Finite Group Databasebeta

← C3⋊D3×AGL(1,𝔽5)2 #72230975

Subgroups — C3⋊D3×AGL(1,𝔽5)2 #72230975

Order 7200.

1344 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 182 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,𝔽5)×C3⋊ΓL(1,𝔽16)C212yesmaximal, complemented
2D5×C3⋊D3×AGL(1,𝔽5)C212yesmaximal
3#72230965C211yesmaximal, characteristic, complemented
4C32×AGL(1,𝔽5)2C211yesmaximal, characteristic, complemented
5D3×AGL(1,𝔽5)2—34nomaximal
6C3⋊D3×D5⋊AGL(1,𝔽5)C211yesmaximal, characteristic
7D52×C3⋊D3C2211yescharacteristic
8#1042⋊(C3⋊D3)C412yescomplemented
9AGL(1,𝔽5)×C3⋊D15C412yescomplemented
10D5×C3⋊ΓL(1,𝔽16)C2212yes—
11C3⋊D3×#1042C412yescomplemented
12#14268426C411yescharacteristic, complemented
13#14268425C411yescharacteristic, complemented
14D5⋊(C3⋊ΓL(1,𝔽16))C2211yescharacteristic
15C3⋊D3×#1044C411yescharacteristic, complemented
16C3⋊D3×#1043C411yescharacteristic, complemented
17C5×C3⋊D3×AGL(1,𝔽5)C412yescomplemented
18C32×D5×AGL(1,𝔽5)C2212yes—
19C32×D5⋊AGL(1,𝔽5)C2211yescharacteristic
20C4×C3⋊D3×AGL(1,𝔽5)—52nomaximal
21D3×D5×AGL(1,𝔽5)—38no—
22D3×D5⋊AGL(1,𝔽5)—34no—
23ΓL(1,𝔽16)⋊AGL(1,𝔽5)—34no—
24AGL(1,𝔽5)×ΓL(1,𝔽16)—38no—
25C3×AGL(1,𝔽5)2D314yescomplemented
26#11744719C2×C411yescharacteristic
27D5×C3⋊D15C2×C412yes—
28C3⋊D3×C5⋊D5C2×C411yescharacteristic
29C5×D5×C3⋊D3C2×C412yes—
30C32×D52C2311yescharacteristic
31C3⋊(C3⋊#1042)C2×C412yescomplemented
32C3⋊(C3⋊#1044)C2×C411yescharacteristic, complemented
33C3⋊(C3⋊#1043)C2×C411yescharacteristic, complemented
34C5×C3⋊ΓL(1,𝔽16)C2×C412yescomplemented
35C32×#1042C2×C412yescomplemented
36C32×#1044C2×C411yescharacteristic, complemented
37C32×#1043C2×C411yescharacteristic, complemented
38C32×C5×AGL(1,𝔽5)C2×C412yescomplemented
39C2×AGL(1,𝔽5)2—91noHall
40C2×C3⋊D3×AGL(1,𝔽5)—52no—
41C2×C3⋊D3×AGL(1,𝔽5)—52no—
42C4×D5×C3⋊D3—52no—
43C4×C3⋊ΓL(1,𝔽16)—52no—
44C32×C4×AGL(1,𝔽5)—52no—
45AGL(1,𝔽5)×C3⋊Dic3—52no—
46AGL(1,𝔽5)×C3⋊Dic3—52no—
47D3×D52—34no—
48D3×#1042—38no—
49ΓL(1,𝔽16)⋊D5—38no—
50D5×ΓL(1,𝔽16)—38no—
51AGL(1,𝔽5)×D15—38no—
52#10597183—34no—
53#10597182—34no—
54D5⋊ΓL(1,𝔽16)—34no—
55D3×#1044—34no—
56D3×#1043—34no—
57C5×D3×AGL(1,𝔽5)—38no—
58C3×D5×AGL(1,𝔽5)C2×D318yes—
59C3×D5⋊AGL(1,𝔽5)C2×D314yes—
60C4×D3×AGL(1,𝔽5)—158no—
61C3⋊(C3⋊(C5⋊D5))C4211yescharacteristic, complemented
62C5×C3⋊D15C4212yescomplemented
63C52×C3⋊D3C4211yescharacteristic, complemented
64C32×C5⋊D5C22×C411yescharacteristic
65C32×C5×D5C22×C412yes—
66C2×D5⋊AGL(1,𝔽5)—91no—
67C2×D5×AGL(1,𝔽5)—92no—
