Group #33000733 — subgroups

Subgroups — #33000733

Order 18688.

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

normal subgroups only

This group has 246 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	#22612211	C₂	1	2	yes	maximal
2	#22612204	C₂	1	2	yes	maximal
3	#22609549	C₂	1	1	yes	maximal, characteristic
4	#22607848	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1710	C₂	1	1	yes	maximal, characteristic
6	#17988992	—	2	1	no	—
7	#17988992	C₂²	1	2	yes	—
8	#17988967	—	2	1	no	—
9	#17988414	—	2	4	no	—
10	#17988412	—	2	4	no	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	#500×C₇₃	C₂²	1	1	yes	characteristic
13	#500×C₇₃	C₂²	1	1	yes	characteristic
14	#487×C₇₃	—	2	1	no	—
15	#487×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	4	2	no	—
18	#14556591	—	2	2	no	—
19	C₂×#11924003	D₄	1	4	yes	—
20	#14556525	—	2	2	no	—
21	#14556525	—	2	2	no	—
22	#14556525	—	4	2	no	—
23	C₄⋊Q₈×C₇₃	—	2	1	no	—
24	C₄⋊Q₈×C₇₃	—	2	1	no	—
25	#107×C₇₃	—	2	2	no	—
26	#107×C₇₃	—	2	2	no	—
27	#103×C₇₃	—	4	2	no	—
28	#103×C₇₃	—	4	2	no	—
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₄₆	—	4	2	no	—
33	Dic₂₉₂	—	8	4	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	#11924003	Q₁₆	1	8	yes	complemented
37	C₂×Q₈×C₇₃	—	4	1	no	—
38	C₂×Q₈×C₇₃	—	4	1	no	—
39	C₂×C₈×C₇₃	D₈	1	2	yes	—
40	C₂×C₈×C₇₃	D₈	1	2	yes	—
41	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₇₃	—	4	1	no	—
43	C₄⋊C₄×C₇₃	—	4	1	no	—
44	C₄⋊C₄×C₇₃	—	4	1	no	—
45	C₄⋊C₄×C₇₃	—	4	1	no	—
46	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₇₃	—	8	2	no	—
48	Q₈×C₇₃	—	8	2	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	—	8	1	no	—
51	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₇₃	—	8	2	no	—
55	Dic₁₄₆	—	8	4	no	—
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₇₃	#493	1	1	yes	characteristic
60	C₄×C₇₃	—	16	1	no	—
61	C₄×C₇₃	—	16	1	no	—
62	C₄×C₇₃	#494	1	2	yes	—
63	C₄×C₇₃	#509	1	2	yes	—
64	C₄×C₇₃	#510	1	2	yes	—
65	Dic₇₃	—	16	2	no	—
66	#12524	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#1713	1	1	yes	characteristic
68	C₂×C₇₃	#2250	1	1	yes	characteristic
69	C₂×C₇₃	#2251	1	1	yes	characteristic
70	#2254	—	73	2	no	—
71	#2252	—	73	2	no	—
72	#1713	—	73	1	no	—
73	#1710	D₇₃	1	1	yes	characteristic
74	#1311	—	73	1	no	—
75	C₇₃	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	73	2	no	—
77	#500	—	146	1	no	—
78	#500	C₂×D₇₃	1	1	yes	characteristic
79	#500	C₂×D₇₃	1	1	yes	characteristic
80	#494	—	146	1	no	—
81	#487	—	2	1	no	—
82	#487	—	2	1	no	—
83	#367	—	146	4	no	—
84	#366	—	146	4	no	—
85	C₄×C₁₆	—	73	2	no	—
86	C₈²	C₂×D₇₃	1	1	yes	characteristic
87	C₂×Q₁₆	—	292	2	no	—
88	C₄⋊Q₈	—	2	1	no	—
89	C₄⋊Q₈	—	2	1	no	—
90	C₄⋊Q₈	—	146	2	no	—
91	C₂×C₁₆	—	73	4	no	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#107	—	292	2	no	—
95	#107	—	146	2	no	—
96	#107	—	146	2	no	—
97	#103	—	4	2	no	—
98	#103	—	4	2	no	—
99	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	292	2	no	—
105	Q₁₆	—	584	4	no	—
106	C₂×C₈	#11924013	1	2	yes	—
107	C₂×C₈	#11924013	1	2	yes	—
108	C₂×C₈	D₄×D₇₃	1	2	yes	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	292	2	no	—
114	C₄⋊C₄	—	292	2	no	—
115	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
116	C₁₆	—	73	8	no	—
117	Q₈	—	8	2	no	—
118	Q₈	—	8	2	no	—
119	Q₈	—	584	4	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	584	2	no	—
123	C₂×C₄	#14556587	1	1	yes	characteristic
124	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
125	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	Q₁₆×D₇₃	1	4	yes	—
129	C₂²	#17988963	1	1	yes	characteristic
130	C₄	—	16	1	no	—
131	C₄	—	16	1	no	—
132	C₄	—	1168	2	no	—
133	C₄	#17988965	1	2	yes	—
134	C₄	#17989022	1	2	yes	—
135	C₄	#17989027	1	2	yes	—
136	C₂	#22609547	1	1	yes	characteristic
137	C₂	#22612198	1	1	yes	characteristic
138	C₂	#22612203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612211

