Finite Group Databasebeta

← #37524913

Subgroups — #37524913

Order 22932.

280 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 47 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
1#26039555C211yesmaximal, characteristic, complemented
2#26039523C211yesmaximal, characteristic, complemented
3#26039518C211yesmaximal, characteristic, complemented
4#10587716⋊D7C311yesmaximal, characteristic, complemented
5#10587716⋊D7C311yesmaximal, characteristic, complemented
6D13×#64425C311yesmaximal, characteristic, complemented
7#21439600C311yesmaximal, characteristic, complemented
8#19153211C2211yescharacteristic, complemented, Hall
9D13×#3862C2×C311yescharacteristic, complemented
10#16838474C2×C311yescharacteristic, complemented
11#16838470C2×C311yescharacteristic, complemented
12#16838470C2×C311yescharacteristic, complemented
13#16838468C2×C311yescharacteristic, complemented
14#16838468C2×C311yescharacteristic, complemented
15#16838454C2×C311yescharacteristic, complemented
16C13×#64425C2×C311yescharacteristic, complemented
17#16838449C2×C311yescharacteristic, complemented
18#16838447C2×C311yescharacteristic, complemented
19C7×#10587716C2×C311yescharacteristic, complemented
20C7×#10587716C2×C311yescharacteristic, complemented
21AGL(1,𝔽7)×D13⋊C3—71nomaximal
22AGL(1,𝔽7)×D13⋊C3—71nomaximal
23D13×C7⋊D7C3211yescharacteristic, complemented, Hall
24C13×#3862C22×C311yescharacteristic, complemented
25#14274262C22×C311yescharacteristic, complemented
26C7×#63270C22×C311yescharacteristic, complemented
27C7×#63270C22×C311yescharacteristic, complemented
28C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)—131nomaximal, Hall
29ΓL(1,𝔽8)⋊(D13⋊C3)—71no—
30ΓL(1,𝔽8)⋊(D13⋊C3)—71no—
31C13⋊C3×AGL(1,𝔽7)—71no—
32C13⋊C3×AGL(1,𝔽7)—71no—
33ΓL(1,𝔽8)×D13⋊C3—71no—
34ΓL(1,𝔽8)×D13⋊C3—71no—
35C7⋊D91C2×C3211yescharacteristic, complemented
36C13×C7⋊D7C2×C3211yescharacteristic, complemented
37C72×D13C2×C3211yescharacteristic, complemented
38#10587716⋊C2—71no—
39#10587716⋊C2—211no—
40#10587717⋊C2—71no—
41#10587717⋊C2—71no—
42#10587717⋊C2—211no—
43#10587717⋊C2—211no—
44D13×AGL(1,𝔽7)—71no—
45D13×AGL(1,𝔽7)—71no—
46D7×D13⋊C3—71no—
47D7×D13⋊C3—71no—
48C2×ΓL(1,𝔽8)2—131no—
49ΓL(1,𝔽8)⋊AGL(1,𝔽7)—131no—
50ΓL(1,𝔽8)⋊AGL(1,𝔽7)—131no—
51ΓL(1,𝔽8)×C13⋊C3—71no—
52ΓL(1,𝔽8)×C13⋊C3—71no—
53C72×C13C22×C3211yescharacteristic, complemented, Hall
54C2×#64425—391no—
55C2×#64424—131no—
56C2×ΓL(1,𝔽8)⋊D7—131no—
57C2×ΓL(1,𝔽8)⋊D7—131no—
58#10587723—71no—
59#10587723—71no—
60#10587723—211no—
61#10587723—211no—
62#10587722—71no—
63#10587722—211no—
64#10587721—71no—
65#10587721—71no—
66#10587721—211no—
67#10587721—211no—
68#10587720—71no—
69#10587720—211no—
70ΓL(1,𝔽8)⋊D13—71no—
71ΓL(1,𝔽8)⋊D13—71no—
72C13×AGL(1,𝔽7)—71no—
73C13×AGL(1,𝔽7)—71no—
74#10587717—71no—
75#10587717—71no—
76#10587717—211no—
77#10587717—211no—
78#10587716AGL(1,𝔽7)11yescharacteristic, complemented
79#10587716AGL(1,𝔽7)11yescharacteristic, complemented
80ΓL(1,𝔽8)×D13—71no—
81ΓL(1,𝔽8)×D13—71no—
82(C13⋊C3)⋊D7—71no—
