Group #33000361 — subgroups

Subgroups — #33000361

Order 18688.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22611839	C₂	1	1	yes	maximal, characteristic
3	#22609326	C₂	1	2	yes	maximal
4	#22607987	C₂	1	2	yes	maximal
5	C₇₃×#1372	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988798	—	2	2	no	—
8	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	#17988413	—	2	1	no	—
10	#17988413	—	2	1	no	—
11	C₄⋊#11924003	C₂²	1	1	yes	characteristic
12	#17988405	—	2	1	no	—
13	#17988362	—	2	1	no	—
14	#17988334	—	2	2	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	#335×C₇₃	C₂²	1	2	yes	—
17	C₂×Dic₂₉₂	—	4	2	no	—
18	#14556591	—	2	2	no	—
19	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
20	C₂×#11924003	—	2	1	no	—
21	#14556525	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	C₄⋊#10596831	D₄	1	1	yes	characteristic
24	#14556520	—	2	1	no	—
25	#14556516	—	4	2	no	—
26	C₈×Dic₇₃	—	2	1	no	—
27	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	2	no	—
30	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₆	—	4	2	no	—
32	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
33	Dic₂₉₂	—	8	4	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
38	#11924007	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	#11924003	—	4	1	no	—
41	C₂×C₈×C₇₃	—	4	2	no	—
42	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
44	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₃	—	2	1	no	—
46	C₁₆×C₇₃	Q₁₆	1	2	yes	complemented
47	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	1	no	—
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	8	2	no	—
53	Dic₁₄₆	—	2	2	no	—
54	Dic₁₄₆	—	8	2	no	—
55	Dic₁₄₆	—	8	2	no	—
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	8	2	no	—
58	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₇₃	#137	1	1	yes	characteristic
60	#10596831	—	4	1	no	—
61	C₂²×C₇₃	#466	1	1	yes	characteristic
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	#462	1	1	yes	characteristic
64	C₄×C₇₃	#473	1	1	yes	characteristic
65	C₄×C₇₃	#509	1	2	yes	—
66	Dic₇₃	—	4	1	no	—
67	Dic₇₃	—	4	1	no	—
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	—	73	1	no	—
75	#1674	—	73	2	no	—
76	#1372	D₇₃	1	1	yes	characteristic
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	—	73	1	no	—
82	#366	—	146	1	no	—
83	#366	—	146	1	no	—
84	C₄⋊C₁₆	—	73	1	no	—
85	C₄⋊C₁₆	—	146	1	no	—
86	#346	—	146	1	no	—
87	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	#335	C₂×D₇₃	1	2	yes	—
89	#332	—	146	2	no	—
90	C₂×Q₁₆	—	292	2	no	—
91	C₄⋊Q₈	—	146	2	no	—
92	C₄×Q₈	—	73	1	no	—
93	C₂×C₁₆	—	146	1	no	—
94	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
95	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
96	#107	—	146	2	no	—
97	#107	—	146	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	73	1	no	—
100	C₄⋊C₈	—	146	1	no	—
101	#103	—	292	2	no	—
102	C₄×C₈	—	146	1	no	—
103	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×Q₈	—	292	2	no	—
105	C₂×Q₈	—	73	1	no	—
106	Q₁₆	—	584	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
110	C₂×C₈	#11924029	1	1	yes	characteristic
111	C₄⋊C₄	—	292	2	no	—
112	C₄⋊C₄	—	292	2	no	—
113	C₄⋊C₄	—	73	1	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄²	—	146	1	no	—
116	C₄²	D₄×D₇₃	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	292	1	no	—
119	C₁₆	Dic₂₉₂	1	2	yes	complemented
120	Q₈	—	584	2	no	—
121	Q₈	—	584	2	no	—
122	Q₈	—	146	2	no	—
123	C₂×C₄	—	584	2	no	—
124	C₂×C₄	—	146	1	no	—
125	C₂×C₄	—	146	1	no	—
126	C₂×C₄	#14556566	1	1	yes	characteristic
127	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
128	C₂×C₄	#14556614	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	8	2	no	—
131	C₈	—	292	1	no	—
132	C₈	C₂×Dic₂₉₂	1	1	yes	characteristic
133	C₈	#14556624	1	1	yes	characteristic
134	C₂²	#17988824	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	292	1	no	—
137	C₄	—	292	1	no	—
138	C₄	—	1168	2	no	—
139	C₄	#17988798	1	1	yes	characteristic
140	C₄	#17988863	1	1	yes	characteristic
141	C₄	#17989024	1	2	yes	—
142	C₂	#22609326	1	1	yes	characteristic
143	C₂	#22612035	1	1	yes	characteristic
144	C₂	#22612057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22611839

