Finite Group Databasebeta

← C22×D3×ΓL(1,𝔽25) #11918443

Subgroups — C22×D3×ΓL(1,𝔽25) #11918443

Order 1152.

3589 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D32C211yesmaximal, characteristic
2C22×#3743C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C22×C3×D3×C8C211yesmaximal, characteristic, complemented
5C22×#3712C211yesmaximal, characteristic, complemented
6C22×#3711C211yesmaximal, characteristic, complemented
7C22×D3×C3⋊C8C211yesmaximal, characteristic, complemented
8C2×D3×ΓL(1,𝔽25)C2124yesmaximal, direct product factor
9C22×D3×M16—31nomaximal
10C23×ΓL(1,𝔽25)—31nomaximal
11C23×D32C411yescharacteristic
12C22×C4×C3⋊D3C2211yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×C3×C4×D3C2211yescharacteristic
15C22×#624C411yescharacteristic
16C22×D3×Dic3C411yescharacteristic
17C22×D3×Dic3C411yescharacteristic
18C2×C4×D32C4112yes—
19C2×C4×D32C22112yes—
20C22×C32×C8C2211yescharacteristic, complemented
21C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×#3743C22112yescomplemented
23C22×C3×C3⋊C8C2211yescharacteristic, complemented
24C22×C3×C3⋊C8C2211yescharacteristic, complemented
25C2×C3×ΓL(1,𝔽25)C22112yescomplemented
26C2×C3×D3×C8C22112yescomplemented
27C2×#3712C22112yescomplemented
28C2×#3711C22112yescomplemented
29C2×D3×C3⋊C8C22112yescomplemented
30D3×ΓL(1,𝔽25)C22164yesdirect product factor
31C23×C4×D3—31no—
32C23×C4×D3—31no—
33C22×C3×M16—31no—
34C23×C3×C8—31no—
35C22×#208—31no—
36C23×C3⋊C8—31no—
37C2×D3×M16—324no—
38C22×ΓL(1,𝔽25)—31no—
39C22×ΓL(1,𝔽25)—31no—
40C22×ΓL(1,𝔽25)—31no—
41C22×ΓL(1,𝔽25)—31no—
42C22×ΓL(1,𝔽25)—31no—
43C22×ΓL(1,𝔽25)—324no—
44C22×ΓL(1,𝔽25)—61no—
45C22×ΓL(1,𝔽25)D311yescharacteristic, direct product factor
46C22×D3×C8—31no—
47C22×D3×C8D311yescharacteristic, complemented
48C23×C3⋊D3C2×C411yescharacteristic
49C23×C3×D3C2×C411yescharacteristic
50C23×C3×D3C2×C411yescharacteristic
51C22×D32—28no—
52C22×D32C2×C4112yes—
53C22×C32×C4C2311yescharacteristic
54C22×C3⋊Dic3C2×C411yescharacteristic
55C2×C4×C3⋊D3C2×C416yes—
56C2×C4×C3⋊D3C2316yes—
57C22×C3×Dic3C2×C411yescharacteristic
58C22×C3×Dic3C2×C411yescharacteristic
59C2×C3×C4×D3C2×C416yes—
60C2×C3×C4×D3C2×C416yes—
61C2×C3×C4×D3C2316yes—
62C2×C3×C4×D3C2316yes—
63C2×#624C2×C4112yes—
64C2×D3×Dic3C2×C4112yes—
65C2×D3×Dic3C2×C4112yes—
66C4×D32C2×C4148yes—
67C4×D32C23116yes—
68C2×C32×C8C2316yescomplemented
69C2×C3⋊(C3⋊C8)C2316yescomplemented
70#3743C23116yescomplemented
71C2×C3×C3⋊C8C2316yescomplemented
72C2×C3×C3⋊C8C2316yescomplemented
73C3×ΓL(1,𝔽25)C23116yescomplemented
74C3×D3×C8C23116yescomplemented
75#3712C23116yescomplemented
76#3711C23116yescomplemented
77D3×C3⋊C8C23116yescomplemented
