Finite Group Databasebeta

← AGL(1,𝔽7)×#10587723 #37524919

Subgroups — AGL(1,𝔽7)×#10587723 #37524919

Order 22932.

248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 49 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
1AGL(1,𝔽7)×#63271C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)×#10587723C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊#10587723C211yesmaximal, characteristic, complemented
4AGL(1,𝔽7)×D91C311yesmaximal, characteristic, complemented
5D7×#10587723C311yesmaximal, characteristic, complemented
6#21439617C311yesmaximal, characteristic, complemented
7#21439609C311yesmaximal, characteristic, complemented
8ΓL(1,𝔽8)×#63271C2211yescharacteristic, complemented, Hall
9ΓL(1,𝔽8)×D91C2×C311yescharacteristic, complemented
10D7×#63271C2×C311yescharacteristic, complemented
11ΓL(1,𝔽8)⋊D91C2×C311yescharacteristic, complemented
12#16838469C2×C311yescharacteristic, complemented
13#16838464C2×C311yescharacteristic, complemented
14#16838463C2×C311yescharacteristic, complemented
15#16838459C2×C311yescharacteristic, complemented
16#16838457C2×C311yescharacteristic, complemented
17#16838452C2×C311yescharacteristic, complemented
18#16838447C2×C311yescharacteristic, complemented
19C7×#10587723C2×C311yescharacteristic, complemented
20C7×C13×AGL(1,𝔽7)C2×C311yescharacteristic, complemented
21C2×C3×#10587723—71nomaximal
22AGL(1,𝔽7)×D13⋊C3—71nomaximal
23D7×D91C3211yescharacteristic, complemented, Hall
24C13⋊#3862C22×C311yescharacteristic, complemented
25#14274262C22×C311yescharacteristic, complemented
26C7×#63271C22×C311yescharacteristic, complemented
27C7×C13×ΓL(1,𝔽8)C22×C311yescharacteristic, complemented
28AGL(1,𝔽7)2—131nomaximal, Hall
29C2×C3×#63271—71no—
30C3×#10587723—71no—
31C3×#10587723—71no—
32ΓL(1,𝔽8)⋊(D13⋊C3)—71no—
33C13⋊C3×AGL(1,𝔽7)—71no—
34ΓL(1,𝔽8)×D13⋊C3—71no—
35C7⋊D91C2×C3211yescharacteristic, complemented
36C7×D91C2×C3211yescharacteristic, complemented
37C7×C13×D7C2×C3211yescharacteristic, complemented
38C2×C3×D91—71no—
39C2×#10587723—71no—
40C2×#10587723—71no—
41C2×#10587723—71no—
42#10587716⋊C2—71no—
43#10587717⋊C2—71no—
44D13×AGL(1,𝔽7)—211no—
45D7×D13⋊C3—71no—
46ΓL(1,𝔽8)⋊AGL(1,𝔽7)—131no—
47ΓL(1,𝔽8)×AGL(1,𝔽7)—131no—
48ΓL(1,𝔽8)×AGL(1,𝔽7)—131no—
49C3×#63271—71no—
50ΓL(1,𝔽8)×C13⋊C3—71no—
51C72×C13C22×C3211yescharacteristic, complemented, Hall
52#64422⋊C2—131no—
53#64423⋊C2—131no—
54D7×AGL(1,𝔽7)—131no—
55D7×AGL(1,𝔽7)—391no—
56C2×C3×C7×C13—71no—
57C3×D91—71no—
58C3×D91—71no—
59C2×#63271—71no—
60C2×#63271—71no—
61C2×#63271—211no—
62#10587723—71no—
63#10587723—71no—
64#10587723—71no—
65#10587723—71no—
66#10587723—71no—
67#10587723—211no—
68#10587723—421no—
69#10587723AGL(1,𝔽7)11yescharacteristic, direct product factor
70#10587722—71no—
71#10587721—71no—
72#10587720—71no—
73ΓL(1,𝔽8)⋊D13—211no—
74C13×AGL(1,𝔽7)AGL(1,𝔽7)11yescharacteristic, complemented
