Finite Group Databasebeta

← C2×C3⋊D3×ΓL(1,𝔽25) #13176612

Subgroups — C2×C3⋊D3×ΓL(1,𝔽25) #13176612

Order 1728.

2060 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
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
4C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5C2×#88788C211yesmaximal, characteristic, complemented
6C2×#88786C211yesmaximal, characteristic, complemented
7C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3⋊D3×ΓL(1,𝔽25)C218yesmaximal, direct product factor
9C2×M16×C3⋊D3—31nomaximal
10C2×D3×ΓL(1,𝔽25)—34nomaximal
11C22×D3×C3⋊D3C411yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6289C411yescharacteristic
16C2×Dic3×C3⋊D3C411yescharacteristic
17C2×D3×C3⋊Dic3C411yescharacteristic
18C4×D3×C3⋊D3C414yes—
19C4×D3×C3⋊D3C2214yes—
20C2×C33×C8C2211yescharacteristic, complemented
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
22#88850C2214yescomplemented
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
24C3×C8×C3⋊D3C2214yescomplemented
25C2×C32×C3⋊C8C2211yescharacteristic, complemented
26C32×ΓL(1,𝔽25)C2214yescomplemented
27#88788C2214yescomplemented
28#88786C2214yescomplemented
29C3⋊D3×C3⋊C8C2214yescomplemented
30C22×C4×C3⋊D3—31no—
31C2×C4×D32—34no—
32C2×C32×M16—31no—
33C2×#3748—31no—
34M16×C3⋊D3—38no—
35C2×#3743—31no—
36C2×#3743—31no—
37C2×#3743—34no—
38C2×#3743—64no—
39C2×C8×C3⋊D3—31no—
40C2×C8×C3⋊D3D311yescharacteristic, complemented
41C2×C3×ΓL(1,𝔽25)D314yescomplemented
42C2×C3×D3×C8—34no—
43C2×#3712—34no—
44C2×#3711—34no—
45C2×D3×C3⋊C8—34no—
46D3×ΓL(1,𝔽25)—332no—
47C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
48C22×C3×C3⋊D3C2×C411yescharacteristic
49C22×C32×D3C2×C411yescharacteristic
50C2×D3×C3⋊D3—24no—
51C2×D3×C3⋊D3C2×C414yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
54C4×C3⋊(C3⋊D3)C2×C412yes—
55C4×C3⋊(C3⋊D3)C2312yes—
56C2×C3×C3⋊Dic3C2×C411yescharacteristic
57C3×C4×C3⋊D3C2×C412yes—
58C3×C4×C3⋊D3C2312yes—
59C2×C32×Dic3C2×C411yescharacteristic
60C32×C4×D3C2×C412yes—
61C32×C4×D3C2312yes—
62#6289C2×C414yes—
63Dic3×C3⋊D3C2×C414yes—
64D3×C3⋊Dic3C2×C414yes—
65C33×C8C2312yescomplemented
66C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
67C3×C3⋊(C3⋊C8)C2312yescomplemented
68C32×C3⋊C8C2312yescomplemented
69C2×D3×M16—94no—
70C22×ΓL(1,𝔽25)—91no—
71C23×C3⋊D3—31no—
72C22×D32—34no—
73C22×C32×C4—31no—
74C22×C3⋊Dic3—31no—
75C2×C4×C3⋊D3—31no—
76C2×C4×C3⋊D3—31no—
77C2×C4×C3⋊D3—34no—
78C2×C4×C3⋊D3—31no—
79C2×C4×C3⋊D3—34no—
80C2×C4×C3⋊D3—34no—
81C2×C4×C3⋊D3—64no—
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×C3×C4×D3—34no—
