Group #33000360 — subgroups

Subgroups — #33000360

Order 18688.

200 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22612206	C₂	1	1	yes	maximal, characteristic
2	#22609325	C₂	1	2	yes	maximal
3	#22608005	C₂	1	1	yes	maximal, characteristic
4	#22607985	C₂	1	2	yes	maximal
5	C₇₃×#2183	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988797	—	2	2	no	—
8	#17988413	—	2	1	no	—
9	#17988413	—	2	1	no	—
10	#17988340	C₂²	1	2	yes	—
11	#17988332	—	2	2	no	—
12	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₃	—	2	1	no	—
14	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#346×C₇₃	—	2	1	no	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924018	—	4	2	no	—
18	#14556591	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556525	—	2	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	#14556514	—	4	2	no	—
23	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	—	2	1	no	—
25	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₃	—	2	1	no	—
28	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₆	—	4	2	no	—
32	#11924018	—	8	4	no	—
33	C₂×#10596831	—	4	2	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
42	C₄²×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	1	no	—
45	C₁₆×C₇₃	—	4	1	no	—
46	C₁₆×C₇₃	Q₁₆	1	2	yes	complemented
47	Q₈×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
53	C₂×Dic₇₃	—	8	2	no	—
54	Dic₁₄₆	—	8	2	no	—
55	Dic₁₄₆	—	8	2	no	—
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	4	1	no	—
58	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₇₃	#137	1	1	yes	characteristic
60	#10596831	—	8	2	no	—
61	C₂²×C₇₃	#466	1	1	yes	characteristic
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	4	1	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	#462	1	1	yes	characteristic
66	C₄×C₇₃	#473	1	1	yes	characteristic
67	C₄×C₇₃	#509	1	2	yes	—
68	Dic₇₃	—	16	2	no	—
69	#12460	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	#1674	1	1	yes	characteristic
71	C₂×C₇₃	#2218	1	1	yes	characteristic
72	C₂×C₇₃	#2222	1	1	yes	characteristic
73	#2252	—	73	1	no	—
74	#2183	D₇₃	1	1	yes	characteristic
75	#1674	—	73	2	no	—
76	#1372	—	73	1	no	—
77	#1364	—	73	2	no	—
78	C₇₃	#12460	1	1	yes	characteristic, complemented, Sylow
79	#500	—	73	2	no	—
80	#462	—	146	2	no	—
81	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
82	#366	—	146	1	no	—
83	#366	—	146	1	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
86	#346	—	2	1	no	—
87	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	#335	—	73	2	no	—
89	#332	—	146	2	no	—
90	C₂×Q₁₆	—	292	2	no	—
91	C₄⋊Q₈	—	146	2	no	—
92	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
95	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
96	#107	—	146	2	no	—
97	#107	—	146	2	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	146	2	no	—
100	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
101	#103	—	292	2	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×Q₈	—	292	2	no	—
105	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
106	Q₁₆	—	584	4	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	292	2	no	—
109	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
110	C₂×C₈	#11924028	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	292	2	no	—
113	C₄⋊C₄	—	292	2	no	—
114	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	4	1	no	—
119	C₁₆	#11924018	1	2	yes	complemented
120	Q₈	—	2	2	no	—
121	Q₈	—	584	2	no	—
122	Q₈	—	584	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	584	2	no	—
126	C₂×C₄	#14556565	1	1	yes	characteristic
127	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
128	C₂×C₄	#14556613	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	4	1	no	—
131	C₈	—	584	2	no	—
132	C₈	C₂×#11924018	1	1	yes	characteristic
133	C₈	#14556621	1	1	yes	characteristic
134	C₂²	#17988823	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	1	no	—
138	C₄	—	1168	2	no	—
139	C₄	#17988797	1	1	yes	characteristic
140	C₄	#17988862	1	1	yes	characteristic
141	C₄	#17989025	1	2	yes	—
142	C₂	#22609325	1	1	yes	characteristic
143	C₂	#22612034	1	1	yes	characteristic
144	C₂	#22612056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22609325

C₂

yes

maximal

#22608005

C₂

yes

maximal, characteristic

#22607985

C₂

yes

maximal

C₇₃×#2183

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

C₂²

yes

—

#17988332

—

C₈×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₄⋊C₁₆×C₇₃

C₂²

yes

characteristic

#346×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

—

—

C₄×Q₈×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

#11924018

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

Q₁₆

yes

complemented

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#466

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

C₇₃

#12460

yes

characteristic, complemented, Sylow

#500

—

#462

—

146

—

C₈×Q₈

C₂×D₇₃

yes

characteristic

#366

—

146

—

#366

—

146

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇₃

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

#332

—

146

—

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

146

—

C₄×Q₈

D₂₉₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

#107

—

146

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

100

C₄⋊C₈

D₂₉₂

yes

characteristic

101

#103

—

292

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₇₃

yes

characteristic

104

C₂×Q₈

—

292

—

105

C₂×Q₈

D₅₈₄

yes

characteristic

106

Q₁₆

—

584

—

107

C₂×C₈

—

108

C₂×C₈

—

292

—

109

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

110

C₂×C₈

#11924028

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

292

—

113

C₄⋊C₄

—

292

—

114

C₄⋊C₄

D₅₈₄

yes

characteristic

115

C₄²

—

116

C₄²

C₂×D₂₉₂

yes

characteristic

117

C₁₆

—

118

C₁₆

—

119

C₁₆

#11924018

yes

complemented

120

Q₈

—

121

Q₈

—

584

—

122

Q₈

—

584

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

584

—

126

C₂×C₄

#14556565

yes

characteristic

127

C₂×C₄

C₂×D₅₈₄

yes

characteristic

128

C₂×C₄

#14556613

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

584

—

132

C₈

C₂×#11924018

yes

characteristic

133

C₈

#14556621

yes

characteristic

134

C₂²

#17988823

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

1168

—

139

C₄

#17988797

yes

characteristic

140

C₄

#17988862

yes

characteristic

141

C₄

#17989025

yes

—

142

C₂

#22609325

yes

characteristic

143

C₂

#22612034

yes

characteristic

144

C₂

#22612056

yes

characteristic

	subgroup	G/H
center	C₂²	#17988823
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2183	C₂
socle, prime radical	C₂²×C₇₃	#466
hypercenter	#2183	D₇₃
hypocenter	C₇₃	#12460
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#17988823

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2183

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
46. C₁₆×C₇₃	Q₁₆	semidirect
78. C₇₃	#12460	semidirect
119. C₁₆	#11924018	semidirect

H ≅ G/N

type

46. C₁₆×C₇₃

Q₁₆

semidirect

78. C₇₃

#12460

semidirect

119. C₁₆

#11924018

semidirect

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

subgroup	index
#22612206	2
#22609325	2
#22608005	2
#22607985	2
C₇₃×#2183	2
#12460	73

subgroup

index

C₇₃×#2183

Subgroups — #33000360

Conjugacy classes of subgroups

Complements

Maximal subgroups