Group #33000775 — subgroups

Subgroups — #33000775

Order 18688.

272 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22612213	C₂	1	2	yes	maximal, complemented
2	#22612155	C₂	1	2	yes	maximal
3	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609523	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988992	—	2	1	no	—
7	#17988992	C₂²	1	1	yes	characteristic
8	#17988992	C₂²	1	2	yes	—
9	#17988967	—	2	1	no	—
10	#17988964	C₂²	1	1	yes	characteristic
11	#17988960	—	2	1	no	—
12	#17988934	—	2	1	no	—
13	#17988415	—	2	4	no	—
14	#17988389	—	2	4	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	4	2	no	—
18	C₂×#11924017	—	4	2	no	—
19	C₂×D₅₈₄	—	2	2	no	—
20	#14556591	—	2	1	no	—
21	#14556591	—	2	2	no	—
22	#14556588	—	2	1	no	—
23	#14556525	—	2	2	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	2	2	no	—
26	#14556525	—	4	2	no	—
27	#14556513	—	4	2	no	—
28	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₄₆	—	4	1	no	—
33	C₂×Dic₁₄₆	—	4	2	no	—
34	C₂×D₂₉₂	—	4	1	no	—
35	C₂×D₂₉₂	—	4	1	no	—
36	C₂×D₂₉₂	—	4	1	no	—
37	Dic₂₉₂	—	8	4	no	—
38	#11924017	—	8	4	no	—
39	D₅₈₄	—	4	4	no	—
40	C₄⋊Dic₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	—	4	1	no	—
42	C₄⋊Dic₇₃	—	4	1	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₂×C₈×C₇₃	D₈	1	2	yes	—
45	C₂×C₈×C₇₃	D₈	1	2	yes	—
46	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
47	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₇₃	—	2	4	no	—
49	C₂²×D₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇₃	—	8	1	no	—
54	C₂×Dic₇₃	—	8	2	no	—
55	D₂₉₂	—	8	2	no	—
56	D₂₉₂	—	8	2	no	—
57	D₂₉₂	—	8	2	no	—
58	Dic₁₄₆	—	8	2	no	—
59	Dic₁₄₆	—	8	4	no	—
60	C₈×C₇₃	—	2	2	no	—
61	C₈×C₇₃	SD₃₂	1	4	yes	—
62	C₈×C₇₃	#137	1	4	yes	—
63	C₂²×C₇₃	#493	1	1	yes	characteristic
64	C₂×D₇₃	—	16	1	no	—
65	C₂×D₇₃	—	16	1	no	—
66	C₂×D₇₃	—	16	1	no	—
67	C₄×C₇₃	#497	1	2	yes	—
68	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
69	C₄×C₇₃	#509	1	2	yes	—
70	Dic₇₃	—	16	1	no	—
71	Dic₇₃	—	16	2	no	—
72	#12532	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#1722	1	1	yes	characteristic
74	C₂×C₇₃	#2248	1	1	yes	characteristic
75	C₂×C₇₃	#2250	1	1	yes	characteristic
76	D₇₃	—	32	1	no	—
77	#2254	—	73	2	no	—
78	#2240	—	73	2	no	—
79	#1713	—	73	1	no	—
80	#1706	—	73	1	no	—
81	#1312	D₇₃	1	1	yes	characteristic, complemented
82	C₇₃	#12532	1	1	yes	characteristic, complemented, Sylow
83	#500	—	73	1	no	—
84	#500	—	73	2	no	—
85	#500	—	146	1	no	—
86	#494	—	146	1	no	—
87	#493	—	73	1	no	—
88	#492	—	146	1	no	—
89	#486	—	146	1	no	—
90	#367	—	146	4	no	—
91	#357	—	146	4	no	—
92	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
93	C₈²	C₂×D₇₃	1	1	yes	characteristic
94	C₂×Q₁₆	—	292	2	no	—
95	C₂×SD₁₆	—	292	2	no	—
96	C₂×D₈	—	146	2	no	—
97	C₄⋊Q₈	—	146	1	no	—
98	C₄⋊Q₈	—	146	2	no	—
99	#127	—	146	1	no	—
100	C₂×C₁₆	D₂₉₂	1	4	yes	—
101	#107	—	292	2	no	—
102	#107	—	146	2	no	—
103	#107	—	146	2	no	—
104	#107	—	146	2	no	—
105	#102	—	292	2	no	—
106	C₄×C₈	D₂₉₂	1	1	yes	characteristic
107	C₄×C₈	D₂₉₂	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
109	C₂×Q₈	—	292	1	no	—
110	C₂×Q₈	—	292	2	no	—
111	C₂×D₄	—	292	1	no	—
112	C₂×D₄	—	292	1	no	—
113	C₂×D₄	—	292	1	no	—
114	Q₁₆	—	584	4	no	—
115	SD₁₆	—	584	4	no	—
116	D₈	—	292	4	no	—
117	C₂×C₈	D₅₈₄	1	2	yes	—
118	C₂×C₈	D₅₈₄	1	2	yes	—
119	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
120	C₄⋊C₄	—	292	2	no	—
121	C₄⋊C₄	—	292	1	no	—
122	C₄⋊C₄	—	292	1	no	—
123	C₄⋊C₄	—	292	2	no	—
124	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	584	1	no	—
127	Q₈	—	584	2	no	—
128	Q₈	—	584	4	no	—
129	D₄	—	584	2	no	—
130	D₄	—	584	2	no	—
131	D₄	—	584	2	no	—
132	C₂×C₄	—	584	1	no	—
133	C₂×C₄	—	584	2	no	—
134	C₂×C₄	#14556588	1	1	yes	characteristic
135	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
136	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#14556536	1	4	yes	—
139	C₈	#14556624	1	4	yes	—
140	C₂²	—	1168	1	no	—
141	C₂²	—	1168	1	no	—
142	C₂²	—	1168	1	no	—
143	C₂²	#17988964	1	1	yes	characteristic
144	C₄	—	1168	1	no	—
145	C₄	—	1168	2	no	—
146	C₄	#17988981	1	2	yes	—
147	C₄	C₂×#14556536	1	2	yes	—
148	C₄	#17989025	1	2	yes	—
149	C₂	—	2336	1	no	—
150	C₂	#22609602	1	1	yes	characteristic
151	C₂	#22612187	1	1	yes	characteristic
152	C₂	#22612197	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612213

