Finite Group Databasebeta

← #37524935

Subgroups — #37524935

Order 22932.

248 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 54 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#26039540C211yesmaximal, characteristic, complemented
2#26039540C211yesmaximal, characteristic, complemented
3#26039524C211yesmaximal, characteristic, complemented
4D7×#10587721C311yesmaximal, characteristic, complemented
5D7×#10587721C311yesmaximal, characteristic, complemented
6C13×#64422⋊C2C311yesmaximal, characteristic, complemented
7#21439612C311yesmaximal, characteristic, complemented
8#19153212C2211yescharacteristic, complemented, Hall
9D7×#63271C2×C311yescharacteristic, complemented
10D7×#63271C2×C311yescharacteristic, complemented
11#16838471C2×C311yescharacteristic, complemented
12#16838471C2×C311yescharacteristic, complemented
13C13×#64422C2×C311yescharacteristic, complemented
14C13×#64422C2×C311yescharacteristic, complemented
15#16838458C2×C311yescharacteristic, complemented
16#16838458C2×C311yescharacteristic, complemented
17C13×#64425C2×C311yescharacteristic, complemented
18#16838448C2×C311yescharacteristic, complemented
19C7×#10587721C2×C311yescharacteristic, complemented
20C7×#10587721C2×C311yescharacteristic, complemented
21C2×C13⋊C3×AGL(1,𝔽7)—71nomaximal
22C2×C13⋊C3×AGL(1,𝔽7)—71nomaximal
23C13×D72C3211yescharacteristic, complemented, Hall
24C13×#3862C22×C311yescharacteristic, complemented
25#14274263C22×C311yescharacteristic, complemented
26C7×#63271C22×C311yescharacteristic, complemented
27C7×#63271C22×C311yescharacteristic, complemented
28AGL(1,𝔽7)2—131nomaximal, Hall
29C2×ΓL(1,𝔽8)×C13⋊C3—71no—
30C2×ΓL(1,𝔽8)×C13⋊C3—71no—
31C13⋊C3×AGL(1,𝔽7)—71no—
32C13⋊C3×AGL(1,𝔽7)—71no—
33C13⋊C3×AGL(1,𝔽7)—71no—
34C13⋊C3×AGL(1,𝔽7)—71no—
35C13×C7⋊D7C2×C3211yescharacteristic, complemented
36C7×C13×D7C2×C3211yescharacteristic, complemented
37C7×C13×D7C2×C3211yescharacteristic, complemented
38C2×#10587721—211no—
39C2×#10587721—211no—
40C2×#10587720—71no—
41C2×#10587720—71no—
42C2×C13×AGL(1,𝔽7)—71no—
43C2×C13×AGL(1,𝔽7)—71no—
44C2×D7×C13⋊C3—71no—
45C2×D7×C13⋊C3—71no—
46ΓL(1,𝔽8)⋊AGL(1,𝔽7)—131no—
47ΓL(1,𝔽8)×AGL(1,𝔽7)—131no—
48ΓL(1,𝔽8)×AGL(1,𝔽7)—131no—
49ΓL(1,𝔽8)×C13⋊C3—71no—
50ΓL(1,𝔽8)×C13⋊C3—71no—
51C72×C13C22×C3211yescharacteristic, complemented, Hall
52#64422⋊C2C13⋊C311yescharacteristic, complemented
53#64423⋊C2—131no—
54D7×AGL(1,𝔽7)—131no—
55D7×AGL(1,𝔽7)—131no—
56C2×#63271—211no—
57C2×#63271—211no—
58C2×#63270—71no—
59C2×#63270—71no—
60C2×C13×ΓL(1,𝔽8)—71no—
61C2×C13×ΓL(1,𝔽8)—71no—
62C2×C7×C13⋊C3—71no—
63C2×C7×C13⋊C3—71no—
64#10587721—211no—
65#10587721—211no—
66#10587721AGL(1,𝔽7)11yescharacteristic, complemented
67#10587721AGL(1,𝔽7)11yescharacteristic, complemented
68#10587720—71no—
69#10587720—71no—
70#10587720—71no—
71#10587720—71no—
72#10587720—421no—
73C13×AGL(1,𝔽7)—71no—
74C13×AGL(1,𝔽7)—71no—