68AGL(1,𝔽5)2—91no—
69AGL(1,𝔽5)2—92no—
70AGL(1,𝔽5)2C3⋊D311yescharacteristic, direct product factor
71C2×D5×C3⋊D3—52no—
72C2×C3⋊ΓL(1,𝔽16)—52no—
73C2×C3⋊ΓL(1,𝔽16)—52no—
74C2×C32×AGL(1,𝔽5)—52no—
75C2×C32×AGL(1,𝔽5)—52no—
76C3⋊D3×AGL(1,𝔽5)—52no—
77C3⋊D3×AGL(1,𝔽5)—52no—
78C3⋊D3×AGL(1,𝔽5)—52no—
79C3⋊D3×AGL(1,𝔽5)—52no—
80C3⋊D3×AGL(1,𝔽5)—52no—
81C3⋊D3×AGL(1,𝔽5)—52no—
82C3⋊D3×AGL(1,𝔽5)—52no—
83C3⋊D3×AGL(1,𝔽5)—201no—
84C3⋊D3×AGL(1,𝔽5)AGL(1,𝔽5)12yesdirect product factor
85C4×C3⋊D15—52no—
86C4×C5×C3⋊D3—52no—
87C32×C4×D5—52no—
88#67137—52no—
89C3⋊D3×Dic5—52no—
90D5×C3⋊Dic3—52no—
91D15⋊D5—34no—
92D5×D15—38no—
93D3×C5⋊D5—34no—
94C5×D3×D5—38no—
95C3×D52C22×D314yes—
96C3⋊#1044—34no—
97C3⋊#1043—34no—
98C3⋊#1042—38no—
99C5×ΓL(1,𝔽16)—38no—
100C3×#1044C4×D314yescomplemented
101C3×#1043C4×D314yescomplemented
102C3×#1042C4×D318yescomplemented
103C3×C5×AGL(1,𝔽5)C4×D318yescomplemented
104C42×C3⋊D3—251noHall
105C2×D3×AGL(1,𝔽5)—158no—
106C2×D3×AGL(1,𝔽5)—158no—
107C4×D3×D5—158no—
108C4×ΓL(1,𝔽16)—158no—
109C3×C4×AGL(1,𝔽5)—58no—
110Dic3×AGL(1,𝔽5)—158no—
111Dic3×AGL(1,𝔽5)—158no—
112C32×C52C2×C4211yescharacteristic, complemented, Hall
113C2×D52—91no—
114C2×#1044—91no—
115C2×#1043—91no—
116C2×#1042—92no—
117C2×C5×AGL(1,𝔽5)—92no—
118D5⋊AGL(1,𝔽5)—91no—
119D5⋊AGL(1,𝔽5)—91no—
120D5⋊AGL(1,𝔽5)—91no—
121D5⋊AGL(1,𝔽5)C2×C3⋊D311yescharacteristic
122D5×AGL(1,𝔽5)—92no—
123D5×AGL(1,𝔽5)—92no—
124D5×AGL(1,𝔽5)—92no—
125D5×AGL(1,𝔽5)C2×C3⋊D312yes—
126C2×C3⋊D15—52no—
127C2×C5×C3⋊D3—52no—
128C2×C32×D5—52no—
129D5×C3⋊D3—52no—
130D5×C3⋊D3—52no—
131D5×C3⋊D3—52no—
132D5×C3⋊D3—201no—
133D5×C3⋊D3C2×AGL(1,𝔽5)12yes—
134C3⋊ΓL(1,𝔽16)—52no—
135C3⋊ΓL(1,𝔽16)—52no—
136C3⋊ΓL(1,𝔽16)—52no—
137C3⋊ΓL(1,𝔽16)—201no—
138C3⋊ΓL(1,𝔽16)C2×AGL(1,𝔽5)12yescomplemented
139C32×AGL(1,𝔽5)—52no—
140C32×AGL(1,𝔽5)—52no—
141C32×AGL(1,𝔽5)—52no—
142C32×AGL(1,𝔽5)—201no—
143C32×AGL(1,𝔽5)C2×AGL(1,𝔽5)12yescomplemented
144C32×C4×C5—52no—
145C3⋊Dic15—52no—
146C5×C3⋊Dic3—52no—
147C32×Dic5—52no—
148C2×C4×AGL(1,𝔽5)—452no—
149C3⋊(C5⋊D5)—34no—
150C5×D15—38no—
151C52×D3—34no—
152C3×C5⋊D5C2×C4×D314yes—
153C3×C5×D5C2×C4×D318yes—
154C2×C4×C3⋊D3—252no—
155C2×C4×C3⋊D3—251no—
156C32×C42—251no—
157C4×C3⋊Dic3—252no—
158C4×C3⋊Dic3—251no—
159C2×D3×D5—158no—
160C2×ΓL(1,𝔽16)—158no—
161C2×ΓL(1,𝔽16)—158no—
162C2×C3×AGL(1,𝔽5)—58no—
163C2×C3×AGL(1,𝔽5)—58no—
164D3×AGL(1,𝔽5)—38no—
165D3×AGL(1,𝔽5)—158no—
166D3×AGL(1,𝔽5)—158no—
167D3×AGL(1,𝔽5)—158no—
168D3×AGL(1,𝔽5)—158no—
169D3×AGL(1,𝔽5)—158no—
