Group #32989053 — subgroups

Subgroups — #32989053

Order 18688.

484 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22610754	C₂	1	3	yes	maximal
2	#22607812	C₂	1	3	yes	maximal, complemented
3	C₇₃×#2844	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556588	C₂²	1	1	yes	characteristic
5	C₂×#14556511	—	2	3	no	—
6	C₂×#14556511	—	2	3	no	—
7	C₂×#14556502	C₂²	1	3	yes	—
8	#400×C₇₃	—	2	3	no	—
9	#395×C₇₃	—	2	3	no	—
10	#382×C₇₃	C₂²	1	3	yes	—
11	C₂²×D₂₉₂	—	2	3	no	—
12	C₂²×#10596831	—	2	3	no	—
13	#14556588	—	2	3	no	—
14	#14556588	Q₈	1	2	yes	complemented
15	#14556511	—	4	6	no	—
16	#14556511	—	4	6	no	—
17	#14556502	—	2	6	no	—
18	C₂²×Q₈×C₇₃	—	2	1	no	—
19	C₂×C₄⋊C₄×C₇₃	—	2	3	no	—
20	C₂×C₄⋊C₄×C₇₃	D₄	1	3	yes	—
21	C₂×C₄⋊C₄×C₇₃	D₄	1	3	yes	—
22	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
23	#95×C₇₃	—	2	3	no	—
24	#95×C₇₃	—	2	1	no	—
25	#95×C₇₃	—	2	6	no	—
26	C₂³×D₇₃	—	4	1	no	—
27	C₂×D₂₉₂	—	2	6	no	—
28	C₂×D₂₉₂	—	4	3	no	—
29	C₂×D₂₉₂	—	4	3	no	—
30	C₂×D₂₉₂	—	4	3	no	—
31	C₂×D₂₉₂	—	4	3	no	—
32	C₂×D₂₉₂	—	4	3	no	—
33	C₂×#10596831	—	2	6	no	—
34	C₂×#10596831	—	4	3	no	—
35	C₂×#10596831	—	4	3	no	—
36	C₂×Q₈×C₇₃	—	8	3	no	—
37	C₂²×C₄×C₇₃	—	2	3	no	—
38	C₂²×C₄×C₇₃	—	2	3	no	—
39	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
40	C₄⋊C₄×C₇₃	—	4	3	no	—
41	C₄⋊C₄×C₇₃	—	4	3	no	—
42	C₄⋊C₄×C₇₃	—	8	3	no	—
43	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₇₃	#41	1	3	yes	—
45	C₂³×C₇₃	#120	1	1	yes	characteristic
46	C₂²×D₇₃	—	4	2	no	—
47	C₂²×D₇₃	—	8	3	no	—
48	C₂²×D₇₃	—	8	3	no	—
49	Q₈×C₇₃	—	8	4	no	—
50	C₂×C₄×C₇₃	—	2	3	no	—
51	C₂×C₄×C₇₃	—	2	3	no	—
52	C₂×C₄×C₇₃	—	4	3	no	—
53	C₂×C₄×C₇₃	—	4	3	no	—
54	C₂×C₄×C₇₃	—	4	6	no	—
55	C₂×C₄×C₇₃	—	4	6	no	—
56	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
57	C₂×C₄×C₇₃	#124	1	3	yes	—
58	D₂₉₂	—	4	6	no	—
59	D₂₉₂	—	4	6	no	—
60	D₂₉₂	—	8	3	no	—
61	D₂₉₂	—	8	3	no	—
62	#10596831	—	4	6	no	—
63	C₂²×C₇₃	#393	1	1	yes	characteristic
64	C₂²×C₇₃	#453	1	3	yes	—
65	C₂²×C₇₃	#454	1	3	yes	—
66	C₂×D₇₃	—	8	6	no	—
67	C₂×D₇₃	—	16	1	no	—
68	C₂×D₇₃	—	16	3	no	—
69	C₄×C₇₃	—	2	3	no	—
70	C₄×C₇₃	—	2	3	no	—
71	C₄×C₇₃	—	8	3	no	—
72	C₄×C₇₃	—	8	3	no	—
73	#10358	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	#1655	1	1	yes	characteristic
75	C₂×C₇₃	#1657	1	3	yes	—
76	C₂×C₇₃	#2027	1	3	yes	—
77	D₇₃	—	16	2	no	—
78	#2844	D₇₃	1	1	yes	characteristic, complemented
79	#1961	—	73	3	no	—
80	#1295	—	73	3	no	—
81	C₇₃	#10358	1	1	yes	characteristic, complemented, Sylow
82	C₂×#127	—	73	1	no	—
83	C₂×#102	—	146	3	no	—
84	C₂×#102	—	146	3	no	—
85	C₂×#97	—	73	3	no	—
86	#400	—	2	3	no	—
87	#395	—	2	3	no	—
88	#382	C₂×D₇₃	1	3	yes	—
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×D₄	—	146	3	no	—
91	C₂²×C₈	—	146	3	no	—
92	#127	—	73	2	no	—
93	#127	—	146	3	no	—
94	C₂×C₄⋊C₄	—	2	3	no	—
95	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	3	yes	—
96	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	3	yes	—
97	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
98	#102	—	292	6	no	—
99	#102	—	292	6	no	—
100	#97	—	146	6	no	—
101	#95	—	2	3	no	—
102	#95	—	2	1	no	—
103	#95	—	2	6	no	—
104	C₂⁴	—	292	1	no	—
105	C₂×Q₈	—	8	3	no	—
106	C₂×D₄	—	292	3	no	—
107	C₂×D₄	—	292	3	no	—
108	C₂×D₄	—	292	3	no	—
109	C₂×D₄	—	292	3	no	—
110	C₂×D₄	—	292	3	no	—
111	C₂×D₄	—	146	6	no	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	C₂×C₂²⋊D₇₃	1	3	yes	—
115	C₂×C₈	—	292	3	no	—
116	C₂×C₈	—	292	3	no	—
117	C₂×C₈	—	146	6	no	—
118	C₄⋊C₄	—	4	3	no	—
119	C₄⋊C₄	—	4	3	no	—
120	C₄⋊C₄	—	8	3	no	—
121	C₄²	Q₈×D₇₃	1	1	yes	characteristic
122	C₄²	#11924029	1	3	yes	—
123	C₂³	—	292	2	no	—
124	C₂³	—	584	3	no	—
125	C₂³	—	584	3	no	—
126	C₂³	#14556560	1	1	yes	characteristic
127	Q₈	—	8	4	no	—
128	D₄	—	292	6	no	—
129	D₄	—	292	6	no	—
130	D₄	—	584	3	no	—
131	D₄	—	584	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	4	3	no	—
135	C₂×C₄	—	4	3	no	—
136	C₂×C₄	—	4	6	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	#14556567	1	3	yes	—
139	C₂×C₄	#14556580	1	3	yes	—
140	C₈	—	292	6	no	—
141	C₂²	—	584	6	no	—
142	C₂²	—	1168	1	no	—
143	C₂²	—	1168	3	no	—
144	C₂²	#17988498	1	1	yes	characteristic
145	C₂²	#17988755	1	3	yes	—
146	C₂²	#17988760	1	3	yes	—
147	C₄	—	2	3	no	—
148	C₄	—	2	3	no	—
149	C₄	—	8	3	no	—
150	C₄	—	8	3	no	—
151	C₂	—	1168	2	no	—
152	C₂	#22609220	1	1	yes	characteristic
153	C₂	#22609229	1	3	yes	—
154	C₂	#22611084	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610754

