Finite Group Databasebeta

← #37525191

Subgroups — #37525191

Order 22932.

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

normal subgroups only

This group has 216 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⋊#16838470C211yesmaximal, characteristic, complemented
2C7×C3⋊#10587716C211yesmaximal, characteristic, complemented
3C3×#16838468C211yesmaximal, characteristic, complemented
4D39⋊(C7⋊D7)C311yesmaximal, characteristic, complemented
5#10587716⋊D7—31nomaximal
6C3×C7×#63270C2211yescharacteristic, complemented, Hall
7C13×C3⋊(C7⋊D7)C2×C311yescharacteristic, complemented
8C72×D39C2×C311yescharacteristic, complemented
9C3×C7⋊D91C2×C311yescharacteristic, complemented
10#16838470—31no—
11#16838468—21no—
12#16838468D311yescharacteristic, complemented
13C7×#10587716—31no—
14(C3⋊(D13⋊C3))⋊D7—71nomaximal
15#10587722⋊D3—71nomaximal
16D13×C7⋊D7—31noHall
17C3×C72×C13C22×C311yescharacteristic, complemented
18C7×#63270—21no—
19C7×#63270C2×D311yescharacteristic, complemented
20D3×ΓL(1,𝔽8)⋊D7—131nomaximal, Hall
21C3⋊#10587720—71no—
22C3⋊#10587716D711yescharacteristic, complemented
23C13⋊C3×D21—71no—
24C7×C3⋊(D13⋊C3)—141no—
25C3×#10587722—71no—
26C3×(C13⋊C3)⋊D7—71no—
27C7⋊D91C3×D311yescharacteristic, complemented
28C13×C7⋊D7—31no—
29C72×D13—31no—
30D21⋊D13—71no—
31D21⋊D13—71no—
32D21⋊D13—212no—
33#10587716⋊C2—211no—
34D7×D13⋊C3—211no—
35C3⋊(ΓL(1,𝔽8)⋊D7)—261no—
36D3×C7×ΓL(1,𝔽8)—131no—
37C3×ΓL(1,𝔽8)⋊D7—131no—
38C3×#63270C2×D711yescharacteristic, complemented
39C3×C7×C13⋊C3—141no—
40C72×C13C2×C3×D311yescharacteristic, complemented, Hall
41D3×C7⋊D7—131no—
42C2×ΓL(1,𝔽8)⋊D7—391no—
43C13×D21—71no—
44C13×D21—71no—
45C13×D21—212no—
46C7×D39—32no—
47C7×D39AGL(1,𝔽7)11yescharacteristic, complemented
48C7×D39C3×D711yescharacteristic, complemented
49C3×D91—71no—
50C3×D91—71no—
51C3×D91—212no—
52#10587722—71no—
53#10587722—141no—
54#10587720—211no—
55#10587716—31no—
56(C13⋊C3)⋊D7—71no—
57(C13⋊C3)⋊D7—141no—
58D7×C13⋊C3—211no—
59C7×D13⋊C3—421no—
60D3×D13⋊C3—491noHall
61C3×C7×ΓL(1,𝔽8)—261noHall
62D7×D13—211no—
63D7×D13—211no—
64D7×D13—632no—
65C3⋊(C7⋊D7)D13⋊C311yescharacteristic, complemented
66D3×C72—131no—
67C3×C7⋊D7—131no—
68C2×C7×ΓL(1,𝔽8)—391no—
69ΓL(1,𝔽8)⋊D7—131no—
70ΓL(1,𝔽8)⋊D7—261no—
71ΓL(1,𝔽8)⋊D7—781no—
72C3×C7×C13—32no—
73C3×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
74C3×C7×C13C2×C3×D711yescharacteristic, complemented
75#63270—21no—
76#63270D3×D711yescharacteristic, complemented
77C7×C13⋊C3—141no—
78C7×C13⋊C3—281no—
79C3×D3×D7—911no—
80D3×AGL(1,𝔽7)—911no—
81C3⋊(D13⋊C3)—141no—
82D3×C13⋊C3—491no—
83C3×D13⋊C3—491no—
84C2×C7⋊D7—391noHall
85D91—71no—
86D91—71no—
87D91—212no—
88C7×D13—31no—
89C7×D13—31no—
90C7×D13—92no—
91C13×D7—211no—
92C13×D7—211no—
93C13×D7—632no—
94D3×D13—491no—
95C2×D13⋊C3—1471no—
96C3×C72C2×D13⋊C311yescharacteristic, complemented
97C7×ΓL(1,𝔽8)—261no—
98C7×ΓL(1,𝔽8)—521no—
99C3×D21—1821no—
100C3×D3×C7—1821no—
101C32×D7—911no—
102C3⋊AGL(1,𝔽7)—1821no—
103D3×ΓL(1,𝔽8)—131no—
104C3×AGL(1,𝔽7)—911no—
105C3×C13⋊C3—141noHall
106C2×C72—391no—
107C7⋊D7—31no—
108C7⋊D7—131no—
109C7×C13—32no—
