Group #32997641 — subgroups

Subgroups — #32997641

Order 18688.

180 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22611746	C₂	1	2	yes	maximal
2	#22611526	C₂	1	1	yes	maximal, characteristic
3	#22608007	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607998	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556530	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
8	#17988541	C₂²	1	1	yes	characteristic
9	#17988408	C₂²	1	2	yes	—
10	#17988368	—	2	2	no	—
11	#17988364	—	2	2	no	—
12	#17988363	C₂²	1	1	yes	characteristic
13	#17988362	C₄	1	1	yes	characteristic
14	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
15	#364×C₇₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924011	D₄	1	1	yes	characteristic
18	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
19	#14556548	Q₈	1	1	yes	characteristic
20	#14556531	—	2	2	no	—
21	#14556530	—	2	2	no	—
22	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
24	C₁₆×D₇₃	—	2	2	no	—
25	#14556502	Q₈	1	1	yes	characteristic
26	#14556501	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₇₃	—	2	2	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
33	#11924011	—	2	1	no	—
34	#11924011	—	2	1	no	—
35	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
36	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
37	C₈×D₇₃	—	2	1	no	—
38	C₈×D₇₃	—	2	1	no	—
39	#11924007	—	2	1	no	—
40	#11924005	—	2	1	no	—
41	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
42	#11924003	—	2	1	no	—
43	#11924003	—	4	2	no	—
44	#11924003	M₁₆	1	2	yes	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₇₃	—	2	1	no	—
49	C₁₆×C₇₃	—	4	2	no	—
50	C₁₆×C₇₃	M₁₆	1	2	yes	—
51	C₂²×D₇₃	#108	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	1	no	—
55	C₂×Dic₇₃	#108	1	1	yes	characteristic
56	C₄×D₇₃	—	2	1	no	—
57	C₄×D₇₃	—	2	1	no	—
58	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₇₃	#135	1	1	yes	characteristic
61	C₈×C₇₃	#135	1	1	yes	characteristic
62	#10596831	—	2	1	no	—
63	#10596831	—	2	1	no	—
64	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
65	C₂×D₇₃	—	2	1	no	—
66	C₂×D₇₃	—	4	1	no	—
67	C₄×C₇₃	—	4	1	no	—
68	C₄×C₇₃	#423	1	1	yes	characteristic
69	C₄×C₇₃	#427	1	1	yes	characteristic
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	—	4	1	no	—
72	#12026	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	2	1	no	—
74	C₂×C₇₃	#1568	1	1	yes	characteristic
75	D₇₃	—	4	1	no	—
76	#2166	—	73	2	no	—
77	#2108	—	73	1	no	—
78	#1373	—	73	1	no	—
79	#1373	D₇₃	1	1	yes	characteristic, complemented
80	#1369	—	73	2	no	—
81	C₇₃	#12026	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₃₂	—	73	1	no	—
83	C₂²×C₁₆	—	73	1	no	—
84	#405	—	73	1	no	—
85	#364	—	73	2	no	—
86	#364	C₂×D₇₃	1	2	yes	—
87	#349	—	146	2	no	—
88	#347	—	146	2	no	—
89	#346	—	73	1	no	—
90	#346	—	73	1	no	—
91	C₄×C₁₆	—	73	1	no	—
92	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
93	C₂×M₁₆	—	73	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	#117	—	73	1	no	—
96	M₃₂	—	2	2	no	—
97	M₃₂	—	146	2	no	—
98	M₃₂	—	146	2	no	—
99	C₂×C₁₆	—	73	1	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	146	2	no	—
102	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
103	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
104	#97	—	73	1	no	—
105	#97	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
108	C₂²×C₄	—	73	1	no	—
109	M₁₆	—	146	1	no	—
110	M₁₆	—	146	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	146	1	no	—
114	C₂×C₈	—	146	1	no	—
115	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
116	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
117	C₄⋊C₄	—	146	1	no	—
118	C₂²⋊C₄	—	146	1	no	—
119	C₄²	—	73	1	no	—
120	C₄²	Q₈×D₇₃	1	1	yes	characteristic
121	C₁₆	—	2	1	no	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	292	2	no	—
124	C₁₆	—	73	2	no	—
125	C₁₆	—	146	1	no	—
126	C₁₆	#11924011	1	2	yes	—
127	C₂³	—	73	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	146	1	no	—
131	C₂×C₄	—	146	1	no	—
132	C₂×C₄	—	146	1	no	—
133	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
134	C₈	—	146	1	no	—
135	C₈	—	146	1	no	—
136	C₈	M₁₆×D₇₃	1	1	yes	characteristic
137	C₈	C₂×#11924011	1	1	yes	characteristic
138	C₈	#14556610	1	1	yes	characteristic
139	C₈	#14556614	1	1	yes	characteristic
140	C₂²	—	292	1	no	—
141	C₂²	—	146	1	no	—
142	C₂²	#108×D₇₃	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	292	1	no	—
145	C₄	—	146	1	no	—
146	C₄	#17988609	1	1	yes	characteristic
147	C₄	#17988624	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	292	1	no	—
150	C₂	#22608693	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611746

