Finite Group Databasebeta

← #37525307

Subgroups — #37525307

Order 22932.

230 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊#16838457C211yesmaximal, characteristic, complemented
2C3⋊#16838447C211yesmaximal, characteristic, complemented
3C3×#16838459C211yesmaximal, characteristic, complemented
4D7×D273C311yesmaximal, characteristic, complemented
5#21439609—31nomaximal
6C3×#14274262C2211yescharacteristic, complemented, Hall
7C3⋊(C7⋊D91)C2×C311yescharacteristic, complemented
8C7×D273C2×C311yescharacteristic, complemented
9C3×C7×C13×D7C2×C311yescharacteristic, complemented
10#16838459—21no—
11#16838459D311yescharacteristic, complemented
12#16838457—31no—
13#16838447—31no—
14C2×C3⋊#10587723—71nomaximal
15#10587721⋊D3—71nomaximal
16D7×D91—31noHall
17C3×C72×C13C22×C311yescharacteristic, complemented
18#14274262—21no—
19#14274262C2×D311yescharacteristic, complemented
20#64423⋊D3—131nomaximal, Hall
21C2×C3×#63271—71no—
22C3⋊#10587723—71no—
23C3⋊#10587723—71no—
24C3⋊#10587723—71no—
25C3⋊#10587723—143no—
26C3⋊#10587717—71no—
27C3×#10587721—71no—
28C7⋊D91—31no—
29C7×D91—31no—
30C7×C13×D7C3×D311yescharacteristic, complemented
31C2×D273—71no—
32D7×D39—71no—
33C2×#10587723—211no—
34#10587717⋊C2—211no—
35C3⋊#64424—131no—
36C3⋊#64423—131no—
37C3×#64423—131no—
38C3×#63271—71no—
39C3×#63271—71no—
40C3×#63271—143no—
41C72×C13C2×C3×D311yescharacteristic, complemented, Hall
42D7×D21—131no—
43#64423⋊C2—391no—
44C2×C3×C7×C13—71no—
45D273—71no—
46D273—71no—
47D273—143no—
48D273AGL(1,𝔽7)11yescharacteristic, complemented
49C7×D39—71no—
50C3×C13×D7AGL(1,𝔽7)11yescharacteristic, complemented
51C2×#63271—71no—
52C2×#63271—141no—
53#10587723—211no—
54#10587723—211no—
55#10587723—211no—
56#10587723—423no—
57#10587721—71no—
58#10587721—141no—
59#10587717—211no—
60C2×C3⋊(D13⋊C3)—491noHall
61C3×#3861—131noHall
62C2×D91—211no—
63D7×D13—211no—
64C3⋊(C7⋊D7)—131no—
65C7×D21—131no—
66C3×C7×D7D13⋊C311yescharacteristic, complemented
67#64424—391no—
68#64423—131no—
69#64423—261no—
70#64423—391no—
71C3×C7×C13—23no—
72C3×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
73C3×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
74#63271—71no—
75#63271—71no—
76#63271—143no—
77#63271—141no—
78#63271—141no—
79#63271—283no—
80C2×C3⋊AGL(1,𝔽7)—911no—
81D3×AGL(1,𝔽7)—911no—
82C2×C3×C13⋊C3—491no—
83C3⋊(D13⋊C3)—491no—
84C3⋊(D13⋊C3)—491no—
85D72—391noHall
86C2×C7×C13—71no—
87D91—31no—
88D91—211no—
89D91—211no—
90D91—423no—
91C7×D13—211no—
92C13×D7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
93C2×D39—491no—
94C2×D13⋊C3—1471no—
95C3×C72C2×D13⋊C311yescharacteristic, complemented
96#3861—131no—
97#3861—261no—
98C2×C3×ΓL(1,𝔽8)—911no—
99C3⋊AGL(1,𝔽7)—911no—
100C3⋊AGL(1,𝔽7)—911no—
101C3⋊AGL(1,𝔽7)—911no—
102C3⋊AGL(1,𝔽7)—1823no—
103D3×ΓL(1,𝔽8)—911no—
104C3×AGL(1,𝔽7)—911no—
105C3×C13⋊C3—491noHall
106C7⋊D7—391no—
107C7×D7—391no—
108C7×D7C3⋊(D13⋊C3)11yescharacteristic, complemented
109C7×C13—23no—
110C7×C13D3×AGL(1,𝔽7)11yescharacteristic, complemented
111C7×C13C2×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
112C2×D21—911no—
113D3×D7—911no—
