Group #33000715 — subgroups

Subgroups — #33000715

Order 18688.

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

normal subgroups only

This group has 316 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
2	#22612183	C₂	1	2	yes	maximal, complemented
3	#22609537	C₂	1	1	yes	maximal, characteristic
4	#22607848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556537	—	2	4	no	—
7	C₂×#14556534	—	2	4	no	—
8	#17988967	C₂²	1	1	yes	characteristic
9	#17988964	C₂²	1	1	yes	characteristic
10	#17988934	—	2	2	no	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	#500×C₇₃	—	2	2	no	—
13	#494×C₇₃	C₂²	1	2	yes	—
14	C₈²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₉₂	—	2	2	no	—
16	C₂×D₅₈₄	—	2	2	no	—
17	#14556591	—	2	1	no	—
18	#14556588	—	2	1	no	—
19	#14556537	—	4	8	no	—
20	#14556534	—	4	8	no	—
21	C₂×#11924003	D₄	1	4	yes	—
22	#14556513	—	4	4	no	—
23	C₂×Q₁₆×C₇₃	—	2	2	no	—
24	C₂×Q₁₆×C₇₃	—	2	2	no	—
25	C₄⋊Q₈×C₇₃	—	2	2	no	—
26	#107×C₇₃	—	4	4	no	—
27	C₄×C₈×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₆	—	4	1	no	—
30	C₂×D₂₉₂	—	4	1	no	—
31	C₂×D₂₉₂	—	4	2	no	—
32	Dic₂₉₂	—	4	4	no	—
33	D₅₈₄	—	4	4	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	#11924003	D₈	1	8	yes	—
36	C₂×Q₈×C₇₃	—	4	2	no	—
37	Q₁₆×C₇₃	—	4	4	no	—
38	Q₁₆×C₇₃	—	4	4	no	—
39	C₂×C₈×C₇₃	D₈	1	4	yes	—
40	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₇₃	—	4	4	no	—
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₇₃	—	8	1	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	C₂×Dic₇₃	—	8	1	no	—
50	D₂₉₂	—	8	2	no	—
51	D₂₉₂	—	8	4	no	—
52	Dic₁₄₆	—	8	2	no	—
53	C₈×C₇₃	—	2	4	no	—
54	C₈×C₇₃	C₂×D₈	1	4	yes	—
55	C₂²×C₇₃	#493	1	1	yes	characteristic
56	C₂×D₇₃	—	16	1	no	—
57	C₂×D₇₃	—	16	2	no	—
58	C₄×C₇₃	—	16	2	no	—
59	C₄×C₇₃	#493	1	2	yes	—
60	C₄×C₇₃	#510	1	4	yes	—
61	Dic₇₃	—	16	1	no	—
62	#12520	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	#1712	1	1	yes	characteristic
64	C₂×C₇₃	#2251	1	2	yes	—
65	D₇₃	—	32	1	no	—
66	#2248	—	73	2	no	—
67	#2247	—	73	2	no	—
68	#1713	D₇₃	1	1	yes	characteristic, complemented
69	#1709	—	73	1	no	—
70	#1311	—	73	1	no	—
71	C₇₃	#12520	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	146	4	no	—
73	C₂×SD₃₂	—	146	4	no	—
74	#500	—	2	2	no	—
75	#494	—	73	1	no	—
76	#494	C₂×D₇₃	1	2	yes	—
77	#493	—	73	1	no	—
78	#486	—	146	2	no	—
79	C₄×C₁₆	—	73	2	no	—
80	C₈²	C₂×D₇₃	1	1	yes	characteristic
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×Q₁₆	—	146	2	no	—
84	C₂×D₈	—	146	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	C₄⋊Q₈	—	146	1	no	—
87	#127	—	146	1	no	—
88	Q₃₂	—	292	8	no	—
89	SD₃₂	—	292	8	no	—
90	C₂×C₁₆	—	73	4	no	—
91	#107	—	4	4	no	—
92	#102	—	292	4	no	—
93	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
94	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	292	1	no	—
97	C₂×D₄	—	292	1	no	—
98	C₂×D₄	—	292	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	292	4	no	—
102	D₈	—	292	4	no	—
103	C₂×C₈	#11924013	1	4	yes	—
104	C₂×C₈	D₄×D₇₃	1	2	yes	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	292	2	no	—
107	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
108	C₁₆	—	73	8	no	—
109	C₂³	—	584	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	Q₈	—	584	2	no	—
113	D₄	—	584	2	no	—
114	D₄	—	584	4	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	584	1	no	—
117	C₂×C₄	#14556587	1	1	yes	characteristic
118	C₂×C₄	C₂×#11924013	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	D₈×D₇₃	1	4	yes	—
121	C₂²	—	1168	1	no	—
122	C₂²	—	1168	2	no	—
123	C₂²	#17988963	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	—	1168	1	no	—
126	C₄	#17988962	1	2	yes	—
127	C₄	#17989028	1	4	yes	—
128	C₂	—	2336	1	no	—
129	C₂	#22609545	1	1	yes	characteristic
130	C₂	#22612200	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612188