75#10587717—71no—
76#10587716—71no—
77ΓL(1,𝔽8)×D13—211no—
78(C13⋊C3)⋊D7—71no—
79D7×C13⋊C3—71no—
80C7×D13⋊C3—71no—
81C2×C3×D13⋊C3—491noHall
82ΓL(1,𝔽8)2—131noHall
83C2×D91—71no—
84D7×D13—211no—
85#64425—131no—
86#64424—131no—
87#64423—131no—
88#64423—131no—
89#64422—131no—
90#64422—131no—
91ΓL(1,𝔽8)⋊D7—131no—
92ΓL(1,𝔽8)⋊D7—391no—
93D7×ΓL(1,𝔽8)—131no—
94D7×ΓL(1,𝔽8)—391no—
95C7×AGL(1,𝔽7)—131no—
96C7×AGL(1,𝔽7)D13⋊C311yescharacteristic, complemented
97C3×C7×C13—71no—
98#63271—71no—
99#63271—71no—
100#63271—71no—
101#63271—421no—
102#63271C2×AGL(1,𝔽7)11yescharacteristic, complemented
103#63270—71no—
104C7×C13⋊C3—71no—
105C13×ΓL(1,𝔽8)C2×AGL(1,𝔽7)11yescharacteristic, complemented
106C2×C3×AGL(1,𝔽7)—911no—
107C2×C3×AGL(1,𝔽7)—911no—
108C2×C3×C13⋊C3—491no—
109C3×D13⋊C3—491no—
110C3×D13⋊C3—491no—
111D72—391noHall
112C2×C7×C13—211no—
113D91—211no—
114D91—211no—
115D91—1261no—
116D91C3×AGL(1,𝔽7)11yescharacteristic, complemented
117C7×D13—211no—
118C13×D7C3×AGL(1,𝔽7)11yescharacteristic, complemented
119C2×C3×D13—1471no—
120C2×D13⋊C3—491no—
121C2×D13⋊C3—491no—
122C2×D13⋊C3—1471no—
123#3862—131no—
124#3861—131no—
125C7×ΓL(1,𝔽8)—131no—
126C7×ΓL(1,𝔽8)C2×D13⋊C311yescharacteristic, complemented
127C2×C3×ΓL(1,𝔽8)—911no—
128C2×C3×ΓL(1,𝔽8)—911no—
129C3×AGL(1,𝔽7)—911no—
130C3×AGL(1,𝔽7)—911no—
131C3×AGL(1,𝔽7)—911no—
132C3×AGL(1,𝔽7)—911no—
133C3×C13⋊C3—491noHall
134C7⋊D7—391no—
135C7×D7—391no—
136C7×D7C3×D13⋊C311yescharacteristic, complemented
137C7×C13—61no—
138C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
139C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
140C2×C3×D7—2731no—
141C2×C3×D7—911no—
142C2×AGL(1,𝔽7)—2731no—
143C2×AGL(1,𝔽7)—2731no—
144C2×AGL(1,𝔽7)—911no—
145C2×AGL(1,𝔽7)—911no—
146C2×AGL(1,𝔽7)—911no—
147C2×AGL(1,𝔽7)—911no—
148C2×C3×C13—71no—
149C3×D13—1471no—
150C3×D13—1471no—
151C2×C13⋊C3—491no—
152C2×C13⋊C3—491no—
153C2×C13⋊C3—1471no—
154D13⋊C3—71no—
155D13⋊C3—491no—
156D13⋊C3—491no—
157D13⋊C3—491no—
158D13⋊C3—491no—
159D13⋊C3—1471no—
160C3×ΓL(1,𝔽8)—911no—
161C3×ΓL(1,𝔽8)—911no—
162C2×D13—1471noHall
163C72C2×C3×D13⋊C311yescharacteristic, complemented, Sylow
164C2×C3×C7—71no—
165C2×C3×C7—911no—
166C3×D7—2731no—
167C3×D7—911no—
168C3×D7—911no—
169C3×D7—911no—
170C2×ΓL(1,𝔽8)—2731no—
171C2×ΓL(1,𝔽8)—2731no—
172C2×ΓL(1,𝔽8)—911no—
173C2×ΓL(1,𝔽8)—911no—
174C2×ΓL(1,𝔽8)—911no—
175C2×ΓL(1,𝔽8)—911no—
176AGL(1,𝔽7)—131no—
177AGL(1,𝔽7)—2731no—
178AGL(1,𝔽7)—2731no—
179AGL(1,𝔽7)—5461no—
180AGL(1,𝔽7)—911no—
181AGL(1,𝔽7)—911no—
182AGL(1,𝔽7)—911no—
183AGL(1,𝔽7)—911no—
184AGL(1,𝔽7)—911no—
185AGL(1,𝔽7)—911no—
186AGL(1,𝔽7)—911no—
187AGL(1,𝔽7)—911no—
188AGL(1,𝔽7)#1058772311yescharacteristic, direct product factor
189C3×C13—71no—
190C13⋊C3—71no—
191C13⋊C3—491no—
192C13⋊C3—491no—
193C22×C32—6371noHall
194C2×D7—2731no—
195C2×D7—8191no—
196C2×C13—211no—
