂	1	1	yes	maximal, characteristic
6	#17988309	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924015	—	2	2	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556588	C₂²	1	1	yes	characteristic
11	#14556515	—	2	2	no	—
12	#14556514	—	2	2	no	—
13	#14556513	—	2	2	no	—
14	#14556511	—	2	2	no	—
15	C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	#107×C₇₃	—	2	2	no	—
20	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	#11924015	—	4	4	no	—
26	C₂×#10596831	D₄	1	2	yes	—
27	C₂×#10596831	D₄	1	2	yes	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₂×Q₈×C₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	D₄	1	2	yes	—
33	C₄⋊C₄×C₇₃	—	2	1	no	—
34	C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	C₂³	1	1	yes	characteristic
38	C₂²×D₇₃	—	4	1	no	—
39	Q₈×C₇₃	—	4	2	no	—
40	Q₈×C₇₃	—	4	2	no	—
41	C₂×C₄×C₇₃	—	4	1	no	—
42	C₂×C₄×C₇₃	—	4	1	no	—
43	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₇₃	—	4	1	no	—
47	D₂₉₂	—	4	2	no	—
48	D₂₉₂	—	4	2	no	—
49	D₂₉₂	—	4	2	no	—
50	Dic₁₄₆	—	4	2	no	—
51	C₈×C₇₃	—	2	2	no	—
52	#10596831	—	2	2	no	—
53	#10596831	—	2	2	no	—
54	C₂²×C₇₃	#127	1	1	yes	characteristic
55	C₂×D₇₃	—	8	1	no	—
56	C₂×D₇₃	—	8	1	no	—
57	C₂×D₇₃	—	8	1	no	—
58	C₄×C₇₃	—	8	1	no	—
59	C₄×C₇₃	—	8	1	no	—
60	C₄×C₇₃	#135	1	2	yes	—
61	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₇₃	#137	1	2	yes	—
63	Dic₇₃	—	8	1	no	—
64	C₂×C₇₃	#495	1	1	yes	characteristic
65	C₂×C₇₃	#496	1	1	yes	characteristic
66	C₂×C₇₃	#497	1	1	yes	characteristic
67	D₇₃	—	16	1	no	—
68	#1719	—	73	1	no	maximal, Sylow
69	C₇₃	#1719	1	1	yes	characteristic, complemented, Sylow
70	#500	D₇₃	1	1	yes	characteristic, complemented
71	#492	—	73	1	no	—
72	#487	—	73	1	no	—
73	#487	—	73	1	no	—
74	#486	—	73	1	no	—
75	#486	—	73	1	no	—
76	#322	—	73	1	no	—
77	C₂×SD₁₆	—	146	2	no	—
78	C₄⋊Q₈	—	73	1	no	—
79	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
81	#127	—	73	1	no	—
82	#107	—	2	2	no	—
83	#103	—	146	2	no	—
84	#103	—	146	2	no	—
85	#102	—	146	2	no	—
86	#102	—	146	2	no	—
87	C₄×C₈	—	73	1	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	146	1	no	—
93	C₂×D₄	—	146	1	no	—
94	C₂×D₄	—	146	1	no	—
95	C₂×D₄	—	146	1	no	—
96	SD₁₆	—	292	4	no	—
97	C₂×C₈	—	73	2	no	—
98	C₂×C₈	—	73	2	no	—
99	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	146	1	no	—
105	C₄⋊C₄	—	146	1	no	—
106	C₄²	C₂²×D₇₃	1	1	yes	characteristic
107	C₂³	—	292	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	292	2	no	—
111	D₄	—	292	2	no	—
112	D₄	—	292	2	no	—
113	D₄	—	292	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	292	1	no	—
117	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	146	2	no	—
122	C₈	—	146	2	no	—
123	C₂²	—	584	1	no	—
124	C₂²	—	584	1	no	—
125	C₂²	—	584	1	no	—
126	C₂²	#14556587	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	584	1	no	—
130	C₄	#14556614	1	2	yes	—
131	C₄	#14556615	1	2	yes	—
132	C₄	#14556622	1	2	yes	—
133	C₂	—	1168	1	no	—
134	C₂	#17988972	1	1	yes	characteristic
135	C₂	#17988975	1	1	yes	characteristic
136	C₂	#17988979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988961

C₂

yes

maximal, characteristic

#17988939

C₂

yes

maximal, characteristic, complemented

#17988937

C₂

yes

maximal, characteristic, complemented

#17988935

C₂

yes

maximal, characteristic

#17988934

C₂

yes

maximal, characteristic

#17988309

C₂

yes

maximal, characteristic, complemented

#500×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

—

#14556591

C₂²

yes

characteristic

#14556588

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#107×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924015

—

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#127

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#135

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

#137

yes

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1719

—

maximal, Sylow

C₇₃

#1719

yes

characteristic, complemented, Sylow

#500

D₇₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂×SD₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

C₂×D₇₃

yes

—

—

—

146

—

#103

—

146

—

#102

—

146

—

#102

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

SD₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₃

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

146

—

105

C₄⋊C₄

—

146

—

106

C₄²

C₂²×D₇₃

yes

characteristic

107

C₂³

—

292

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

292

—

111

D₄

—

292

—

112

D₄

—

292

—

113

D₄

—

292

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

292

—

117

C₂×C₄

D₄×D₇₃

yes

characteristic

118

C₂×C₄

D₄×D₇₃

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

120

C₈

—

121

C₈

—

146

—

122

C₈

—

146

—

123

C₂²

—

584

—

124

C₂²

—

584

—

125

C₂²

—

584

—

126

C₂²

#14556587

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

584

—

130

C₄

#14556614

yes

—

131

C₄

#14556615

yes

—

132

C₄

#14556622

yes

—

133

C₂

—

1168

—

134

C₂

#17988972

yes

characteristic

135

C₂

#17988975

yes

characteristic

136

C₂

#17988979

yes

characteristic

	subgroup	G/H
center	C₂²	#14556587
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#500×C₇₃	C₂
socle	C₂²×C₇₃	#127
hypercenter	#500	D₇₃
hypocenter	C₇₃	#1719
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14556587

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#500×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #17988939	C₂	semidirect
3. #17988937	C₂	semidirect
6. #17988309	C₂	semidirect
7. #500×C₇₃	C₂	semidirect
69. C₇₃	#1719	semidirect
70. #500	D₇₃	semidirect

H ≅ G/N

type

2. #17988939

C₂

semidirect

3. #17988937

C₂

semidirect

6. #17988309

C₂

semidirect

7. #500×C₇₃

C₂

semidirect

69. C₇₃

#1719

semidirect

70. #500

D₇₃

semidirect

Subgroups — #22609587

Conjugacy classes of subgroups

Complements