Finite Group Databasebeta

← C37×#36572 #22657825

Subgroups — C37×#36572 #22657825

Order 9472.

2446 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#2848C213yesmaximal, complemented
2C37×#2564C213yesmaximal, complemented
3C37×#2549C213yesmaximal, complemented
4C37×#2505C216yesmaximal, complemented
5C37×#2450C216yesmaximal, complemented
6C37×#2368C213yesmaximal, complemented
7C37×#2349C213yesmaximal, complemented
8C4×#120×C37C213yesmaximal
9C22×#95×C37C211yesmaximal, characteristic, complemented
10C24×C4×C37C2213yes—
11C2×C22⋊Q8×C37C2213yes—
12C2×C22⋊Q8×C37C2216yes—
13C2×#120×C37C2211yescharacteristic
14C2×C4×D4×C37C2213yes—
15C2×C4×D4×C37C2216yes—
16C37×#400C2216yescomplemented
17C37×#400C2213yescomplemented
18C37×#399C2211yescharacteristic, complemented
19C37×#398C2216yescomplemented
20C37×#396C2212yescomplemented
21C37×#395C2213yescomplemented
22C37×#393C2213yescomplemented
23C37×#391C2216yescomplemented
24C37×#389C2216yescomplemented
25C37×#387C2216yescomplemented
26C37×#387C2216yescomplemented
27C37×#386C2213yes—
28C37×#386C2216yes—
29C37×#385C2216yes—
30C37×#385C2216yes—
31C37×#385C2213yes—
32C37×#385C2216yes—
33C37×#385C2216yes—
34C37×#385C2216yescomplemented
35C37×#382C2213yescomplemented
36C37×#380C2216yes—
37C37×#380C2213yes—
38C37×#380C2213yescomplemented
39C37×#379C2213yes—
40C4×C4⋊C4×C37C2213yescomplemented
41C4×C22⋊C4×C37C2216yes—
42C4×C22⋊C4×C37C2213yes—
43C2×#95×C37C2213yes—
44C2×#95×C37C2216yes—
45C2×#95×C37C2213yes—
46C2×#95×C37C2211yescharacteristic, complemented
47C25×C37C2311yescharacteristic
48C22×Q8×C37C2313yes—
49C22×D4×C37C2313yes—
50C23×C4×C37—212no—
51C23×C4×C37—412no—
52C23×C4×C37C2316yes—
53C23×C4×C37C2316yes—
54C23×C4×C37C2313yes—
55C23×C4×C37C2313yes—
56C23×C4×C37C2313yes—
57C22⋊Q8×C37—212no—
58C22⋊Q8×C37—412no—
59#120×C37Q818yescomplemented
60C4×D4×C37—212no—
61C4×D4×C37—412no—
62C2×C4⋊C4×C37C2316yes—
63C2×C4⋊C4×C37C2313yes—
64C2×C4⋊C4×C37C2313yes—
65C2×C4⋊C4×C37C2316yes—
66C2×C4⋊C4×C37C2316yes—
67C2×C4⋊C4×C37C2316yes—
68C2×C4⋊C4×C37C2316yes—
69C2×C22⋊C4×C37C2313yes—
70C2×C22⋊C4×C37C2316yes—
71C2×C22⋊C4×C37C2316yes—
72C2×C22⋊C4×C37C2313yes—
73C2×C22⋊C4×C37C2313yes—
74C2×C22⋊C4×C37C2316yes—
75C2×C22⋊C4×C37C2313yes—
76C2×C42×C37C2313yes—
77C2×C42×C37C2316yes—
78#95×C37C2311yescharacteristic
79#95×C37C2313yes—
80#95×C37C2316yes—
81#95×C37C2313yes—
82#95×C37C2311yescharacteristic
83#95×C37C2312yes—
84#95×C37C2313yes—
85#95×C37C2316yes—
86#95×C37C2316yes—
87#95×C37C2316yes—
88#95×C37C2313yes—
89#95×C37C2316yes—
90#95×C37C2316yes—
91C24×C37—26no—
92C24×C37—26no—
93C24×C37C2×Q814yes—
94C24×C37C2411yescharacteristic
95C24×C37C2413yes—
96C2×Q8×C37—26no—
97C2×Q8×C37—46no—
98C2×D4×C37—26no—
