Group #32977154 — subgroups

Subgroups — #32977154

Order 18688.

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

normal subgroups only

This group has 366 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	#22608679	C₂	1	2	yes	maximal
2	#22608223	C₂	1	2	yes	maximal
3	#22607764	C₂	1	2	yes	maximal
4	C₄×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17988640	C₂²	1	4	yes	—
6	#17988340	C₄	1	4	yes	complemented
7	#17988338	C₄	1	4	yes	complemented
8	#17988309	—	2	4	no	—
9	C₂×#97×C₇₃	C₄	1	2	yes	—
10	C₂×#97×C₇₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₄³×C₇₃	C₂²	1	1	yes	characteristic
14	C₄⋊#10596831	C₂×C₄	1	8	yes	—
15	C₂²⋊#10596831	—	2	4	no	—
16	C₄×#10596831	—	2	4	no	—
17	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₃	—	2	2	no	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
23	#97×C₇₃	—	2	4	no	—
24	#97×C₇₃	—	2	4	no	—
25	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	Q₈	1	2	yes	—
29	C₂×#10596831	—	2	8	no	—
30	C₂²×C₄×C₇₃	—	2	2	no	—
31	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	#41	1	2	yes	—
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₄²	1	2	yes	—
39	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
41	C₄²×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	—	4	4	no	—
43	C₄²×C₇₃	M₁₆	1	4	yes	—
44	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
45	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂³×C₇₃	#117	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	4	no	—
49	C₂×C₄×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	—	4	4	no	—
51	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	—	2	1	no	—
59	C₈×C₇₃	—	2	4	no	—
60	C₈×C₇₃	#97	1	4	yes	—
61	#10596831	—	4	8	no	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₇₃	#432	1	2	yes	—
65	C₄×C₇₃	—	2	2	no	—
66	C₄×C₇₃	—	4	4	no	—
67	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
68	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
69	C₄×C₇₃	#423	1	2	yes	—
70	C₄×C₇₃	#444	1	4	yes	—
71	#8269	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	#1767	1	1	yes	characteristic
74	C₂×C₇₃	#1773	1	2	yes	—
75	C₄×#97	D₇₃	1	1	yes	characteristic
76	#1564	—	73	2	no	—
77	#1452	—	73	2	no	—
78	#1276	—	73	2	no	—
79	C₇₃	#8269	1	1	yes	characteristic, complemented, Sylow
80	#431	—	73	4	no	—
81	C₂×#97	Dic₇₃	1	2	yes	—
82	C₂×#97	Dic₇₃	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
85	C₄³	C₂×D₇₃	1	1	yes	characteristic
86	#335	—	73	4	no	—
87	#334	—	73	4	no	—
88	#322	—	146	4	no	—
89	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
91	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
94	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
95	C₄⋊C₈	—	73	8	no	—
96	C₂²⋊C₈	—	146	4	no	—
97	#97	—	2	4	no	—
98	#97	—	2	4	no	—
99	C₄×C₈	—	146	4	no	—
100	C₄×C₈	Dic₁₄₆	1	2	yes	—
101	C₄×C₈	C₄×D₇₃	1	2	yes	—
102	C₄×C₈	C₄×D₇₃	1	2	yes	—
103	C₄×C₈	D₂₉₂	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
106	C₂²×C₄	#11924028	1	2	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	146	8	no	—
111	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
112	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
114	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
115	C₂×C₈	#11924012	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#11924012	1	4	yes	—
119	C₄²	C₂×C₄×D₇₃	1	2	yes	—
120	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
121	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
122	C₂³	#14556550	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
128	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
129	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
130	C₂×C₄	C₂×#11924012	1	2	yes	—
131	C₂×C₄	C₂×#11924012	1	2	yes	—
132	C₈	—	2	4	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	292	8	no	—
136	C₈	#14556502	1	4	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
139	C₂²	#17988644	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂×#14556502	1	1	yes	characteristic
143	C₄	C₄×#11924012	1	1	yes	characteristic
144	C₄	#17988611	1	2	yes	—
145	C₄	#17988710	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#22609784	1	1	yes	characteristic
148	C₂	#22609815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22608679

