Group #33241486 — subgroups

Subgroups — #33241486

Order 18688.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#22620445	C₂	1	2	yes	maximal
2	#22612217	C₂	1	8	yes	maximal, complemented
3	#22611715	C₂	1	4	yes	maximal
4	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989144	—	2	2	no	—
6	#17988992	C₂²	1	4	yes	—
7	#17988989	—	2	2	no	—
8	#17988989	C₂²	1	4	yes	—
9	#17988985	—	2	2	no	—
10	#17988625	—	2	2	no	—
11	#17988625	C₂²	1	4	yes	—
12	#17988415	C₂²	1	8	yes	—
13	#17988413	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#346×C₇₃	C₂²	1	4	yes	—
17	#14556591	—	2	4	no	—
18	#14556583	—	2	4	no	—
19	#14556570	—	4	4	no	—
20	C₄×Dic₁₄₆	—	4	4	no	—
21	#14556550	—	2	4	no	—
22	#14556525	—	2	4	no	—
23	#14556525	C₂³	1	8	yes	—
24	#14556523	—	2	4	no	—
25	#14556523	—	2	4	no	—
26	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₃	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
31	C₄×C₈×C₇₃	D₄	1	2	yes	—
32	C₄×C₈×C₇₃	C₂³	1	2	yes	—
33	C₂×Dic₁₄₆	—	4	4	no	—
34	C₄⋊Dic₇₃	—	2	8	no	—
35	C₄⋊Dic₇₃	—	4	4	no	—
36	#11924007	—	4	8	no	—
37	C₂²⋊Dic₇₃	—	4	4	no	—
38	C₄×Dic₇₃	—	4	4	no	—
39	C₂²×C₄×C₇₃	D₈	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
42	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
46	C₄²×C₇₃	D₈	1	2	yes	—
47	C₄²×C₇₃	C₂×D₄	1	2	yes	—
48	C₁₆×C₇₃	C₂×Q₈	1	8	yes	—
49	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₇₃	—	4	8	no	—
57	Dic₁₄₆	—	8	8	no	—
58	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
59	C₈×C₇₃	C₂×Q₁₆	1	4	yes	—
60	C₈×C₇₃	C₂²×Q₈	1	2	yes	—
61	C₂²×C₇₃	—	2	2	no	—
62	C₂²×C₇₃	#500	1	2	yes	—
63	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
64	C₄×C₇₃	—	2	4	no	—
65	C₄×C₇₃	#500	1	2	yes	—
66	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
68	Dic₇₃	—	8	8	no	—
69	#33728	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	C₂×#500	1	1	yes	characteristic
72	C₂×C₇₃	#3414	1	2	yes	—
73	#3160	—	73	2	no	—
74	#2255	—	73	8	no	—
75	#2158	—	73	4	no	—
76	C₂×#346	D₇₃	1	1	yes	characteristic
77	C₇₃	#33728	1	1	yes	characteristic, complemented, Sylow
78	#533	—	146	2	no	—
79	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
80	#500	—	73	4	no	—
81	#499	—	73	4	no	—
82	#499	—	146	2	no	—
83	#498	—	146	2	no	—
84	#427	—	73	4	no	—
85	#427	—	146	2	no	—
86	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
87	#367	—	73	8	no	—
88	#366	—	73	8	no	—
89	#346	C₂×D₇₃	1	4	yes	—
90	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
92	C₄⋊Q₈	—	146	4	no	—
93	#125	—	146	4	no	—
94	C₂²⋊Q₈	—	292	4	no	—
95	C₄×Q₈	—	292	4	no	—
96	#117	—	146	4	no	—
97	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₄₆	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
100	#107	—	73	8	no	—
101	#107	—	146	4	no	—
102	#106	—	146	4	no	—
103	#106	—	146	4	no	—
104	C₄×C₈	D₂₉₂	1	2	yes	—
105	C₄×C₈	C₂²×D₇₃	1	2	yes	—
106	C₂×Q₈	—	292	4	no	—
107	C₂²×C₄	D₅₈₄	1	2	yes	—
108	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
109	C₂×C₈	Dic₂₉₂	1	4	yes	—
110	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
111	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
113	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
114	C₄⋊C₄	—	292	8	no	—
115	C₄⋊C₄	—	292	4	no	—
116	C₄⋊C₄	—	146	8	no	—
117	C₂²⋊C₄	—	292	4	no	—
118	C₄²	—	292	4	no	—
119	C₄²	D₅₈₄	1	2	yes	—
120	C₄²	C₂×D₂₉₂	1	2	yes	—
121	C₁₆	C₂×Dic₁₄₆	1	8	yes	—
122	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
123	Q₈	—	584	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	292	8	no	—
126	C₂×C₄	#14556591	1	2	yes	—
127	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
128	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
130	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
131	C₈	#14556591	1	2	yes	—
132	C₈	C₂×Dic₂₉₂	1	4	yes	—
133	C₈	C₂²×Dic₁₄₆	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#17988992	1	2	yes	—
136	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	584	8	no	—
139	C₄	#17988992	1	2	yes	—
140	C₄	C₂×#14556591	1	1	yes	characteristic
141	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#17988992	1	1	yes	characteristic
144	C₂	#22623067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22620445

