Group #33000639 — subgroups

Subgroups — #33000639

Order 18688.

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

normal subgroups only

This group has 262 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	#22612206	C₂	1	1	yes	maximal, characteristic
2	#22612158	C₂	1	2	yes	maximal
3	#22609525	C₂	1	2	yes	maximal
4	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988983	—	2	2	no	—
8	#17988939	—	2	2	no	—
9	#17988413	—	2	2	no	—
10	#17988391	—	2	4	no	—
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₇₃	—	2	2	no	—
14	#494×C₇₃	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	#14556591	—	2	2	no	—
17	C₂×#11924003	D₄	1	2	yes	—
18	#14556525	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556522	—	4	4	no	—
21	#14556514	—	4	4	no	—
22	C₄×#10596831	D₄	1	1	yes	characteristic
23	C₄×#10596831	D₄	1	1	yes	characteristic
24	C₂×Q₁₆×C₇₃	—	2	2	no	—
25	C₂×Q₁₆×C₇₃	—	2	2	no	—
26	C₄⋊Q₈×C₇₃	—	2	2	no	—
27	Q₃₂×C₇₃	—	4	4	no	—
28	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₆	—	4	2	no	—
31	C₂×#10596831	D₈	1	2	yes	—
32	C₂×#10596831	D₈	1	2	yes	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	#11924003	—	2	2	no	—
36	C₂×Q₈×C₇₃	—	4	2	no	—
37	Q₁₆×C₇₃	—	4	4	no	—
38	Q₁₆×C₇₃	—	4	4	no	—
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	4	2	no	—
42	C₄⋊C₄×C₇₃	—	4	2	no	—
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	SD₁₆	1	4	yes	—
45	Q₈×C₇₃	—	8	2	no	—
46	Q₈×C₇₃	—	8	2	no	—
47	C₂×C₄×C₇₃	—	8	2	no	—
48	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₇₃	—	8	2	no	—
52	Dic₁₄₆	—	8	4	no	—
53	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
54	C₈×C₇₃	#135	1	2	yes	—
55	#10596831	—	2	2	no	—
56	#10596831	Q₃₂	1	4	yes	complemented
57	C₂²×C₇₃	#493	1	1	yes	characteristic
58	C₄×C₇₃	—	16	2	no	—
59	C₄×C₇₃	#492	1	1	yes	characteristic
60	C₄×C₇₃	#495	1	1	yes	characteristic
61	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
62	C₄×C₇₃	#508	1	2	yes	—
63	Dic₇₃	—	16	2	no	—
64	#12504	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	#1706	1	1	yes	characteristic
66	C₂×C₇₃	#2247	1	1	yes	characteristic
67	C₂×C₇₃	#2249	1	1	yes	characteristic
68	#2252	—	73	1	no	—
69	#2247	D₇₃	1	1	yes	characteristic
70	#2241	—	73	2	no	—
71	#1707	—	73	2	no	—
72	C₈×C₁₆	—	73	1	no	—
73	C₇₃	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	2	2	no	—
75	#500	—	73	2	no	—
76	#498	—	146	2	no	—
77	#494	C₂×D₇₃	1	2	yes	—
78	#487	—	146	2	no	—
79	#366	—	146	2	no	—
80	#358	—	146	4	no	—
81	C₄×C₁₆	—	73	1	no	—
82	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
83	C₈²	—	73	1	no	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	146	2	no	—
88	Q₃₂	—	4	4	no	—
89	C₂×C₁₆	—	73	2	no	—
90	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
91	#107	—	146	2	no	—
92	#107	—	146	2	no	—
93	#106	—	292	4	no	—
94	#103	—	292	4	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	292	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	C₂×C₈	—	73	2	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	292	2	no	—
109	C₄⋊C₄	—	292	2	no	—
110	C₄²	D₄×D₇₃	1	1	yes	characteristic
111	C₁₆	—	146	2	no	—
112	C₁₆	#11924016	1	4	yes	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	584	4	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	584	2	no	—
118	C₂×C₄	#14556587	1	1	yes	characteristic
119	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
120	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
121	C₈	—	73	4	no	—
122	C₈	—	146	2	no	—
123	C₈	C₂×#11924016	1	2	yes	—
124	C₈	#14556610	1	2	yes	—
125	C₂²	#17988962	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	1168	2	no	—
128	C₄	#17988959	1	1	yes	characteristic
129	C₄	#17988970	1	1	yes	characteristic
130	C₄	Q₃₂×D₇₃	1	2	yes	—
131	C₄	#17989020	1	2	yes	—
132	C₂	#22609520	1	1	yes	characteristic
133	C₂	#22612182	1	1	yes	characteristic
134	C₂	#22612193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22612158

C₂

yes

maximal

#22609525

C₂

yes

maximal

C₁₆×#10596831

C₂

yes

maximal, characteristic

C₇₃×#2247

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

—

#494×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#14556591

—

C₂×#11924003

D₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×#10596831

D₈

yes

—

C₂×#10596831

D₈

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#135

yes

—

#10596831

—

#10596831

Q₃₂

yes

complemented

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×Q₃₂

yes

—

C₄×C₇₃

#508

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

#2241

—

#1707

—

C₈×C₁₆

—

C₇₃

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#500

—

#498

—

146

—

#494

C₂×D₇₃

yes

—

#487

—

146

—

#366

—

146

—

#358

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

—

#107

—

146

—

#107

—

146

—

#106

—

292

—

#103

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₇₃

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

292

—

109

C₄⋊C₄

—

292

—

110

C₄²

D₄×D₇₃

yes

characteristic

111

C₁₆

—

146

—

112

C₁₆

#11924016

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

584

—

116

C₂×C₄

—

117

C₂×C₄

—

584

—

118

C₂×C₄

#14556587

yes

characteristic

119

C₂×C₄

D₈×D₇₃

yes

characteristic

120

C₂×C₄

D₈×D₇₃

yes

characteristic

121

C₈

—

122

C₈

—

146

—

123

C₈

C₂×#11924016

yes

—

124

C₈

#14556610

yes

—

125

C₂²

#17988962

yes

characteristic

126

C₄

—

127

C₄

—

1168

—

128

C₄

#17988959

yes

characteristic

129

C₄

#17988970

yes

characteristic

130

C₄

Q₃₂×D₇₃

yes

—

131

C₄

#17989020

yes

—

132

C₂

#22609520

yes

characteristic

133

C₂

#22612182

yes

characteristic

134

C₂

#22612193

yes

characteristic

	subgroup	G/H
center	C₂²	#17988962
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2247	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#2247	D₇₃
hypocenter	C₇₃	#12504
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

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
56. #10596831	Q₃₂	semidirect
73. C₇₃	#12504	semidirect

H ≅ G/N

type

56. #10596831

Q₃₂

semidirect

73. C₇₃

#12504

semidirect

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

subgroup	index
#22612206	2
#22612158	2
#22609525	2
C₁₆×#10596831	2
C₇₃×#2247	2
#12504	73

subgroup

index

#22612206

#22612158

#22609525

C₁₆×#10596831

C₇₃×#2247

#12504

Subgroups — #33000639

Conjugacy classes of subgroups

Complements

Maximal subgroups