Finite Group Databasebeta

← #37524949

Subgroups — #37524949

Order 22932.

312 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C13⋊C3×#64423C211yesmaximal, characteristic, complemented
2(C13⋊C3)⋊#64423C211yesmaximal, characteristic, complemented
3(C13⋊C3)⋊#64424C211yesmaximal, characteristic, complemented
4D7×(C13⋊C3)⋊D7C311yesmaximal, characteristic, complemented
5#64423⋊D13C311yesmaximal, characteristic, complemented
6#21439609C311yesmaximal, characteristic, complemented
7#21439608C311yesmaximal, characteristic, complemented
8C13⋊C3×#3861C2211yescharacteristic, complemented, Hall
9(C13⋊C3)⋊(C7⋊D7)C2×C311yescharacteristic, complemented
10#16838460C2×C311yescharacteristic, complemented
11#16838459C2×C311yescharacteristic, complemented
12#16838458C2×C311yescharacteristic, complemented
13#16838457C2×C311yescharacteristic, complemented
14#16838456C2×C311yescharacteristic, complemented
15C13×#64423C2×C311yescharacteristic, complemented
16#16838450C2×C311yescharacteristic, complemented
17#16838447C2×C311yescharacteristic, complemented
18#16838446C2×C311yescharacteristic, complemented
19C7×D7×C13⋊C3C2×C311yescharacteristic, complemented
20C7×(C13⋊C3)⋊D7C2×C311yescharacteristic, complemented
21C2×ΓL(1,𝔽8)⋊(D13⋊C3)—71nomaximal
22AGL(1,𝔽7)×D13⋊C3—71nomaximal
23D7×D91C3211yescharacteristic, complemented, Hall
24#14274263C22×C311yescharacteristic, complemented
25#14274262C22×C311yescharacteristic, complemented
26C13×#3861C22×C311yescharacteristic, complemented
27C72×C13⋊C3C22×C311yescharacteristic, complemented
28C3×#64423⋊C2—131nomaximal, Hall
29C2×ΓL(1,𝔽8)×C13⋊C3—71no—
30ΓL(1,𝔽8)⋊(D13⋊C3)—71no—
31ΓL(1,𝔽8)⋊(D13⋊C3)—71no—
32ΓL(1,𝔽8)⋊(D13⋊C3)—71no—
33ΓL(1,𝔽8)⋊(D13⋊C3)—143no—
34C13⋊C3×AGL(1,𝔽7)—71no—
35ΓL(1,𝔽8)×D13⋊C3—71no—
36C7⋊D91C2×C3211yescharacteristic, complemented
37C7×D91C2×C3211yescharacteristic, complemented
38C7×C13×D7C2×C3211yescharacteristic, complemented
39C2×#10587723—71no—
40C2×#10587722—71no—
41C2×ΓL(1,𝔽8)⋊D13—71no—
42C2×(C13⋊C3)⋊D7—211no—
43#10587716⋊C2—71no—
44#10587717⋊C2—71no—
45D13×AGL(1,𝔽7)—71no—
46D7×D13⋊C3—71no—
47C3×#64424—131no—
48C3×#64423—131no—
49C3×#64423—131no—
50ΓL(1,𝔽8)×C13⋊C3—71no—
51ΓL(1,𝔽8)×C13⋊C3—71no—
52ΓL(1,𝔽8)×C13⋊C3—143no—
53C72×C13C22×C3211yescharacteristic, complemented, Hall
54C3×D72—131no—
55#64423⋊C2—131no—
56#64423⋊C2—131no—
57#64423⋊C2—391no—
58C2×#63271—71no—
59C2×#63270—71no—
60C2×C13×ΓL(1,𝔽8)—71no—
61C2×C7×C13⋊C3—211no—
62#10587723—71no—
63#10587723—71no—
64#10587723—71no—
65#10587723—143no—
66#10587722—71no—
67#10587722—71no—
68#10587722—71no—
69#10587722—143no—
70#10587721—71no—
71#10587720—71no—
72ΓL(1,𝔽8)⋊D13—71no—
73ΓL(1,𝔽8)⋊D13—71no—
74ΓL(1,𝔽8)⋊D13—71no—
75ΓL(1,𝔽8)⋊D13—143no—
76C13×AGL(1,𝔽7)—71no—
77#10587717—71no—
78#10587716—71no—
79ΓL(1,𝔽8)×D13—71no—
80(C13⋊C3)⋊D7—211no—
81(C13⋊C3)⋊D7—211no—
82(C13⋊C3)⋊D7—423no—