83(C13⋊C3)⋊D7—71no—
84D7×C13⋊C3—71no—
85D7×C13⋊C3—71no—
86C7×D13⋊C3—71no—
87C7×D13⋊C3—71no—
88C2×C3×D13⋊C3—491noHall
89ΓL(1,𝔽8)2—131noHall
90D7×D13—71no—
91D7×D13—211no—
92D7×D13—211no—
93D7×D13—631no—
94C2×#3862—391no—
95C2×#3861—131no—
96C2×C7×ΓL(1,𝔽8)—131no—
97C2×C7×ΓL(1,𝔽8)—131no—
98#64425—391no—
99#64425D13⋊C311yescharacteristic, complemented
100#64424—131no—
101#64424—131no—
102ΓL(1,𝔽8)⋊D7—131no—
103ΓL(1,𝔽8)⋊D7—131no—
104ΓL(1,𝔽8)⋊D7—131no—
105ΓL(1,𝔽8)⋊D7—131no—
106#63271—71no—
107#63271—71no—
108#63271—211no—
109#63271—211no—
110#63270C2×AGL(1,𝔽7)11yescharacteristic, complemented
111#63270C2×AGL(1,𝔽7)11yescharacteristic, complemented
112C7×C13⋊C3—71no—
113C7×C13⋊C3—71no—
114C13×ΓL(1,𝔽8)—71no—
115C13×ΓL(1,𝔽8)—71no—
116C2×C3×AGL(1,𝔽7)—911no—
117C2×C3×AGL(1,𝔽7)—911no—
118C2×C3×C13⋊C3—491no—
119C3×D13⋊C3—491no—
120C3×D13⋊C3—491no—
121C2×C7⋊D7—391noHall
122D91—71no—
123D91—211no—
124D91—631no—
125D91—631no—
126C7×D13—31no—
127C7×D13—31no—
128C7×D13C3×AGL(1,𝔽7)11yescharacteristic, complemented
129C7×D13C3×AGL(1,𝔽7)11yescharacteristic, complemented
130C13×D7—71no—
131C13×D7—211no—
132C13×D7—631no—
133C13×D7—631no—
134C2×C3×D13—491no—
135C2×D13⋊C3—491no—
136C2×D13⋊C3—1471no—
137C2×D13⋊C3—1471no—
138#3862C2×D13⋊C311yescharacteristic, complemented
139#3861—131no—
140C7×ΓL(1,𝔽8)—131no—
141C7×ΓL(1,𝔽8)—131no—
142C2×C3×ΓL(1,𝔽8)—911no—
143C2×C3×ΓL(1,𝔽8)—911no—
144C3×AGL(1,𝔽7)—911no—
145C3×AGL(1,𝔽7)—911no—
146C3×AGL(1,𝔽7)—911no—
147C3×AGL(1,𝔽7)—911no—
148C3×C13⋊C3—491noHall
149C2×C72—391no—
150C7⋊D7—391no—
151C7⋊D7C3×D13⋊C311yescharacteristic, complemented
152C7×C13—31no—
153C7×C13—31no—
154C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
155C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
156C2×C3×D7—911no—
157C2×C3×D7—911no—
158C2×AGL(1,𝔽7)—2731no—
159C2×AGL(1,𝔽7)—2731no—
160C2×AGL(1,𝔽7)—2731no—
161C2×AGL(1,𝔽7)—2731no—
162C2×AGL(1,𝔽7)—2731no—
163C2×AGL(1,𝔽7)—2731no—
164C2×AGL(1,𝔽7)—911no—
165C2×AGL(1,𝔽7)—911no—
166C2×C3×C13—491no—
167C3×D13—491no—
168C3×D13—491no—
169C2×C13⋊C3—491no—
170C2×C13⋊C3—1471no—
171C2×C13⋊C3—1471no—
172D13⋊C3—71no—
173D13⋊C3—71no—
174D13⋊C3—491no—
175D13⋊C3—491no—
176D13⋊C3—1471no—
177D13⋊C3—1471no—
178C3×ΓL(1,𝔽8)—911no—
179C3×ΓL(1,𝔽8)—911no—
180C2×D13—4411noHall
181C72C2×C3×D13⋊C311yescharacteristic, complemented, Sylow
182C2×C3×C7—911no—
183C2×C3×C7—911no—
184C3×D7—911no—
185C3×D7—911no—
186C3×D7—911no—
187C3×D7—911no—
188C2×ΓL(1,𝔽8)—131no—
189C2×ΓL(1,𝔽8)—131no—
190C2×ΓL(1,𝔽8)—2731no—
191C2×ΓL(1,𝔽8)—2731no—
192C2×ΓL(1,𝔽8)—2731no—
193C2×ΓL(1,𝔽8)—2731no—
194C2×ΓL(1,𝔽8)—911no—
195C2×ΓL(1,𝔽8)—911no—
196AGL(1,𝔽7)—71no—
197AGL(1,𝔽7)—71no—
198AGL(1,𝔽7)—2731no—
199AGL(1,𝔽7)—2731no—
200AGL(1,𝔽7)—2731no—
201AGL(1,𝔽7)—2731no—
202AGL(1,𝔽7)—2731no—
203AGL(1,𝔽7)—2731no—
204AGL(1,𝔽7)—2731no—
205AGL(1,𝔽7)—2731no—
206AGL(1,𝔽7)—2731no—
207AGL(1,𝔽7)—911no—
208AGL(1,𝔽7)—911no—
209AGL(1,𝔽7)—911no—
210AGL(1,𝔽7)—911no—
211AGL(1,𝔽7)—911no—
212C3×C13—491no—
213C13⋊C3—71no—
214C13⋊C3—71no—
215C13⋊C3—491no—