84C2×C3×C4×D3C2×D314yes—
85C2×#624—34no—
86C2×D3×Dic3—34no—
87C2×D3×Dic3—34no—
88C4×D32—316no—
89C4×D32—316no—
90C32×M16—34no—
91C2×C32×C8—24no—
92C2×C32×C8—31no—
93C2×C32×C8C2×D314yescomplemented
94C2×C32×C8C2×D311yescharacteristic, complemented
95#3748—34no—
96C2×C3⋊(C3⋊C8)—31no—
97C2×C3⋊(C3⋊C8)—34no—
98C2×C3⋊(C3⋊C8)—64no—
99C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
100#3743—34no—
101#3743—316no—
102#3743—34no—
103#3743—616no—
104C8×C3⋊D3—34no—
105C8×C3⋊D3C2×D314yescomplemented
106C2×C3×C3⋊C8—34no—
107C2×C3×C3⋊C8C2×D314yescomplemented
108C3×ΓL(1,𝔽25)C2×D3116yescomplemented
109C3×D3×C8—316no—
110#3712—316no—
111#3711—316no—
112D3×C3⋊C8—316no—
113C22×C33C22×C411yescharacteristic
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)C22×C412yes—
116C2×C3×C3⋊D3M1614yescomplemented
117C2×C3×C3⋊D3C22×C412yes—
118C2×C32×D3—22no—
119C2×C32×D3C22×C412yes—
120D3×C3⋊D3—28no—
121C33×C4C22×C411yescharacteristic
122C33×C4C2411yescharacteristic
123C3⋊(C3⋊Dic3)C22×C412yes—
124C3×C3⋊Dic3C22×C412yes—
125C32×Dic3C22×C412yes—
126C22×C4×D3—94no—
127C22×C4×D3—91no—
128C2×C3×M16—34no—
129C22×C3×C8—91no—
130C2×#208—94no—
131C22×C3⋊C8—91no—
132D3×M16—932no—
133C2×ΓL(1,𝔽25)—94no—
134C2×ΓL(1,𝔽25)—91no—
135C2×ΓL(1,𝔽25)—94no—
136C2×ΓL(1,𝔽25)—91no—
137C2×ΓL(1,𝔽25)—91no—
138C2×ΓL(1,𝔽25)—98no—
139C2×ΓL(1,𝔽25)—184no—
140C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, direct product factor
141C2×D3×C8—34no—
142C2×D3×C8—94no—
143C23×C32—31no—
144C22×C3⋊D3—31no—
145C22×C3⋊D3—34no—
146C22×C3⋊D3—34no—
147C22×C3⋊D3—64no—
148C22×C3⋊D3—64no—
149C22×C3⋊D3C4×D311yescharacteristic
150C22×C3×D3—34no—
151C22×C3×D3C4×D314yes—
152C2×D32—316no—
153C2×D32—616no—
154C2×C32×C4—24no—
155C2×C32×C4—31no—
156C2×C32×C4—32no—
157C2×C32×C4—32no—
158C2×C32×C4C22×D314yes—
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—34no—
162C2×C3⋊Dic3—34no—
163C2×C3⋊Dic3—64no—
164C2×C3⋊Dic3C4×D311yescharacteristic
165C4×C3⋊D3—34no—
166C4×C3⋊D3—32no—
167C4×C3⋊D3—32no—
168C4×C3⋊D3—34no—
169C4×C3⋊D3—38no—
170C4×C3⋊D3—38no—
171C4×C3⋊D3—68no—
172C4×C3⋊D3—68no—
173C4×C3⋊D3C4×D312yes—
174C4×C3⋊D3C22×D312yes—
175C2×C3×Dic3—34no—
176C2×C3×Dic3C4×D314yes—
177C3×C4×D3—38no—
178C3×C4×D3—38no—
179C3×C4×D3C4×D318yes—
180C3×C4×D3C22×D318yes—
181#624—316no—
182D3×Dic3—316no—
183D3×Dic3—316no—
184C32×C8—28no—
185C32×C8—32no—
186C32×C8C22×D318yescomplemented
187C32×C8C22×D312yescomplemented
188C3⋊(C3⋊C8)—32no—
189C3⋊(C3⋊C8)—38no—
190C3⋊(C3⋊C8)—68no—
191C3⋊(C3⋊C8)C22×D312yescomplemented