83(C13⋊C3)⋊D7AGL(1,𝔽7)11yescharacteristic, complemented
84D7×C13⋊C3AGL(1,𝔽7)11yescharacteristic, complemented
85C7×D13⋊C3—211no—
86C2×C3×D13⋊C3—491noHall
87C3×#3861—131noHall
88C2×D91—211no—
89D7×D13—71no—
90C3×C7⋊D7—131no—
91C3×C7×D7—131no—
92C3×C7×D7—131no—
93#64424—131no—
94#64424—131no—
95#64424—391no—
96#64423—131no—
97#64423—131no—
98#64423—131no—
99#64423—131no—
100#64423—391no—
101#64423D13⋊C311yescharacteristic, complemented
102#63271—71no—
103#63271—71no—
104#63271—143no—
105#63270—71no—
106#63270—71no—
107#63270—143no—
108C7×C13⋊C3—23no—
109C7×C13⋊C3C2×AGL(1,𝔽7)11yescharacteristic, complemented
110C7×C13⋊C3C2×AGL(1,𝔽7)11yescharacteristic, complemented
111C13×ΓL(1,𝔽8)—71no—
112C13×ΓL(1,𝔽8)—71no—
113C13×ΓL(1,𝔽8)—143no—
114C2×C3×AGL(1,𝔽7)—911no—
115C2×C3×AGL(1,𝔽7)—911no—
116C2×C3×C13⋊C3—491no—
117C3×D13⋊C3—491no—
118C3×D13⋊C3—491no—
119D72—391noHall
120C2×C7×C13—211no—
121D91—211no—
122D91—211no—
123D91—423no—
124D91C3×AGL(1,𝔽7)11yescharacteristic, complemented
125C7×D13—211no—
126C13×D7C3×AGL(1,𝔽7)11yescharacteristic, complemented
127C2×C3×D13—491no—
128C2×D13⋊C3—491no—
129C2×D13⋊C3—491no—
130C2×D13⋊C3—1471no—
131C3×C72—131no—
132#3861—131no—
133#3861—131no—
134#3861C2×D13⋊C311yescharacteristic, complemented
135C2×C3×ΓL(1,𝔽8)—911no—
136C2×C3×ΓL(1,𝔽8)—911no—
137C3×AGL(1,𝔽7)—911no—
138C3×AGL(1,𝔽7)—911no—
139C3×AGL(1,𝔽7)—911no—
140C3×AGL(1,𝔽7)—911no—
141C3×AGL(1,𝔽7)—1823no—
142C3×C13⋊C3—491noHall
143C7⋊D7—391no—
144C7×D7—391no—
145C7×D7C3×D13⋊C311yescharacteristic, complemented
146C7×C13—23no—
147C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
148C7×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
149C2×C3×D7—2731no—
150C2×C3×D7—2731no—
151C2×AGL(1,𝔽7)—2731no—
152C2×AGL(1,𝔽7)—911no—
153C2×AGL(1,𝔽7)—911no—
154C2×AGL(1,𝔽7)—911no—
155C2×AGL(1,𝔽7)—911no—
156C2×AGL(1,𝔽7)—911no—
157C2×C3×C13—491no—
158C3×D13—491no—
159C3×D13—491no—
160C2×C13⋊C3—211no—
161C2×C13⋊C3—491no—
162C2×C13⋊C3—491no—
163D13⋊C3—71no—
164D13⋊C3—491no—
165D13⋊C3—491no—
166D13⋊C3—491no—
167D13⋊C3—491no—
168D13⋊C3—1471no—
169C3×ΓL(1,𝔽8)—911no—
170C3×ΓL(1,𝔽8)—911no—
171C3×ΓL(1,𝔽8)—1823no—
172C2×D13—1471noHall
173C72C2×C3×D13⋊C311yescharacteristic, complemented, Sylow
174C2×C3×C7—2731no—
175C2×C3×C7—2731no—
176C3×D7—131no—
177C3×D7—131no—
178C3×D7—2731no—
179C3×D7—5463no—
180C3×D7—911no—
181C2×ΓL(1,𝔽8)—71no—
182C2×ΓL(1,𝔽8)—2731no—
183C2×ΓL(1,𝔽8)—911no—
184C2×ΓL(1,𝔽8)—911no—
185C2×ΓL(1,𝔽8)—911no—
186C2×ΓL(1,𝔽8)—911no—
187AGL(1,𝔽7)—71no—
188AGL(1,𝔽7)—2731no—
189AGL(1,𝔽7)—2731no—
190AGL(1,𝔽7)—2731no—
191AGL(1,𝔽7)—5463no—
192AGL(1,𝔽7)—911no—
193AGL(1,𝔽7)—911no—
194AGL(1,𝔽7)—911no—
195AGL(1,𝔽7)—911no—
196AGL(1,𝔽7)—911no—
197AGL(1,𝔽7)—911no—
198AGL(1,𝔽7)—911no—
199AGL(1,𝔽7)—911no—
200AGL(1,𝔽7)—1823no—
201AGL(1,𝔽7)—1823no—
202C3×C13—491no—
203C13⋊C3—491no—
204C13⋊C3—491no—