75C13×AGL(1,𝔽7)—71no—
76C13×AGL(1,𝔽7)—71no—
77D7×C13⋊C3—71no—
78D7×C13⋊C3—71no—
79D7×C13⋊C3—71no—
80D7×C13⋊C3—71no—
81C22×C3×C13⋊C3—491noHall
82ΓL(1,𝔽8)2—131noHall
83C2×C13×D7—211no—
84C2×C13×D7—211no—
85#64425C2×C13⋊C311yescharacteristic, complemented
86#64424—131no—
87#64423—131no—
88#64423—131no—
89#64422C2×C13⋊C311yescharacteristic, complemented
90#64422C2×C13⋊C311yescharacteristic, complemented
91ΓL(1,𝔽8)⋊D7—131no—
92ΓL(1,𝔽8)⋊D7—131no—
93D7×ΓL(1,𝔽8)—131no—
94D7×ΓL(1,𝔽8)—131no—
95C7×AGL(1,𝔽7)—131no—
96C7×AGL(1,𝔽7)—131no—
97#63271C2×AGL(1,𝔽7)11yescharacteristic, complemented
98#63271C2×AGL(1,𝔽7)11yescharacteristic, complemented
99#63270—71no—
100#63270—71no—
101#63270—421no—
102C7×C13⋊C3—71no—
103C7×C13⋊C3—71no—
104C13×ΓL(1,𝔽8)—71no—
105C13×ΓL(1,𝔽8)—71no—
106C2×C3×AGL(1,𝔽7)—911no—
107C2×C3×AGL(1,𝔽7)—911no—
108C2×C3×C13⋊C3—491no—
109C2×C3×C13⋊C3—491no—
110C2×C3×C13⋊C3—491no—
111D72C3×C13⋊C311yescharacteristic, complemented, Hall
112C2×C7×C13—211no—
113C2×C7×C13—211no—
114C13×D7—211no—
115C13×D7—211no—
116C13×D7—1261no—
117C13×D7C3×AGL(1,𝔽7)11yescharacteristic, complemented
118C13×D7C3×AGL(1,𝔽7)11yescharacteristic, complemented
119C22×C3×C13—491no—
120C22×C13⋊C3—491no—
121C22×C13⋊C3—491no—
122C22×C13⋊C3—1471no—
123#3862C22×C13⋊C311yescharacteristic, complemented
124#3861—131no—
125C7×ΓL(1,𝔽8)—131no—
126C7×ΓL(1,𝔽8)—131no—
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⋊D7C2×C3×C13⋊C311yescharacteristic, complemented
135C7×D7C2×C3×C13⋊C311yescharacteristic, complemented
136C7×D7C2×C3×C13⋊C311yescharacteristic, complemented
137C7×C13—61no—
138C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
139C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
140C2×C3×D7—911no—
141C2×C3×D7—911no—
142C2×AGL(1,𝔽7)—71no—
143C2×AGL(1,𝔽7)—71no—
144C2×AGL(1,𝔽7)—2731no—
145C2×AGL(1,𝔽7)—2731no—
146C2×AGL(1,𝔽7)—911no—
147C2×AGL(1,𝔽7)—911no—
148C2×C3×C13—491no—
149C2×C3×C13—491no—
150C2×C3×C13—491no—
151C2×C13⋊C3—71no—
152C2×C13⋊C3—71no—
153C2×C13⋊C3—491no—
154C2×C13⋊C3—491no—
155C2×C13⋊C3—491no—
156C2×C13⋊C3—491no—
157C2×C13⋊C3—1471no—
158C2×C13⋊C3—1471no—
159C2×C13⋊C3—1471no—
160C3×ΓL(1,𝔽8)—911no—
161C3×ΓL(1,𝔽8)—911no—
162C22×C13—4411noHall
163C72C22×C3×C13⋊C311yescharacteristic, complemented, Sylow
164C2×C3×C7—911no—
165C2×C3×C7—911no—
166C3×D7—911no—
167C3×D7—911no—
168C3×D7—911no—
169C3×D7—911no—
170C2×ΓL(1,𝔽8)—71no—
171C2×ΓL(1,𝔽8)—71no—
172C2×ΓL(1,𝔽8)—2731no—
173C2×ΓL(1,𝔽8)—2731no—
174C2×ΓL(1,𝔽8)—911no—
175C2×ΓL(1,𝔽8)—911no—
176AGL(1,𝔽7)—71no—
177AGL(1,𝔽7)—71no—
178AGL(1,𝔽7)—71no—
179AGL(1,𝔽7)—71no—
180AGL(1,𝔽7)—131no—
181AGL(1,𝔽7)—131no—
182AGL(1,𝔽7)—2731no—
183AGL(1,𝔽7)—5461no—
184AGL(1,𝔽7)—911no—
185AGL(1,𝔽7)—911no—
186AGL(1,𝔽7)—911no—
187AGL(1,𝔽7)—911no—
188AGL(1,𝔽7)—911no—
189C3×C13—491no—
190C13⋊C3—71no—
191C13⋊C3—71no—
192C13⋊C3—491no—
193C22×C32—6371noHall