192C3×C3⋊C8—38no—
193C3×C3⋊C8C22×D318yescomplemented
194C22×M16—271noSylow
195C2×C33C2×M1612yescomplemented
196C2×C33C23×C411yescharacteristic
197C3⋊(C3⋊D3)—22no—
198C3×C3⋊D3C2×M1614yescomplemented
199C32×D3—22no—
200C23×D3—94no—
201C23×D3—91no—
202C22×C3×C4—34no—
203C22×C3×C4—91no—
204C22×Dic3—94no—
205C22×Dic3—91no—
206C2×C4×D3—34no—
207C2×C4×D3—916no—
208C2×C4×D3—916no—
209C2×C4×D3—94no—
210C2×C4×D3—94no—
211C2×C4×D3—91no—
212C2×C4×D3—94no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×C4×D3—94no—
216C2×C4×D3—94no—
217C2×C4×D3—184no—
218C2×C4×D3C2×C3⋊D311yescharacteristic
219C3×M16—316no—
220C2×C3×C8—24no—
221C2×C3×C8—34no—
222C2×C3×C8—91no—
223C2×C3×C8—94no—
224C2×C3×C8D3214yescomplemented
225C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
226#208—916no—
227C2×C3⋊C8—34no—
228C2×C3⋊C8—94no—
229C2×C3⋊C8—91no—
230C2×C3⋊C8—94no—
231C2×C3⋊C8—184no—
232C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
233ΓL(1,𝔽25)—916no—
234ΓL(1,𝔽25)—94no—
235ΓL(1,𝔽25)—916no—
236ΓL(1,𝔽25)—94no—
237ΓL(1,𝔽25)—94no—
238ΓL(1,𝔽25)—1816no—
239ΓL(1,𝔽25)C2×C3⋊D314yesdirect product factor
240D3×C8—316no—
241D3×C8—916no—
242C22×C32—24no—
243C22×C32—32no—
244C22×C32—62no—
245C22×C32C2×C4×D314yes—
246C22×C32C2×C4×D311yescharacteristic
247C2×C3⋊D3—32no—
248C2×C3⋊D3—38no—
249C2×C3⋊D3—68no—
250C2×C3⋊D3—62no—
251C2×C3⋊D3—68no—
252C2×C3⋊D3—68no—
253C2×C3⋊D3—128no—
254C2×C3⋊D3ΓL(1,𝔽25)14yesdirect product factor
255C2×C3⋊D3C2×C4×D312yes—
256C2×C3×D3—28no—
257C2×C3×D3—316no—
258C2×C3×D3—38no—
259C2×C3×D3C2×C4×D318yes—
260D32—632no—
261C32×C4—24no—
262C32×C4—24no—
263C32×C4—32no—
264C32×C4C2×C4×D314yes—
265C32×C4C2×C4×D311yescharacteristic
266C32×C4C23×D314yes—
267C32×C4C23×D311yescharacteristic
268C3⋊Dic3—32no—
269C3⋊Dic3—38no—
270C3⋊Dic3—68no—
271C3⋊Dic3C2×C4×D312yes—
272C3×Dic3—38no—
273C3×Dic3C2×C4×D318yes—
274C23×C4—271no—
275C2×M16—31no—
276C2×M16—271no—
277C2×M16—278no—
278C2×M16—271no—
279C2×M16—271no—
280C22×C8—91no—
281C22×C8—271no—
282C33C22×M1611yescharacteristic, complemented, Sylow
283C23×C3—34no—
284C23×C3—91no—
285C22×D3—34no—
286C22×D3—916no—
287C22×D3—94no—
288C22×D3—91no—
289C22×D3—94no—
290C22×D3—184no—
291C22×D3—184no—
292C22×D3—1816no—
293C22×D3C4×C3⋊D311yescharacteristic
294C2×C3×C4—24no—
295C2×C3×C4—34no—
296C2×C3×C4—38no—
297C2×C3×C4—38no—
298C2×C3×C4—91no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4C2×D3214yes—
302C2×C3×C4C22×C3⋊D311yescharacteristic
303C2×Dic3—34no—
304C2×Dic3—916no—
305C2×Dic3—94no—
306C2×Dic3—91no—
307C2×Dic3—94no—