C₂

yes

maximal

#22612183

C₂

yes

maximal, complemented

#22609537

C₂

yes

maximal, characteristic

#22607848

C₂

yes

maximal, characteristic, complemented

C₇₃×#1713

C₂

yes

maximal, characteristic, complemented

C₂×#14556537

—

C₂×#14556534

—

#17988967

C₂²

yes

characteristic

#17988964

C₂²

yes

characteristic

#17988934

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

—

#494×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×D₅₈₄

—

—

—

—

—

D₄

yes

—

#14556513

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

Dic₂₉₂

—

D₅₈₄

—

C₄⋊Dic₇₃

—

#11924003

D₈

yes

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#493

yes

—

C₄×C₇₃

#510

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2248

—

#2247

—

#1713

D₇₃

yes

characteristic, complemented

#1709

—

#1311

—

C₇₃

#12520

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

146

—

C₂×SD₃₂

—

146

—

#500

—

#494

—

#494

C₂×D₇₃

yes

—

#493

—

#486

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

#127

—

146

—

Q₃₂

—

292

—

SD₃₂

—

292

—

C₂×C₁₆

—

#107

—

#102

—

292

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

292

—

102

D₈

—

292

—

103

C₂×C₈

#11924013

yes

—

104

C₂×C₈

D₄×D₇₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

292

—

107

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

108

C₁₆

—

109

C₂³

—

584

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

584

—

113

D₄

—

584

—

114

D₄

—

584

—

115

C₂×C₄

—

116

C₂×C₄

—

584

—

117

C₂×C₄

#14556587

yes

characteristic

118

C₂×C₄

C₂×#11924013

yes

—

119

C₈

—

120

C₈

D₈×D₇₃

yes

—

121

C₂²

—

1168

—

122

C₂²

—

1168

—

123

C₂²

#17988963

yes

characteristic

124

C₄

—

125

C₄

—

1168

—

126

C₄

#17988962

yes

—

127

C₄

#17989028

yes

—

128

C₂

—

2336

—

129

C₂

#22609545

yes

characteristic

130

C₂

#22612200

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1713	C₂
socle	C₂²×C₇₃	#493
prime radical	#17988964	C₂²
hypercenter	#1713	D₇₃
hypocenter	C₇₃	#12520
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1713

C₂

C₂²×C₇₃

C₂²

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
2. #22612183	C₂	semidirect
4. #22607848	C₂	semidirect
5. C₇₃×#1713	C₂	semidirect
68. #1713	D₇₃	semidirect
71. C₇₃	#12520	semidirect

H ≅ G/N

type

2. #22612183

C₂

semidirect

4. #22607848

C₂

semidirect

5. C₇₃×#1713

C₂

semidirect

68. #1713

D₇₃

semidirect

71. C₇₃

#12520

semidirect

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

subgroup	index
#22612188	2
#22612183	2
#22609537	2
#22607848	2
C₇₃×#1713	2
#12520	73

subgroup

index

C₇₃×#1713

Subgroups — #33000715

Conjugacy classes of subgroups

Complements

Maximal subgroups