Finite Group Databasebeta

← C31×#60140 #21516899

Subgroups — C31×#60140 #21516899

Order 7936.

2722 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#131C212yesmaximal, complemented
2C31×#3449C211yesmaximal, characteristic
3C31×#2932C214yesmaximal, complemented
4C31×#2931C214yesmaximal, complemented
5C31×#2930C218yesmaximal, complemented
6C31×#2929C218yesmaximal, complemented
7C31×#2905C211yesmaximal, characteristic, complemented
8C31×#2879C212yesmaximal, complemented
9C2×C31×#405C211yesmaximal, characteristic, complemented
10C22×#41×C31C2212yes—
11C2×C31×#131—216no—
12C2×C31×#131C2214yes—
13C2×C31×#131C2214yes—
14C2×C31×#131C2214yescomplemented
15C2×C31×#131C2214yescomplemented
16C22×M16×C31C2211yescharacteristic
17C22×M16×C31C2212yes—
18C22×M16×C31C2214yescomplemented
19C23×C8×C31C2211yescharacteristic
20C23×C8×C31C2214yescomplemented
21C31×#537C414yes—
22C31×#536C414yes—
23C31×#535C414yes—
24C31×#534C414yes—
25C31×#532C2214yes—
26C31×#525C2218yes—
27C2×C31×#117C2211yescharacteristic
28C31×#436C22116yescomplemented
29C31×#435C22132yescomplemented
30C8×D4×C31C22116yescomplemented
31C31×#424C2218yescomplemented
32C31×#423C2214yescomplemented
33C31×#408C2218yes—
34C31×#408C2218yescomplemented
35C31×#407C2218yescomplemented
36C31×#405C2214yescomplemented
37C31×#405C2214yescomplemented
38C2×C31×#97C2212yescomplemented
39C2×C4×C8×C31C2212yescomplemented
40C2×#41×C31—28no—
41C2×#41×C31—28no—
42C2×#41×C31C2×C414yes—
43C2×#41×C31C2×C414yes—
44C2×#41×C31C2314yes—
45C2×#41×C31C2314yes—
46C22×Q8×C31C2×C412yes—
47C22×D4×C31C2×C414yes—
48C22×D4×C31C2×C412yes—
49C23×C4×C31C2314yes—
50C23×C4×C31C2311yescharacteristic
51C31×#131—216no—
52C31×#131—216no—
53C31×#131—216no—
54C31×#131—216no—
55C2×M16×C31—28no—
56C2×M16×C31—24no—
57C2×M16×C31—216no—
58C2×M16×C31C2314yes—
59C2×M16×C31C2314yes—
60C2×M16×C31C2314yes—
61C2×M16×C31C2318yes—
62C2×M16×C31C2318yes—
63C22×C8×C31—216no—
64C22×C8×C31D418yescomplemented
65C22×C8×C31C2312yes—
66C22×C8×C31C2314yes—
67C22×C8×C31C2318yes—
68C22×C8×C31C2318yes—
69C22×C8×C31C2311yescharacteristic
70C22×C8×C31C2312yes—
71C22×C8×C31C2311yescharacteristic
72C22×C8×C31C2312yes—
73C31×C4⋊Q8C2×C414yes—
74C31×#127C2×C414yes—
75C31×#124C2×C418yes—
76C31×#123C2×C4116yes—
77C31×C22⋊Q8C2×C4116yes—
78C31×#121C2×C4116yes—
79C31×#121C2×C4116yes—
80C31×#120C2×C4116yes—
81C4×D4×C31C23116yes—
82C31×#117C2×C414yes—
83C31×#117C2314yes—
84C2×C4⋊C4×C31C2×C412yes—
85C2×C22⋊C4×C31C2×C412yes—
86C2×C42×C31C2312yes—
87C31×C4⋊C8C23116yes—
88C31×C22⋊C8C23132yes—
89C31×#97C2318yes—
90C4×C8×C31C2314yes—
91C4×C8×C31C2314yes—
92C24×C31C22×C414yes—
93C24×C31C22×C411yescharacteristic
94#41×C31—216no—
95#41×C31—216no—
96#41×C31—28no—
97#41×C31—28no—
98#41×C31—28no—
99#41×C31—28no—
100C2×Q8×C31—28no—
101C2×Q8×C31C22×C414yes—
102C2×Q8×C31C22×C414yes—
103C2×D4×C31—216no—
104C2×D4×C31—28no—
105C2×D4×C31C22×C418yes—
106C2×D4×C31C22×C418yes—
