Group #33000609 — subgroups

Subgroups — #33000609

Order 18688.

212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22612074	C₂	1	2	yes	maximal
2	#22609549	C₂	1	1	yes	maximal, characteristic
3	#22608829	C₂	1	1	yes	maximal, characteristic
4	#22607985	C₂	1	2	yes	maximal
5	C₇₃×#1367	C₂	1	1	yes	maximal, characteristic
6	#17988992	—	2	1	no	—
7	#17988992	C₂²	1	2	yes	—
8	#17988967	—	2	1	no	—
9	#17988717	—	2	1	no	—
10	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
11	#17988415	—	2	2	no	—
12	#17988392	—	2	2	no	—
13	#17988340	C₂²	1	1	yes	characteristic
14	#17988319	—	2	1	no	—
15	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	4	1	no	—
18	C₂×Dic₂₉₂	—	4	1	no	—
19	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
20	#14556591	—	2	2	no	—
21	C₄×Dic₁₄₆	—	2	1	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556525	—	4	2	no	—
25	#14556525	D₄	1	1	yes	characteristic
26	C₄⋊#10596831	—	2	1	no	—
27	#14556520	—	4	1	no	—
28	#14556516	—	2	1	no	—
29	#14556501	—	4	1	no	—
30	C₂×C₁₆×C₇₃	—	2	2	no	—
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₄₆	—	2	1	no	—
35	C₂×Dic₁₄₆	—	4	2	no	—
36	Dic₂₉₂	—	2	2	no	—
37	Dic₂₉₂	—	8	2	no	—
38	Dic₂₉₂	—	8	2	no	—
39	C₂×#10596831	—	4	1	no	—
40	C₄⋊Dic₇₃	—	2	1	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	#11924007	—	4	1	no	—
44	C₄×Dic₇₃	—	4	1	no	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
50	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₁₆×C₇₃	—	4	2	no	—
52	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	—	8	2	no	—
58	Dic₁₄₆	—	4	1	no	—
59	Dic₁₄₆	—	4	1	no	—
60	Dic₁₄₆	—	8	2	no	—
61	Dic₁₄₆	—	8	2	no	—
62	C₈×C₇₃	—	2	2	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	SD₃₂	1	2	yes	—
66	C₈×C₇₃	#137	1	2	yes	—
67	#10596831	—	8	1	no	—
68	C₂²×C₇₃	#459	1	1	yes	characteristic
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	#470	1	1	yes	characteristic
71	C₄×C₇₃	#473	1	1	yes	characteristic
72	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
73	C₄×C₇₃	#509	1	1	yes	characteristic
74	Dic₇₃	—	8	1	no	—
75	Dic₇₃	—	8	1	no	—
76	Dic₇₃	—	16	2	no	—
77	#12498	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	#1689	1	1	yes	characteristic
79	C₂×C₇₃	#2207	1	1	yes	characteristic
80	C₂×C₇₃	#2214	1	1	yes	characteristic
81	#2225	—	73	2	no	—
82	#1713	—	73	1	no	—
83	#1591	—	73	1	no	—
84	#1367	D₇₃	1	1	yes	characteristic
85	#1364	—	73	2	no	—
86	C₇₃	#12498	1	1	yes	characteristic, complemented, Sylow
87	#500	—	73	2	no	—
88	#500	—	146	1	no	—
89	#494	—	146	1	no	—
90	#446	—	146	1	no	—
91	C₄×Q₁₆	—	73	1	no	—
92	#367	—	146	2	no	—
93	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
94	#358	—	146	2	no	—
95	#335	—	73	1	no	—
96	Q₈⋊C₈	—	146	1	no	—
97	C₈²	C₂×D₇₃	1	1	yes	characteristic
98	C₂×Q₁₆	—	292	1	no	—
99	C₂×Q₁₆	—	292	1	no	—
100	C₂×Q₁₆	—	73	1	no	—
101	C₄⋊Q₈	—	146	2	no	—
102	C₄×Q₈	—	146	1	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#107	—	292	2	no	—
105	#107	—	73	1	no	—
106	#107	—	146	2	no	—
107	#107	—	146	2	no	—
108	C₄⋊C₈	—	292	1	no	—
109	C₄⋊C₈	—	146	1	no	—
110	#103	—	146	1	no	—
111	#97	—	292	1	no	—
112	C₄×C₈	D₂₉₂	1	1	yes	characteristic
113	C₄×C₈	D₂₉₂	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
115	C₂×Q₈	—	292	2	no	—
116	C₂×Q₈	—	146	1	no	—
117	Q₁₆	—	584	2	no	—
118	Q₁₆	—	584	2	no	—
119	Q₁₆	—	146	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	292	1	no	—
122	C₂×C₈	D₅₈₄	1	1	yes	characteristic
123	C₂×C₈	D₅₈₄	1	1	yes	characteristic
124	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
125	C₂×C₈	#11924029	1	1	yes	characteristic
126	C₄⋊C₄	—	292	1	no	—
127	C₄⋊C₄	—	292	2	no	—
128	C₄⋊C₄	—	292	2	no	—
129	C₄⋊C₄	—	146	1	no	—
130	C₄²	—	292	1	no	—
131	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	Q₈	—	292	1	no	—
134	Q₈	—	292	1	no	—
135	Q₈	—	584	2	no	—
136	Q₈	—	584	2	no	—
137	C₂×C₄	—	292	1	no	—
138	C₂×C₄	—	292	1	no	—
139	C₂×C₄	—	584	2	no	—
140	C₂×C₄	#14556566	1	1	yes	characteristic
141	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
142	C₂×C₄	#14556619	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	584	1	no	—
147	C₈	#14556536	1	2	yes	—
148	C₈	#14556620	1	2	yes	—
149	C₂²	#17988783	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	584	1	no	—
152	C₄	—	584	1	no	—
153	C₄	—	1168	2	no	—
154	C₄	#17988846	1	1	yes	characteristic
155	C₄	#17988863	1	1	yes	characteristic
156	C₄	C₂×#14556536	1	1	yes	characteristic
157	C₄	#17989025	1	1	yes	characteristic
158	C₂	#22609421	1	1	yes	characteristic
159	C₂	#22611972	1	1	yes	characteristic
160	C₂	#22612013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612074