78C23×M16—91noSylow
79C24×D3—31no—
80C24×D3—31no—
81C23×C3×C4—31no—
82C23×C3×C4—31no—
83C23×Dic3—31no—
84C23×Dic3—31no—
85C22×C4×D3—31no—
86C22×C4×D3—31no—
87C22×C4×D3—31no—
88C22×C4×D3—31no—
89C22×C4×D3—31no—
90C22×C4×D3—31no—
91C22×C4×D3—312no—
92C22×C4×D3—312no—
93C22×C4×D3—312no—
94C22×C4×D3—312no—
95C22×C4×D3—61no—
96C22×C4×D3C2×D311yescharacteristic
97C22×C4×D3C2×D311yescharacteristic
98C2×C3×M16—312no—
99C22×C3×C8—21no—
100C22×C3×C8—31no—
101C22×C3×C8—31no—
102C22×C3×C8—312no—
103C22×C3×C8C2×D311yescharacteristic, complemented
104C22×C3×C8C2×D311yescharacteristic, complemented
105C2×#208—312no—
106C22×C3⋊C8—31no—
107C22×C3⋊C8—31no—
108C22×C3⋊C8—312no—
109C22×C3⋊C8—61no—
110C22×C3⋊C8C2×D311yescharacteristic, complemented
111C22×C3⋊C8C2×D311yescharacteristic, complemented
112D3×M16—364no—
113C2×ΓL(1,𝔽25)—312no—
114C2×ΓL(1,𝔽25)—312no—
115C2×ΓL(1,𝔽25)—312no—
116C2×ΓL(1,𝔽25)—312no—
117C2×ΓL(1,𝔽25)—312no—
118C2×ΓL(1,𝔽25)—364no—
119C2×ΓL(1,𝔽25)—612no—
120C2×ΓL(1,𝔽25)C2×D3112yesdirect product factor
121C2×D3×C8—312no—
122C2×D3×C8C2×D3112yescomplemented
123C23×C32C22×C411yescharacteristic
124C22×C3⋊D3—24no—
125C22×C3⋊D3C22×C416yes—
126C22×C3×D3—24no—
127C22×C3×D3M1618yescomplemented
128C22×C3×D3C22×C416yes—
129C22×C3×D3C22×C416yes—
130C2×D32—248no—
131C2×D32C22×C4116yes—
132C2×C32×C4C22×C413yes—
133C2×C32×C4C2413yes—
134C2×C3⋊Dic3C22×C416yes—
135C4×C3⋊D3C22×C4112yes—
136C4×C3⋊D3C2414yes—
137C2×C3×Dic3C22×C416yes—
138C2×C3×Dic3C22×C416yes—
139C3×C4×D3C22×C4112yes—
140C3×C4×D3C22×C4112yes—
141C3×C4×D3C2414yes—
142C3×C4×D3C2414yes—
143#624C22×C4116yes—
144D3×Dic3C22×C4116yes—
145D3×Dic3C22×C4116yes—
146C32×C8C2414yescomplemented
147C3⋊(C3⋊C8)C2414yescomplemented
148C3×C3⋊C8C2414yescomplemented
149C3×C3⋊C8C2414yescomplemented
150C24×C4—91no—
151C22×M16—31no—
152C22×M16—91no—
153C22×M16—924no—
154C22×M16—91no—
155C22×M16—91no—
156C23×C8—31no—
157C23×C8—91no—
158C24×C3—31no—
159C24×C3—31no—
160C23×D3—31no—
161C23×D3—31no—
162C23×D3—312no—
163C23×D3—312no—
164C23×D3—61no—
165C23×D3—68no—
166C23×D3—68no—
167C23×D3C4×D311yescharacteristic
168C23×D3C4×D311yescharacteristic
169C22×C3×C4—21no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—36no—
173C22×C3×C4—36no—
174C22×C3×C4—36no—
175C22×C3×C4—36no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C22×Dic3—31no—
179C22×Dic3—31no—
180C22×Dic3—312no—
181C22×Dic3—312no—
182C22×Dic3—61no—
183C22×Dic3C4×D311yescharacteristic
184C22×Dic3C4×D311yescharacteristic
185C2×C4×D3—312no—
186C2×C4×D3—312no—
187C2×C4×D3—36no—
188C2×C4×D3—36no—
189C2×C4×D3—36no—
190C2×C4×D3—36no—
191C2×C4×D3—312no—
192C2×C4×D3—312no—
