Group #33006516 — subgroups

Subgroups — #33006516

Order 18688.

180 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22611526	C₂	1	1	yes	maximal, characteristic
2	#22608126	C₂	1	2	yes	maximal, complemented
3	#22608068	C₂	1	2	yes	maximal
4	#22608065	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556530	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
8	#17988541	C₄	1	1	yes	characteristic
9	C₂×#14556497	C₂²	1	2	yes	—
10	#17988363	C₄	1	1	yes	characteristic
11	#17988362	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924011	C₈	1	1	yes	characteristic
16	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
17	#14556548	C₈	1	1	yes	characteristic
18	#14556530	—	2	2	no	—
19	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
21	C₁₆×D₇₃	—	2	2	no	—
22	#14556502	C₈	1	1	yes	characteristic
23	#14556501	C₈	1	1	yes	characteristic
24	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	#14556497	—	2	2	no	—
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₃₂×C₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
31	#11924011	—	2	2	no	—
32	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
33	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
34	C₈×D₇₃	—	2	1	no	—
35	C₈×D₇₃	—	2	1	no	—
36	#11924007	C₁₆	1	2	yes	—
37	#11924005	C₁₆	1	2	yes	—
38	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
39	#11924003	—	2	1	no	—
40	#11924003	—	2	1	no	—
41	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₁₆×C₇₃	M₁₆	1	2	yes	—
45	C₁₆×C₇₃	#41	1	2	yes	—
46	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
51	C₄×D₇₃	—	2	1	no	—
52	C₄×D₇₃	—	2	1	no	—
53	C₈×C₇₃	M₃₂	1	2	yes	—
54	C₈×C₇₃	#117	1	1	yes	characteristic
55	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
56	#10596831	—	2	1	no	—
57	#10596831	—	2	1	no	—
58	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	2	2	no	—
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	#431	1	1	yes	characteristic
63	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	2	2	no	—
66	#13641	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#2162	1	1	yes	characteristic
68	C₂×C₇₃	#2258	1	2	yes	—
69	D₇₃	—	4	1	no	—
70	#2108	—	73	1	no	—
71	C₄⋊C₃₂	—	73	2	no	—
72	C₂²⋊C₃₂	—	73	2	no	—
73	#1397	—	73	1	no	—
74	C₄×C₃₂	D₇₃	1	1	yes	characteristic, complemented
75	C₇₃	#13641	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₃₂	—	73	1	no	—
77	C₂²×C₁₆	—	73	1	no	—
78	#405	—	73	1	no	—
79	C₂×C₃₂	—	73	2	no	—
80	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
81	#346	—	73	1	no	—
82	#346	—	73	1	no	—
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
85	C₂×M₁₆	—	73	1	no	—
86	C₂²×C₈	—	73	1	no	—
87	#117	—	73	1	no	—
88	M₃₂	—	146	2	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	146	2	no	—
92	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
93	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
94	#97	—	73	1	no	—
95	#97	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
98	C₃₂	—	2	2	no	—
99	C₃₂	—	146	2	no	—
100	C₂²×C₄	—	73	1	no	—
101	M₁₆	—	146	2	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
108	C₄⋊C₄	—	73	2	no	—
109	C₂²⋊C₄	—	73	2	no	—
110	C₄²	—	73	1	no	—
111	C₄²	C₈×D₇₃	1	1	yes	characteristic
112	C₁₆	—	146	1	no	—
113	C₁₆	—	146	1	no	—
114	C₁₆	#11924011	1	2	yes	—
115	C₁₆	#11924028	1	2	yes	—
116	C₂³	—	73	1	no	—
117	C₂×C₄	—	73	2	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	146	1	no	—
120	C₂×C₄	—	146	1	no	—
121	C₂×C₄	C₁₆×D₇₃	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
123	C₈	—	146	1	no	—
124	C₈	—	146	1	no	—
125	C₈	#14556530	1	2	yes	—
126	C₈	#14556548	1	1	yes	characteristic
127	C₈	C₂×#11924011	1	1	yes	characteristic
128	C₂²	—	146	2	no	—
129	C₂²	—	146	1	no	—
130	C₂²	C₂×C₁₆×D₇₃	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	146	2	no	—
133	C₄	—	146	1	no	—
134	C₄	#17988638	1	1	yes	characteristic
135	C₄	C₂×#14556530	1	1	yes	characteristic
136	C₂	—	292	1	no	—
137	C₂	#22611728	1	1	yes	characteristic
138	C₂	#22612224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22611526

