Group #33000713 — subgroups

Subgroups — #33000713

Order 18688.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612188	C₂	1	2	yes	maximal, complemented
2	#22612180	C₂	1	2	yes	maximal, complemented
3	#22609546	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556534	—	2	4	no	—
7	C₂×#14556532	—	2	4	no	—
8	#17988964	—	2	2	no	—
9	#17988964	C₂²	1	2	yes	—
10	C₄×#11924003	C₂²	1	2	yes	—
11	#494×C₇₃	C₂²	1	1	yes	characteristic
12	#493×C₇₃	C₂²	1	1	yes	characteristic
13	#486×C₇₃	—	2	2	no	—
14	C₈²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×D₅₈₄	—	2	2	no	—
16	C₂×D₅₈₄	—	2	2	no	—
17	C₂×D₅₈₄	—	4	4	no	—
18	#14556588	—	2	2	no	—
19	#14556534	—	4	8	no	—
20	#14556532	—	4	8	no	—
21	C₂×#11924003	D₄	1	4	yes	—
22	C₂×Q₁₆×C₇₃	—	2	2	no	—
23	C₂×D₈×C₇₃	—	2	2	no	—
24	C₄⋊Q₈×C₇₃	—	2	1	no	—
25	#127×C₇₃	—	2	1	no	—
26	#102×C₇₃	—	4	4	no	—
27	C₄×C₈×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×D₂₉₂	—	4	2	no	—
30	C₂×D₂₉₂	—	4	4	no	—
31	D₅₈₄	—	4	4	no	—
32	D₅₈₄	—	4	4	no	—
33	D₅₈₄	—	8	8	no	—
34	#11924003	D₈	1	8	yes	complemented
35	C₂×Q₈×C₇₃	—	4	1	no	—
36	C₂×D₄×C₇₃	—	4	1	no	—
37	C₂×D₄×C₇₃	—	4	2	no	—
38	Q₁₆×C₇₃	—	4	4	no	—
39	D₈×C₇₃	—	4	4	no	—
40	C₂×C₈×C₇₃	D₈	1	4	yes	—
41	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₇₃	—	4	2	no	—
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₇₃	—	8	1	no	—
45	C₂²×D₇₃	—	8	2	no	—
46	Q₈×C₇₃	—	8	2	no	—
47	D₄×C₇₃	—	8	2	no	—
48	D₄×C₇₃	—	8	4	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
52	D₂₉₂	—	8	2	no	—
53	D₂₉₂	—	8	2	no	—
54	D₂₉₂	—	8	8	no	—
55	C₈×C₇₃	—	2	4	no	—
56	C₈×C₇₃	C₂×D₈	1	4	yes	—
57	C₂²×C₇₃	—	16	1	no	—
58	C₂²×C₇₃	—	16	2	no	—
59	C₂²×C₇₃	#493	1	1	yes	characteristic
60	C₂×D₇₃	—	16	2	no	—
61	C₂×D₇₃	—	16	4	no	—
62	C₄×C₇₃	—	16	1	no	—
63	C₄×C₇₃	#493	1	2	yes	—
64	C₄×C₇₃	#509	1	4	yes	—
65	#12519	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	32	1	no	—
67	C₂×C₇₃	#1712	1	1	yes	characteristic
68	C₂×C₇₃	#2250	1	2	yes	—
69	D₇₃	—	32	2	no	—
70	#2248	—	73	2	no	—
71	#2246	—	73	2	no	—
72	#1712	—	73	1	no	—
73	#1709	D₇₃	1	1	yes	characteristic, complemented
74	#1311	—	73	1	no	—
75	C₇₃	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	146	4	no	—
77	C₂×D₁₆	—	146	4	no	—
78	#494	C₂×D₇₃	1	1	yes	characteristic
79	#493	—	73	2	no	—
80	#493	—	146	2	no	—
81	#493	C₂×D₇₃	1	1	yes	characteristic
82	#486	—	2	2	no	—
83	C₄×C₁₆	—	73	2	no	—
84	C₈²	C₂×D₇₃	1	1	yes	characteristic
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	292	4	no	—
88	C₂×D₈	—	146	2	no	—
89	C₂×D₈	—	146	2	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#127	—	146	2	no	—
93	SD₃₂	—	292	8	no	—
94	D₁₆	—	292	8	no	—
95	C₂×C₁₆	—	73	4	no	—
96	#102	—	4	4	no	—
97	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
98	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	292	2	no	—
103	C₂×D₄	—	292	4	no	—
104	Q₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	292	4	no	—
107	D₈	—	292	4	no	—
108	D₈	—	584	8	no	—
109	C₂×C₈	#11924013	1	4	yes	—
110	C₂×C₈	D₄×D₇₃	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
113	C₁₆	—	73	8	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	584	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	4	no	—
119	D₄	—	584	2	no	—
120	D₄	—	584	2	no	—
121	D₄	—	584	8	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#14556587	1	1	yes	characteristic
124	C₂×C₄	C₂×#11924013	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₇₃	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	1168	2	no	—
130	C₂²	—	1168	4	no	—
131	C₂²	#17988963	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#17988962	1	2	yes	—
134	C₄	#17989022	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	2336	2	no	—
137	C₂	#22609545	1	1	yes	characteristic
138	C₂	#22612198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612188