193C2×C4×D3—348no—
194C2×C4×D3—348no—
195C2×C4×D3—316no—
196C2×C4×D3—316no—
197C2×C4×D3—66no—
198C2×C4×D3—66no—
199C2×C4×D3C4×D316yes—
200C2×C4×D3C4×D316yes—
201C2×C4×D3C22×D316yes—
202C2×C4×D3C22×D316yes—
203C3×M16—316no—
204C2×C3×C8—26no—
205C2×C3×C8—36no—
206C2×C3×C8—36no—
207C2×C3×C8—316no—
208C2×C3×C8C22×D316yescomplemented
209C2×C3×C8C22×D316yescomplemented
210#208—316no—
211C2×C3⋊C8—36no—
212C2×C3⋊C8—36no—
213C2×C3⋊C8—316no—
214C2×C3⋊C8—66no—
215C2×C3⋊C8C22×D316yescomplemented
216C2×C3⋊C8C22×D316yescomplemented
217ΓL(1,𝔽25)—316no—
218ΓL(1,𝔽25)—316no—
219ΓL(1,𝔽25)—316no—
220ΓL(1,𝔽25)—316no—
221ΓL(1,𝔽25)—316no—
222ΓL(1,𝔽25)—616no—
223ΓL(1,𝔽25)C22×D3116yesdirect product factor
224D3×C8—316no—
225D3×C8C22×D3116yescomplemented
226C22×C32C2×M1614yescomplemented
227C22×C32C23×C413yes—
228C2×C3⋊D3—212no—
229C2×C3⋊D3C23×C414yes—
230C2×C3×D3—212no—
231C2×C3×D3C2×M16124yescomplemented
232C2×C3×D3C23×C414yes—
233C2×C3×D3C23×C414yes—
234D32—232no—
235C32×C4C23×C413yes—
236C32×C4C2511yescharacteristic
237C3⋊Dic3C23×C414yes—
238C3×Dic3C23×C414yes—
239C3×Dic3C23×C414yes—
240C25—91no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—912no—
244C23×C4—91no—
245C23×C4—912no—
246C23×C4—91no—
247C23×C4—91no—
248C23×C4—91no—
249C2×M16—312no—
250C2×M16—912no—
251C2×M16—964no—
252C2×M16—912no—
253C2×M16—912no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8—312no—
257C22×C8—912no—
258C22×C8—91no—
259C22×C8D3211yescharacteristic, complemented
260C23×C3—21no—
261C23×C3—38no—
262C23×C3—36no—
263C23×C3—36no—
264C23×C3—64no—
265C23×C3C2×C4×D311yescharacteristic
266C23×C3C2×C4×D311yescharacteristic
267C22×D3—24no—
268C22×D3—36no—
269C22×D3—36no—
270C22×D3—316no—
271C22×D3—316no—
272C22×D3—66no—
273C22×D3—648no—
274C22×D3—64no—
275C22×D3—64no—
276C22×D3—648no—
277C22×D3—124no—
278C22×D3ΓL(1,𝔽25)18yesdirect product factor
279C22×D3C2×C4×D316yes—
280C22×D3C2×C4×D316yes—
281C2×C3×C4—23no—
282C2×C3×C4—23no—
283C2×C3×C4—36no—
284C2×C3×C4—36no—
285C2×C3×C4—312no—
286C2×C3×C4—312no—
287C2×C3×C4—34no—
288C2×C3×C4—34no—
289C2×C3×C4C2×C4×D313yes—
290C2×C3×C4C2×C4×D313yes—
291C2×C3×C4C23×D313yes—
292C2×C3×C4C23×D313yes—
293C2×Dic3—36no—
294C2×Dic3—36no—
295C2×Dic3—316no—
296C2×Dic3—316no—
297C2×Dic3—66no—
298C2×Dic3C2×C4×D316yes—
299C2×Dic3C2×C4×D316yes—
300C4×D3—316no—
301C4×D3—316no—
302C4×D3—312no—
303C4×D3—312no—
304C4×D3—34no—
305C4×D3—34no—
306C4×D3—316no—
307C4×D3—316no—
308C4×D3—612no—
309C4×D3—64no—
310C4×D3C2×C4×D3112yes—
311C4×D3C2×C4×D3112yes—
312C4×D3C23×D314yes—
313C4×D3C23×D314yes—
314C3×C8—24no—
315C3×C8—34no—
316C3×C8—34no—
317C3×C8C23×D314yescomplemented
