Finite Group Databasebeta

← C31×#8250 #21465009

Subgroups — C31×#8250 #21465009

Order 7936.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2966C211yesmaximal, characteristic
2C31×#1746C211yesmaximal, characteristic, complemented
3C31×#1524C211yesmaximal, characteristic
4C31×#1500C211yesmaximal, characteristic
5C8×M16×C31C211yesmaximal, characteristic, complemented
6C31×#1302C211yesmaximal, characteristic, complemented
7C31×#1300C211yesmaximal, characteristic, complemented
8C31×#529C2211yescharacteristic
9C31×#446C411yescharacteristic, complemented
10C8×Q8×C31C411yescharacteristic, complemented
11C31×#436C411yescharacteristic, complemented
12C31×#435—21no—
13C8×D4×C31—21no—
14C8×D4×C31C411yescharacteristic, complemented
15C31×#433—21no—
16C31×#432C2211yescharacteristic
17C31×#431C2211yescharacteristic
18C31×#405—21no—
19C4×M16×C31C2211yescharacteristic
20C2×C31×#97C2211yescharacteristic
21C2×C4×C8×C31C2211yescharacteristic
22C31×#376C2211yescharacteristic
23C31×#336—21no—
24C31×#336—21no—
25C31×#330C412yescomplemented
26C31×#329C412yescomplemented
27C31×#328C412yes—
28C31×#327C412yes—
29C31×#322—21no—
30C82×C31—21no—
31C2×M16×C31—21no—
32C2×M16×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C22×C8×C31D411yescharacteristic
36C22×C8×C31D411yescharacteristic
37C31×#126—21no—
38C31×#125C2×C411yescharacteristic
39C31×#124C2×C411yescharacteristic
40C31×#123—21no—
41C31×C22⋊Q8C2×C411yescharacteristic
42C31×#121C2×C411yescharacteristic
43C4×Q8×C31C2×C411yescharacteristic
44C4×D4×C31—21no—
45C4×D4×C31C2×C411yescharacteristic
46C31×#117—21no—
47C2×C42×C31—21no—
48C2×C42×C31C2311yescharacteristic
49C31×C4⋊C8—21no—
50C31×C4⋊C8—21no—
51C31×C4⋊C8—21no—
52C31×C4⋊C8C2×C411yescharacteristic
53C31×C4⋊C8C2×C411yescharacteristic
54C31×C22⋊C8—21no—
55C31×C22⋊C8—21no—
56C31×C22⋊C8—21no—
57C31×C22⋊C8C2×C412yes—
58C31×#97—21no—
59C31×#97—21no—
60C31×#97—21no—
61C31×#97C2×C412yes—
62C31×#97C2×C412yes—
63C4×C8×C31—21no—
64C4×C8×C31—21no—
65C4×C8×C31—21no—
66C4×C8×C31C2×C411yescharacteristic
67C4×C8×C31C2×C411yescharacteristic
68C4×C8×C31D411yescharacteristic
69C4×C8×C31D411yescharacteristic
70C31×#95—21no—
71C31×#95C2×C412yes—
72C2×Q8×C31C4211yescharacteristic
73C2×D4×C31—21no—
74C2×D4×C31C4211yescharacteristic
75C22×C4×C31—21no—
76C22×C4×C31—22no—
77C22×C4×C31C22×C411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31#4111yescharacteristic
80M16×C31—24no—
81M16×C31—42no—
82C2×C8×C31—22no—
83C2×C8×C31—21no—
84C2×C8×C31—21no—
85C2×C8×C31—22no—
86C2×C8×C31—22no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C2×C8×C31—42no—
90C2×C8×C31—41no—
91C2×C8×C31—41no—
92C2×C8×C31C22⋊C411yescharacteristic
93C2×C8×C31C22⋊C411yescharacteristic
94C2×C8×C31C22⋊C411yescharacteristic
95C2×C8×C31C22⋊C411yescharacteristic
96C4⋊C4×C31—21no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31C4211yescharacteristic
101C4⋊C4×C31C4211yescharacteristic
102C22⋊C4×C31—21no—
103C22⋊C4×C31—21no—
104C22⋊C4×C31—21no—
105C22⋊C4×C31—21no—
106C22⋊C4×C31—21no—
107C42×C31—21no—
108C42×C31—41no—
109C42×C31—41no—
