Group #33000619 — subgroups

Subgroups — #33000619

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	#22612084	C₂	1	2	yes	maximal
2	#22609549	C₂	1	1	yes	maximal, characteristic
3	#22608829	C₂	1	1	yes	maximal, characteristic
4	#22607988	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	#17988413	—	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	2	no	—
18	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
19	#14556591	—	2	2	no	—
20	C₄×Dic₁₄₆	—	2	1	no	—
21	#14556525	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	4	1	no	—
24	#14556525	—	4	1	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	4	no	—
38	C₂×#10596831	—	4	1	no	—
39	C₄⋊Dic₇₃	—	2	1	no	—
40	C₄⋊Dic₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	#11924007	—	4	1	no	—
43	C₄×Dic₇₃	—	4	1	no	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
46	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
49	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×Dic₇₃	—	4	1	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	8	2	no	—
57	Dic₁₄₆	—	4	1	no	—
58	Dic₁₄₆	—	4	1	no	—
59	Dic₁₄₆	—	8	4	no	—
60	C₈×C₇₃	—	2	2	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	Q₃₂	1	2	yes	—
64	C₈×C₇₃	#137	1	2	yes	—
65	#10596831	—	8	1	no	—
66	C₂²×C₇₃	#459	1	1	yes	characteristic
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	#470	1	1	yes	characteristic
69	C₄×C₇₃	#473	1	1	yes	characteristic
70	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
71	C₄×C₇₃	#510	1	1	yes	characteristic
72	Dic₇₃	—	8	1	no	—
73	Dic₇₃	—	8	1	no	—
74	Dic₇₃	—	16	2	no	—
75	#12500	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	#1689	1	1	yes	characteristic
77	C₂×C₇₃	#2209	1	1	yes	characteristic
78	C₂×C₇₃	#2213	1	1	yes	characteristic
79	#2227	—	73	2	no	—
80	#1713	—	73	1	no	—
81	#1591	—	73	1	no	—
82	#1367	D₇₃	1	1	yes	characteristic
83	#1365	—	73	2	no	—
84	C₇₃	#12500	1	1	yes	characteristic, complemented, Sylow
85	#500	—	73	2	no	—
86	#500	—	146	1	no	—
87	#494	—	146	1	no	—
88	#446	—	146	1	no	—
89	C₄×Q₁₆	—	73	1	no	—
90	#366	—	146	2	no	—
91	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
92	#358	—	146	2	no	—
93	#335	—	73	1	no	—
94	Q₈⋊C₈	—	146	1	no	—
95	C₈²	C₂×D₇₃	1	1	yes	characteristic
96	C₂×Q₁₆	—	292	2	no	—
97	C₂×Q₁₆	—	73	1	no	—
98	C₄⋊Q₈	—	146	2	no	—
99	C₄×Q₈	—	146	1	no	—
100	C₂×C₁₆	—	2	2	no	—
101	#107	—	292	1	no	—
102	#107	—	292	1	no	—
103	#107	—	73	1	no	—
104	#107	—	146	2	no	—
105	#107	—	146	2	no	—
106	C₄⋊C₈	—	292	1	no	—
107	C₄⋊C₈	—	146	1	no	—
108	#103	—	146	1	no	—
109	#97	—	292	1	no	—
110	C₄×C₈	D₂₉₂	1	1	yes	characteristic
111	C₄×C₈	D₂₉₂	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
113	C₂×Q₈	—	292	2	no	—
114	C₂×Q₈	—	146	1	no	—
115	Q₁₆	—	584	4	no	—
116	Q₁₆	—	146	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	292	1	no	—
119	C₂×C₈	D₅₈₄	1	1	yes	characteristic
120	C₂×C₈	D₅₈₄	1	1	yes	characteristic
121	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
122	C₂×C₈	#11924029	1	1	yes	characteristic
123	C₄⋊C₄	—	292	1	no	—
124	C₄⋊C₄	—	292	2	no	—
125	C₄⋊C₄	—	292	2	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₄²	—	292	1	no	—
128	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	Q₈	—	292	1	no	—
131	Q₈	—	292	1	no	—
132	Q₈	—	584	4	no	—
133	C₂×C₄	—	292	1	no	—
134	C₂×C₄	—	292	1	no	—
135	C₂×C₄	—	584	2	no	—
136	C₂×C₄	#14556566	1	1	yes	characteristic
137	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
138	C₂×C₄	#14556619	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	584	1	no	—
143	C₈	Dic₅₈₄	1	2	yes	—
144	C₈	#14556620	1	2	yes	—
145	C₂²	#17988783	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	584	1	no	—
148	C₄	—	584	1	no	—
149	C₄	—	1168	2	no	—
150	C₄	#17988846	1	1	yes	characteristic
151	C₄	#17988863	1	1	yes	characteristic
152	C₄	C₂×Dic₅₈₄	1	1	yes	characteristic
153	C₄	#17989030	1	1	yes	characteristic
154	C₂	#22609421	1	1	yes	characteristic
155	C₂	#22611982	1	1	yes	characteristic
156	C₂	#22612008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612084