170D3×AGL(1,𝔽5)—158no—
171D3×AGL(1,𝔽5)—158no—
172D3×AGL(1,𝔽5)—604no—
173C4×D15—158no—
174C4×C5×D3—158no—
175C3×C4×D5—58no—
176#1200—158no—
177D3×Dic5—158no—
178D5×Dic3—158no—
179C2×C5⋊D5—91no—
180C2×C5×D5—92no—
181D52—92no—
182D52—91no—
183D52C22×C3⋊D311yescharacteristic
184#1044—91no—
185#1044C4×C3⋊D311yescharacteristic, complemented
186#1043—91no—
187#1043C4×C3⋊D311yescharacteristic, complemented
188#1042—92no—
189#1042C4×C3⋊D312yescomplemented
190C5×AGL(1,𝔽5)—92no—
191C5×AGL(1,𝔽5)C4×C3⋊D312yescomplemented
192C42×D3—754no—
193C2×C32×C5—52no—
194C3⋊D15—52no—
195C3⋊D15—201no—
196C3⋊D15C4×AGL(1,𝔽5)12yescomplemented
197C5×C3⋊D3—41no—
198C5×C3⋊D3—52no—
199C5×C3⋊D3C4×AGL(1,𝔽5)12yescomplemented
200C32×D5—52no—
201C32×D5—201no—
202C32×D5C22×AGL(1,𝔽5)12yes—
203C22×AGL(1,𝔽5)—452no—
204C22×AGL(1,𝔽5)—452no—
205C2×C4×D5—452no—
206C4×AGL(1,𝔽5)—52no—
207C4×AGL(1,𝔽5)—452no—
208C4×AGL(1,𝔽5)—452no—
209C4×AGL(1,𝔽5)—452no—
210C3×C52C42×D314yescomplemented
211C22×C3⋊D3—251no—
212C2×C32×C4—252no—
213C2×C32×C4—251no—
214C2×C3⋊Dic3—252no—
215C2×C3⋊Dic3—251no—
216C4×C3⋊D3—52no—
217C4×C3⋊D3—252no—
218C4×C3⋊D3—252no—
219C4×C3⋊D3—252no—
220C4×C3⋊D3—251no—
221C4×C3⋊D3—251no—
222C4×C3⋊D3—251no—
223C4×C3⋊D3—251no—
224C2×D15—158no—
225C2×C5×D3—158no—
226C2×C3×D5—58no—
227D3×D5—38no—
228D3×D5—158no—
229D3×D5—158no—
230D3×D5—158no—
231D3×D5—604no—
232ΓL(1,𝔽16)—38no—
233ΓL(1,𝔽16)—158no—
234ΓL(1,𝔽16)—158no—
235ΓL(1,𝔽16)—158no—
236ΓL(1,𝔽16)—604no—
237C3×AGL(1,𝔽5)—58no—
238C3×AGL(1,𝔽5)—58no—
239C3×AGL(1,𝔽5)—58no—
240C3×AGL(1,𝔽5)—204no—
241C3×AGL(1,𝔽5)D3×AGL(1,𝔽5)18yescomplemented
242C3×C4×C5—58no—
243Dic15—158no—
244C3×Dic5—58no—
245C5×Dic3—158no—
246C2×C52—91no—
247C5⋊D5—91no—
248C5⋊D5C2×C4×C3⋊D311yescharacteristic
249C5×D5—92no—
250C5×D5C2×C4×C3⋊D312yes—
251C2×C4×D3—758no—
252C2×C4×D3—754no—
253C3×C42—254no—
254C4×Dic3—754no—
255C4×Dic3—758no—
256C32×C5—41no—
257C32×C5C2×C4×AGL(1,𝔽5)12yescomplemented
258C22×D5—452no—
259C2×AGL(1,𝔽5)—52no—
260C2×AGL(1,𝔽5)—52no—
261C2×AGL(1,𝔽5)—92no—
262C2×AGL(1,𝔽5)—452no—
263C2×AGL(1,𝔽5)—452no—
264C2×AGL(1,𝔽5)—452no—
265C2×AGL(1,𝔽5)—452no—
266C2×AGL(1,𝔽5)—452no—
267C2×AGL(1,𝔽5)—452no—
268C2×AGL(1,𝔽5)—452no—
269C2×AGL(1,𝔽5)—452no—
270C2×AGL(1,𝔽5)—452no—
271C2×AGL(1,𝔽5)—1801no—
272C2×C4×C5—452no—
273C2×Dic5—452no—
274C4×D5—52no—
275C4×D5—452no—
276C4×D5—452no—
277C4×D5—452no—
278C22×C32—251no—
279C2×C3⋊D3—52no—
280C2×C3⋊D3—251no—
281C2×C3⋊D3—252no—
282C2×C3⋊D3—251no—
283C32×C4—52no—