C₂

yes

maximal

#22611526

C₂

yes

maximal, characteristic

#22608007

C₂

yes

maximal, characteristic, complemented

#22607998

C₂

yes

maximal, complemented

C₇₃×#1373

C₂

yes

maximal, characteristic, complemented

C₂×#14556530

C₄

yes

characteristic

C₂×C₁₆×D₇₃

C₄

yes

characteristic

#17988541

C₂²

yes

characteristic

#17988408

C₂²

yes

—

#17988368

—

#17988364

—

#17988363

C₂²

yes

characteristic

#17988362

C₄

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

characteristic

#364×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#11924011

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₁₆×D₇₃

—

#14556502

Q₈

yes

characteristic

#14556501

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₄⋊C₄

yes

—

—

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

#11924007

—

#11924005

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

—

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

—

C₂²×D₇₃

#108

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#108

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

C₂×#108

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#12026

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1568

yes

characteristic

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

#1369

—

C₇₃

#12026

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#364

—

#364

C₂×D₇₃

yes

—

#349

—

146

—

#347

—

146

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

146

—

102

C₂×C₁₆

C₄×D₇₃

yes

characteristic

103

C₂×C₁₆

C₄×D₇₃

yes

characteristic

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂²×D₇₃

yes

characteristic

108

C₂²×C₄

—

109

M₁₆

—

146

—

110

M₁₆

—

146

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

146

—

114

C₂×C₈

—

146

—

115

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

116

C₂×C₈

D₄×D₇₃

yes

characteristic

117

C₄⋊C₄

—

146

—

118

C₂²⋊C₄

—

146

—

119

C₄²

—

120

C₄²

Q₈×D₇₃

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

292

—

124

C₁₆

—

125

C₁₆

—

146

—

126

C₁₆

#11924011

yes

—

127

C₂³

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

134

C₈

—

146

—

135

C₈

—

146

—

136

C₈

M₁₆×D₇₃

yes

characteristic

137

C₈

C₂×#11924011

yes

characteristic

138

C₈

#14556610

yes

characteristic

139

C₈

#14556614

yes

characteristic

140

C₂²

—

292

—

141

C₂²

—

146

—

142

C₂²

#108×D₇₃

yes

characteristic

143

C₄

—

144

C₄

—

292

—

145

C₄

—

146

—

146

C₄

#17988609

yes

characteristic

147

C₄

#17988624

yes

characteristic

148

C₂

—

149

C₂

—

292

—

150

C₂

#22608693

yes

characteristic

	subgroup	G/H
center	C₄	#17988624
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1373	C₂
socle	C₂×C₇₃	#1568
prime radical	C₂²×D₇₃	#108
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1373	D₇₃
hypocenter	C₇₃	#12026
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₄

#17988624

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1373

C₂

C₂×C₇₃

C₂²×D₇₃

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
3. #22608007	C₂	semidirect
4. #22607998	C₂	semidirect
5. C₇₃×#1373	C₂	semidirect
79. #1373	D₇₃	semidirect
81. C₇₃	#12026	semidirect

H ≅ G/N

type

3. #22608007

C₂

semidirect

4. #22607998

C₂

semidirect

5. C₇₃×#1373

C₂

semidirect

79. #1373

D₇₃

semidirect

81. C₇₃

#12026

semidirect

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

subgroup	index
#22611746	2
#22611526	2
#22608007	2
#22607998	2
C₇₃×#1373	2
#12026	73

subgroup

index

C₇₃×#1373

Subgroups — #32997641

Conjugacy classes of subgroups

Complements

Maximal subgroups