Group #33000756 — subgroups

Subgroups — #33000756

Order 18688.

318 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612188	C₂	1	2	yes	maximal, complemented
2	#22612156	C₂	1	2	yes	maximal, complemented
3	#22609546	C₂	1	1	yes	maximal, characteristic
4	#22607849	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556534	—	2	4	no	—
7	#17988964	—	2	1	no	—
8	#17988964	—	2	1	no	—
9	#17988964	C₂²	1	2	yes	—
10	#17988387	—	2	4	no	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	#500×C₇₃	C₂²	1	1	yes	characteristic
13	#498×C₇₃	—	2	1	no	—
14	#494×C₇₃	C₂²	1	1	yes	characteristic
15	#487×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₅₈₄	—	2	2	no	—
18	C₂×D₅₈₄	—	2	2	no	—
19	C₂×D₅₈₄	—	4	2	no	—
20	C₂×D₅₈₄	—	4	2	no	—
21	#14556588	—	2	2	no	—
22	#14556534	—	4	8	no	—
23	C₂×#11924003	D₄	1	4	yes	—
24	C₂×Q₁₆×C₇₃	—	2	2	no	—
25	C₄⋊Q₈×C₇₃	—	2	1	no	—
26	C₄⋊Q₈×C₇₃	—	2	1	no	—
27	#107×C₇₃	—	2	2	no	—
28	#106×C₇₃	—	4	2	no	—
29	#103×C₇₃	—	4	2	no	—
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×D₂₉₂	—	4	2	no	—
34	C₂×D₂₉₂	—	4	2	no	—
35	C₂×D₂₉₂	—	4	2	no	—
36	D₅₈₄	—	4	4	no	—
37	D₅₈₄	—	4	4	no	—
38	D₅₈₄	—	8	4	no	—
39	D₅₈₄	—	8	4	no	—
40	#11924003	—	2	4	no	—
41	C₂×Q₈×C₇₃	—	4	1	no	—
42	C₂×Q₈×C₇₃	—	4	1	no	—
43	Q₁₆×C₇₃	—	4	4	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	C₄⋊C₄×C₇₃	—	4	1	no	—
48	C₄⋊C₄×C₇₃	—	4	1	no	—
49	C₄⋊C₄×C₇₃	—	4	1	no	—
50	C₄⋊C₄×C₇₃	—	4	1	no	—
51	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×D₇₃	—	8	2	no	—
53	Q₈×C₇₃	—	8	2	no	—
54	Q₈×C₇₃	—	8	2	no	—
55	C₂×C₄×C₇₃	—	8	1	no	—
56	C₂×C₄×C₇₃	—	8	1	no	—
57	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	D₂₉₂	—	8	2	no	—
61	D₂₉₂	—	8	2	no	—
62	D₂₉₂	—	8	4	no	—
63	D₂₉₂	—	8	4	no	—
64	C₈×C₇₃	—	2	2	no	—
65	C₈×C₇₃	SD₃₂	1	4	yes	complemented
66	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
67	C₂²×C₇₃	#493	1	1	yes	characteristic
68	C₂×D₇₃	—	16	2	no	—
69	C₂×D₇₃	—	16	2	no	—
70	C₂×D₇₃	—	16	2	no	—
71	C₄×C₇₃	—	16	1	no	—
72	C₄×C₇₃	—	16	1	no	—
73	C₄×C₇₃	#496	1	2	yes	—
74	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
75	C₄×C₇₃	#509	1	2	yes	—
76	#12529	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	#1721	1	1	yes	characteristic
78	C₂×C₇₃	#2248	1	1	yes	characteristic
79	C₂×C₇₃	#2250	1	1	yes	characteristic
80	D₇₃	—	32	2	no	—
81	#2248	—	73	2	no	—
82	#2240	—	73	2	no	—
83	#1712	—	73	1	no	—
84	#1707	D₇₃	1	1	yes	characteristic, complemented
85	#1312	—	73	1	no	—
86	C₇₃	#12529	1	1	yes	characteristic, complemented, Sylow
87	C₂×SD₃₂	—	146	4	no	—
88	#500	C₂×D₇₃	1	1	yes	characteristic
89	#498	—	2	1	no	—
90	#494	C₂×D₇₃	1	1	yes	characteristic
91	#493	—	73	2	no	—
92	#493	—	146	1	no	—
93	#493	—	146	1	no	—
94	#487	—	2	1	no	—
95	#357	—	146	4	no	—
96	C₄×C₁₆	—	73	2	no	—
97	C₈²	C₂×D₇₃	1	1	yes	characteristic
98	C₂×Q₁₆	—	2	2	no	—
99	C₂×D₈	—	292	2	no	—
100	C₂×D₈	—	292	2	no	—
101	C₂×D₈	—	146	2	no	—
102	C₂×D₈	—	146	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	2	1	no	—
105	#127	—	146	2	no	—
106	SD₃₂	—	292	8	no	—
107	C₂×C₁₆	—	73	4	no	—
108	#107	—	2	2	no	—
109	#106	—	4	2	no	—
110	#103	—	4	2	no	—
111	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	—	4	1	no	—
116	C₂×D₄	—	292	2	no	—
117	C₂×D₄	—	292	2	no	—
118	C₂×D₄	—	292	2	no	—
119	Q₁₆	—	4	4	no	—
120	D₈	—	292	4	no	—
121	D₈	—	292	4	no	—
122	D₈	—	584	4	no	—
123	D₈	—	584	4	no	—
124	C₂×C₈	#11924013	1	2	yes	—
125	C₂×C₈	#11924013	1	2	yes	—
126	C₂×C₈	D₄×D₇₃	1	2	yes	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	4	1	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
132	C₁₆	—	146	4	no	—
133	C₂³	—	584	2	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	8	2	no	—
136	D₄	—	584	2	no	—
137	D₄	—	584	2	no	—
138	D₄	—	584	4	no	—
139	D₄	—	584	4	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	#14556587	1	1	yes	characteristic
143	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
144	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#14556534	1	4	yes	complemented
147	C₈	#14556618	1	4	yes	—
148	C₂²	—	1168	2	no	—
149	C₂²	—	1168	2	no	—
150	C₂²	—	1168	2	no	—
151	C₂²	#17988963	1	1	yes	characteristic
152	C₄	—	16	1	no	—
153	C₄	—	16	1	no	—
154	C₄	#17988973	1	2	yes	—
155	C₄	C₂×#14556534	1	2	yes	—
156	C₄	#17989022	1	2	yes	—
157	C₂	—	2336	2	no	—
158	C₂	#22609595	1	1	yes	characteristic
159	C₂	#22612188	1	1	yes	characteristic
160	C₂	#22612198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612188