C₂

yes

maximal

#22612204

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic

#22607848

C₂

yes

maximal, characteristic

C₇₃×#1710

C₂

yes

maximal, characteristic

#17988992

—

#17988992

C₂²

yes

—

—

—

—

C₂²

yes

—

#500×C₇₃

C₂²

yes

characteristic

#500×C₇₃

C₂²

yes

characteristic

#487×C₇₃

—

#487×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

#14556591

—

C₂×#11924003

D₄

yes

—

#14556525

—

#14556525

—

#14556525

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#103×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

Q₁₆

yes

complemented

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×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

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#494

yes

—

C₄×C₇₃

#509

yes

—

C₄×C₇₃

#510

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

D₇₃

yes

characteristic

#1311

—

C₇₃

#12524

yes

characteristic, complemented, Sylow

#500

—

#500

—

146

—

#500

C₂×D₇₃

yes

characteristic

#500

C₂×D₇₃

yes

characteristic

#494

—

146

—

#487

—

#487

—

#367

—

146

—

#366

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

292

—

#107

—

146

—

#107

—

146

—

#103

—

#103

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

100

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

101

C₄×C₈

C₂²×D₇₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

292

—

105

Q₁₆

—

584

—

106

C₂×C₈

#11924013

yes

—

107

C₂×C₈

#11924013

yes

—

108

C₂×C₈

D₄×D₇₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

292

—

114

C₄⋊C₄

—

292

—

115

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

116

C₁₆

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

584

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

584

—

123

C₂×C₄

#14556587

yes

characteristic

124

C₂×C₄

C₂×#11924013

yes

characteristic

125

C₂×C₄

C₂×#11924013

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

Q₁₆×D₇₃

yes

—

129

C₂²

#17988963

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

1168

—

133

C₄

#17988965

yes

—

134

C₄

#17989022

yes

—

135

C₄

#17989027

yes

—

136

C₂

#22609547

yes

characteristic

137

C₂

#22612198

yes

characteristic

138

C₂

#22612203

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1710	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1710	D₇₃
hypocenter	C₇₃	#12524
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1710

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
36. #11924003	Q₁₆	semidirect
75. C₇₃	#12524	semidirect

H ≅ G/N

type

36. #11924003

Q₁₆

semidirect

75. C₇₃

#12524

semidirect

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

subgroup	index
#22612211	2
#22612204	2
#22609549	2
#22607848	2
C₇₃×#1710	2
#12524	73

subgroup

index

C₇₃×#1710

Subgroups — #33000733

Conjugacy classes of subgroups

Complements

Maximal subgroups