C₂

yes

maximal

#22612217

C₂

yes

maximal, complemented

#22611715

C₂

yes

maximal

C₂×#346×C₇₃

C₂

yes

maximal, characteristic

#17989144

—

#17988992

C₂²

yes

—

#17988989

—

#17988989

C₂²

yes

—

#17988985

—

#17988625

—

#17988625

C₂²

yes

—

#17988415

C₂²

yes

—

#17988413

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

—

#14556591

—

#14556583

—

#14556570

—

C₄×Dic₁₄₆

—

#14556550

—

#14556525

—

#14556525

C₂³

yes

—

#14556523

—

#14556523

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

Q₈

yes

complemented

C₂×C₁₆×C₇₃

C₂³

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

—

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

Q₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₁₆×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₄⋊Q₈

yes

—

C₈×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

C₂²×Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#500

yes

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#500

yes

—

C₄×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²×Q₁₆

yes

characteristic

Dic₇₃

—

#33728

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×#500

yes

characteristic

C₂×C₇₃

#3414

yes

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#33728

yes

characteristic, complemented, Sylow

#533

—

146

—

C₂²×C₁₆

C₂×D₇₃

yes

—

#500

—

#499

—

#499

—

146

—

#498

—

146

—

#427

—

#427

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

—

—

C₂×D₇₃

yes

—

C₂²×C₈

D₂₉₂

yes

characteristic

C₂²×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

#125

—

146

—

C₂²⋊Q₈

—

292

—

C₄×Q₈

—

292

—

#117

—

146

—

C₂×C₄²

D₂₉₂

yes

characteristic

C₂×C₁₆

Dic₁₄₆

yes

complemented

C₂×C₁₆

C₂²×D₇₃

yes

—

100

#107

—

101

#107

—

146

—

102

#106

—

146

—

103

#106

—

146

—

104

C₄×C₈

D₂₉₂

yes

—

105

C₄×C₈

C₂²×D₇₃

yes

—

106

C₂×Q₈

—

292

—

107

C₂²×C₄

D₅₈₄

yes

—

108

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

109

C₂×C₈

Dic₂₉₂

yes

—

110

C₂×C₈

C₂×D₂₉₂

yes

characteristic

111

C₂×C₈

C₂×D₂₉₂

yes

—

112

C₂×C₈

C₂×Dic₁₄₆

yes

—

113

C₂×C₈

C₂³×D₇₃

yes

characteristic

114

C₄⋊C₄

—

292

—

115

C₄⋊C₄

—

292

—

116

C₄⋊C₄

—

146

—

117

C₂²⋊C₄

—

292

—

118

C₄²

—

292

—

119

C₄²

D₅₈₄

yes

—

120

C₄²

C₂×D₂₉₂

yes

—

121

C₁₆

C₂×Dic₁₄₆

yes

—

122

C₂³

C₂×D₅₈₄

yes

characteristic

123

Q₈

—

584

—

124

C₂×C₄

—

125

C₂×C₄

—

292

—

126

C₂×C₄

#14556591

yes

—

127

C₂×C₄

C₂×D₅₈₄

yes

—

128

C₂×C₄

C₂×D₅₈₄

yes

characteristic

129

C₂×C₄

C₂×Dic₂₉₂

yes

—

130

C₂×C₄

C₂²×D₂₉₂

yes

characteristic

131

C₈

#14556591

yes

—

132

C₈

C₂×Dic₂₉₂

yes

—

133

C₈

C₂²×Dic₁₄₆

yes

—

134

C₂²

—

135

C₂²

#17988992

yes

—

136

C₂²

C₂²×D₅₈₄

yes

characteristic

137

C₄

—

138

C₄

—

584

—

139

C₄

#17988992

yes

—

140

C₄

C₂×#14556591

yes

characteristic

141

C₄

C₂²×Dic₂₉₂

yes

characteristic

142

C₂

—

143

C₂

C₂×#17988992

yes

characteristic

144

C₂

#22623067

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#33728
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, cosocle

C₂×C₈×C₇₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₃

Fitting

C₂×#346×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₈

prime radical

C₂³×C₇₃

C₂×D₈

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
2. #22612217	C₂	semidirect
29. C₂×C₁₆×C₇₃	Q₈	semidirect
77. C₇₃	#33728	semidirect
98. C₂×C₁₆	Dic₁₄₆	semidirect

H ≅ G/N

type

2. #22612217

C₂

semidirect

29. C₂×C₁₆×C₇₃

Q₈

semidirect

77. C₇₃

#33728

semidirect

98. C₂×C₁₆

Dic₁₄₆

semidirect

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

subgroup	index
#22620445	2
#22612217	2
#22611715	2
C₂×#346×C₇₃	2
#33728	73

subgroup

index

C₂×#346×C₇₃

Subgroups — #33241486

Conjugacy classes of subgroups

Complements

Maximal subgroups