C₂

yes

maximal

#22607812

C₂

yes

maximal, complemented

C₇₃×#2844

C₂

yes

maximal, characteristic, complemented

C₂×#14556588

C₂²

yes

—

—

C₂²

yes

—

#400×C₇₃

—

#395×C₇₃

—

#382×C₇₃

C₂²

yes

—

C₂²×D₂₉₂

—

C₂²×#10596831

—

#14556588

—

#14556588

Q₈

yes

—

—

—

C₂²×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×C₇₃

—

#95×C₇₃

—

#95×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#120

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#124

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

#10596831

—

C₂²×C₇₃

#393

yes

characteristic

C₂²×C₇₃

#453

yes

—

C₂²×C₇₃

#454

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2027

yes

—

D₇₃

—

#2844

D₇₃

yes

characteristic, complemented

#1961

—

#1295

—

C₇₃

#10358

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

146

—

C₂×#102

—

146

—

C₂×#97

—

#400

—

#395

—

#382

C₂×D₇₃

yes

—

C₂²×Q₈

—

C₂²×D₄

—

146

—

C₂²×C₈

—

146

—

#127

—

#127

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₇₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₇₃

yes

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

#102

—

292

—

#102

—

292

—

100

#97

—

146

—

101

#95

—

102

#95

—

103

#95

—

104

C₂⁴

—

292

—

105

C₂×Q₈

—

106

C₂×D₄

—

292

—

107

C₂×D₄

—

292

—

108

C₂×D₄

—

292

—

109

C₂×D₄

—

292

—

110

C₂×D₄

—

292

—

111

C₂×D₄

—

146

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

115

C₂×C₈

—

292

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

146

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

Q₈×D₇₃

yes

characteristic

122

C₄²

#11924029

yes

—

123

C₂³

—

292

—

124

C₂³

—

584

—

125

C₂³

—

584

—

126

C₂³

#14556560

yes

characteristic

127

Q₈

—

128

D₄

—

292

—

129

D₄

—

292

—

130

D₄

—

584

—

131

D₄

—

584

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14556567

yes

—

139

C₂×C₄

#14556580

yes

—

140

C₈

—

292

—

141

C₂²

—

584

—

142

C₂²

—

1168

—

143

C₂²

—

1168

—

144

C₂²

#17988498

yes

characteristic

145

C₂²

#17988755

yes

—

146

C₂²

#17988760

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

1168

—

152

C₂

#22609220

yes

characteristic

153

C₂

#22609229

yes

—

154

C₂

#22611084

yes

—

	subgroup	G/H
center	C₂³	#14556560
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#2844	C₂
socle	C₂³×C₇₃	#120
prime radical	C₂×#14556588	C₂²
hypercenter	#2844	D₇₃
hypocenter	C₇₃	#10358
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#14556560

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#2844

C₂

C₂³×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
2. #22607812	C₂	semidirect
3. C₇₃×#2844	C₂	semidirect
14. #14556588	Q₈	semidirect
78. #2844	D₇₃	semidirect
81. C₇₃	#10358	semidirect

H ≅ G/N

type

2. #22607812

C₂

semidirect

3. C₇₃×#2844

C₂

semidirect

14. #14556588

Q₈

semidirect

78. #2844

D₇₃

semidirect

81. C₇₃

#10358

semidirect

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

subgroup	index
#22610754	2
#22607812	2
C₇₃×#2844	2
#10358	73

subgroup

index

#22610754

#22607812

C₇₃×#2844

#10358

Subgroups — #32989053

Conjugacy classes of subgroups

Complements

Maximal subgroups