99C2×D4×C37—26no—
100C2×D4×C37C2×Q8112yes—
101C22×C4×C37—26no—
102C22×C4×C37—212no—
103C22×C4×C37—212no—
104C22×C4×C37—212no—
105C22×C4×C37—212no—
106C22×C4×C37—26no—
107C22×C4×C37—46no—
108C22×C4×C37—46no—
109C22×C4×C37—46no—
110C22×C4×C37—46no—
111C22×C4×C37—46no—
112C22×C4×C37—448no—
113C22×C4×C37#41112yes—
114C22×C4×C37C2416yes—
115C22×C4×C37C2413yes—
116C22×C4×C37C2416yes—
117C22×C4×C37C2413yes—
118C22×C4×C37C2413yes—
119C22×C4×C37C2413yes—
120C22×C4×C37C2413yes—
121C4⋊C4×C37—26no—
122C4⋊C4×C37—26no—
123C4⋊C4×C37—212no—
124C4⋊C4×C37—46no—
125C4⋊C4×C37—46no—
126C4⋊C4×C37—46no—
127C4⋊C4×C37—46no—
128C22⋊C4×C37—26no—
129C22⋊C4×C37—212no—
130C22⋊C4×C37—212no—
131C22⋊C4×C37—26no—
132C22⋊C4×C37—26no—
133C22⋊C4×C37—46no—
134C22⋊C4×C37C2×Q8112yes—
135C42×C37—26no—
136C42×C37—46no—
137C23×C37—224no—
138C23×C37—28no—
139C23×C37—224no—
140C23×C37—26no—
141C23×C37—26no—
142C23×C37—23no—
143C23×C37—23no—
144C23×C37C22×Q812yes—
145C23×C37C22×Q816yes—
146C23×C37C2×#4116yes—
147C23×C37H(5,𝔽2)16yes—
148C23×C37C2511yescharacteristic
149Q8×C37—86no—
150D4×C37—224no—
151C2×C4×C37—23no—
152C2×C4×C37—23no—
153C2×C4×C37—26no—
154C2×C4×C37—26no—
155C2×C4×C37—23no—
156C2×C4×C37—26no—
157C2×C4×C37—23no—
158C2×C4×C37—26no—
159C2×C4×C37—26no—
160C2×C4×C37—26no—
161C2×C4×C37—26no—
162C2×C4×C37—26no—
163C2×C4×C37—26no—
164C2×C4×C37—26no—
165C2×C4×C37—412no—
166C2×C4×C37—424no—
167C2×C4×C37—424no—
168C2×C4×C37—412no—
169C2×C4×C37—86no—
170C2×C4×C37C22×Q816yes—
171C2×C4×C37C2×#4116yes—
172C2×C4×C37C2×#4116yes—
173#36572C3711yesmaximal, characteristic, direct product factor, Sylow
174C22×C37—232no—
175C22×C37—26no—
176C22×C37—212no—
177C22×C37—26no—
178C22×C37—212no—
179C22×C37—26no—
180C22×C37—41no—
181C22×C37—43no—
182C22×C37#554112yes—
183C22×C37C23×Q811yescharacteristic
184C22×C37C22×#4113yes—
185C22×C37C2×H(5,𝔽2)13yes—
186C4×C37—26no—
187C4×C37—26no—
188C4×C37—26no—
189C4×C37—43no—
190C4×C37—46no—
191C4×C37—43no—
192C4×C37—46no—
193#2848C2×C3713yescomplemented
194#2564C2×C3713yescomplemented
195#2549C2×C3713yescomplemented
196#2505C2×C3716yescomplemented
197#2450C2×C3716yescomplemented
198#2368C2×C3713yescomplemented
199#2349C2×C3713yescomplemented
200C4×#120C2×C3713yes—
201C22×#95C2×C3711yescharacteristic, complemented
202C2×C37—212no—
203C2×C37—42no—
204C2×C37#344411yescharacteristic
205C2×C37C2×#55413yes—
206C2×C37#349013yes—
207C24×C4C22×C3713yes—
208C2×C22⋊Q8C22×C3713yes—
209C2×C22⋊Q8C22×C3716yes—
210C2×#120C22×C3711yescharacteristic
211C2×C4×D4C22×C3713yes—
212C2×C4×D4C22×C3716yes—
213#400C22×C3713yescomplemented
214#400C22×C3716yescomplemented
215#399C22×C3711yescharacteristic, complemented
216#398C22×C3716yescomplemented