216C22×C32—6371noHall
217C2×D7—8191no—
218C2×D7—8191no—
219C2×D7—8191no—
220C2×D7—8191no—
221C2×C13—4411no—
222D13—4411no—
223D13ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic, complemented
224C3×C7—911no—
225C3×C7—911no—
226ΓL(1,𝔽8)—71no—
227ΓL(1,𝔽8)—71no—
228ΓL(1,𝔽8)—131no—
229ΓL(1,𝔽8)—131no—
230ΓL(1,𝔽8)—2731no—
231ΓL(1,𝔽8)—2731no—
232ΓL(1,𝔽8)—911no—
233ΓL(1,𝔽8)—911no—
234C2×C32—6371no—
235C2×C32—6371no—
236C2×C32—6371no—
237C2×C7—391no—
238C2×C7—391no—
239C2×C7—1171no—
240C2×C7—1171no—
241D7—71no—
242D7—211no—
243D7—211no—
244D7—8191no—
245D7—8191no—
246D7—8191no—
247D7—8191no—
248D7—631no—
249C13C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
250C22×C3—19111no—
251C22×C3—19111no—
252C22×C3—19111no—
253C22×C3—6371no—
254C32—6371noSylow
255C7—31no—
256C7—31no—
257C7AGL(1,𝔽7)×D13⋊C311yescharacteristic, complemented
258C7AGL(1,𝔽7)×D13⋊C311yescharacteristic, complemented
259C2×C3—491no—
260C2×C3—911no—
261C2×C3—911no—
262C2×C3—19111no—
263C2×C3—19111no—
264C2×C3—19111no—
265C2×C3—19111no—
266C2×C3—19111no—
267C2×C3—6371no—
268C2×C3—6371no—
269C2×C3—6371no—
270C2×C3—6371no—
271C22—57331noSylow
272C3—491no—
273C3—911no—
274C3—911no—
275C3—6371no—
276C2—391no—
277C2—57331no—
278C2—4411no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C13C22×C32
LCSG ▷ C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
1. #26039555C2semidirect
2. #26039523C2semidirect
3. #26039518C2semidirect
4. #10587716⋊D7C3semidirect
5. #10587716⋊D7C3semidirect
6. D13×#64425C3semidirect
7. #21439600C3semidirect
8. #19153211C22semidirect
9. D13×#3862C2×C3semidirect
10. #16838474C2×C3semidirect
11. #16838470C2×C3semidirect
12. #16838470C2×C3semidirect
13. #16838468C2×C3semidirect
14. #16838468C2×C3semidirect
15. #16838454C2×C3semidirect
16. C13×#64425C2×C3semidirect
17. #16838449C2×C3semidirect
18. #16838447C2×C3semidirect
19. C7×#10587716C2×C3semidirect
20. C7×#10587716C2×C3semidirect
23. D13×C7⋊D7C32semidirect
24. C13×#3862C22×C3semidirect
25. #14274262C22×C3semidirect
26. C7×#63270C22×C3semidirect
27. C7×#63270C22×C3semidirect
35. C7⋊D91C2×C32semidirect
36. C13×C7⋊D7C2×C32semidirect
37. C72×D13C2×C32semidirect
53. C72×C13C22×C32semidirect
78. #10587716AGL(1,𝔽7)semidirect
79. #10587716AGL(1,𝔽7)semidirect
99. #64425D13⋊C3semidirect
110. #63270C2×AGL(1,𝔽7)semidirect
111. #63270C2×AGL(1,𝔽7)semidirect
128. C7×D13C3×AGL(1,𝔽7)semidirect
129. C7×D13C3×AGL(1,𝔽7)semidirect
138. #3862C2×D13⋊C3semidirect
151. C7⋊D7C3×D13⋊C3semidirect
154. C7×C13C2×C3×AGL(1,𝔽7)semidirect
155. C7×C13C2×C3×AGL(1,𝔽7)semidirect
181. C72C2×C3×D13⋊C3semidirect
223. D13ΓL(1,𝔽8)⋊AGL(1,𝔽7)semidirect
249. C13C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)semidirect
257. C7AGL(1,𝔽7)×D13⋊C3semidirect
258. C7AGL(1,𝔽7)×D13⋊C3semidirect

Maximal subgroups

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

subgroupindex
#260395552
#260395232
#260395182
#10587716⋊D73
D13×#644253
#214396003
AGL(1,𝔽7)×D13⋊C37
C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)13