110C7×C13D3×AGL(1,𝔽7)11yescharacteristic, complemented
111C7×C13C3×D3×D711yescharacteristic, complemented
112C2×C3×D7—2731no—
113D3×D7—2732no—
114D3×D7—911no—
115D3×D7—911no—
116C2×AGL(1,𝔽7)—2731no—
117D39ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
118C3×D13—491no—
119D3×C13—491no—
120C2×C13⋊C3—1471no—
121D13⋊C3—211no—
122D13⋊C3—491no—
123D13⋊C3—981no—
124C32×C7—3641no—
125C3×ΓL(1,𝔽8)—261no—
126C2×D13—1471noHall
127C72D3×D13⋊C311yescharacteristic, complemented, Sylow
128C2×C3×C7—5461no—
129D21—71no—
130D21—71no—
131D21—212no—
132C3×D7—2732no—
133C3×D7—5461no—
134C3×D7—911no—
135C3×D7—911no—
136C3×D7—911no—
137C3×D7—1821no—
138D3×C7—131no—
139D3×C7—131no—
140D3×C7—392no—
141C2×ΓL(1,𝔽8)—391no—
142AGL(1,𝔽7)—5461no—
143AGL(1,𝔽7)—911no—
144AGL(1,𝔽7)—1821no—
145C3×C13C2×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
146C13⋊C3—71no—
147C13⋊C3—281no—
148C2×C3×D3—6371noHall
149C2×D7—2731no—
150C2×D7—2731no—
151C2×D7—8192no—
152C2×C13—1471no—
153D13—31no—
154D13—491no—
155C3×C7—32no—
156C3×C7—3641no—
157C3×C7—1821no—
158C3×C7D7×D13⋊C311yescharacteristic, complemented
159C3×C7#10587716⋊C211yescharacteristic, complemented
160ΓL(1,𝔽8)—261no—
161ΓL(1,𝔽8)—261no—
162C2×C32—6371no—
163C3×D3—1821no—
164C3×D3—12741no—
165C2×C7—391no—
166C2×C7—391no—
167C2×C7—1172no—
168D7—2732no—
169D7—211no—
170D7—211no—
171D7—632no—
172D7—911no—
173D7—911no—
174C13D3×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Sylow
175C22×C3—19111no—
176C2×D3—6371no—
177C32—3641noSylow
178C7—32no—
179C7#10587722⋊D311yescharacteristic, complemented
180C7(C3⋊(D13⋊C3))⋊D711yescharacteristic, complemented
181C2×C3—5461no—
182C2×C3—6371no—
183C2×C3—6371no—
184C2×C3—38221no—
185C2×C3—12741no—
186D3—131no—
187D3—491no—
188C22—19111noSylow
189C3—3641no—
190C3—3641no—
191C3#10587716⋊D711yescharacteristic, complemented
192C2—391no—
193C2—6371no—
194C2—1471no—
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⋊#16838470C2semidirect
2. C7×C3⋊#10587716C2semidirect
3. C3×#16838468C2semidirect
4. D39⋊(C7⋊D7)C3semidirect
6. C3×C7×#63270C22semidirect
7. C13×C3⋊(C7⋊D7)C2×C3semidirect
8. C72×D39C2×C3semidirect
9. C3×C7⋊D91C2×C3semidirect
12. #16838468D3semidirect
17. C3×C72×C13C22×C3semidirect
19. C7×#63270C2×D3semidirect
22. C3⋊#10587716D7semidirect
27. C7⋊D91C3×D3semidirect
38. C3×#63270C2×D7semidirect
40. C72×C13C2×C3×D3semidirect
47. C7×D39AGL(1,𝔽7)semidirect
48. C7×D39C3×D7semidirect
65. C3⋊(C7⋊D7)D13⋊C3semidirect
73. C3×C7×C13C2×AGL(1,𝔽7)semidirect
74. C3×C7×C13C2×C3×D7semidirect
76. #63270D3×D7semidirect
96. C3×C72C2×D13⋊C3semidirect
110. C7×C13D3×AGL(1,𝔽7)semidirect
111. C7×C13C3×D3×D7semidirect
117. D39ΓL(1,𝔽8)⋊D7semidirect
127. C72D3×D13⋊C3semidirect
145. C3×C13C2×ΓL(1,𝔽8)⋊D7semidirect
158. C3×C7D7×D13⋊C3semidirect
159. C3×C7#10587716⋊C2semidirect
174. C13D3×ΓL(1,𝔽8)⋊D7semidirect
179. C7#10587722⋊D3semidirect
180. C7(C3⋊(D13⋊C3))⋊D7semidirect
191. C3#10587716⋊D7semidirect

Maximal subgroups

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

subgroupindex
C3⋊#168384702
C7×C3⋊#105877162
C3×#168384682
D39⋊(C7⋊D7)3
#10587716⋊D73
(C3⋊(D13⋊C3))⋊D77
#10587722⋊D37
D3×ΓL(1,𝔽8)⋊D713