284C32×C4—252no—
285C32×C4—251no—
286C32×C4—251no—
287C3⋊Dic3—52no—
288C3⋊Dic3—252no—
289C3⋊Dic3—251no—
290C3⋊Dic3—251no—
291C2×C42—2251noSylow
292C2×C3×C5—58no—
293D15—38no—
294D15—158no—
295D15—604no—
296C3×D5—58no—
297C3×D5—204no—
298C3×D5C2×D3×AGL(1,𝔽5)18yes—
299C5×D3—38no—
300C5×D3—124no—
301C5×D3—158no—
302C52C42×C3⋊D311yescharacteristic, complemented, Sylow
303C22×D3—754no—
304C2×C3×C4—258no—
305C2×C3×C4—254no—
306C2×Dic3—758no—
307C2×Dic3—754no—
308C4×D3—158no—
309C4×D3—758no—
310C4×D3—758no—
311C4×D3—758no—
312C4×D3—754no—
313C4×D3—754no—
314C4×D3—754no—
315C4×D3—754no—
316C22×C5—452no—
317C2×D5—52no—
318C2×D5—92no—
319C2×D5—452no—
320C2×D5—452no—
321C2×D5—452no—
322C2×D5—452no—
323C2×D5—1801no—
324AGL(1,𝔽5)—52no—
325AGL(1,𝔽5)—52no—
326AGL(1,𝔽5)—52no—
327AGL(1,𝔽5)—92no—
328AGL(1,𝔽5)—201no—
329AGL(1,𝔽5)—452no—
330AGL(1,𝔽5)—452no—
331AGL(1,𝔽5)—452no—
332AGL(1,𝔽5)—1801no—
333AGL(1,𝔽5)C3⋊D3×AGL(1,𝔽5)12yesdirect product factor
334C4×C5—52no—
335C4×C5—452no—
336Dic5—52no—
337Dic5—452no—
338C2×C32—52no—
339C2×C32—251no—
340C3⋊D3—52no—
341C3⋊D3—251no—
342C3⋊D3AGL(1,𝔽5)211yescharacteristic, direct product factor
343C22×C4—2252no—
344C22×C4—2251no—
345C42—251no—
346C42—2252no—
347C42—2251no—
348C3×C5—44no—
349C3×C5C4×D3×AGL(1,𝔽5)18yescomplemented
350C22×C3—254no—
351C2×D3—158no—
352C2×D3—754no—
353C2×D3—754no—
354C2×D3—758no—
355C3×C4—58no—
356C3×C4—258no—
357C3×C4—254no—
358C3×C4—254no—
359Dic3—158no—
360Dic3—758no—
361Dic3—754no—
362Dic3—754no—
363C2×C5—52no—
364C2×C5—92no—
365C2×C5—361no—
366C2×C5—452no—
367D5—52no—
368D5—92no—
369D5—201no—
370D5—452no—
371D5—1801no—
372D5C2×C3⋊D3×AGL(1,𝔽5)12yes—
373C32C2×AGL(1,𝔽5)211yescharacteristic, complemented, Sylow
374C23—2251no—
375C2×C4—252no—
376C2×C4—251no—
377C2×C4—452no—
378C2×C4—2252no—
379C2×C4—2252no—
380C2×C4—2252no—
381C2×C4—2251no—
382C2×C4—2251no—
383C2×C4—2251no—
384C2×C4—2251no—
385C2×C4—2252no—
386C2×C4—2251no—
387C2×C3—58no—
388C2×C3—254no—
389D3—34no—
390D3—158no—
391D3—754no—
392C5—41no—
393C5C4×C3⋊D3×AGL(1,𝔽5)12yescomplemented
394C22—251no—
395C22—452no—
396C22—2252no—
397C22—2251no—
398C22—2251no—
399C4—52no—
400C4—252no—
401C4—251no—
402C4—251no—
403C4—452no—
404C4—2252no—
405C4—2251no—
406C4—2251no—
407C3D3×AGL(1,𝔽5)214yescomplemented
408C2—52no—
409C2—91no—
410C2—251no—
411C2—452no—
412C2—2251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C52C2×C42
prime radicalD52×C3⋊D3C22
cosocleC32×D52C23
LCSG ▷ C32×C52
UCSC1

