Finite Group Databasebeta

← C31×#36607 #21493366

Subgroups — C31×#36607 #21493366

Order 7936.

2524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2847C213yesmaximal, complemented
2C31×#2569C213yesmaximal, complemented
3C31×#2543C213yesmaximal, complemented
4C31×#2510C216yesmaximal, complemented
5C31×#2444C216yesmaximal, complemented
6C31×#2364C213yesmaximal, complemented
7C31×#2353C213yesmaximal, complemented
8C4×C31×#120C213yesmaximal, complemented
9C22×C31×#95C211yesmaximal, characteristic, complemented
10C24×C4×C31C2213yes—
11C2×C31×#121C2213yes—
12C2×C31×#121C2216yes—
13C2×C31×#120C2213yes—
14C2×C4×D4×C31C2213yes—
15C2×C4×D4×C31C2216yes—
16C31×#401C2212yescomplemented
17C31×#400C2216yescomplemented
18C31×#398C2213yescomplemented
19C31×#397C2216yescomplemented
20C31×#397C2213yescomplemented
21C31×#394C2213yescomplemented
22C31×#392C2211yescharacteristic, complemented
23C31×#388C2216yescomplemented
24C31×#388C2216yescomplemented
25C31×#387C2216yescomplemented
26C31×#387C2216yescomplemented
27C31×#386C2216yes—
28C31×#386C2216yes—
29C31×#386C2216yes—
30C31×#386C2213yes—
31C31×#386C2216yes—
32C31×#386C2216yescomplemented
33C31×#385C2216yes—
34C31×#385C2213yes—
35C31×#383C2213yescomplemented
36C31×#381C2213yescomplemented
37C31×#380C2213yes—
38C31×#380C2216yes—
39C31×#379C2211yescharacteristic
40C4×C22⋊C4×C31C2213yes—
41C4×C22⋊C4×C31C2216yes—
42C4×C22⋊C4×C31C2213yescomplemented
43C31×#376C2211yescharacteristic, complemented
44C2×C31×#95C2213yes—
45C2×C31×#95C2213yes—
46C2×C31×#95C2216yes—
47C25×C31C2311yescharacteristic
48C22×D4×C31C2313yes—
49C22×D4×C31C2316yes—
50C22×D4×C31C2313yes—
51C23×C4×C31—212no—
52C23×C4×C31—412no—
53C23×C4×C31C2316yes—
54C23×C4×C31C2316yes—
55C23×C4×C31C2313yes—
56C23×C4×C31C2313yes—
57C23×C4×C31C2313yes—
58C31×#121—212no—
59C31×#121—412no—
60C31×#120—212no—
61C4×D4×C31—212no—
62C4×D4×C31—412no—
63C2×C4⋊C4×C31C2313yes—
64C2×C4⋊C4×C31C2313yes—
65C2×C4⋊C4×C31C2316yes—
66C2×C4⋊C4×C31C2316yes—
67C2×C22⋊C4×C31C2313yes—
68C2×C22⋊C4×C31C2313yes—
69C2×C22⋊C4×C31C2313yes—
70C2×C22⋊C4×C31C2313yes—
71C2×C22⋊C4×C31C2316yes—
72C2×C22⋊C4×C31C2316yes—
73C2×C22⋊C4×C31C2316yes—
74C2×C22⋊C4×C31C2316yes—
75C2×C22⋊C4×C31C2316yes—
76C2×C42×C31C2313yes—
77C2×C42×C31C2316yes—
78C31×#95C2312yes—
79C31×#95C2313yes—
80C31×#95C2316yes—
81C31×#95C2316yes—
82C31×#95C2316yes—
83C31×#95C2311yescharacteristic
84C31×#95C2316yes—
85C31×#95C2316yes—
86C31×#95C2311yescharacteristic
87C31×#95C2313yes—
88C31×#95C2316yes—
89C31×#95C2313yes—
90C31×#95C2313yes—
91C24×C31—26no—
92C24×C31—26no—
93C24×C31#4114yes—
94C24×C31C2413yes—
95C24×C31C2413yes—
96C2×D4×C31—26no—
97C2×D4×C31—26no—
98C2×D4×C31—212no—
99C2×D4×C31—212no—
100C2×D4×C31—212no—