107C2×D4×C31C22×C414yes—
108C2×D4×C31C22×C414yes—
109C2×D4×C31C22×C4116yes—
110C22×C4×C31—28no—
111C22×C4×C31—28no—
112C22×C4×C31C22×C418yes—
113C22×C4×C31C22×C414yes—
114C22×C4×C31C22×C414yes—
115C22×C4×C31C22×C411yescharacteristic
116C22×C4×C31C22×C412yes—
117C22×C4×C31C2×D414yes—
118C22×C4×C31#4114yes—
119C22×C4×C31C2418yes—
120C22×C4×C31C2411yescharacteristic
121C22×C4×C31C2412yes—
122M16×C31—28no—
123M16×C31—28no—
124M16×C31—28no—
125M16×C31—216no—
126M16×C31—216no—
127C2×C8×C31—28no—
128C2×C8×C31—24no—
129C2×C8×C31—216no—
130C2×C8×C31—216no—
131C2×C8×C31C2×D418yes—
132C2×C8×C31C2×D418yes—
133C2×C8×C31C2×D414yes—
134C2×C8×C31C2×D414yes—
135C2×C8×C31C2414yes—
136C2×C8×C31C2418yes—
137C2×C8×C31C2411yescharacteristic
138C2×C8×C31C2412yes—
139C2×C8×C31C2411yescharacteristic
140C4⋊C4×C31C22×C418yes—
141C4⋊C4×C31C22×C4116yes—
142C22⋊C4×C31C22×C418yes—
143C22⋊C4×C31C22×C4132yes—
144C42×C31C22×C414yes—
145C42×C31C2414yes—
146#60140C3111yesmaximal, characteristic, direct product factor, Sylow
147C23×C31—28no—
148C23×C31—42no—
149C23×C31—48no—
150C23×C31C4×D414yes—
151C23×C31C23×C418yes—
152C23×C31C23×C412yes—
153C23×C31C23×C411yescharacteristic
154Q8×C31—28no—
155Q8×C31—28no—
156D4×C31—216no—
157D4×C31—216no—
158D4×C31—28no—
159D4×C31—28no—
160D4×C31—416no—
161C2×C4×C31—28no—
162C2×C4×C31—216no—
163C2×C4×C31—28no—
164C2×C4×C31—28no—
165C2×C4×C31—44no—
166C2×C4×C31C4×D418yes—
167C2×C4×C31C4×D414yes—
168C2×C4×C31C23×C418yes—
169C2×C4×C31C23×C418yes—
170C2×C4×C31C23×C411yescharacteristic
171C2×C4×C31C23×C412yes—
172C2×C4×C31C22×D414yes—
173C2×C4×C31C22×D412yes—
174C2×C4×C31C2×#4112yes—
175C2×C4×C31C2×#4114yes—
176C2×C4×C31C2511yescharacteristic
177C8×C31—28no—
178C8×C31—24no—
179C8×C31C22×D412yes—
180C8×C31C22×D414yes—
181C8×C31C22×D412yes—
182C22×#131C2×C3112yescomplemented
183#3449C2×C3111yescharacteristic
184#2932C2×C3114yescomplemented
185#2931C2×C3114yescomplemented
186#2930C2×C3118yescomplemented
187#2929C2×C3118yescomplemented
188#2905C2×C3111yescharacteristic, complemented
189#2879C2×C3112yescomplemented
190C2×#405C2×C3111yescharacteristic, complemented
191C22×C31—22no—
192C22×C31—24no—
193C22×C31—28no—
194C22×C31—216no—
195C22×C31—44no—
196C22×C31—416no—
197C22×C31C2×C4×D414yes—
198C22×C31C2×C4×D412yes—
199C22×C31C24×C411yescharacteristic
200C4×C31—28no—
201C4×C31—28no—
202C4×C31C2×C4×D414yes—
203C4×C31C2×C4×D412yes—
204C4×C31C23×D411yescharacteristic
205C4×C31C22×#4111yescharacteristic
206C22×#41C22×C3112yes—
207C2×#131—216no—
208C2×#131C22×C3114yes—
209C2×#131C22×C3114yes—
210C2×#131C22×C3114yescomplemented
211C2×#131C22×C3114yescomplemented
212C22×M16C22×C3111yescharacteristic
213C22×M16C22×C3112yes—
214C22×M16C22×C3114yescomplemented
215C23×C8C22×C3111yescharacteristic
216C23×C8C22×C3114yescomplemented
217#537C4×C3114yes—
218#536C4×C3114yes—
219#535C4×C3114yes—
220#534C4×C3114yes—
221#532C22×C3114yes—
222#525C22×C3118yes—