308C2×Dic3—184no—
309C2×Dic3C4×C3⋊D311yescharacteristic
310C4×D3—38no—
311C4×D3—38no—
312C4×D3—916no—
313C4×D3—916no—
314C4×D3—94no—
315C4×D3—98no—
316C4×D3—92no—
317C4×D3—98no—
318C4×D3—92no—
319C4×D3—94no—
320C4×D3—188no—
321C4×D3—188no—
322C4×D3C4×C3⋊D312yes—
323C4×D3C22×C3⋊D312yes—
324C3×C8—28no—
325C3×C8—38no—
326C3×C8—92no—
327C3×C8C2×D3218yescomplemented
328C3×C8C22×C3⋊D312yescomplemented
329C3⋊C8—38no—
330C3⋊C8—98no—
331C3⋊C8—92no—
332C3⋊C8—188no—
333C3⋊C8C22×C3⋊D312yescomplemented
334C2×C32—28no—
335C2×C32—24no—
336C2×C32—62no—
337C2×C32C2×ΓL(1,𝔽25)12yescomplemented
338C2×C32D3×M1618yescomplemented
339C2×C32C22×C4×D314yes—
340C2×C32C22×C4×D311yescharacteristic
341C3⋊D3—62no—
342C3⋊D3—68no—
343C3⋊D3—128no—
344C3⋊D3C2×ΓL(1,𝔽25)14yesdirect product factor
345C3×D3—28no—
346C3×D3—316no—
347C24—271no—
348C22×C4—31no—
349C22×C4—91no—
350C22×C4—274no—
351C22×C4—271no—
352C22×C4—274no—
353C22×C4—271no—
354C22×C4—271no—
355C22×C4—271no—
356M16—34no—
357M16—274no—
358M16—274no—
359M16—274no—
360C2×C8—31no—
361C2×C8—91no—
362C2×C8—94no—
363C2×C8—274no—
364C2×C8—271no—
365C2×C8D3×C3⋊D311yescharacteristic, complemented
366C22×C3—24no—
367C22×C3—38no—
368C22×C3—68no—
369C22×C3—92no—
370C22×C3—94no—
371C22×C3C4×D3214yes—
372C22×C3C2×C4×C3⋊D311yescharacteristic
373C2×D3—22no—
374C2×D3—316no—
375C2×D3—38no—
376C2×D3—98no—
377C2×D3—92no—
378C2×D3—188no—
379C2×D3—188no—
380C2×D3—1832no—
381C2×D3—188no—
382C2×D3—182no—
383C2×D3—368no—
384C2×D3C2×C4×C3⋊D312yes—
385C3×C4—24no—
386C3×C4—24no—
387C3×C4—38no—
388C3×C4—92no—
389C3×C4C4×D3214yes—
390C3×C4C2×C4×C3⋊D311yescharacteristic
391C3×C4C22×D3214yes—
392C3×C4C23×C3⋊D311yescharacteristic
393Dic3—38no—
394Dic3—98no—
395Dic3—92no—
396Dic3—188no—
397Dic3C2×C4×C3⋊D312yes—
398C32—24no—
399C32C22×ΓL(1,𝔽25)11yescharacteristic, complemented
400C32C2×D3×M1614yescomplemented
401C23—31no—
402C23—91no—
403C23—274no—
404C23—271no—
405C23—544no—
406C2×C4—31no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—91no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—274no—
413C2×C4—274no—
414C2×C4—274no—
415C2×C4—271no—
416C2×C4—272no—
417C2×C4—272no—
418C2×C4C2×D3×C3⋊D311yescharacteristic
419C8—32no—
420C8—92no—
421C8—272no—
422C8C2×D3×C3⋊D312yescomplemented
423C2×C3—24no—
424C2×C3—28no—
425C2×C3—68no—
426C2×C3—94no—
427C2×C3D3×ΓL(1,𝔽25)18yescomplemented
428C2×C3M16×C3⋊D312yescomplemented
429C2×C3C2×C4×D3214yes—
430C2×C3C22×C4×C3⋊D311yescharacteristic
431D3—22no—
432D3—316no—
433D3—188no—
434D3—182no—
435D3—368no—
436C22—32no—