114C2×AGL(1,𝔽7)—2731no—
115C2×AGL(1,𝔽7)—2731no—
116C2×C3×C13—71no—
117D39—71no—
118D39—491no—
119C2×C13⋊C3—491no—
120C2×C13⋊C3—981no—
121D13⋊C3—1471no—
122D13⋊C3—1471no—
123C3×ΓL(1,𝔽8)—911no—
124C3×ΓL(1,𝔽8)—911no—
125C3×ΓL(1,𝔽8)—1823no—
126C2×D13—1471noHall
127C72C2×C3⋊(D13⋊C3)11yescharacteristic, complemented, Sylow
128C2×C3×C7—71no—
129D21—131no—
130D21—911no—
131D21—911no—
132D21—1823no—
133C3×D7#1058772311yescharacteristic, complemented
134D3×C7—911no—
135C2×ΓL(1,𝔽8)—2731no—
136C2×ΓL(1,𝔽8)—911no—
137C2×ΓL(1,𝔽8)—1821no—
138AGL(1,𝔽7)—2731no—
139AGL(1,𝔽7)—2731no—
140AGL(1,𝔽7)—2731no—
141AGL(1,𝔽7)—5463no—
142AGL(1,𝔽7)—911no—
143AGL(1,𝔽7)—1821no—
144C3×C13#64423⋊C211yescharacteristic, complemented
145C13⋊C3—491no—
146C13⋊C3—981no—
147C2×C3×D3—6371noHall
148C2×D7—2731no—
149C2×D7—2731no—
150C2×C13—71no—
151D13—211no—
152D13—1471no—
153C3×C7—23no—
154C3×C7#10587717⋊C211yescharacteristic, complemented
155C3×C7C2×#1058772311yescharacteristic, complemented
156ΓL(1,𝔽8)—911no—
157ΓL(1,𝔽8)—911no—
158ΓL(1,𝔽8)—3643no—
159ΓL(1,𝔽8)—1823no—
160ΓL(1,𝔽8)—1821no—
161ΓL(1,𝔽8)—1821no—
162C2×C32—6371no—
163C3×D3—6371no—
164C3×D3—6371no—
165C2×C7—71no—
166C2×C7—2731no—
167D7—2731no—
168D7—2731no—
169D7—5463no—
170D7—391no—
171D7C3⋊#1058772311yescharacteristic, complemented
172C13#64423⋊D311yescharacteristic, complemented, Sylow
173C22×C3—19111no—
174C2×D3—6371no—
175C32—6371noSylow
176C7—23no—
177C7#10587721⋊D311yescharacteristic, complemented
178C7C2×C3⋊#1058772311yescharacteristic, complemented
179C2×C3—71no—
180C2×C3—19111no—
181C2×C3—19111no—
182C2×C3—6371no—
183C2×C3—12741no—
184D3—911no—
185D3—6371no—
186C22—19111noSylow
187C3—6371no—
188C3—12741no—
189C3#2143960911yescharacteristic, complemented
190C2—71no—
191C2—2731no—
192C2—19111no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C13C22×C3
LCSG ▷ C3×C72×C13
UCSC1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3⋊#16838457C2semidirect
2. C3⋊#16838447C2semidirect
3. C3×#16838459C2semidirect
4. D7×D273C3semidirect
6. C3×#14274262C22semidirect
7. C3⋊(C7⋊D91)C2×C3semidirect
8. C7×D273C2×C3semidirect
9. C3×C7×C13×D7C2×C3semidirect
11. #16838459D3semidirect
17. C3×C72×C13C22×C3semidirect
19. #14274262C2×D3semidirect
30. C7×C13×D7C3×D3semidirect
41. C72×C13C2×C3×D3semidirect
48. D273AGL(1,𝔽7)semidirect
50. C3×C13×D7AGL(1,𝔽7)semidirect
66. C3×C7×D7D13⋊C3semidirect
72. C3×C7×C13C2×AGL(1,𝔽7)semidirect
73. C3×C7×C13C2×AGL(1,𝔽7)semidirect
92. C13×D7C3⋊AGL(1,𝔽7)semidirect
95. C3×C72C2×D13⋊C3semidirect
108. C7×D7C3⋊(D13⋊C3)semidirect
110. C7×C13D3×AGL(1,𝔽7)semidirect
111. C7×C13C2×C3⋊AGL(1,𝔽7)semidirect
127. C72C2×C3⋊(D13⋊C3)semidirect
133. C3×D7#10587723semidirect
144. C3×C13#64423⋊C2semidirect
154. C3×C7#10587717⋊C2semidirect
155. C3×C7C2×#10587723semidirect
171. D7C3⋊#10587723semidirect
172. C13#64423⋊D3semidirect
177. C7#10587721⋊D3semidirect
178. C7C2×C3⋊#10587723semidirect
189. C3#21439609semidirect

Maximal subgroups

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

subgroupindex
C3⋊#168384572
C3⋊#168384472
C3×#168384592
D7×D2733
#214396093
C2×C3⋊#105877237
#10587721⋊D37
#64423⋊D313