223C2×#117C22×C3111yescharacteristic
224#436C22×C31116yescomplemented
225#435C22×C31132yescomplemented
226C8×D4C22×C31116yescomplemented
227#424C22×C3118yescomplemented
228#423C22×C3114yescomplemented
229#408C22×C3118yes—
230#408C22×C3118yescomplemented
231#407C22×C3118yescomplemented
232#405C22×C3114yescomplemented
233#405C22×C3114yescomplemented
234C2×#97C22×C3112yescomplemented
235C2×C4×C8C22×C3112yescomplemented
236C2×C31—24no—
237C2×C31—22no—
238C2×C31—48no—
239C2×C31C22×C4×D411yescharacteristic
240C2×C31#357312yes—
241C2×#41—28no—
242C2×#41—28no—
243C2×#41C2×C4×C3114yes—
244C2×#41C2×C4×C3114yes—
245C2×#41C23×C3114yes—
246C2×#41C23×C3114yes—
247C22×Q8C2×C4×C3112yes—
248C22×D4C2×C4×C3114yes—
249C22×D4C2×C4×C3112yes—
250C23×C4C23×C3114yes—
251C23×C4C23×C3111yescharacteristic
252#131—216no—
253#131—216no—
254#131—216no—
255#131—216no—
256C2×M16—28no—
257C2×M16—24no—
258C2×M16—216no—
259C2×M16C23×C3114yes—
260C2×M16C23×C3114yes—
261C2×M16C23×C3114yes—
262C2×M16C23×C3118yes—
263C2×M16C23×C3118yes—
264C22×C8—216no—
265C22×C8D4×C3118yescomplemented
266C22×C8C23×C3114yes—
267C22×C8C23×C3118yes—
268C22×C8C23×C3118yes—
269C22×C8C23×C3112yes—
270C22×C8C23×C3112yes—
271C22×C8C23×C3112yes—
272C22×C8C23×C3111yescharacteristic
273C22×C8C23×C3111yescharacteristic
274C4⋊Q8C2×C4×C3114yes—
275#127C2×C4×C3114yes—
276#124C2×C4×C3118yes—
277#123C2×C4×C31116yes—
278C22⋊Q8C2×C4×C31116yes—
279#121C2×C4×C31116yes—
280#121C2×C4×C31116yes—
281#120C2×C4×C31116yes—
282C4×D4C23×C31116yes—
283#117C2×C4×C3114yes—
284#117C23×C3114yes—
285C2×C4⋊C4C2×C4×C3112yes—
286C2×C22⋊C4C2×C4×C3112yes—
287C2×C42C23×C3112yes—
288C4⋊C8C23×C31116yes—
289C22⋊C8C23×C31132yes—
290#97C23×C3118yes—
291C4×C8C23×C3114yes—
292C4×C8C23×C3114yes—
293C31#6014011yescharacteristic, direct product factor, Sylow
294C24C22×C4×C3114yes—
295C24C22×C4×C3111yescharacteristic
296#41—216no—
297#41—216no—
298#41—28no—
299#41—28no—
300#41—28no—
301#41—28no—
302C2×Q8—28no—
303C2×Q8C22×C4×C3114yes—
304C2×Q8C22×C4×C3114yes—
305C2×D4—216no—
306C2×D4—28no—
307C2×D4C22×C4×C3118yes—
308C2×D4C22×C4×C3118yes—
309C2×D4C22×C4×C3114yes—
310C2×D4C22×C4×C3114yes—
311C2×D4C22×C4×C31116yes—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4C22×C4×C3118yes—
315C22×C4C22×C4×C3114yes—
316C22×C4C22×C4×C3114yes—
317C22×C4C22×C4×C3111yescharacteristic
318C22×C4C22×C4×C3112yes—
319C22×C4C2×D4×C3114yes—
320C22×C4#41×C3114yes—
321C22×C4C24×C3118yes—
322C22×C4C24×C3111yescharacteristic
323C22×C4C24×C3112yes—
324M16—28no—
325M16—28no—
326M16—28no—
327M16—216no—
328M16—216no—
329C2×C8—28no—
330C2×C8—24no—
331C2×C8—216no—
332C2×C8—216no—
333C2×C8C2×D4×C3118yes—
334C2×C8C2×D4×C3118yes—
335C2×C8C2×D4×C3114yes—
336C2×C8C2×D4×C3114yes—
337C2×C8C24×C3114yes—
338C2×C8C24×C3118yes—
339C2×C8C24×C3111yescharacteristic
340C2×C8C24×C3112yes—
341C2×C8C24×C3111yescharacteristic
342C4⋊C4C22×C4×C3118yes—
343C4⋊C4C22×C4×C31116yes—