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic

#22608829

C₂

yes

maximal, characteristic

#22607988

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

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

—

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

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

Q₃₂

yes

—

C₈×C₇₃

#137

yes

—

#10596831

—

C₂²×C₇₃

#459

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×Q₃₂

yes

characteristic

C₄×C₇₃

#510

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

D₇₃

yes

characteristic

#1365

—

C₇₃

#12500

yes

characteristic, complemented, Sylow

#500

—

#500

—

146

—

#494

—

146

—

#446

—

146

—

C₄×Q₁₆

—

#366

—

146

—

C₄⋊C₁₆

C₂×D₇₃

yes

—

#358

—

146

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

C₄⋊Q₈

—

146

—

C₄×Q₈

—

146

—

100

C₂×C₁₆

—

101

#107

—

292

—

102

#107

—

292

—

103

#107

—

104

#107

—

146

—

105

#107

—

146

—

106

C₄⋊C₈

—

292

—

107

C₄⋊C₈

—

146

—

108

#103

—

146

—

109

#97

—

292

—

110

C₄×C₈

D₂₉₂

yes

characteristic

111

C₄×C₈

D₂₉₂

yes

characteristic

112

C₄×C₈

C₂²×D₇₃

yes

characteristic

113

C₂×Q₈

—

292

—

114

C₂×Q₈

—

146

—

115

Q₁₆

—

584

—

116

Q₁₆

—

146

—

117

C₂×C₈

—

118

C₂×C₈

—

292

—

119

C₂×C₈

D₅₈₄

yes

characteristic

120

C₂×C₈

D₅₈₄

yes

characteristic

121

C₂×C₈

D₄×D₇₃

yes

characteristic

122

C₂×C₈

#11924029

yes

characteristic

123

C₄⋊C₄

—

292

—

124

C₄⋊C₄

—

292

—

125

C₄⋊C₄

—

292

—

126

C₄⋊C₄

—

146

—

127

C₄²

—

292

—

128

C₄²

C₂×D₂₉₂

yes

characteristic

129

C₁₆

—

130

Q₈

—

292

—

131

Q₈

—

292

—

132

Q₈

—

584

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

584

—

136

C₂×C₄

#14556566

yes

characteristic

137

C₂×C₄

C₂×D₅₈₄

yes

characteristic

138

C₂×C₄

#14556619

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

584

—

143

C₈

Dic₅₈₄

yes

—

144

C₈

#14556620

yes

—

145

C₂²

#17988783

yes

characteristic

146

C₄

—

147

C₄

—

584

—

148

C₄

—

584

—

149

C₄

—

1168

—

150

C₄

#17988846

yes

characteristic

151

C₄

#17988863

yes

characteristic

152

C₄

C₂×Dic₅₈₄

yes

characteristic

153

C₄

#17989030

yes

characteristic

154

C₂

#22609421

yes

characteristic

155

C₂

#22611982

yes

characteristic

156

C₂

#22612008

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	#12500
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
84. C₇₃	#12500	semidirect

H ≅ G/N

type

84. C₇₃

#12500

semidirect

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

subgroup	index
#22612084	2
#22609549	2
#22608829	2
#22607988	2
C₇₃×#1367	2
#12500	73

subgroup

index

C₇₃×#1367

Subgroups — #33000619

Conjugacy classes of subgroups

Complements

Maximal subgroups