C₂

yes

maximal, complemented

#22612156

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic

#22607849

C₂

yes

maximal, characteristic, complemented

C₇₃×#1707

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

#17988387

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

C₂²

yes

characteristic

#498×C₇₃

—

#494×C₇₃

C₂²

yes

characteristic

#487×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

#14556534

—

C₂×#11924003

D₄

yes

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#106×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

complemented

C₈×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#496

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

C₄×C₇₃

#509

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

#1312

—

C₇₃

#12529

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

146

—

#500

C₂×D₇₃

yes

characteristic

#498

—

#494

C₂×D₇₃

yes

characteristic

#493

—

#493

—

146

—

#493

—

146

—

#487

—

#357

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

292

—

100

C₂×D₈

—

292

—

101

C₂×D₈

—

146

—

102

C₂×D₈

—

146

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

#127

—

146

—

106

SD₃₂

—

292

—

107

C₂×C₁₆

—

108

#107

—

109

#106

—

110

#103

—

111

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

112

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

113

C₄×C₈

C₂²×D₇₃

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×D₄

—

292

—

117

C₂×D₄

—

292

—

118

C₂×D₄

—

292

—

119

Q₁₆

—

120

D₈

—

292

—

121

D₈

—

292

—

122

D₈

—

584

—

123

D₈

—

584

—

124

C₂×C₈

#11924013

yes

—

125

C₂×C₈

#11924013

yes

—

126

C₂×C₈

D₄×D₇₃

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

132

C₁₆

—

146

—

133

C₂³

—

584

—

134

Q₈

—

135

Q₈

—

136

D₄

—

584

—

137

D₄

—

584

—

138

D₄

—

584

—

139

D₄

—

584

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#14556587

yes

characteristic

143

C₂×C₄

C₂×#11924013

yes

characteristic

144

C₂×C₄

C₂×#11924013

yes

characteristic

145

C₈

—

146

C₈

#14556534

yes

complemented

147

C₈

#14556618

yes

—

148

C₂²

—

1168

—

149

C₂²

—

1168

—

150

C₂²

—

1168

—

151

C₂²

#17988963

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

#17988973

yes

—

155

C₄

C₂×#14556534

yes

—

156

C₄

#17989022

yes

—

157

C₂

—

2336

—

158

C₂

#22609595

yes

characteristic

159

C₂

#22612188

yes

characteristic

160

C₂

#22612198

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1707	C₂
socle	C₂²×C₇₃	#493
prime radical	#22609546	C₂
hypercenter	#1707	D₇₃
hypocenter	C₇₃	#12529
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1707

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

N	H ≅ G/N	type
1. #22612188	C₂	semidirect
2. #22612156	C₂	semidirect
4. #22607849	C₂	semidirect
5. C₇₃×#1707	C₂	semidirect
65. C₈×C₇₃	SD₃₂	semidirect
84. #1707	D₇₃	semidirect
86. C₇₃	#12529	semidirect
146. C₈	#14556534	semidirect

H ≅ G/N

type

1. #22612188

C₂

semidirect

2. #22612156

C₂

semidirect

4. #22607849

C₂

semidirect

5. C₇₃×#1707

C₂

semidirect

65. C₈×C₇₃

SD₃₂

semidirect

84. #1707

D₇₃

semidirect

86. C₇₃

#12529

semidirect

146. C₈

#14556534

semidirect

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

subgroup	index
#22612188	2
#22612156	2
#22609546	2
#22607849	2
C₇₃×#1707	2
#12529	73

subgroup

index

C₇₃×#1707

Subgroups — #33000756

Conjugacy classes of subgroups

Complements

Maximal subgroups