Group #33000655 — subgroups

Subgroups — #33000655

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	#22612181	C₂	1	1	yes	maximal, characteristic
2	#22612156	C₂	1	2	yes	maximal, complemented
3	#22609524	C₂	1	2	yes	maximal, complemented
4	#22607847	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₆₈	—	2	2	no	—
7	#17988964	C₂²	1	2	yes	—
8	#17988961	—	2	2	no	—
9	#17988935	—	2	2	no	—
10	#17988387	—	2	4	no	—
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	#500×C₇₃	C₂²	1	2	yes	—
14	#367×C₇₃	—	2	2	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924015	—	4	4	no	—
17	C₂×D₅₈₄	—	2	2	no	—
18	C₂×D₅₈₄	—	2	2	no	—
19	#14556588	—	2	2	no	—
20	D₁₁₆₈	—	4	4	no	—
21	C₂×#11924003	D₄	1	2	yes	—
22	#14556511	—	4	4	no	—
23	C₄×#10596831	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₃	—	2	2	no	—
25	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
26	#107×C₇₃	—	2	2	no	—
27	#107×C₇₃	—	2	2	no	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×D₂₉₂	—	4	2	no	—
30	C₂×D₂₉₂	—	4	2	no	—
31	C₂×D₂₉₂	—	4	2	no	—
32	#11924015	—	8	8	no	—
33	D₅₈₄	—	4	4	no	—
34	D₅₈₄	—	4	4	no	—
35	C₂×#10596831	D₈	1	4	yes	—
36	#11924003	—	2	2	no	—
37	C₂×Q₈×C₇₃	—	4	2	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	2	no	—
41	C₄⋊C₄×C₇₃	—	4	2	no	—
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	SD₁₆	1	4	yes	complemented
44	C₂²×D₇₃	—	8	2	no	—
45	Q₈×C₇₃	—	8	4	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	4	no	—
52	D₂₉₂	—	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	#493	1	1	yes	characteristic
57	C₂×D₇₃	—	16	2	no	—
58	C₂×D₇₃	—	16	2	no	—
59	C₂×D₇₃	—	16	2	no	—
60	C₄×C₇₃	—	16	2	no	—
61	C₄×C₇₃	#492	1	1	yes	characteristic
62	C₄×C₇₃	#495	1	1	yes	characteristic
63	C₄×C₇₃	#509	1	4	yes	—
64	#12506	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	#1706	1	1	yes	characteristic
66	C₂×C₇₃	#2250	1	2	yes	—
67	D₇₃	—	32	2	no	—
68	#2254	D₇₃	1	1	yes	characteristic, complemented
69	#2246	—	73	1	no	—
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	—	146	2	no	—
75	#500	C₂×D₇₃	1	2	yes	—
76	#493	—	73	2	no	—
77	#492	—	146	2	no	—
78	#486	—	146	2	no	—
79	#367	—	2	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	—	146	2	no	—
86	C₂×D₈	—	146	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#127	—	146	2	no	—
89	D₁₆	—	292	4	no	—
90	C₂×C₁₆	—	73	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	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	—	4	2	no	—
98	C₂×D₄	—	292	2	no	—
99	C₂×D₄	—	292	2	no	—
100	C₂×D₄	—	292	2	no	—
101	SD₁₆	—	584	8	no	—
102	D₈	—	292	4	no	—
103	D₈	—	292	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	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₇₃	1	1	yes	characteristic
110	C₁₆	—	146	2	no	—
111	C₁₆	#11924015	1	4	yes	complemented
112	C₂³	—	584	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	584	2	no	—
115	D₄	—	584	2	no	—
116	D₄	—	584	4	no	—
117	D₄	—	584	4	no	—
118	C₂×C₄	—	8	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×#11924015	1	2	yes	—
123	C₈	#14556614	1	2	yes	—
124	C₂²	—	1168	2	no	—
125	C₂²	—	1168	2	no	—
126	C₂²	—	1168	2	no	—
127	C₂²	#17988962	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#17988961	1	1	yes	characteristic
130	C₄	#17988972	1	1	yes	characteristic
131	C₄	#17989024	1	4	yes	—
132	C₂	—	2336	2	no	—
133	C₂	#22609524	1	1	yes	characteristic
134	C₂	#22612196	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612181

