Group #33000651 — subgroups

Subgroups — #33000651

Order 18688.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22612213	C₂	1	1	yes	maximal, characteristic, complemented
2	#22612154	C₂	1	2	yes	maximal, complemented
3	#22609520	C₂	1	2	yes	maximal, complemented
4	#22607847	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2246	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988959	—	2	2	no	—
8	#17988933	—	2	2	no	—
9	#17988415	—	2	2	no	—
10	#17988388	—	2	4	no	—
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₇₃	—	2	2	no	—
14	#493×C₇₃	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924016	—	4	4	no	—
17	#14556591	—	2	2	no	—
18	C₂×#11924003	D₄	1	2	yes	—
19	#14556525	—	2	2	no	—
20	#14556525	—	2	2	no	—
21	#14556512	—	4	4	no	—
22	C₄×#10596831	D₄	1	2	yes	—
23	C₂×D₈×C₇₃	—	2	2	no	—
24	C₂×D₈×C₇₃	—	2	2	no	—
25	#127×C₇₃	—	2	2	no	—
26	D₁₆×C₇₃	—	4	4	no	—
27	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₆	—	4	2	no	—
30	#11924016	—	8	8	no	—
31	C₂×#10596831	D₈	1	4	yes	—
32	C₄⋊Dic₇₃	—	4	2	no	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	#11924003	—	2	2	no	—
35	C₂×D₄×C₇₃	—	4	2	no	—
36	C₂×D₄×C₇₃	—	4	2	no	—
37	C₂×D₄×C₇₃	—	4	2	no	—
38	D₈×C₇₃	—	4	4	no	—
39	D₈×C₇₃	—	4	4	no	—
40	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	SD₁₆	1	4	yes	complemented
44	C₂³×C₇₃	—	8	2	no	—
45	D₄×C₇₃	—	8	2	no	—
46	D₄×C₇₃	—	8	2	no	—
47	D₄×C₇₃	—	8	4	no	—
48	D₄×C₇₃	—	8	4	no	—
49	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
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	4	no	—
56	C₂²×C₇₃	—	16	2	no	—
57	C₂²×C₇₃	—	16	2	no	—
58	C₂²×C₇₃	—	16	2	no	—
59	C₂²×C₇₃	#493	1	1	yes	characteristic
60	C₄×C₇₃	#492	1	1	yes	characteristic
61	C₄×C₇₃	#495	1	1	yes	characteristic
62	C₄×C₇₃	#509	1	4	yes	—
63	Dic₇₃	—	16	2	no	—
64	#12506	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	—	32	2	no	—
66	C₂×C₇₃	#1706	1	1	yes	characteristic
67	C₂×C₇₃	#2250	1	2	yes	—
68	#2254	—	73	1	no	—
69	#2246	D₇₃	1	1	yes	characteristic
70	#2240	—	73	2	no	—
71	#1706	—	73	2	no	—
72	#1311	—	73	1	no	—
73	C₇₃	#12506	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₁₆	—	2	2	no	—
75	#500	—	73	2	no	—
76	#493	C₂×D₇₃	1	2	yes	—
77	#492	—	146	2	no	—
78	#486	—	146	2	no	—
79	#367	—	146	2	no	—
80	#357	—	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×SD₁₆	—	292	4	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₄⋊Q₈	—	146	2	no	—
88	#127	—	2	2	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	73	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
92	#107	—	146	2	no	—
93	#107	—	146	2	no	—
94	#102	—	292	4	no	—
95	C₄×C₈	—	73	2	no	—
96	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
97	C₂×Q₈	—	292	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	2	no	—
101	SD₁₆	—	584	8	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	73	4	no	—
105	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
107	C₄⋊C₄	—	292	2	no	—
108	C₄⋊C₄	—	292	2	no	—
109	C₄²	D₄×D₇₃	1	1	yes	characteristic
110	C₁₆	—	146	2	no	—
111	C₁₆	#11924016	1	4	yes	complemented
112	C₂³	—	8	2	no	—
113	Q₈	—	584	4	no	—
114	D₄	—	8	2	no	—
115	D₄	—	8	2	no	—
116	D₄	—	8	4	no	—
117	D₄	—	8	4	no	—
118	C₂×C₄	—	584	2	no	—
119	C₂×C₄	#14556587	1	1	yes	characteristic
120	C₂×C₄	D₈×D₇₃	1	2	yes	—
121	C₈	—	146	4	no	—
122	C₈	C₂×#11924016	1	2	yes	—
123	C₈	#14556610	1	2	yes	—
124	C₂²	—	16	2	no	—
125	C₂²	—	16	2	no	—
126	C₂²	—	16	2	no	—
127	C₂²	#17988962	1	1	yes	characteristic
128	C₄	—	1168	2	no	—
129	C₄	#17988959	1	1	yes	characteristic
130	C₄	#17988970	1	1	yes	characteristic
131	C₄	#17989021	1	4	yes	—
132	C₂	—	32	2	no	—
133	C₂	#22609520	1	1	yes	characteristic
134	C₂	#22612194	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612213

