Group #33000707 — subgroups

Subgroups — #33000707

Order 18688.

330 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22612187	C₂	1	2	yes	maximal, complemented
2	#22612184	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612181	C₂	1	1	yes	maximal, characteristic
4	#22609537	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₈₄	—	2	2	no	—
7	C₂×#14556536	—	2	4	no	—
8	C₂×D₁₁₆₈	—	2	2	no	—
9	#17988967	C₂²	1	2	yes	—
10	#17988964	C₂²	1	2	yes	—
11	#17988934	—	2	4	no	—
12	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₈²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₉₂	—	2	2	no	—
16	C₂×Dic₂₉₂	—	2	2	no	—
17	C₂×D₅₈₄	—	2	2	no	—
18	C₂×D₅₈₄	—	2	2	no	—
19	#14556591	—	2	2	no	—
20	#14556588	—	2	2	no	—
21	Dic₅₈₄	—	4	4	no	—
22	#14556536	—	4	8	no	—
23	D₁₁₆₈	—	4	4	no	—
24	#14556513	—	4	8	no	—
25	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
27	C₄×C₈×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₆	—	4	2	no	—
30	C₂×D₂₉₂	—	4	2	no	—
31	C₂×D₂₉₂	—	4	4	no	—
32	Dic₂₉₂	—	4	4	no	—
33	Dic₂₉₂	—	4	4	no	—
34	D₅₈₄	—	4	4	no	—
35	D₅₈₄	—	4	4	no	—
36	C₄⋊Dic₇₃	—	4	4	no	—
37	C₂×C₈×C₇₃	D₈	1	4	yes	—
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	D₈	1	4	yes	—
42	C₁₆×C₇₃	D₈	1	4	yes	—
43	C₂²×D₇₃	—	8	2	no	—
44	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
46	C₂×Dic₇₃	—	8	2	no	—
47	D₂₉₂	—	8	2	no	—
48	D₂₉₂	—	8	2	no	—
49	D₂₉₂	—	8	8	no	—
50	Dic₁₄₆	—	8	2	no	—
51	Dic₁₄₆	—	8	2	no	—
52	C₈×C₇₃	—	2	4	no	—
53	C₈×C₇₃	C₂×D₈	1	2	yes	—
54	C₈×C₇₃	C₂×D₈	1	2	yes	—
55	C₂²×C₇₃	#493	1	1	yes	characteristic
56	C₂×D₇₃	—	16	2	no	—
57	C₂×D₇₃	—	16	4	no	—
58	C₄×C₇₃	#493	1	1	yes	characteristic
59	C₄×C₇₃	#493	1	1	yes	characteristic
60	C₄×C₇₃	#508	1	4	yes	—
61	Dic₇₃	—	16	2	no	—
62	#12518	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	#1712	1	1	yes	characteristic
64	C₂×C₇₃	#2249	1	2	yes	—
65	D₇₃	—	32	2	no	—
66	#2248	—	73	2	no	—
67	#2247	—	73	1	no	—
68	#2246	—	73	1	no	—
69	#1709	—	73	2	no	—
70	C₈×C₁₆	D₇₃	1	1	yes	characteristic, complemented
71	C₇₃	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	146	2	no	—
73	C₂×SD₃₂	—	146	4	no	—
74	C₂×D₁₆	—	146	2	no	—
75	#494	—	73	2	no	—
76	#493	—	73	2	no	—
77	#486	—	146	4	no	—
78	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
79	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
80	C₈²	C₂×D₇₃	1	1	yes	characteristic
81	C₂×Q₁₆	—	146	2	no	—
82	C₂×Q₁₆	—	146	2	no	—
83	C₂×D₈	—	146	2	no	—
84	C₂×D₈	—	146	2	no	—
85	C₄⋊Q₈	—	146	2	no	—
86	#127	—	146	2	no	—
87	Q₃₂	—	292	4	no	—
88	SD₃₂	—	292	8	no	—
89	D₁₆	—	292	4	no	—
90	C₂×C₁₆	D₂₉₂	1	2	yes	—
91	C₂×C₁₆	D₂₉₂	1	2	yes	—
92	#102	—	292	8	no	—
93	C₄×C₈	D₂₉₂	1	2	yes	—
94	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
95	C₂×Q₈	—	292	2	no	—
96	C₂×D₄	—	292	2	no	—
97	C₂×D₄	—	292	4	no	—
98	Q₁₆	—	292	4	no	—
99	Q₁₆	—	292	4	no	—
100	D₈	—	292	4	no	—
101	D₈	—	292	4	no	—
102	C₂×C₈	D₅₈₄	1	4	yes	—
103	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
105	C₄⋊C₄	—	292	4	no	—
106	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
107	C₁₆	D₅₈₄	1	4	yes	—
108	C₁₆	D₅₈₄	1	4	yes	—
109	C₂³	—	584	2	no	—
110	Q₈	—	584	2	no	—
111	Q₈	—	584	2	no	—
112	D₄	—	584	2	no	—
113	D₄	—	584	2	no	—
114	D₄	—	584	8	no	—
115	C₂×C₄	—	584	2	no	—
116	C₂×C₄	#14556588	1	1	yes	characteristic
117	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₅₈₄	1	2	yes	—
120	C₈	C₂×D₅₈₄	1	2	yes	—
121	C₂²	—	1168	2	no	—
122	C₂²	—	1168	4	no	—
123	C₂²	#17988964	1	1	yes	characteristic
124	C₄	—	1168	2	no	—
125	C₄	#17988964	1	1	yes	characteristic
126	C₄	#17988964	1	1	yes	characteristic
127	C₄	#17989019	1	4	yes	—
128	C₂	—	2336	2	no	—
129	C₂	#22609546	1	1	yes	characteristic
130	C₂	#22612192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612187

