Group #33000792 — subgroups

Subgroups — #33000792

Order 18688.

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

normal subgroups only

This group has 250 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
2	#22612159	C₂	1	2	yes	maximal
3	#22609549	C₂	1	1	yes	maximal, characteristic
4	#22609528	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1312	C₂	1	1	yes	maximal, characteristic
6	#17988992	—	2	1	no	—
7	#17988992	C₂²	1	1	yes	characteristic
8	#17988992	C₂²	1	2	yes	—
9	#17988985	—	2	1	no	—
10	#17988967	—	2	1	no	—
11	#17988967	C₂²	1	1	yes	characteristic
12	#17988938	—	2	1	no	—
13	#17988415	—	2	4	no	—
14	#17988392	—	2	4	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	4	2	no	—
19	#14556591	—	2	1	no	—
20	#14556591	—	2	1	no	—
21	#14556591	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	4	2	no	—
26	#14556523	—	4	2	no	—
27	#14556516	—	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	1	no	—
34	C₂×Dic₁₄₆	—	4	2	no	—
35	Dic₂₉₂	—	4	4	no	—
36	Dic₂₉₂	—	8	4	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	1	no	—
39	C₄⋊Dic₇₃	—	4	1	no	—
40	C₄⋊Dic₇₃	—	4	1	no	—
41	C₄⋊Dic₇₃	—	4	1	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	C₂×C₈×C₇₃	D₈	1	2	yes	—
44	C₂×C₈×C₇₃	D₈	1	2	yes	—
45	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
46	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₇₃	—	8	1	no	—
52	C₂×Dic₇₃	—	8	2	no	—
53	C₂×Dic₇₃	—	8	1	no	—
54	Dic₁₄₆	—	8	2	no	—
55	Dic₁₄₆	—	8	4	no	—
56	Dic₁₄₆	—	8	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	SD₃₂	1	4	yes	—
59	C₈×C₇₃	#137	1	4	yes	—
60	C₂²×C₇₃	#493	1	1	yes	characteristic
61	C₄×C₇₃	#497	1	2	yes	—
62	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
63	C₄×C₇₃	#510	1	2	yes	—
64	Dic₇₃	—	16	1	no	—
65	Dic₇₃	—	16	2	no	—
66	Dic₇₃	—	16	1	no	—
67	#12535	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	#1722	1	1	yes	characteristic
69	C₂×C₇₃	#2248	1	1	yes	characteristic
70	C₂×C₇₃	#2251	1	1	yes	characteristic
71	#2254	—	73	2	no	—
72	#2241	—	73	2	no	—
73	#1713	—	73	1	no	—
74	#1707	—	73	1	no	—
75	#1312	D₇₃	1	1	yes	characteristic
76	C₇₃	#12535	1	1	yes	characteristic, complemented, Sylow
77	#500	—	73	1	no	—
78	#500	—	73	2	no	—
79	#500	—	146	1	no	—
80	#498	—	146	1	no	—
81	#494	—	73	1	no	—
82	#494	—	146	1	no	—
83	#487	—	146	1	no	—
84	#367	—	146	4	no	—
85	#358	—	146	4	no	—
86	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
87	C₈²	C₂×D₇₃	1	1	yes	characteristic
88	C₂×Q₁₆	—	292	2	no	—
89	C₂×Q₁₆	—	146	2	no	—
90	C₄⋊Q₈	—	146	1	no	—
91	C₄⋊Q₈	—	146	1	no	—
92	C₄⋊Q₈	—	146	2	no	—
93	C₂×C₁₆	D₂₉₂	1	4	yes	—
94	#107	—	292	2	no	—
95	#107	—	146	2	no	—
96	#107	—	146	2	no	—
97	#107	—	146	2	no	—
98	#106	—	292	2	no	—
99	#103	—	292	2	no	—
100	C₄×C₈	D₂₉₂	1	1	yes	characteristic
101	C₄×C₈	D₂₉₂	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
103	C₂×Q₈	—	292	1	no	—
104	C₂×Q₈	—	292	1	no	—
105	C₂×Q₈	—	292	2	no	—
106	Q₁₆	—	292	4	no	—
107	Q₁₆	—	584	4	no	—
108	C₂×C₈	D₅₈₄	1	2	yes	—
109	C₂×C₈	D₅₈₄	1	2	yes	—
110	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
111	C₄⋊C₄	—	292	2	no	—
112	C₄⋊C₄	—	292	1	no	—
113	C₄⋊C₄	—	292	1	no	—
114	C₄⋊C₄	—	292	1	no	—
115	C₄⋊C₄	—	292	1	no	—
116	C₄⋊C₄	—	292	2	no	—
117	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	Q₈	—	584	2	no	—
120	Q₈	—	584	4	no	—
121	Q₈	—	584	2	no	—
122	C₂×C₄	—	584	1	no	—
123	C₂×C₄	—	584	2	no	—
124	C₂×C₄	—	584	1	no	—
125	C₂×C₄	#14556588	1	1	yes	characteristic
126	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
127	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	#14556536	1	4	yes	—
130	C₈	#14556624	1	4	yes	—
131	C₂²	#17988964	1	1	yes	characteristic
132	C₄	—	1168	1	no	—
133	C₄	—	1168	2	no	—
134	C₄	—	1168	1	no	—
135	C₄	#17988981	1	2	yes	—
136	C₄	C₂×#14556536	1	2	yes	—
137	C₄	#17989030	1	2	yes	—
138	C₂	#22609602	1	1	yes	characteristic
139	C₂	#22612187	1	1	yes	characteristic
140	C₂	#22612202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612213