318C3×C8C23×D314yescomplemented
319C3⋊C8—34no—
320C3⋊C8—34no—
321C3⋊C8—64no—
322C3⋊C8C23×D314yescomplemented
323C3⋊C8C23×D314yescomplemented
324C2×C32C22×M1616yescomplemented
325C2×C32C24×C411yescharacteristic
326C3⋊D3—24no—
327C3×D3—24no—
328C3×D3C22×M1618yescomplemented
329C24—31no—
330C24—31no—
331C24—912no—
332C24—91no—
333C24—188no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—36no—
337C22×C4—36no—
338C22×C4—36no—
339C22×C4—36no—
340C22×C4—916no—
341C22×C4—912no—
342C22×C4—948no—
343C22×C4—912no—
344C22×C4—912no—
345C22×C4—91no—
346C22×C4—96no—
347C22×C4—96no—
348C22×C4C2×D3211yescharacteristic
349M16—316no—
350M16—916no—
351M16—916no—
352M16—916no—
353C2×C8—36no—
354C2×C8—36no—
355C2×C8—316no—
356C2×C8—916no—
357C2×C8—96no—
358C2×C8C2×D3216yescomplemented
359C22×C3—24no—
360C22×C3—23no—
361C22×C3—34no—
362C22×C3—324no—
363C22×C3—34no—
364C22×C3—612no—
365C22×C3C2×ΓL(1,𝔽25)14yescomplemented
366C22×C3D3×M1614yescomplemented
367C22×C3C22×C4×D313yes—
368C22×C3C22×C4×D313yes—
369C2×D3—212no—
370C2×D3—34no—
371C2×D3—34no—
372C2×D3—64no—
373C2×D3—632no—
374C2×D3—612no—
375C2×D3—612no—
376C2×D3—632no—
377C2×D3—1212no—
378C2×D3C2×ΓL(1,𝔽25)124yesdirect product factor
379C2×D3C22×C4×D314yes—
380C2×D3C22×C4×D314yes—
381C3×C4—23no—
382C3×C4—21no—
383C3×C4—34no—
384C3×C4—34no—
385C3×C4C22×C4×D313yes—
386C3×C4C22×C4×D313yes—
387C3×C4C24×D311yescharacteristic
388C3×C4C24×D311yescharacteristic
389Dic3—34no—
390Dic3—34no—
391Dic3—64no—
392Dic3C22×C4×D314yes—
393Dic3C22×C4×D314yes—
394C32C23×M1611yescharacteristic, complemented, Sylow
395C23—38no—
396C23—36no—
397C23—36no—
398C23—64no—
399C23—916no—
400C23—96no—
401C23—1848no—
402C23—184no—
403C23C4×D3211yescharacteristic
404C2×C4—36no—
405C2×C4—36no—
406C2×C4—312no—
407C2×C4—312no—
408C2×C4—34no—
409C2×C4—34no—
410C2×C4—916no—
411C2×C4—916no—
412C2×C4—916no—
413C2×C4—96no—
414C2×C4—912no—
415C2×C4—94no—
416C2×C4C4×D3213yes—
417C2×C4C22×D3213yes—
418C8—34no—
419C8—34no—
420C8—94no—
421C8C22×D3214yescomplemented
422C2×C3—21no—
423C2×C3—26no—
424C2×C3—38no—
425C2×C3—64no—
426C2×C3C22×ΓL(1,𝔽25)16yescomplemented
427C2×C3C2×D3×M1616yescomplemented
428C2×C3C23×C4×D311yescharacteristic
429C2×C3C23×C4×D311yescharacteristic
430D3—24no—
431D3—64no—
432D3—64no—
433D3—124no—
434D3C22×ΓL(1,𝔽25)18yesdirect product factor
435C22—34no—
436C22—324no—
437C22—34no—
438C22—612no—
439C22—94no—
440C22—1832no—
441C22—1812no—
442C22D3×ΓL(1,𝔽25)14yesdirect product factor
443C22C2×C4×D3213yes—
444C4—34no—
445C4—34no—
446C4—94no—
447C4C2×C4×D3213yes—
448C4C23×D3211yescharacteristic
449C3—21no—
450C3C23×ΓL(1,𝔽25)11yescharacteristic, complemented