C₂

yes

maximal, complemented

#22612180

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic, complemented

#22607848

C₂

yes

maximal, characteristic, complemented

C₇₃×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

C₄×#11924003

C₂²

yes

—

#494×C₇₃

C₂²

yes

characteristic

#493×C₇₃

C₂²

yes

characteristic

#486×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

—

—

—

D₄

yes

—

C₂×Q₁₆×C₇₃

—

C₂×D₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#127×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924003

D₈

yes

complemented

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#493

yes

—

C₄×C₇₃

#509

yes

—

#12519

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1712

yes

characteristic

C₂×C₇₃

#2250

yes

—

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

#1311

—

C₇₃

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

146

—

C₂×D₁₆

—

146

—

#494

C₂×D₇₃

yes

characteristic

#493

—

#493

—

146

—

#493

C₂×D₇₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

292

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

#127

—

#127

—

146

—

SD₃₂

—

292

—

D₁₆

—

292

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

292

—

103

C₂×D₄

—

292

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

292

—

107

D₈

—

292

—

108

D₈

—

584

—

109

C₂×C₈

#11924013

yes

—

110

C₂×C₈

D₄×D₇₃

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

584

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

584

—

120

D₄

—

584

—

121

D₄

—

584

—

122

C₂×C₄

—

123

C₂×C₄

#14556587

yes

characteristic

124

C₂×C₄

C₂×#11924013

yes

—

125

C₈

—

126

C₈

D₈×D₇₃

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

1168

—

130

C₂²

—

1168

—

131

C₂²

#17988963

yes

characteristic

132

C₄

—

133

C₄

#17988962

yes

—

134

C₄

#17989022

yes

—

135

C₂

—

136

C₂

—

2336

—

137

C₂

#22609545

yes

characteristic

138

C₂

#22612198

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1709	C₂
socle	C₂²×C₇₃	#493
hypercenter	#1709	D₇₃
hypocenter	C₇₃	#12519
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1709

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #22612188	C₂	semidirect
2. #22612180	C₂	semidirect
3. #22609546	C₂	semidirect
4. #22607848	C₂	semidirect
5. C₇₃×#1709	C₂	semidirect
34. #11924003	D₈	semidirect
73. #1709	D₇₃	semidirect
75. C₇₃	#12519	semidirect

H ≅ G/N

type

1. #22612188

C₂

semidirect

2. #22612180

C₂

semidirect

3. #22609546

C₂

semidirect

4. #22607848

C₂

semidirect

5. C₇₃×#1709

C₂

semidirect

34. #11924003

D₈

semidirect

73. #1709

D₇₃

semidirect

75. C₇₃

#12519

semidirect

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

subgroup	index
#22612188	2
#22612180	2
#22609546	2
#22607848	2
C₇₃×#1709	2
#12519	73

subgroup

index

C₇₃×#1709

Subgroups — #33000713

Conjugacy classes of subgroups

Complements

Maximal subgroups