C₂

yes

maximal

#22608223

C₂

yes

maximal

#22607764

C₂

yes

maximal

C₄×#97×C₇₃

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

#17988340

C₄

yes

complemented

#17988338

C₄

yes

complemented

#17988309

—

C₂×#97×C₇₃

C₄

yes

—

C₂×#97×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄³×C₇₃

C₂²

yes

characteristic

C₄⋊#10596831

C₂×C₄

yes

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#97×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂×#10596831

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

M₁₆

yes

—

C₄²×C₇₃

C₂²×C₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

#117

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#97

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#432

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

C₄×M₁₆

yes

characteristic

C₄×C₇₃

#423

yes

—

C₄×C₇₃

#444

yes

—

#8269

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1767

yes

characteristic

C₂×C₇₃

#1773

yes

—

C₄×#97

D₇₃

yes

—

—

—

C₇₃

#8269

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

Dic₇₃

yes

—

C₂×#97

Dic₇₃

yes

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄³

C₂×D₇₃

yes

—

—

—

146

—

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂²×C₈

C₂×Dic₇₃

yes

—

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₃

yes

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

146

—

#97

—

#97

—

C₄×C₈

—

146

—

100

C₄×C₈

Dic₁₄₆

yes

—

101

C₄×C₈

C₄×D₇₃

yes

—

102

C₄×C₈

C₄×D₇₃

yes

—

103

C₄×C₈

D₂₉₂

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

106

C₂²×C₄

#11924028

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

146

—

111

C₂×C₈

C₄×Dic₇₃

yes

characteristic

112

C₂×C₈

C₄×Dic₇₃

yes

characteristic

113

C₂×C₈

C₄×Dic₇₃

yes

—

114

C₂×C₈

C₄⋊Dic₇₃

yes

—

115

C₂×C₈

#11924012

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#11924012

yes

—

119

C₄²

C₂×C₄×D₇₃

yes

—

120

C₄²

C₂×D₂₉₂

yes

characteristic

121

C₄²

C₂×Dic₁₄₆

yes

characteristic

122

C₂³

#14556550

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×Dic₇₃

yes

characteristic

127

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

128

C₂×C₄

C₄×D₂₉₂

yes

—

129

C₂×C₄

C₄×Dic₁₄₆

yes

—

130

C₂×C₄

C₂×#11924012

yes

—

131

C₂×C₄

C₂×#11924012

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

292

—

136

C₈

#14556502

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₇₃

yes

characteristic

139

C₂²

#17988644

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#14556502

yes

characteristic

143

C₄

C₄×#11924012

yes

characteristic

144

C₄

#17988611

yes

—

145

C₄

#17988710

yes

—

146

C₂

—

147

C₂

#22609784

yes

characteristic

148

C₂

#22609815

yes

—

	subgroup	G/H
center	C₄²	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#97×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#97	D₇₃
hypocenter	C₇₃	#8269
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

subgroup

G/H

center

C₄²

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₄×#97×C₇₃

C₂

socle

C₂²×C₇₃

C₄×C₄⋊C₄

prime radical

C₂³×C₇₃

#117

cosocle

C₂×C₄²×C₇₃

C₂³

hypercenter

C₄×#97

D₇₃

hypocenter

C₇₃

#8269

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

N	H ≅ G/N	type
6. #17988340	C₄	semidirect
7. #17988338	C₄	semidirect
79. C₇₃	#8269	semidirect

H ≅ G/N

type

6. #17988340

C₄

semidirect

7. #17988338

C₄

semidirect

79. C₇₃

#8269

semidirect

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

subgroup	index
#22608679	2
#22608223	2
#22607764	2
C₄×#97×C₇₃	2
#8269	73

subgroup

index

C₄×#97×C₇₃

Subgroups — #32977154

Conjugacy classes of subgroups

Complements

Maximal subgroups