Group #33002436 — subgroups

Subgroups — #33002436

Order 18688.

306 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#17988417	C₂	1	4	yes	maximal
2	#22610103	C₂	1	2	yes	maximal
3	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556527	—	2	2	no	—
5	C₂×#14556527	C₂²	1	4	yes	—
6	#17988611	—	2	2	no	—
7	#17988417	—	2	8	no	—
8	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
10	#346×C₇₃	C₄	1	4	yes	—
11	C₂×#11924012	—	2	4	no	—
12	#14556527	—	2	8	no	—
13	#14556527	—	4	4	no	—
14	C₄⋊#10596831	—	4	4	no	—
15	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
20	C₂×C₁₆×C₇₃	Q₈	1	4	yes	—
21	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₃	Q₈	1	2	yes	—
23	#11924012	—	4	4	no	—
24	#11924012	—	4	4	no	—
25	C₂×#10596831	—	4	4	no	—
26	C₂²×C₄×C₇₃	Q₁₆	1	2	yes	—
27	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₇₃	D₈	1	2	yes	—
30	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
31	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
33	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
34	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₄²×C₇₃	D₈	1	2	yes	—
37	C₁₆×C₇₃	C₄⋊C₄	1	8	yes	—
38	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	4	2	no	—
40	C₂×C₄×C₇₃	#107	1	4	yes	—
41	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
47	C₈×C₇₃	#107	1	4	yes	—
48	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
50	#10596831	—	8	4	no	—
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
53	C₂²×C₇₃	#494	1	1	yes	characteristic
54	C₂²×C₇₃	#500	1	1	yes	characteristic
55	C₄×C₇₃	—	2	4	no	—
56	C₄×C₇₃	#384	1	1	yes	characteristic
57	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
58	C₄×C₇₃	#493	1	1	yes	characteristic
59	C₄×C₇₃	#500	1	1	yes	characteristic
60	#12910	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	—	2	1	no	—
62	C₂×C₇₃	—	2	1	no	—
63	C₂×C₇₃	#1845	1	1	yes	characteristic
64	C₂×C₇₃	#2161	1	2	yes	—
65	C₂×#368	—	73	4	no	—
66	C₂×#346	D₇₃	1	1	yes	characteristic
67	#1848	—	73	2	no	—
68	C₇₃	#12910	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
70	C₂×#108	—	73	4	no	—
71	C₂×#108	—	146	2	no	—
72	#423	—	146	2	no	—
73	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
74	#368	—	146	8	no	—
75	#346	Dic₇₃	1	4	yes	—
76	C₂×M₁₆	—	146	4	no	—
77	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
79	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
80	C₂×C₁₆	Dic₁₄₆	1	4	yes	—
81	C₂×C₁₆	D₂₉₂	1	4	yes	—
82	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
83	#108	—	292	4	no	—
84	#108	—	146	8	no	—
85	C₄⋊C₈	—	292	4	no	—
86	C₄×C₈	Dic₁₄₆	1	2	yes	—
87	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
88	C₂²×C₄	Dic₂₉₂	1	2	yes	—
89	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
90	M₁₆	—	292	4	no	—
91	M₁₆	—	292	4	no	—
92	C₂×C₈	—	292	4	no	—
93	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
94	C₂×C₈	D₅₈₄	1	2	yes	—
95	C₂×C₈	Dic₂₉₂	1	2	yes	—
96	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
97	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
98	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
100	C₄²	C₄⋊Dic₇₃	1	2	yes	—
101	C₄²	D₅₈₄	1	2	yes	—
102	C₁₆	C₄⋊Dic₇₃	1	8	yes	—
103	C₂³	C₂×Dic₂₉₂	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#14556525	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
107	C₂×C₄	#14556588	1	1	yes	characteristic
108	C₂×C₄	#14556591	1	1	yes	characteristic
109	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
110	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
112	C₈	—	584	4	no	—
113	C₈	#14556525	1	4	yes	—
114	C₈	C₂×C₄⋊Dic₇₃	1	2	yes	—
115	C₈	#14556591	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#14556525	1	1	yes	characteristic
118	C₂²	#17988967	1	1	yes	characteristic
119	C₂²	#17988992	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#17988457	1	1	yes	characteristic
122	C₄	C₂×#14556525	1	1	yes	characteristic
123	C₄	#17988964	1	1	yes	characteristic
124	C₄	#17988992	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	#22610091	1	1	yes	characteristic
128	C₂	#22611726	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988417