C₂

yes

maximal, characteristic, complemented

#22612154

C₂

yes

maximal, complemented

#22609520

C₂

yes

maximal, complemented

#22607847

C₂

yes

maximal, characteristic, complemented

C₇₃×#2246

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₂×D₁₆×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

D₄

yes

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

#11924016

—

C₂×#10596831

D₈

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

complemented

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#135

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12506

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1706

yes

characteristic

C₂×C₇₃

#2250

yes

—

#2254

—

#2246

D₇₃

yes

—

—

—

C₇₃

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

#500

—

#493

C₂×D₇₃

yes

—

#492

—

146

—

#486

—

146

—

#367

—

146

—

#357

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

C₂×SD₁₆

—

292

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

146

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

—

#107

—

146

—

#107

—

146

—

#102

—

292

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

SD₁₆

—

584

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₇₃

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

107

C₄⋊C₄

—

292

—

108

C₄⋊C₄

—

292

—

109

C₄²

D₄×D₇₃

yes

characteristic

110

C₁₆

—

146

—

111

C₁₆

#11924016

yes

complemented

112

C₂³

—

113

Q₈

—

584

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

584

—

119

C₂×C₄

#14556587

yes

characteristic

120

C₂×C₄

D₈×D₇₃

yes

—

121

C₈

—

146

—

122

C₈

C₂×#11924016

yes

—

123

C₈

#14556610

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#17988962

yes

characteristic

128

C₄

—

1168

—

129

C₄

#17988959

yes

characteristic

130

C₄

#17988970

yes

characteristic

131

C₄

#17989021

yes

—

132

C₂

—

133

C₂

#22609520

yes

characteristic

134

C₂

#22612194

yes

—

	subgroup	G/H
center	C₂²	#17988962
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting, prime radical	C₇₃×#2246	C₂
socle	C₂²×C₇₃	#493
hypercenter	#2246	D₇₃
hypocenter	C₇₃	#12506
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#17988962

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting, prime radical

C₇₃×#2246

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #22612213	C₂	semidirect
2. #22612154	C₂	semidirect
3. #22609520	C₂	semidirect
4. #22607847	C₂	semidirect
43. C₁₆×C₇₃	SD₁₆	semidirect
73. C₇₃	#12506	semidirect
111. C₁₆	#11924016	semidirect

H ≅ G/N

type

1. #22612213

C₂

semidirect

2. #22612154

C₂

semidirect

3. #22609520

C₂

semidirect

4. #22607847

C₂

semidirect

43. C₁₆×C₇₃

SD₁₆

semidirect

73. C₇₃

#12506

semidirect

111. C₁₆

#11924016

semidirect

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

subgroup	index
#22612213	2
#22612154	2
#22609520	2
#22607847	2
C₇₃×#2246	2
#12506	73

subgroup

index

C₇₃×#2246

Subgroups — #33000651

Conjugacy classes of subgroups

Complements

Maximal subgroups