451C3C22×D3×M1611yescharacteristic, complemented
452C2—38no—
453C2—64no—
454C2—184no—
455C2C2×D3×ΓL(1,𝔽25)16yesdirect product factor
456C2C22×C4×D3211yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutatorC2×C32C24×C4
FrattiniC4C23×D32
FittingC22×C32×C8C22
socleC23×C32C22×C4
prime radicalC23×D32C4
cosocleC32×C4C25
hypercenterC22×C8D32
hypocenterC32C23×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×#3743C2semidirect
3. C22×C3×ΓL(1,𝔽25)C2semidirect
4. C22×C3×D3×C8C2semidirect
5. C22×#3712C2semidirect
6. C22×#3711C2semidirect
7. C22×D3×C3⋊C8C2semidirect
8. C2×D3×ΓL(1,𝔽25)C2direct (N × H)
20. C22×C32×C8C22semidirect
21. C22×C3⋊(C3⋊C8)C22semidirect
22. C2×#3743C22semidirect
23. C22×C3×C3⋊C8C22semidirect
24. C22×C3×C3⋊C8C22semidirect
25. C2×C3×ΓL(1,𝔽25)C22semidirect
26. C2×C3×D3×C8C22semidirect
27. C2×#3712C22semidirect
28. C2×#3711C22semidirect
29. C2×D3×C3⋊C8C22semidirect
30. D3×ΓL(1,𝔽25)C22direct (N × H)
45. C22×ΓL(1,𝔽25)D3direct (N × H)
47. C22×D3×C8D3semidirect
68. C2×C32×C8C23semidirect
69. C2×C3⋊(C3⋊C8)C23semidirect
70. #3743C23semidirect
71. C2×C3×C3⋊C8C23semidirect
72. C2×C3×C3⋊C8C23semidirect
73. C3×ΓL(1,𝔽25)C23semidirect
74. C3×D3×C8C23semidirect
75. #3712C23semidirect
76. #3711C23semidirect
77. D3×C3⋊C8C23semidirect
103. C22×C3×C8C2×D3semidirect
104. C22×C3×C8C2×D3semidirect
110. C22×C3⋊C8C2×D3semidirect
111. C22×C3⋊C8C2×D3semidirect
120. C2×ΓL(1,𝔽25)C2×D3direct (N × H)
122. C2×D3×C8C2×D3semidirect
127. C22×C3×D3M16semidirect
146. C32×C8C24semidirect
147. C3⋊(C3⋊C8)C24semidirect
148. C3×C3⋊C8C24semidirect
149. C3×C3⋊C8C24semidirect
208. C2×C3×C8C22×D3semidirect
209. C2×C3×C8C22×D3semidirect
215. C2×C3⋊C8C22×D3semidirect
216. C2×C3⋊C8C22×D3semidirect
223. ΓL(1,𝔽25)C22×D3direct (N × H)
225. D3×C8C22×D3semidirect
226. C22×C32C2×M16semidirect
231. C2×C3×D3C2×M16semidirect
259. C22×C8D32semidirect
278. C22×D3ΓL(1,𝔽25)direct (N × H)
317. C3×C8C23×D3semidirect
318. C3×C8C23×D3semidirect
322. C3⋊C8C23×D3semidirect
323. C3⋊C8C23×D3semidirect
324. C2×C32C22×M16semidirect
328. C3×D3C22×M16semidirect
358. C2×C8C2×D32semidirect
365. C22×C3C2×ΓL(1,𝔽25)semidirect
366. C22×C3D3×M16semidirect
378. C2×D3C2×ΓL(1,𝔽25)direct (N × H)
394. C32C23×M16semidirect
421. C8C22×D32semidirect
426. C2×C3C22×ΓL(1,𝔽25)semidirect
427. C2×C3C2×D3×M16semidirect
434. D3C22×ΓL(1,𝔽25)direct (N × H)
442. C22D3×ΓL(1,𝔽25)direct (N × H)
450. C3C23×ΓL(1,𝔽25)semidirect
451. C3C22×D3×M16semidirect
455. C2C2×D3×ΓL(1,𝔽25)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D322
C22×#37432
C22×C3×ΓL(1,𝔽25)2
C22×C3×D3×C82
C22×#37122
C22×#37112
C22×D3×C3⋊C82
C2×D3×ΓL(1,𝔽25)2
C22×D3×M163
C23×ΓL(1,𝔽25)3