C₂

yes

maximal

#22610103

C₂

yes

maximal

C₂×#346×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556527

—

C₂×#14556527

C₂²

yes

—

#17988611

—

#17988417

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₄

yes

—

—

—

—

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₂²×C₄×C₇₃

Q₁₆

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

D₈

yes

—

C₁₆×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₈×C₇₃

#107

yes

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1845

yes

characteristic

C₂×C₇₃

#2161

yes

—

C₂×#368

—

C₂×#346

D₇₃

yes

characteristic

#1848

—

C₇₃

#12910

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₃

yes

—

C₂×#108

—

C₂×#108

—

146

—

#423

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#368

—

146

—

#346

Dic₇₃

yes

—

C₂×M₁₆

—

146

—

C₂²×C₈

D₂₉₂

yes

characteristic

C₂²×C₈

C₂²×D₇₃

yes

characteristic

C₂×C₄²

D₂₉₂

yes

characteristic

C₂×C₁₆

Dic₁₄₆

yes

—

C₂×C₁₆

D₂₉₂

yes

—

C₂×C₁₆

C₂×Dic₇₃

yes

—

#108

—

292

—

#108

—

146

—

C₄⋊C₈

—

292

—

C₄×C₈

Dic₁₄₆

yes

—

C₄×C₈

C₂×Dic₇₃

yes

—

C₂²×C₄

Dic₂₉₂

yes

—

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

M₁₆

—

292

—

M₁₆

—

292

—

C₂×C₈

—

292

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

D₅₈₄

yes

—

C₂×C₈

Dic₂₉₂

yes

—

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

C₂×C₈

C₂×D₂₉₂

yes

—

C₂×C₈

C₂×Dic₁₄₆

yes

—

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

100

C₄²

C₄⋊Dic₇₃

yes

—

101

C₄²

D₅₈₄

yes

—

102

C₁₆

C₄⋊Dic₇₃

yes

—

103

C₂³

C₂×Dic₂₉₂

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14556525

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

107

C₂×C₄

#14556588

yes

characteristic

108

C₂×C₄

#14556591

yes

characteristic

109

C₂×C₄

C₂×D₅₈₄

yes

characteristic

110

C₂×C₄

C₂×D₅₈₄

yes

characteristic

111

C₂×C₄

C₂×Dic₂₉₂

yes

characteristic

112

C₈

—

584

—

113

C₈

#14556525

yes

—

114

C₈

C₂×C₄⋊Dic₇₃

yes

—

115

C₈

#14556591

yes

—

116

C₂²

—

117

C₂²

C₂×#14556525

yes

characteristic

118

C₂²

#17988967

yes

characteristic

119

C₂²

#17988992

yes

characteristic

120

C₄

—

121

C₄

#17988457

yes

characteristic

122

C₄

C₂×#14556525

yes

characteristic

123

C₄

#17988964

yes

characteristic

124

C₄

#17988992

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#22610091

yes

characteristic

128

C₂

#22611726

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×C₇₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#12910
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#346×C₇₃

C₂

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂³×C₇₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
68. C₇₃	#12910	semidirect

H ≅ G/N

type

68. C₇₃

#12910

semidirect

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

subgroup	index
C₂×#17988417	2
#22610103	2
C₂×#346×C₇₃	2
#12910	73

subgroup

index

C₂×#17988417

#22610103

C₂×#346×C₇₃

#12910

Subgroups — #33002436

Conjugacy classes of subgroups

Complements

Maximal subgroups