Complements

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

NH ≅ G/Ntype
1. AGL(1,𝔽5)×C3⋊ΓL(1,𝔽16)C2semidirect
3. #72230965C2semidirect
4. C32×AGL(1,𝔽5)2C2semidirect
8. #1042⋊(C3⋊D3)C4semidirect
9. AGL(1,𝔽5)×C3⋊D15C4semidirect
11. C3⋊D3×#1042C4semidirect
12. #14268426C4semidirect
13. #14268425C4semidirect
15. C3⋊D3×#1044C4semidirect
16. C3⋊D3×#1043C4semidirect
17. C5×C3⋊D3×AGL(1,𝔽5)C4semidirect
25. C3×AGL(1,𝔽5)2D3semidirect
31. C3⋊(C3⋊#1042)C2×C4semidirect
32. C3⋊(C3⋊#1044)C2×C4semidirect
33. C3⋊(C3⋊#1043)C2×C4semidirect
34. C5×C3⋊ΓL(1,𝔽16)C2×C4semidirect
35. C32×#1042C2×C4semidirect
36. C32×#1044C2×C4semidirect
37. C32×#1043C2×C4semidirect
38. C32×C5×AGL(1,𝔽5)C2×C4semidirect
61. C3⋊(C3⋊(C5⋊D5))C42semidirect
62. C5×C3⋊D15C42semidirect
63. C52×C3⋊D3C42semidirect
70. AGL(1,𝔽5)2C3⋊D3direct (N × H)
84. C3⋊D3×AGL(1,𝔽5)AGL(1,𝔽5)direct (N × H)
100. C3×#1044C4×D3semidirect
101. C3×#1043C4×D3semidirect
102. C3×#1042C4×D3semidirect
103. C3×C5×AGL(1,𝔽5)C4×D3semidirect
112. C32×C52C2×C42semidirect
138. C3⋊ΓL(1,𝔽16)C2×AGL(1,𝔽5)semidirect
143. C32×AGL(1,𝔽5)C2×AGL(1,𝔽5)semidirect
185. #1044C4×C3⋊D3semidirect
187. #1043C4×C3⋊D3semidirect
189. #1042C4×C3⋊D3semidirect
191. C5×AGL(1,𝔽5)C4×C3⋊D3semidirect
196. C3⋊D15C4×AGL(1,𝔽5)semidirect
199. C5×C3⋊D3C4×AGL(1,𝔽5)semidirect
210. C3×C52C42×D3semidirect
241. C3×AGL(1,𝔽5)D3×AGL(1,𝔽5)semidirect
257. C32×C5C2×C4×AGL(1,𝔽5)semidirect
302. C52C42×C3⋊D3semidirect
333. AGL(1,𝔽5)C3⋊D3×AGL(1,𝔽5)direct (N × H)
342. C3⋊D3AGL(1,𝔽5)2direct (N × H)
349. C3×C5C4×D3×AGL(1,𝔽5)semidirect
373. C32C2×AGL(1,𝔽5)2semidirect
393. C5C4×C3⋊D3×AGL(1,𝔽5)semidirect
407. C3D3×AGL(1,𝔽5)2semidirect

Maximal subgroups

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

subgroupindex
AGL(1,𝔽5)×C3⋊ΓL(1,𝔽16)2
D5×C3⋊D3×AGL(1,𝔽5)2
#722309652
C32×AGL(1,𝔽5)22
D3×AGL(1,𝔽5)23
C3⋊D3×D5⋊AGL(1,𝔽5)2
C4×C3⋊D3×AGL(1,𝔽5)5