217#396C22×C3712yescomplemented
218#395C22×C3713yescomplemented
219#393C22×C3713yescomplemented
220#391C22×C3716yescomplemented
221#389C22×C3716yescomplemented
222#387C22×C3716yescomplemented
223#387C22×C3716yescomplemented
224#386C22×C3716yes—
225#386C22×C3713yes—
226#385C22×C3716yes—
227#385C22×C3716yes—
228#385C22×C3716yes—
229#385C22×C3713yes—
230#385C22×C3716yes—
231#385C22×C3716yescomplemented
232#382C22×C3713yescomplemented
233#380C22×C3716yes—
234#380C22×C3713yes—
235#380C22×C3713yescomplemented
236#379C22×C3713yes—
237C4×C4⋊C4C22×C3713yescomplemented
238C4×C22⋊C4C22×C3716yes—
239C4×C22⋊C4C22×C3713yes—
240C2×#95C22×C3713yes—
241C2×#95C22×C3713yes—
242C2×#95C22×C3716yes—
243C2×#95C22×C3711yescharacteristic, complemented
244C37#3657211yescharacteristic, direct product factor, Sylow
245C25C23×C3711yescharacteristic
246C22×Q8C23×C3713yes—
247C22×D4C23×C3713yes—
248C23×C4—212no—
249C23×C4—412no—
250C23×C4C23×C3716yes—
251C23×C4C23×C3716yes—
252C23×C4C23×C3713yes—
253C23×C4C23×C3713yes—
254C23×C4C23×C3713yes—
255C22⋊Q8—212no—
256C22⋊Q8—412no—
257#120Q8×C3718yescomplemented
258C4×D4—212no—
259C4×D4—412no—
260C2×C4⋊C4C23×C3713yes—
261C2×C4⋊C4C23×C3716yes—
262C2×C4⋊C4C23×C3713yes—
263C2×C4⋊C4C23×C3716yes—
264C2×C4⋊C4C23×C3716yes—
265C2×C4⋊C4C23×C3716yes—
266C2×C4⋊C4C23×C3716yes—
267C2×C22⋊C4C23×C3716yes—
268C2×C22⋊C4C23×C3713yes—
269C2×C22⋊C4C23×C3713yes—
270C2×C22⋊C4C23×C3713yes—
271C2×C22⋊C4C23×C3716yes—
272C2×C22⋊C4C23×C3716yes—
273C2×C22⋊C4C23×C3713yes—
274C2×C42C23×C3713yes—
275C2×C42C23×C3716yes—
276#95C23×C3711yescharacteristic
277#95C23×C3713yes—
278#95C23×C3716yes—
279#95C23×C3713yes—
280#95C23×C3711yescharacteristic
281#95C23×C3712yes—
282#95C23×C3713yes—
283#95C23×C3716yes—
284#95C23×C3716yes—
285#95C23×C3716yes—
286#95C23×C3713yes—
287#95C23×C3716yes—
288#95C23×C3716yes—
289C24—26no—
290C24—26no—
291C24C2×Q8×C3714yes—
292C24C24×C3711yescharacteristic
293C24C24×C3713yes—
294C2×Q8—26no—
295C2×Q8—46no—
296C2×D4—26no—
297C2×D4—26no—
298C2×D4C2×Q8×C37112yes—
299C22×C4—26no—
300C22×C4—212no—
301C22×C4—212no—
302C22×C4—212no—
303C22×C4—212no—
304C22×C4—26no—
305C22×C4—46no—
306C22×C4—46no—
307C22×C4—46no—
308C22×C4—46no—
309C22×C4—46no—
310C22×C4—448no—
311C22×C4#41×C37112yes—
312C22×C4C24×C3716yes—
313C22×C4C24×C3713yes—
314C22×C4C24×C3716yes—
315C22×C4C24×C3713yes—
316C22×C4C24×C3713yes—
317C22×C4C24×C3713yes—
318C22×C4C24×C3713yes—
319C4⋊C4—26no—
320C4⋊C4—26no—
321C4⋊C4—212no—
322C4⋊C4—46no—
323C4⋊C4—46no—
324C4⋊C4—46no—
325C4⋊C4—46no—
326C22⋊C4—26no—
327C22⋊C4—212no—
328C22⋊C4—212no—
329C22⋊C4—26no—
330C22⋊C4—26no—
331C22⋊C4—46no—
332C22⋊C4C2×Q8×C37112yes—
333C42—26no—
334C42—46no—
335C23—224no—
336C23—28no—