344C22⋊C4C22×C4×C3118yes—
345C22⋊C4C22×C4×C31132yes—
346C42C22×C4×C3114yes—
347C42C24×C3114yes—
348C23—28no—
349C23—42no—
350C23—48no—
351C23C4×D4×C3114yes—
352C23C23×C4×C3118yes—
353C23C23×C4×C3112yes—
354C23C23×C4×C3111yescharacteristic
355Q8—28no—
356Q8—28no—
357D4—216no—
358D4—216no—
359D4—28no—
360D4—28no—
361D4—416no—
362C2×C4—28no—
363C2×C4—216no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—44no—
367C2×C4C4×D4×C3118yes—
368C2×C4C4×D4×C3114yes—
369C2×C4C23×C4×C3118yes—
370C2×C4C23×C4×C3118yes—
371C2×C4C23×C4×C3111yescharacteristic
372C2×C4C23×C4×C3112yes—
373C2×C4C22×D4×C3114yes—
374C2×C4C22×D4×C3112yes—
375C2×C4C2×#41×C3112yes—
376C2×C4C2×#41×C3114yes—
377C2×C4C25×C3111yescharacteristic
378C8—28no—
379C8—24no—
380C8C22×D4×C3112yes—
381C8C22×D4×C3114yes—
382C8C22×D4×C3112yes—
383C22—22no—
384C22—24no—
385C22—28no—
386C22—216no—
387C22—44no—
388C22—416no—
389C22C2×C4×D4×C3114yes—
390C22C2×C4×D4×C3112yes—
391C22C24×C4×C3111yescharacteristic
392C4—28no—
393C4—28no—
394C4C2×C4×D4×C3114yes—
395C4C2×C4×D4×C3112yes—
396C4C23×D4×C3111yescharacteristic
397C4C22×#41×C3111yescharacteristic
398C2—24no—
399C2—22no—
400C2—48no—
401C2C22×C4×D4×C3111yescharacteristic
402C2C31×#357312yes—
subgroupG/H
centerC2×C8×C31C24
commutatorC22C24×C4×C31
Frattini, cosocleC2×C4C25×C31
socleC22×C31C24×C4
prime radicalC31×#3449C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×#131C2semidirect
3. C31×#2932C2semidirect
4. C31×#2931C2semidirect
5. C31×#2930C2semidirect
6. C31×#2929C2semidirect
7. C31×#2905C2semidirect
8. C31×#2879C2semidirect
9. C2×C31×#405C2semidirect
14. C2×C31×#131C22semidirect
15. C2×C31×#131C22semidirect
18. C22×M16×C31C22semidirect
20. C23×C8×C31C22semidirect
28. C31×#436C22semidirect
29. C31×#435C22semidirect
30. C8×D4×C31C22semidirect
31. C31×#424C22semidirect
32. C31×#423C22semidirect
34. C31×#408C22semidirect
35. C31×#407C22semidirect
36. C31×#405C22semidirect
37. C31×#405C22semidirect
38. C2×C31×#97C22semidirect
39. C2×C4×C8×C31C22semidirect
64. C22×C8×C31D4semidirect
146. #60140C31direct (N × H)
182. C22×#131C2×C31semidirect
184. #2932C2×C31semidirect
185. #2931C2×C31semidirect
186. #2930C2×C31semidirect
187. #2929C2×C31semidirect
188. #2905C2×C31semidirect
189. #2879C2×C31semidirect
190. C2×#405C2×C31semidirect
210. C2×#131C22×C31semidirect
211. C2×#131C22×C31semidirect
214. C22×M16C22×C31semidirect
216. C23×C8C22×C31semidirect
224. #436C22×C31semidirect
225. #435C22×C31semidirect
226. C8×D4C22×C31semidirect
227. #424C22×C31semidirect
228. #423C22×C31semidirect
230. #408C22×C31semidirect
231. #407C22×C31semidirect
232. #405C22×C31semidirect
233. #405C22×C31semidirect
234. C2×#97C22×C31semidirect
235. C2×C4×C8C22×C31semidirect
265. C22×C8D4×C31semidirect
293. C31#60140direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1312
C31×#34492
C31×#29322
C31×#29312
C31×#29302
C31×#29292
C31×#29052
C31×#28792
C2×C31×#4052
#6014031