194C2×D7—211no—
195C2×D7—211no—
196C2×C13—71no—
197C2×C13—211no—
198C2×C13—4411no—
199C3×C7—911no—
200C3×C7—911no—
201ΓL(1,𝔽8)—71no—
202ΓL(1,𝔽8)—71no—
203ΓL(1,𝔽8)—131no—
204ΓL(1,𝔽8)—131no—
205ΓL(1,𝔽8)—5461no—
206ΓL(1,𝔽8)—911no—
207ΓL(1,𝔽8)—911no—
208C2×C32—6371no—
209C2×C32—6371no—
210C2×C32—6371no—
211C2×C7—211no—
212C2×C7—211no—
213D7—71no—
214D7—211no—
215D7—1261no—
216D7C13⋊C3×AGL(1,𝔽7)11yescharacteristic, complemented
217D7C13⋊C3×AGL(1,𝔽7)11yescharacteristic, complemented
218C13AGL(1,𝔽7)211yescharacteristic, complemented, Sylow
219C22×C3—491no—
220C22×C3—19111no—
221C22×C3—19111no—
222C22×C3—6371no—
223C32—6371noSylow
224C7—61no—
225C7C2×C13⋊C3×AGL(1,𝔽7)11yescharacteristic, complemented
226C7C2×C13⋊C3×AGL(1,𝔽7)11yescharacteristic, complemented
227C2×C3—491no—
228C2×C3—491no—
229C2×C3—491no—
230C2×C3—911no—
231C2×C3—911no—
232C2×C3—19111no—
233C2×C3—19111no—
234C2×C3—19111no—
235C2×C3—19111no—
236C2×C3—6371no—
237C2×C3—6371no—
238C2×C3—6371no—
239C22—4411noSylow
240C3—491no—
241C3—911no—
242C3—911no—
243C3—6371no—
244C2—211no—
245C2—211no—
246C2—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. #26039540C2semidirect
2. #26039540C2semidirect
3. #26039524C2semidirect
4. D7×#10587721C3semidirect
5. D7×#10587721C3semidirect
6. C13×#64422⋊C2C3semidirect
7. #21439612C3semidirect
8. #19153212C22semidirect
9. D7×#63271C2×C3semidirect
10. D7×#63271C2×C3semidirect
11. #16838471C2×C3semidirect
12. #16838471C2×C3semidirect
13. C13×#64422C2×C3semidirect
14. C13×#64422C2×C3semidirect
15. #16838458C2×C3semidirect
16. #16838458C2×C3semidirect
17. C13×#64425C2×C3semidirect
18. #16838448C2×C3semidirect
19. C7×#10587721C2×C3semidirect
20. C7×#10587721C2×C3semidirect
23. C13×D72C32semidirect
24. C13×#3862C22×C3semidirect
25. #14274263C22×C3semidirect
26. C7×#63271C22×C3semidirect
27. C7×#63271C22×C3semidirect
35. C13×C7⋊D7C2×C32semidirect
36. C7×C13×D7C2×C32semidirect
37. C7×C13×D7C2×C32semidirect
51. C72×C13C22×C32semidirect
52. #64422⋊C2C13⋊C3semidirect
66. #10587721AGL(1,𝔽7)semidirect
67. #10587721AGL(1,𝔽7)semidirect
85. #64425C2×C13⋊C3semidirect
89. #64422C2×C13⋊C3semidirect
90. #64422C2×C13⋊C3semidirect
97. #63271C2×AGL(1,𝔽7)semidirect
98. #63271C2×AGL(1,𝔽7)semidirect
111. D72C3×C13⋊C3semidirect
117. C13×D7C3×AGL(1,𝔽7)semidirect
118. C13×D7C3×AGL(1,𝔽7)semidirect
123. #3862C22×C13⋊C3semidirect
134. C7⋊D7C2×C3×C13⋊C3semidirect
135. C7×D7C2×C3×C13⋊C3semidirect
136. C7×D7C2×C3×C13⋊C3semidirect
138. C7×C13C2×C3×AGL(1,𝔽7)semidirect
139. C7×C13C2×C3×AGL(1,𝔽7)semidirect
163. C72C22×C3×C13⋊C3semidirect
216. D7C13⋊C3×AGL(1,𝔽7)semidirect
217. D7C13⋊C3×AGL(1,𝔽7)semidirect
218. C13AGL(1,𝔽7)2semidirect
225. C7C2×C13⋊C3×AGL(1,𝔽7)semidirect
226. C7C2×C13⋊C3×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
#260395402
#260395242
D7×#105877213
C13×#64422⋊C23
#214396123
C2×C13⋊C3×AGL(1,𝔽7)7
AGL(1,𝔽7)213