197D13—211no—
198D13—4411no—
199C3×C7—71no—
200C3×C7—911no—
201ΓL(1,𝔽8)—131no—
202ΓL(1,𝔽8)—5461no—
203ΓL(1,𝔽8)—911no—
204ΓL(1,𝔽8)—911no—
205ΓL(1,𝔽8)—911no—
206ΓL(1,𝔽8)—911no—
207ΓL(1,𝔽8)C2×#1058772311yescharacteristic, complemented
208C2×C32—6371no—
209C2×C32—6371no—
210C2×C32—6371no—
211C2×C7—2731no—
212C2×C7—211no—
213D7—131no—
214D7—2731no—
215D7—2731no—
216D7—16381no—
217D7C3×#1058772311yescharacteristic, complemented
218C13AGL(1,𝔽7)211yescharacteristic, complemented, Sylow
219C22×C3—19111no—
220C22×C3—19111no—
221C22×C3—6371no—
222C22×C3—6371no—
223C32—6371noSylow
224C7—61no—
225C7AGL(1,𝔽7)×D13⋊C311yescharacteristic, complemented
226C7C2×C3×#1058772311yescharacteristic, complemented
227C2×C3—71no—
228C2×C3—911no—
229C2×C3—19111no—
230C2×C3—19111no—
231C2×C3—19111no—
232C2×C3—6371no—
233C2×C3—6371no—
234C2×C3—6371no—
235C2×C3—6371no—
236C2×C3—6371no—
237C2×C3—6371no—
238C2×C3—6371no—
239C22—57331noSylow
240C3—71no—
241C3—911no—
242C3—6371no—
243C3—6371no—
244C2—71no—
245C2—2731no—
246C2—57331no—
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. AGL(1,𝔽7)×#63271C2semidirect
2. ΓL(1,𝔽8)×#10587723C2semidirect
3. ΓL(1,𝔽8)⋊#10587723C2semidirect
4. AGL(1,𝔽7)×D91C3semidirect
5. D7×#10587723C3semidirect
6. #21439617C3semidirect
7. #21439609C3semidirect
8. ΓL(1,𝔽8)×#63271C22semidirect
9. ΓL(1,𝔽8)×D91C2×C3semidirect
10. D7×#63271C2×C3semidirect
11. ΓL(1,𝔽8)⋊D91C2×C3semidirect
12. #16838469C2×C3semidirect
13. #16838464C2×C3semidirect
14. #16838463C2×C3semidirect
15. #16838459C2×C3semidirect
16. #16838457C2×C3semidirect
17. #16838452C2×C3semidirect
18. #16838447C2×C3semidirect
19. C7×#10587723C2×C3semidirect
20. C7×C13×AGL(1,𝔽7)C2×C3semidirect
23. D7×D91C32semidirect
24. C13⋊#3862C22×C3semidirect
25. #14274262C22×C3semidirect
26. C7×#63271C22×C3semidirect
27. C7×C13×ΓL(1,𝔽8)C22×C3semidirect
35. C7⋊D91C2×C32semidirect
36. C7×D91C2×C32semidirect
37. C7×C13×D7C2×C32semidirect
51. C72×C13C22×C32semidirect
69. #10587723AGL(1,𝔽7)direct (N × H)
74. C13×AGL(1,𝔽7)AGL(1,𝔽7)semidirect
96. C7×AGL(1,𝔽7)D13⋊C3semidirect
102. #63271C2×AGL(1,𝔽7)semidirect
105. C13×ΓL(1,𝔽8)C2×AGL(1,𝔽7)semidirect
116. D91C3×AGL(1,𝔽7)semidirect
118. C13×D7C3×AGL(1,𝔽7)semidirect
126. C7×ΓL(1,𝔽8)C2×D13⋊C3semidirect
136. C7×D7C3×D13⋊C3semidirect
138. C7×C13C2×C3×AGL(1,𝔽7)semidirect
139. C7×C13C2×C3×AGL(1,𝔽7)semidirect
163. C72C2×C3×D13⋊C3semidirect
188. AGL(1,𝔽7)#10587723direct (N × H)
207. ΓL(1,𝔽8)C2×#10587723semidirect
217. D7C3×#10587723semidirect
218. C13AGL(1,𝔽7)2semidirect
225. C7AGL(1,𝔽7)×D13⋊C3semidirect
226. C7C2×C3×#10587723semidirect

Maximal subgroups

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

subgroupindex
AGL(1,𝔽7)×#632712
ΓL(1,𝔽8)×#105877232
ΓL(1,𝔽8)⋊#105877232
AGL(1,𝔽7)×D913
D7×#105877233
#214396173
#214396093
C2×C3×#105877237
AGL(1,𝔽7)×D13⋊C37
AGL(1,𝔽7)213