C₂

yes

maximal, characteristic

#22612156

C₂

yes

maximal, complemented

#22609524

C₂

yes

maximal, complemented

#22607847

C₂

yes

maximal, characteristic, complemented

C₇₃×#2254

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₆₈

—

#17988964

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

#500×C₇₃

C₂²

yes

—

#367×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#11924015

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

D₁₁₆₈

—

C₂×#11924003

D₄

yes

—

#14556511

—

C₄×#10596831

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924015

—

D₅₈₄

—

D₅₈₄

—

C₂×#10596831

D₈

yes

—

#11924003

—

C₂×Q₈×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

complemented

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#135

yes

—

#10596831

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2254

D₇₃

yes

characteristic, complemented

—

—

—

—

C₇₃

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

146

—

#500

C₂×D₇₃

yes

—

#493

—

#492

—

146

—

#486

—

146

—

#367

—

#357

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

C₂×SD₁₆

—

292

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

#127

—

146

—

D₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

—

#107

—

#107

—

#102

—

292

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

100

C₂×D₄

—

292

—

101

SD₁₆

—

584

—

102

D₈

—

292

—

103

D₈

—

292

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₇₃

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₇₃

yes

characteristic

110

C₁₆

—

146

—

111

C₁₆

#11924015

yes

complemented

112

C₂³

—

584

—

113

Q₈

—

114

D₄

—

584

—

115

D₄

—

584

—

116

D₄

—

584

—

117

D₄

—

584

—

118

C₂×C₄

—

119

C₂×C₄

#14556587

yes

characteristic

120

C₂×C₄

D₈×D₇₃

yes

—

121

C₈

—

146

—

122

C₈

C₂×#11924015

yes

—

123

C₈

#14556614

yes

—

124

C₂²

—

1168

—

125

C₂²

—

1168

—

126

C₂²

—

1168

—

127

C₂²

#17988962

yes

characteristic

128

C₄

—

129

C₄

#17988961

yes

characteristic

130

C₄

#17988972

yes

characteristic

131

C₄

#17989024

yes

—

132

C₂

—

2336

—

133

C₂

#22609524

yes

characteristic

134

C₂

#22612196

yes

—

	subgroup	G/H
center	C₂²	#17988962
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2254	C₂
socle	C₂²×C₇₃	#493
prime radical	#22612181	C₂
hypercenter	#2254	D₇₃
hypocenter	C₇₃	#12506
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#17988962

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2254

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #22612156	C₂	semidirect
3. #22609524	C₂	semidirect
4. #22607847	C₂	semidirect
5. C₇₃×#2254	C₂	semidirect
43. C₁₆×C₇₃	SD₁₆	semidirect
68. #2254	D₇₃	semidirect
73. C₇₃	#12506	semidirect
111. C₁₆	#11924015	semidirect

H ≅ G/N

type

2. #22612156

C₂

semidirect

3. #22609524

C₂

semidirect

4. #22607847

C₂

semidirect

5. C₇₃×#2254

C₂

semidirect

43. C₁₆×C₇₃

SD₁₆

semidirect

68. #2254

D₇₃

semidirect

73. C₇₃

#12506

semidirect

111. C₁₆

#11924015

semidirect

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

subgroup	index
#22612181	2
#22612156	2
#22609524	2
#22607847	2
C₇₃×#2254	2
#12506	73

subgroup

index

C₇₃×#2254

Subgroups — #33000655

Conjugacy classes of subgroups

Complements

Maximal subgroups