101C2×D4×C31—26no—
102C2×D4×C31—26no—
103C2×D4×C31—46no—
104C22×C4×C31—26no—
105C22×C4×C31—212no—
106C22×C4×C31—212no—
107C22×C4×C31—212no—
108C22×C4×C31—212no—
109C22×C4×C31—26no—
110C22×C4×C31—46no—
111C22×C4×C31—46no—
112C22×C4×C31—46no—
113C22×C4×C31—46no—
114C22×C4×C31—46no—
115C22×C4×C31—448no—
116C22×C4×C31#41112yes—
117C22×C4×C31C2416yes—
118C22×C4×C31C2411yescharacteristic
119C22×C4×C31C2413yes—
120C22×C4×C31C2416yes—
121C22×C4×C31C2413yes—
122C22×C4×C31C2413yes—
123C22×C4×C31C2413yes—
124C4⋊C4×C31—26no—
125C4⋊C4×C31—26no—
126C4⋊C4×C31—46no—
127C4⋊C4×C31—46no—
128C22⋊C4×C31—26no—
129C22⋊C4×C31—26no—
130C22⋊C4×C31—212no—
131C22⋊C4×C31—26no—
132C22⋊C4×C31—26no—
133C22⋊C4×C31—212no—
134C22⋊C4×C31—46no—
135C22⋊C4×C31—46no—
136C22⋊C4×C31—46no—
137C42×C31—26no—
138C42×C31—46no—
139#36607C3111yesmaximal, characteristic, direct product factor, Sylow
140C23×C31—224no—
141C23×C31—28no—
142C23×C31—224no—
143C23×C31—26no—
144C23×C31—26no—
145C23×C31—23no—
146C23×C31—26no—
147C23×C31—23no—
148C23×C31—26no—
149C23×C31—23no—
150C23×C31C2×#4116yes—
151C23×C31C2×#4116yes—
152C23×C31H(5,𝔽2)16yes—
153C23×C31C2511yescharacteristic
154D4×C31—412no—
155D4×C31—424no—
156D4×C31—86no—
157C2×C4×C31—26no—
158C2×C4×C31—26no—
159C2×C4×C31—23no—
160C2×C4×C31—23no—
161C2×C4×C31—23no—
162C2×C4×C31—26no—
163C2×C4×C31—23no—
164C2×C4×C31—26no—
165C2×C4×C31—26no—
166C2×C4×C31—26no—
167C2×C4×C31—26no—
168C2×C4×C31—23no—
169C2×C4×C31—26no—
170C2×C4×C31—26no—
171C2×C4×C31—412no—
172C2×C4×C31—424no—
173C2×C4×C31—424no—
174C2×C4×C31—412no—
175C2×C4×C31—86no—
176C2×C4×C31C2×#4116yes—
177C2×C4×C31C2×#4116yes—
178#2847C2×C3113yescomplemented
179#2569C2×C3113yescomplemented
180#2543C2×C3113yescomplemented
181#2510C2×C3116yescomplemented
182#2444C2×C3116yescomplemented
183#2364C2×C3113yescomplemented
184#2353C2×C3113yescomplemented
185C4×#120C2×C3113yescomplemented
186C22×#95C2×C3111yescharacteristic, complemented
187C22×C31—232no—
188C22×C31—26no—
189C22×C31—212no—
190C22×C31—26no—
191C22×C31—212no—
192C22×C31—43no—
193C22×C31—43no—
194C22×C31—46no—
195C22×C31—43no—
196C22×C31—46no—
197C22×C31#553112yes—
198C22×C31C22×#4113yes—
199C22×C31C22×#4111yescharacteristic
200C22×C31C2×H(5,𝔽2)13yes—
201C4×C31—26no—
202C4×C31—26no—
203C4×C31—41no—
204C4×C31—43no—
205C4×C31—46no—
206C4×C31—46no—
207C4×C31—43no—
208C24×C4C22×C3113yes—
209C2×#121C22×C3113yes—
210C2×#121C22×C3116yes—
211C2×#120C22×C3113yes—
212C2×C4×D4C22×C3113yes—
213C2×C4×D4C22×C3116yes—
214#401C22×C3112yescomplemented
215#400C22×C3116yescomplemented
216#398C22×C3113yescomplemented
217#397C22×C3116yescomplemented
218#397C22×C3113yescomplemented
219#394C22×C3113yescomplemented
220#392C22×C3111yescharacteristic, complemented