205C13⋊C3#64423⋊C211yescharacteristic, complemented
206C22×C32—6371noHall
207C2×D7—2731no—
208C2×D7—8191no—
209C2×C13—211no—
210D13—71no—
211D13—1471no—
212C3×C7—131no—
213C3×C7—131no—
214C3×C7—263no—
215ΓL(1,𝔽8)—71no—
216ΓL(1,𝔽8)—71no—
217ΓL(1,𝔽8)—143no—
218ΓL(1,𝔽8)—911no—
219ΓL(1,𝔽8)—911no—
220ΓL(1,𝔽8)—911no—
221ΓL(1,𝔽8)—911no—
222ΓL(1,𝔽8)—1823no—
223ΓL(1,𝔽8)—1823no—
224C2×C32—6371no—
225C2×C32—6371no—
226C2×C32—6371no—
227C2×C7—211no—
228C2×C7—8191no—
229D7—391no—
230D7—8191no—
231D7—8191no—
232D7—16383no—
233D7ΓL(1,𝔽8)⋊(D13⋊C3)11yescharacteristic, complemented
234C13C3×#64423⋊C211yescharacteristic, complemented, Sylow
235C22×C3—19111no—
236C22×C3—19111no—
237C22×C3—6371no—
238C22×C3—6371no—
239C32—6371noSylow
240C7—23no—
241C7AGL(1,𝔽7)×D13⋊C311yescharacteristic, complemented
242C7C2×ΓL(1,𝔽8)⋊(D13⋊C3)11yescharacteristic, complemented
243C2×C3—2731no—
244C2×C3—2731no—
245C2×C3—491no—
246C2×C3—19111no—
247C2×C3—19111no—
248C2×C3—19111no—
249C2×C3—6371no—
250C2×C3—6371no—
251C2×C3—6371no—
252C2×C3—6371no—
253C2×C3—6371no—
254C2×C3—6371no—
255C22—57331noSylow
256C3—131no—
257C3—491no—
258C3—6371no—
259C3—6371no—
260C2—211no—
261C2—8191no—
262C2—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. C13⋊C3×#64423C2semidirect
2. (C13⋊C3)⋊#64423C2semidirect
3. (C13⋊C3)⋊#64424C2semidirect
4. D7×(C13⋊C3)⋊D7C3semidirect
5. #64423⋊D13C3semidirect
6. #21439609C3semidirect
7. #21439608C3semidirect
8. C13⋊C3×#3861C22semidirect
9. (C13⋊C3)⋊(C7⋊D7)C2×C3semidirect
10. #16838460C2×C3semidirect
11. #16838459C2×C3semidirect
12. #16838458C2×C3semidirect
13. #16838457C2×C3semidirect
14. #16838456C2×C3semidirect
15. C13×#64423C2×C3semidirect
16. #16838450C2×C3semidirect
17. #16838447C2×C3semidirect
18. #16838446C2×C3semidirect
19. C7×D7×C13⋊C3C2×C3semidirect
20. C7×(C13⋊C3)⋊D7C2×C3semidirect
23. D7×D91C32semidirect
24. #14274263C22×C3semidirect
25. #14274262C22×C3semidirect
26. C13×#3861C22×C3semidirect
27. C72×C13⋊C3C22×C3semidirect
36. C7⋊D91C2×C32semidirect
37. C7×D91C2×C32semidirect
38. C7×C13×D7C2×C32semidirect
53. C72×C13C22×C32semidirect
83. (C13⋊C3)⋊D7AGL(1,𝔽7)semidirect
84. D7×C13⋊C3AGL(1,𝔽7)semidirect
101. #64423D13⋊C3semidirect
109. C7×C13⋊C3C2×AGL(1,𝔽7)semidirect
110. C7×C13⋊C3C2×AGL(1,𝔽7)semidirect
124. D91C3×AGL(1,𝔽7)semidirect
126. C13×D7C3×AGL(1,𝔽7)semidirect
134. #3861C2×D13⋊C3semidirect
145. C7×D7C3×D13⋊C3semidirect
147. C7×C13C2×C3×AGL(1,𝔽7)semidirect
148. C7×C13C2×C3×AGL(1,𝔽7)semidirect
173. C72C2×C3×D13⋊C3semidirect
205. C13⋊C3#64423⋊C2semidirect
233. D7ΓL(1,𝔽8)⋊(D13⋊C3)semidirect
234. C13C3×#64423⋊C2semidirect
241. C7AGL(1,𝔽7)×D13⋊C3semidirect
242. C7C2×ΓL(1,𝔽8)⋊(D13⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C13⋊C3×#644232
(C13⋊C3)⋊#644232
(C13⋊C3)⋊#644242
D7×(C13⋊C3)⋊D73
#64423⋊D133
#214396093
#214396083
C2×ΓL(1,𝔽8)⋊(D13⋊C3)7
AGL(1,𝔽7)×D13⋊C37
C3×#64423⋊C213