C₂

yes

maximal

#22612159

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic

#22609528

C₂

yes

maximal, characteristic

C₇₃×#1312

C₂

yes

maximal, characteristic

#17988992

—

#17988992

C₂²

yes

characteristic

#17988992

C₂²

yes

—

#17988985

—

#17988967

—

#17988967

C₂²

yes

—

—

—

C₄×C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

—

—

—

—

—

—

—

—

—

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×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

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

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#137

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

#497

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

C₄×C₇₃

#510

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#12535

yes

characteristic, complemented, Sylow

#500

—

#500

—

#500

—

146

—

#498

—

146

—

#494

—

#494

—

146

—

#487

—

146

—

#367

—

146

—

#358

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

292

—

#107

—

146

—

#107

—

146

—

#107

—

146

—

#106

—

292

—

#103

—

292

—

100

C₄×C₈

D₂₉₂

yes

characteristic

101

C₄×C₈

D₂₉₂

yes

characteristic

102

C₄×C₈

C₂²×D₇₃

yes

characteristic

103

C₂×Q₈

—

292

—

104

C₂×Q₈

—

292

—

105

C₂×Q₈

—

292

—

106

Q₁₆

—

292

—

107

Q₁₆

—

584

—

108

C₂×C₈

D₅₈₄

yes

—

109

C₂×C₈

D₅₈₄

yes

—

110

C₂×C₈

C₂×D₂₉₂

yes

—

111

C₄⋊C₄

—

292

—

112

C₄⋊C₄

—

292

—

113

C₄⋊C₄

—

292

—

114

C₄⋊C₄

—

292

—

115

C₄⋊C₄

—

292

—

116

C₄⋊C₄

—

292

—

117

C₄²

C₂×D₂₉₂

yes

characteristic

118

C₁₆

—

119

Q₈

—

584

—

120

Q₈

—

584

—

121

Q₈

—

584

—

122

C₂×C₄

—

584

—

123

C₂×C₄

—

584

—

124

C₂×C₄

—

584

—

125

C₂×C₄

#14556588

yes

characteristic

126

C₂×C₄

C₂×D₅₈₄

yes

characteristic

127

C₂×C₄

C₂×D₅₈₄

yes

characteristic

128

C₈

—

129

C₈

#14556536

yes

—

130

C₈

#14556624

yes

—

131

C₂²

#17988964

yes

characteristic

132

C₄

—

1168

—

133

C₄

—

1168

—

134

C₄

—

1168

—

135

C₄

#17988981

yes

—

136

C₄

C₂×#14556536

yes

—

137

C₄

#17989030

yes

—

138

C₂

#22609602

yes

characteristic

139

C₂

#22612187

yes

characteristic

140

C₂

#22612202

yes

characteristic

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1312	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1312	D₇₃
hypocenter	C₇₃	#12535
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

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
76. C₇₃	#12535	semidirect

H ≅ G/N

type

76. C₇₃

#12535

semidirect

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

subgroup	index
#22612213	2
#22612159	2
#22609549	2
#22609528	2
C₇₃×#1312	2
#12535	73

subgroup

index

C₇₃×#1312

Subgroups — #33000792

Conjugacy classes of subgroups

Complements

Maximal subgroups