221#388C22×C3116yescomplemented
222#388C22×C3116yescomplemented
223#387C22×C3116yescomplemented
224#387C22×C3116yescomplemented
225#386C22×C3116yes—
226#386C22×C3116yes—
227#386C22×C3116yes—
228#386C22×C3113yes—
229#386C22×C3116yes—
230#386C22×C3116yescomplemented
231#385C22×C3116yes—
232#385C22×C3113yes—
233#383C22×C3113yescomplemented
234#381C22×C3113yescomplemented
235#380C22×C3113yes—
236#380C22×C3116yes—
237#379C22×C3111yescharacteristic
238C4×C22⋊C4C22×C3113yes—
239C4×C22⋊C4C22×C3116yes—
240C4×C22⋊C4C22×C3113yescomplemented
241#376C22×C3111yescharacteristic, complemented
242C2×#95C22×C3113yes—
243C2×#95C22×C3113yes—
244C2×#95C22×C3116yes—
245C2×C31—212no—
246C2×C31—83no—
247C2×C31#344411yescharacteristic
248C2×C31C2×#55313yes—
249C2×C31#349013yes—
250C25C23×C3111yescharacteristic
251C22×D4C23×C3113yes—
252C22×D4C23×C3116yes—
253C22×D4C23×C3113yes—
254C23×C4—212no—
255C23×C4—412no—
256C23×C4C23×C3116yes—
257C23×C4C23×C3116yes—
258C23×C4C23×C3113yes—
259C23×C4C23×C3113yes—
260C23×C4C23×C3113yes—
261#121—212no—
262#121—412no—
263#120—212no—
264C4×D4—212no—
265C4×D4—412no—
266C2×C4⋊C4C23×C3113yes—
267C2×C4⋊C4C23×C3113yes—
268C2×C4⋊C4C23×C3116yes—
269C2×C4⋊C4C23×C3116yes—
270C2×C22⋊C4C23×C3113yes—
271C2×C22⋊C4C23×C3113yes—
272C2×C22⋊C4C23×C3113yes—
273C2×C22⋊C4C23×C3113yes—
274C2×C22⋊C4C23×C3116yes—
275C2×C22⋊C4C23×C3116yes—
276C2×C22⋊C4C23×C3116yes—
277C2×C22⋊C4C23×C3116yes—
278C2×C22⋊C4C23×C3116yes—
279C2×C42C23×C3113yes—
280C2×C42C23×C3116yes—
281#95C23×C3111yescharacteristic
282#95C23×C3113yes—
283#95C23×C3116yes—
284#95C23×C3111yescharacteristic
285#95C23×C3113yes—
286#95C23×C3112yes—
287#95C23×C3113yes—
288#95C23×C3116yes—
289#95C23×C3116yes—
290#95C23×C3116yes—
291#95C23×C3113yes—
292#95C23×C3116yes—
293#95C23×C3116yes—
294C31#3660711yescharacteristic, direct product factor, Sylow
295C24—26no—
296C24—26no—
297C24#41×C3114yes—
298C24C24×C3113yes—
299C24C24×C3113yes—
300C2×D4—26no—
301C2×D4—26no—
302C2×D4—212no—
303C2×D4—212no—
304C2×D4—212no—
305C2×D4—26no—
306C2×D4—26no—
307C2×D4—46no—
308C22×C4—26no—
309C22×C4—212no—
310C22×C4—212no—
311C22×C4—212no—
312C22×C4—212no—
313C22×C4—26no—
314C22×C4—46no—
315C22×C4—46no—
316C22×C4—46no—
317C22×C4—46no—
318C22×C4—46no—
319C22×C4—448no—
320C22×C4#41×C31112yes—
321C22×C4C24×C3116yes—
322C22×C4C24×C3111yescharacteristic
323C22×C4C24×C3113yes—
324C22×C4C24×C3116yes—
325C22×C4C24×C3113yes—
326C22×C4C24×C3113yes—
327C22×C4C24×C3113yes—
328C4⋊C4—26no—
329C4⋊C4—26no—
330C4⋊C4—46no—
331C4⋊C4—46no—
332C22⋊C4—26no—
333C22⋊C4—26no—
334C22⋊C4—212no—
335C22⋊C4—26no—
336C22⋊C4—26no—
337C22⋊C4—212no—
338C22⋊C4—46no—
339C22⋊C4—46no—
340C22⋊C4—46no—
341C42—26no—
342C42—46no—
343C23—224no—