337C23—224no—
338C23—26no—
339C23—26no—
340C23—23no—
341C23—23no—
342C23C22×Q8×C3712yes—
343C23C22×Q8×C3716yes—
344C23C2×#41×C3716yes—
345C23H(5,𝔽2)×C3716yes—
346C23C25×C3711yescharacteristic
347Q8—86no—
348D4—224no—
349C2×C4—23no—
350C2×C4—23no—
351C2×C4—26no—
352C2×C4—26no—
353C2×C4—23no—
354C2×C4—26no—
355C2×C4—23no—
356C2×C4—26no—
357C2×C4—26no—
358C2×C4—26no—
359C2×C4—26no—
360C2×C4—26no—
361C2×C4—26no—
362C2×C4—26no—
363C2×C4—412no—
364C2×C4—424no—
365C2×C4—424no—
366C2×C4—412no—
367C2×C4—86no—
368C2×C4C22×Q8×C3716yes—
369C2×C4C2×#41×C3716yes—
370C2×C4C2×#41×C3716yes—
371C22—232no—
372C22—26no—
373C22—212no—
374C22—26no—
375C22—212no—
376C22—26no—
377C22—41no—
378C22—43no—
379C22C37×#554112yes—
380C22C23×Q8×C3711yescharacteristic
381C22C22×#41×C3713yes—
382C22C2×H(5,𝔽2)×C3713yes—
383C4—26no—
384C4—26no—
385C4—26no—
386C4—43no—
387C4—46no—
388C4—43no—
389C4—46no—
390C2—212no—
391C2—42no—
392C2C37×#344411yescharacteristic
393C2C2×C37×#55413yes—
394C2C37×#349013yes—
subgroupG/H
center, socleC23×C37C25
commutator, Frattini, cosocleC23C25×C37
prime radicalC2×#120×C37C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#2848C2semidirect
2. C37×#2564C2semidirect
3. C37×#2549C2semidirect
4. C37×#2505C2semidirect
5. C37×#2450C2semidirect
6. C37×#2368C2semidirect
7. C37×#2349C2semidirect
9. C22×#95×C37C2semidirect
16. C37×#400C22semidirect
17. C37×#400C22semidirect
18. C37×#399C22semidirect
19. C37×#398C22semidirect
20. C37×#396C22semidirect
21. C37×#395C22semidirect
22. C37×#393C22semidirect
23. C37×#391C22semidirect
24. C37×#389C22semidirect
25. C37×#387C22semidirect
26. C37×#387C22semidirect
34. C37×#385C22semidirect
35. C37×#382C22semidirect
38. C37×#380C22semidirect
40. C4×C4⋊C4×C37C22semidirect
46. C2×#95×C37C22semidirect
59. #120×C37Q8semidirect
173. #36572C37direct (N × H)
193. #2848C2×C37semidirect
194. #2564C2×C37semidirect
195. #2549C2×C37semidirect
196. #2505C2×C37semidirect
197. #2450C2×C37semidirect
198. #2368C2×C37semidirect
199. #2349C2×C37semidirect
201. C22×#95C2×C37semidirect
213. #400C22×C37semidirect
214. #400C22×C37semidirect
215. #399C22×C37semidirect
216. #398C22×C37semidirect
217. #396C22×C37semidirect
218. #395C22×C37semidirect
219. #393C22×C37semidirect
220. #391C22×C37semidirect
221. #389C22×C37semidirect
222. #387C22×C37semidirect
223. #387C22×C37semidirect
231. #385C22×C37semidirect
232. #382C22×C37semidirect
235. #380C22×C37semidirect
237. C4×C4⋊C4C22×C37semidirect
243. C2×#95C22×C37semidirect
244. C37#36572direct (N × H)
257. #120Q8×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#28482
C37×#25642
C37×#25492
C37×#25052
C37×#24502
C37×#23682
C37×#23492
C4×#120×C372
C22×#95×C372
#3657237