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic

#22608829

C₂

yes

maximal, characteristic

#22607985

C₂

yes

maximal

C₇₃×#1367

C₂

yes

maximal, characteristic

#17988992

—

#17988992

C₂²

yes

—

#17988967

—

#17988717

—

C₄×Dic₂₉₂

C₂²

yes

—

—

C₂²

yes

characteristic

#17988319

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

D₄

yes

characteristic

#14556591

—

C₄×Dic₁₄₆

—

—

—

—

D₄

yes

—

—

—

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#137

yes

—

#10596831

—

C₂²×C₇₃

#459

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

C₄×C₇₃

#509

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

D₇₃

yes

characteristic

#1364

—

C₇₃

#12498

yes

characteristic, complemented, Sylow

#500

—

#500

—

146

—

#494

—

146

—

#446

—

146

—

C₄×Q₁₆

—

#367

—

146

—

C₄⋊C₁₆

C₂×D₇₃

yes

—

#358

—

146

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

292

—

100

C₂×Q₁₆

—

101

C₄⋊Q₈

—

146

—

102

C₄×Q₈

—

146

—

103

C₂×C₁₆

—

104

#107

—

292

—

105

#107

—

106

#107

—

146

—

107

#107

—

146

—

108

C₄⋊C₈

—

292

—

109

C₄⋊C₈

—

146

—

110

#103

—

146

—

111

#97

—

292

—

112

C₄×C₈

D₂₉₂

yes

characteristic

113

C₄×C₈

D₂₉₂

yes

characteristic

114

C₄×C₈

C₂²×D₇₃

yes

characteristic

115

C₂×Q₈

—

292

—

116

C₂×Q₈

—

146

—

117

Q₁₆

—

584

—

118

Q₁₆

—

584

—

119

Q₁₆

—

146

—

120

C₂×C₈

—

121

C₂×C₈

—

292

—

122

C₂×C₈

D₅₈₄

yes

characteristic

123

C₂×C₈

D₅₈₄

yes

characteristic

124

C₂×C₈

D₄×D₇₃

yes

characteristic

125

C₂×C₈

#11924029

yes

characteristic

126

C₄⋊C₄

—

292

—

127

C₄⋊C₄

—

292

—

128

C₄⋊C₄

—

292

—

129

C₄⋊C₄

—

146

—

130

C₄²

—

292

—

131

C₄²

C₂×D₂₉₂

yes

characteristic

132

C₁₆

—

133

Q₈

—

292

—

134

Q₈

—

292

—

135

Q₈

—

584

—

136

Q₈

—

584

—

137

C₂×C₄

—

292

—

138

C₂×C₄

—

292

—

139

C₂×C₄

—

584

—

140

C₂×C₄

#14556566

yes

characteristic

141

C₂×C₄

C₂×D₅₈₄

yes

characteristic

142

C₂×C₄

#14556619

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

584

—

147

C₈

#14556536

yes

—

148

C₈

#14556620

yes

—

149

C₂²

#17988783

yes

characteristic

150

C₄

—

151

C₄

—

584

—

152

C₄

—

584

—

153

C₄

—

1168

—

154

C₄

#17988846

yes

characteristic

155

C₄

#17988863

yes

characteristic

156

C₄

C₂×#14556536

yes

characteristic

157

C₄

#17989025

yes

characteristic

158

C₂

#22609421

yes

characteristic

159

C₂

#22611972

yes

characteristic

160

C₂

#22612013

yes

characteristic

	subgroup	G/H
center	C₂²	#17988783
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1367	C₂
socle, prime radical	C₂²×C₇₃	#459
hypercenter	#1367	D₇₃
hypocenter	C₇₃	#12498
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#17988783

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1367

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
86. C₇₃	#12498	semidirect

H ≅ G/N

type

86. C₇₃

#12498

semidirect

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

subgroup	index
#22612074	2
#22609549	2
#22608829	2
#22607985	2
C₇₃×#1367	2
#12498	73

subgroup

index

C₇₃×#1367

Subgroups — #33000609

Conjugacy classes of subgroups

Complements

Maximal subgroups