C₂

yes

maximal, characteristic

#22608126

C₂

yes

maximal, complemented

#22608068

C₂

yes

maximal

#22608065

C₂

yes

maximal, characteristic, complemented

C₄×C₃₂×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14556530

C₄

yes

characteristic

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

#17988541

C₄

yes

characteristic

C₂×#14556497

C₂²

yes

—

#17988363

C₄

yes

characteristic

#17988362

C₄

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#11924011

C₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

C₈

yes

characteristic

#14556530

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₁₆×D₇₃

—

#14556502

C₈

yes

characteristic

#14556501

C₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

#14556497

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

—

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

#11924011

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

#11924007

C₁₆

yes

—

#11924005

C₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

#11924003

—

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

M₁₆

yes

—

C₁₆×C₇₃

#41

yes

—

C₂²×D₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

M₃₂

yes

—

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

—

#10596831

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2108

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

—

C₄×C₃₂

D₇₃

yes

characteristic, complemented

C₇₃

#13641

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₇₃

yes

characteristic

C₃₂

—

C₃₂

—

146

—

100

C₂²×C₄

—

101

M₁₆

—

146

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₈×D₇₃

yes

characteristic

107

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₈×D₇₃

yes

characteristic

112

C₁₆

—

146

—

113

C₁₆

—

146

—

114

C₁₆

#11924011

yes

—

115

C₁₆

#11924028

yes

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

C₁₆×D₇₃

yes

—

122

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

123

C₈

—

146

—

124

C₈

—

146

—

125

C₈

#14556530

yes

—

126

C₈

#14556548

yes

characteristic

127

C₈

C₂×#11924011

yes

characteristic

128

C₂²

—

146

—

129

C₂²

—

146

—

130

C₂²

C₂×C₁₆×D₇₃

yes

characteristic

131

C₄

—

132

C₄

—

146

—

133

C₄

—

146

—

134

C₄

#17988638

yes

characteristic

135

C₄

C₂×#14556530

yes

characteristic

136

C₂

—

292

—

137

C₂

#22611728

yes

characteristic

138

C₂

#22612224

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₃
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄×C₃₂×C₇₃	C₂
prime radical	C₂²×D₇₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	C₄×C₃₂	D₇₃
hypocenter	C₇₃	#13641
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₃

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₄×C₃₂×C₇₃

C₂

prime radical

C₂²×D₇₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

hypercenter

C₄×C₃₂

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
2. #22608126	C₂	semidirect
4. #22608065	C₂	semidirect
5. C₄×C₃₂×C₇₃	C₂	semidirect
74. C₄×C₃₂	D₇₃	semidirect
75. C₇₃	#13641	semidirect

H ≅ G/N

type

2. #22608126

C₂

semidirect

4. #22608065

C₂

semidirect

5. C₄×C₃₂×C₇₃

C₂

semidirect

74. C₄×C₃₂

D₇₃

semidirect

75. C₇₃

#13641

semidirect

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

subgroup	index
#22611526	2
#22608126	2
#22608068	2
#22608065	2
C₄×C₃₂×C₇₃	2
#13641	73

subgroup

index

C₄×C₃₂×C₇₃

Subgroups — #33006516

Conjugacy classes of subgroups

Complements

Maximal subgroups