C₂

yes

maximal, complemented

#22612155

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic, complemented

#22609523

C₂

yes

maximal, characteristic

C₇₃×#1312

C₂

yes

maximal, characteristic, complemented

#17988992

—

#17988992

C₂²

yes

characteristic

#17988992

C₂²

yes

—

#17988967

—

#17988964

C₂²

yes

—

—

—

—

C₄×C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×#11924017

—

C₂×D₅₈₄

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#137

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#497

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

C₄×C₇₃

#509

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#12532

yes

characteristic, complemented, Sylow

#500

—

#500

—

#500

—

146

—

#494

—

146

—

#493

—

#492

—

146

—

#486

—

146

—

#367

—

146

—

#357

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×SD₁₆

—

292

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

#127

—

146

—

100

C₂×C₁₆

D₂₉₂

yes

—

101

#107

—

292

—

102

#107

—

146

—

103

#107

—

146

—

104

#107

—

146

—

105

#102

—

292

—

106

C₄×C₈

D₂₉₂

yes

characteristic

107

C₄×C₈

D₂₉₂

yes

characteristic

108

C₄×C₈

C₂²×D₇₃

yes

characteristic

109

C₂×Q₈

—

292

—

110

C₂×Q₈

—

292

—

111

C₂×D₄

—

292

—

112

C₂×D₄

—

292

—

113

C₂×D₄

—

292

—

114

Q₁₆

—

584

—

115

SD₁₆

—

584

—

116

D₈

—

292

—

117

C₂×C₈

D₅₈₄

yes

—

118

C₂×C₈

D₅₈₄

yes

—

119

C₂×C₈

C₂×D₂₉₂

yes

—

120

C₄⋊C₄

—

292

—

121

C₄⋊C₄

—

292

—

122

C₄⋊C₄

—

292

—

123

C₄⋊C₄

—

292

—

124

C₄²

C₂×D₂₉₂

yes

characteristic

125

C₁₆

—

126

C₂³

—

584

—

127

Q₈

—

584

—

128

Q₈

—

584

—

129

D₄

—

584

—

130

D₄

—

584

—

131

D₄

—

584

—

132

C₂×C₄

—

584

—

133

C₂×C₄

—

584

—

134

C₂×C₄

#14556588

yes

characteristic

135

C₂×C₄

C₂×D₅₈₄

yes

characteristic

136

C₂×C₄

C₂×D₅₈₄

yes

characteristic

137

C₈

—

138

C₈

#14556536

yes

—

139

C₈

#14556624

yes

—

140

C₂²

—

1168

—

141

C₂²

—

1168

—

142

C₂²

—

1168

—

143

C₂²

#17988964

yes

characteristic

144

C₄

—

1168

—

145

C₄

—

1168

—

146

C₄

#17988981

yes

—

147

C₄

C₂×#14556536

yes

—

148

C₄

#17989025

yes

—

149

C₂

—

2336

—

150

C₂

#22609602

yes

characteristic

151

C₂

#22612187

yes

characteristic

152

C₂

#22612197

yes

characteristic

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1312	C₂
socle	C₂²×C₇₃	#493
prime radical	#17988964	C₂²
hypercenter	#1312	D₇₃
hypocenter	C₇₃	#12532
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1312

C₂

C₂²×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #22612213	C₂	semidirect
3. #22609549	C₂	semidirect
5. C₇₃×#1312	C₂	semidirect
81. #1312	D₇₃	semidirect
82. C₇₃	#12532	semidirect

H ≅ G/N

type

1. #22612213

C₂

semidirect

3. #22609549

C₂

semidirect

5. C₇₃×#1312

C₂

semidirect

81. #1312

D₇₃

semidirect

82. C₇₃

#12532

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22612213	2
#22612155	2
#22609549	2
#22609523	2
C₇₃×#1312	2
#12532	73

subgroup

index

C₇₃×#1312

Subgroups — #33000775

Conjugacy classes of subgroups

Complements

Maximal subgroups