C₂

yes

maximal, complemented

#22612184

C₂

yes

maximal, characteristic, complemented

#22612181

C₂

yes

maximal, characteristic

#22609537

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₈₄

—

C₂×#14556536

—

C₂×D₁₁₆₈

—

#17988967

C₂²

yes

—

#17988964

C₂²

yes

—

#17988934

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556591

—

#14556588

—

Dic₅₈₄

—

#14556536

—

D₁₁₆₈

—

#14556513

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

D₈

yes

—

C₁₆×C₇₃

D₈

yes

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₈

yes

—

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

—

C₈×C₁₆

D₇₃

yes

characteristic, complemented

C₇₃

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

146

—

C₂×SD₃₂

—

146

—

C₂×D₁₆

—

146

—

#494

—

#493

—

#486

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

146

—

#127

—

146

—

Q₃₂

—

292

—

SD₃₂

—

292

—

D₁₆

—

292

—

C₂×C₁₆

D₂₉₂

yes

—

C₂×C₁₆

D₂₉₂

yes

—

#102

—

292

—

C₄×C₈

D₂₉₂

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

292

—

Q₁₆

—

292

—

100

D₈

—

292

—

101

D₈

—

292

—

102

C₂×C₈

D₅₈₄

yes

—

103

C₂×C₈

C₂×D₂₉₂

yes

characteristic

104

C₂×C₈

C₂×D₂₉₂

yes

characteristic

105

C₄⋊C₄

—

292

—

106

C₄²

C₂×D₂₉₂

yes

characteristic

107

C₁₆

D₅₈₄

yes

—

108

C₁₆

D₅₈₄

yes

—

109

C₂³

—

584

—

110

Q₈

—

584

—

111

Q₈

—

584

—

112

D₄

—

584

—

113

D₄

—

584

—

114

D₄

—

584

—

115

C₂×C₄

—

584

—

116

C₂×C₄

#14556588

yes

characteristic

117

C₂×C₄

C₂×D₅₈₄

yes

—

118

C₈

—

119

C₈

C₂×D₅₈₄

yes

—

120

C₈

C₂×D₅₈₄

yes

—

121

C₂²

—

1168

—

122

C₂²

—

1168

—

123

C₂²

#17988964

yes

characteristic

124

C₄

—

1168

—

125

C₄

#17988964

yes

characteristic

126

C₄

#17988964

yes

characteristic

127

C₄

#17989019

yes

—

128

C₂

—

2336

—

129

C₂

#22609546

yes

characteristic

130

C₂

#22612192

yes

—

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₈×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#493
prime radical	#22612181	C₂
hypercenter	C₈×C₁₆	D₇₃
hypocenter	C₇₃	#12518
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₈×C₁₆×C₇₃

C₂

C₂²×C₇₃

C₂

C₈×C₁₆

D₇₃

hypocenter

C₇₃

#12518

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #22612187	C₂	semidirect
2. #22612184	C₂	semidirect
4. #22609537	C₂	semidirect
5. C₈×C₁₆×C₇₃	C₂	semidirect
70. C₈×C₁₆	D₇₃	semidirect
71. C₇₃	#12518	semidirect

H ≅ G/N

type

1. #22612187

C₂

semidirect

2. #22612184

C₂

semidirect

4. #22609537

C₂

semidirect

5. C₈×C₁₆×C₇₃

C₂

semidirect

70. C₈×C₁₆

D₇₃

semidirect

71. C₇₃

#12518

semidirect

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

subgroup	index
#22612187	2
#22612184	2
#22612181	2
#22609537	2
C₈×C₁₆×C₇₃	2
#12518	73

subgroup

index

C₈×C₁₆×C₇₃

Subgroups — #33000707

Conjugacy classes of subgroups

Complements

Maximal subgroups