Group #33000708 — subgroups

Subgroups — #33000708

Order 18688.

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

normal subgroups only

This group has 330 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	#22612181	C₂	1	1	yes	maximal, characteristic
3	#22609538	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556534	—	2	4	no	—
7	C₂×D₁₁₆₈	—	2	2	no	—
8	#17988964	C₂²	1	2	yes	—
9	#17988935	—	2	4	no	—
10	C₄×#11924003	C₂²	1	1	yes	characteristic
11	C₈×#10596831	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₇₃	—	2	2	no	—
13	#494×C₇₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×D₅₈₄	—	2	2	no	—
16	C₂×D₅₈₄	—	2	2	no	—
17	#14556588	—	2	2	no	—
18	#14556534	—	4	8	no	—
19	D₁₁₆₈	—	4	4	no	—
20	C₂×#11924003	D₄	1	2	yes	—
21	#14556511	—	4	8	no	—
22	C₄×#10596831	D₄	1	2	yes	—
23	C₂×Q₁₆×C₇₃	—	2	2	no	—
24	C₂×Q₁₆×C₇₃	—	2	2	no	—
25	C₄⋊Q₈×C₇₃	—	2	2	no	—
26	Q₃₂×C₇₃	—	4	4	no	—
27	C₂×C₁₆×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	C₂×#10596831	D₈	1	4	yes	—
34	#11924003	D₈	1	4	yes	—
35	C₂×Q₈×C₇₃	—	4	2	no	—
36	Q₁₆×C₇₃	—	4	4	no	—
37	Q₁₆×C₇₃	—	4	4	no	—
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	4	4	no	—
41	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₃	D₈	1	4	yes	—
43	C₂²×D₇₃	—	8	2	no	—
44	Q₈×C₇₃	—	8	2	no	—
45	Q₈×C₇₃	—	8	2	no	—
46	C₂×C₄×C₇₃	—	8	2	no	—
47	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
49	D₂₉₂	—	8	2	no	—
50	D₂₉₂	—	8	2	no	—
51	D₂₉₂	—	8	8	no	—
52	C₈×C₇₃	C₂×D₈	1	2	yes	—
53	C₈×C₇₃	C₂×D₈	1	2	yes	—
54	#10596831	—	2	4	no	—
55	C₂²×C₇₃	#493	1	1	yes	characteristic
56	C₂×D₇₃	—	16	2	no	—
57	C₂×D₇₃	—	16	4	no	—
58	C₄×C₇₃	—	16	2	no	—
59	C₄×C₇₃	#493	1	1	yes	characteristic
60	C₄×C₇₃	#493	1	1	yes	characteristic
61	C₄×C₇₃	#508	1	4	yes	—
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	D₇₃	1	1	yes	characteristic, complemented
68	#2246	—	73	1	no	—
69	#1709	—	73	2	no	—
70	C₈×C₁₆	—	73	1	no	—
71	C₇₃	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	2	2	no	—
73	C₂×SD₃₂	—	146	4	no	—
74	C₂×D₁₆	—	146	2	no	—
75	#494	C₂×D₇₃	1	2	yes	—
76	#493	—	73	2	no	—
77	#486	—	146	4	no	—
78	C₄×C₁₆	—	73	1	no	—
79	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
80	C₈²	—	73	1	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	146	2	no	—
84	C₂×D₈	—	146	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	146	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	292	8	no	—
89	D₁₆	—	292	4	no	—
90	C₂×C₁₆	—	73	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
92	#102	—	292	8	no	—
93	C₄×C₈	—	73	2	no	—
94	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	292	2	no	—
97	C₂×D₄	—	292	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	292	4	no	—
101	D₈	—	292	4	no	—
102	C₂×C₈	—	73	4	no	—
103	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₇₃	1	1	yes	characteristic
107	C₁₆	—	73	4	no	—
108	C₁₆	#11924013	1	4	yes	—
109	C₂³	—	584	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	584	2	no	—
113	D₄	—	584	2	no	—
114	D₄	—	584	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#14556587	1	1	yes	characteristic
117	C₂×C₄	D₈×D₇₃	1	2	yes	—
118	C₈	—	146	4	no	—
119	C₈	D₈×D₇₃	1	2	yes	—
120	C₈	C₂×#11924013	1	2	yes	—
121	C₂²	—	1168	2	no	—
122	C₂²	—	1168	4	no	—
123	C₂²	#17988962	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#17988962	1	1	yes	characteristic
126	C₄	#17988963	1	1	yes	characteristic
127	C₄	#17989020	1	4	yes	—
128	C₂	—	2336	2	no	—
129	C₂	#22609545	1	1	yes	characteristic
130	C₂	#22612193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612188