437C22—62no—
438C22—92no—
439C22—94no—
440C22—272no—
441C22—548no—
442C22—542no—
443C22C4×D3×C3⋊D311yescharacteristic
444C4—32no—
445C4—92no—
446C4—272no—
447C4C4×D3×C3⋊D311yescharacteristic
448C4C22×D3×C3⋊D311yescharacteristic
449C3—24no—
450C3C2×D3×ΓL(1,𝔽25)14yescomplemented
451C3C2×M16×C3⋊D311yescharacteristic, complemented
452C2—62no—
453C2—94no—
454C2—542no—
455C2C3⋊D3×ΓL(1,𝔽25)12yesdirect product factor
456C2C2×C4×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC22×D3×C3⋊D3C4
cosocleC33×C4C24
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88850C2semidirect
3. C2×C3×C8×C3⋊D3C2semidirect
4. C2×C32×ΓL(1,𝔽25)C2semidirect
5. C2×#88788C2semidirect
6. C2×#88786C2semidirect
7. C2×C3⋊D3×C3⋊C8C2semidirect
8. C3⋊D3×ΓL(1,𝔽25)C2direct (N × H)
20. C2×C33×C8C22semidirect
21. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. #88850C22semidirect
23. C2×C3×C3⋊(C3⋊C8)C22semidirect
24. C3×C8×C3⋊D3C22semidirect
25. C2×C32×C3⋊C8C22semidirect
26. C32×ΓL(1,𝔽25)C22semidirect
27. #88788C22semidirect
28. #88786C22semidirect
29. C3⋊D3×C3⋊C8C22semidirect
40. C2×C8×C3⋊D3D3semidirect
41. C2×C3×ΓL(1,𝔽25)D3semidirect
65. C33×C8C23semidirect
66. C3⋊(C3⋊(C3⋊C8))C23semidirect
67. C3×C3⋊(C3⋊C8)C23semidirect
68. C32×C3⋊C8C23semidirect
93. C2×C32×C8C2×D3semidirect
94. C2×C32×C8C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
105. C8×C3⋊D3C2×D3semidirect
107. C2×C3×C3⋊C8C2×D3semidirect
108. C3×ΓL(1,𝔽25)C2×D3semidirect
116. C2×C3×C3⋊D3M16semidirect
140. C2×ΓL(1,𝔽25)C3⋊D3direct (N × H)
186. C32×C8C22×D3semidirect
187. C32×C8C22×D3semidirect
191. C3⋊(C3⋊C8)C22×D3semidirect
193. C3×C3⋊C8C22×D3semidirect
195. C2×C33C2×M16semidirect
198. C3×C3⋊D3C2×M16semidirect
224. C2×C3×C8D32semidirect
225. C2×C3×C8C2×C3⋊D3semidirect
232. C2×C3⋊C8C2×C3⋊D3semidirect
239. ΓL(1,𝔽25)C2×C3⋊D3direct (N × H)
254. C2×C3⋊D3ΓL(1,𝔽25)direct (N × H)
282. C33C22×M16semidirect
327. C3×C8C2×D32semidirect
328. C3×C8C22×C3⋊D3semidirect
333. C3⋊C8C22×C3⋊D3semidirect
337. C2×C32C2×ΓL(1,𝔽25)semidirect
338. C2×C32D3×M16semidirect
344. C3⋊D3C2×ΓL(1,𝔽25)direct (N × H)
365. C2×C8D3×C3⋊D3semidirect
399. C32C22×ΓL(1,𝔽25)semidirect
400. C32C2×D3×M16semidirect
422. C8C2×D3×C3⋊D3semidirect
427. C2×C3D3×ΓL(1,𝔽25)semidirect
428. C2×C3M16×C3⋊D3semidirect
450. C3C2×D3×ΓL(1,𝔽25)semidirect
451. C3C2×M16×C3⋊D3semidirect
455. C2C3⋊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
C2×C4×D3×C3⋊D32
C2×#888502
C2×C3×C8×C3⋊D32
C2×C32×ΓL(1,𝔽25)2
C2×#887882
C2×#887862
C2×C3⋊D3×C3⋊C82
C3⋊D3×ΓL(1,𝔽25)2
C2×M16×C3⋊D33
C2×D3×ΓL(1,𝔽25)3