C₂

yes

maximal, characteristic

#22609326

C₂

yes

maximal

#22607987

C₂

yes

maximal

C₇₃×#1372

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

#17988798

—

C₈×Dic₁₄₆

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#335×C₇₃

C₂²

yes

—

C₂×Dic₂₉₂

—

#14556591

—

C₄×Dic₁₄₆

D₄

yes

—

—

—

D₄

yes

characteristic

#14556520

—

#14556516

—

C₈×Dic₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₈

yes

characteristic

Dic₂₉₂

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₈

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

#11924003

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

Q₁₆

yes

complemented

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#466

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

D₇₃

yes

characteristic

#1364

—

C₇₃

#12460

yes

characteristic, complemented, Sylow

#500

—

#462

—

146

—

C₈×Q₈

—

#366

—

146

—

#366

—

146

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

#346

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

C₂×D₇₃

yes

—

#332

—

146

—

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

146

—

C₄×Q₈

—

C₂×C₁₆

—

146

—

C₂×C₁₆

D₂₉₂

yes

characteristic

C₂×C₁₆

D₂₉₂

yes

characteristic

#107

—

146

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

146

—

101

#103

—

292

—

102

C₄×C₈

—

146

—

103

C₄×C₈

C₂²×D₇₃

yes

characteristic

104

C₂×Q₈

—

292

—

105

C₂×Q₈

—

106

Q₁₆

—

584

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

C₂×D₂₉₂

yes

characteristic

110

C₂×C₈

#11924029

yes

characteristic

111

C₄⋊C₄

—

292

—

112

C₄⋊C₄

—

292

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄²

—

146

—

116

C₄²

D₄×D₇₃

yes

characteristic

117

C₁₆

—

118

C₁₆

—

292

—

119

C₁₆

Dic₂₉₂

yes

complemented

120

Q₈

—

584

—

121

Q₈

—

584

—

122

Q₈

—

146

—

123

C₂×C₄

—

584

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

#14556566

yes

characteristic

127

C₂×C₄

D₈×D₇₃

yes

characteristic

128

C₂×C₄

#14556614

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

292

—

132

C₈

C₂×Dic₂₉₂

yes

characteristic

133

C₈

#14556624

yes

characteristic

134

C₂²

#17988824

yes

characteristic

135

C₄

—

136

C₄

—

292

—

137

C₄

—

292

—

138

C₄

—

1168

—

139

C₄

#17988798

yes

characteristic

140

C₄

#17988863

yes

characteristic

141

C₄

#17989024

yes

—

142

C₂

#22609326

yes

characteristic

143

C₂

#22612035

yes

characteristic

144

C₂

#22612057

yes

characteristic

	subgroup	G/H
center	C₂²	#17988824
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1372	C₂
socle, prime radical	C₂²×C₇₃	#466
hypercenter	#1372	D₇₃
hypocenter	C₇₃	#12460
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#17988824

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1372

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
46. C₁₆×C₇₃	Q₁₆	semidirect
78. C₇₃	#12460	semidirect
119. C₁₆	Dic₂₉₂	semidirect

H ≅ G/N

type

46. C₁₆×C₇₃

Q₁₆

semidirect

78. C₇₃

#12460

semidirect

119. C₁₆

Dic₂₉₂

semidirect

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

subgroup	index
#22612206	2
#22611839	2
#22609326	2
#22607987	2
C₇₃×#1372	2
#12460	73

subgroup

index

C₇₃×#1372

Subgroups — #33000361

Conjugacy classes of subgroups

Complements

Maximal subgroups