C₂

yes

maximal, complemented

#22612181

C₂

yes

maximal, characteristic

#22609538

C₂

yes

maximal, complemented

C₁₆×#10596831

C₂

yes

maximal, characteristic, complemented

C₇₃×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#14556534

—

C₂×D₁₁₆₈

—

#17988964

C₂²

yes

—

#17988935

—

C₄×#11924003

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

—

#494×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

#14556534

—

D₁₁₆₈

—

C₂×#11924003

D₄

yes

—

#14556511

—

C₄×#10596831

D₄

yes

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

C₂×#10596831

D₈

yes

—

#11924003

D₈

yes

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

D₈

yes

—

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

C₂×D₈

yes

—

C₈×C₇₃

C₂×D₈

yes

—

#10596831

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2248

—

#2247

D₇₃

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₇₃

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

146

—

C₂×D₁₆

—

146

—

#494

C₂×D₇₃

yes

—

#493

—

#486

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

#127

—

146

—

Q₃₂

—

SD₃₂

—

292

—

D₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

—

#102

—

292

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

Q₁₆

—

100

D₈

—

292

—

101

D₈

—

292

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₇₃

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₇₃

yes

characteristic

107

C₁₆

—

108

C₁₆

#11924013

yes

—

109

C₂³

—

584

—

110

Q₈

—

111

Q₈

—

112

D₄

—

584

—

113

D₄

—

584

—

114

D₄

—

584

—

115

C₂×C₄

—

116

C₂×C₄

#14556587

yes

characteristic

117

C₂×C₄

D₈×D₇₃

yes

—

118

C₈

—

146

—

119

C₈

D₈×D₇₃

yes

—

120

C₈

C₂×#11924013

yes

—

121

C₂²

—

1168

—

122

C₂²

—

1168

—

123

C₂²

#17988962

yes

characteristic

124

C₄

—

125

C₄

#17988962

yes

characteristic

126

C₄

#17988963

yes

characteristic

127

C₄

#17989020

yes

—

128

C₂

—

2336

—

129

C₂

#22609545

yes

characteristic

130

C₂

#22612193

yes

—

	subgroup	G/H
center	C₂²	#17988962
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2247	C₂
socle	C₂²×C₇₃	#493
prime radical	#22612181	C₂
hypercenter	#2247	D₇₃
hypocenter	C₇₃	#12518
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#17988962

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2247

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #22612188	C₂	semidirect
3. #22609538	C₂	semidirect
4. C₁₆×#10596831	C₂	semidirect
5. C₇₃×#2247	C₂	semidirect
67. #2247	D₇₃	semidirect
71. C₇₃	#12518	semidirect

H ≅ G/N

type

1. #22612188

C₂

semidirect

3. #22609538

C₂

semidirect

4. C₁₆×#10596831

C₂

semidirect

5. C₇₃×#2247

C₂

semidirect

67. #2247

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
#22612188	2
#22612181	2
#22609538	2
C₁₆×#10596831	2
C₇₃×#2247	2
#12518	73

subgroup

index

#22612188

#22612181

#22609538

C₁₆×#10596831

C₇₃×#2247

#12518

Subgroups — #33000708

Conjugacy classes of subgroups

Complements

Maximal subgroups