344C23—28no—
345C23—224no—
346C23—26no—
347C23—26no—
348C23—23no—
349C23—26no—
350C23—23no—
351C23—26no—
352C23—23no—
353C23C2×#41×C3116yes—
354C23C2×#41×C3116yes—
355C23C31×H(5,𝔽2)16yes—
356C23C25×C3111yescharacteristic
357D4—412no—
358D4—424no—
359D4—86no—
360C2×C4—26no—
361C2×C4—26no—
362C2×C4—23no—
363C2×C4—23no—
364C2×C4—23no—
365C2×C4—26no—
366C2×C4—23no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—23no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—412no—
375C2×C4—424no—
376C2×C4—424no—
377C2×C4—412no—
378C2×C4—86no—
379C2×C4C2×#41×C3116yes—
380C2×C4C2×#41×C3116yes—
381C22—232no—
382C22—26no—
383C22—212no—
384C22—26no—
385C22—212no—
386C22—43no—
387C22—43no—
388C22—46no—
389C22—43no—
390C22—46no—
391C22C31×#553112yes—
392C22C22×#41×C3113yes—
393C22C22×#41×C3111yescharacteristic
394C22C2×C31×H(5,𝔽2)13yes—
395C4—26no—
396C4—26no—
397C4—41no—
398C4—43no—
399C4—46no—
400C4—46no—
401C4—43no—
402C2—212no—
403C2—83no—
404C2C31×#344411yescharacteristic
405C2C2×C31×#55313yes—
406C2C31×#349013yes—
subgroupG/H
center, socleC23×C31C25
commutator, Frattini, cosocleC23C25×C31
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2847C2semidirect
2. C31×#2569C2semidirect
3. C31×#2543C2semidirect
4. C31×#2510C2semidirect
5. C31×#2444C2semidirect
6. C31×#2364C2semidirect
7. C31×#2353C2semidirect
8. C4×C31×#120C2semidirect
9. C22×C31×#95C2semidirect
16. C31×#401C22semidirect
17. C31×#400C22semidirect
18. C31×#398C22semidirect
19. C31×#397C22semidirect
20. C31×#397C22semidirect
21. C31×#394C22semidirect
22. C31×#392C22semidirect
23. C31×#388C22semidirect
24. C31×#388C22semidirect
25. C31×#387C22semidirect
26. C31×#387C22semidirect
32. C31×#386C22semidirect
35. C31×#383C22semidirect
36. C31×#381C22semidirect
42. C4×C22⋊C4×C31C22semidirect
43. C31×#376C22semidirect
139. #36607C31direct (N × H)
178. #2847C2×C31semidirect
179. #2569C2×C31semidirect
180. #2543C2×C31semidirect
181. #2510C2×C31semidirect
182. #2444C2×C31semidirect
183. #2364C2×C31semidirect
184. #2353C2×C31semidirect
185. C4×#120C2×C31semidirect
186. C22×#95C2×C31semidirect
214. #401C22×C31semidirect
215. #400C22×C31semidirect
216. #398C22×C31semidirect
217. #397C22×C31semidirect
218. #397C22×C31semidirect
219. #394C22×C31semidirect
220. #392C22×C31semidirect
221. #388C22×C31semidirect
222. #388C22×C31semidirect
223. #387C22×C31semidirect
224. #387C22×C31semidirect
230. #386C22×C31semidirect
233. #383C22×C31semidirect
234. #381C22×C31semidirect
240. C4×C22⋊C4C22×C31semidirect
241. #376C22×C31semidirect
294. C31#36607direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28472
C31×#25692
C31×#25432
C31×#25102
C31×#24442
C31×#23642
C31×#23532
C4×C31×#1202
C22×C31×#952
#3660731