110C42×C31C22×C411yescharacteristic
111C42×C31C22×C411yescharacteristic
112C42×C31C2×D411yescharacteristic
113C42×C31#4111yescharacteristic
114#8250C3111yesmaximal, characteristic, direct product factor, Sylow
115C23×C31—21no—
116C23×C31C4×D411yescharacteristic
117Q8×C31—41no—
118D4×C31—41no—
119D4×C31—42no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31—42no—
132C2×C4×C31C2×C4211yescharacteristic
133C2×C4×C31C2×C22⋊C411yescharacteristic
134C2×C4×C31#11711yescharacteristic
135C2×C4×C31C4×D411yescharacteristic
136C2×C4×C31C4×D411yescharacteristic
137C2×C4×C31C4×D411yescharacteristic
138C8×C31—24no—
139C8×C31—21no—
140C8×C31—21no—
141C8×C31—21no—
142C8×C31—21no—
143C8×C31—42no—
144C8×C31—42no—
145C8×C31—41no—
146C8×C31—41no—
147#2966C2×C3111yescharacteristic
148#1746C2×C3111yescharacteristic, complemented
149#1524C2×C3111yescharacteristic
150#1500C2×C3111yescharacteristic
151C8×M16C2×C3111yescharacteristic, complemented
152#1302C2×C3111yescharacteristic, complemented
153#1300C2×C3111yescharacteristic, complemented
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31C4×C22⋊C411yescharacteristic
160C22×C31#44312yes—
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—21no—
164C4×C31—21no—
165C4×C31—21no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31—82no—
170C4×C31#42111yescharacteristic
171C4×C31#42111yescharacteristic
172#529C22×C3111yescharacteristic
173#446C4×C3111yescharacteristic, complemented
174C8×Q8C4×C3111yescharacteristic, complemented
175#436C4×C3111yescharacteristic, complemented
176#435—21no—
177C8×D4—21no—
178C8×D4C4×C3111yescharacteristic, complemented
179#433—21no—
180#432C22×C3111yescharacteristic
181#431C22×C3111yescharacteristic
182#405—21no—
183C4×M16C22×C3111yescharacteristic
184C2×#97C22×C3111yescharacteristic
185C2×C4×C8C22×C3111yescharacteristic
186#376C22×C3111yescharacteristic
187#336—21no—
188#336—21no—
189#330C4×C3112yescomplemented
190#329C4×C3112yescomplemented
191#328C4×C3112yes—
192#327C4×C3112yes—
193#322—21no—
194C82—21no—
195C2×C31—22no—
196C2×C31—81no—
197C2×C31#175711yescharacteristic
198C2×C31#175911yescharacteristic
199C2×C31#176411yescharacteristic
200C2×M16—21no—
201C2×M16—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8D4×C3111yescharacteristic
205C22×C8D4×C3111yescharacteristic
206#126—21no—
207#125C2×C4×C3111yescharacteristic
208#124C2×C4×C3111yescharacteristic
209#123—21no—
210C22⋊Q8C2×C4×C3111yescharacteristic
211#121C2×C4×C3111yescharacteristic
212C4×Q8C2×C4×C3111yescharacteristic
213C4×D4—21no—
214C4×D4C2×C4×C3111yescharacteristic
215#117—21no—
216C2×C42—21no—
217C2×C42C23×C3111yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8C2×C4×C3111yescharacteristic
222C4⋊C8C2×C4×C3111yescharacteristic
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8C2×C4×C3112yes—
227#97—21no—
228#97—21no—
229#97—21no—
230#97C2×C4×C3112yes—
231#97C2×C4×C3112yes—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8C2×C4×C3111yescharacteristic
236C4×C8C2×C4×C3111yescharacteristic
237C4×C8D4×C3111yescharacteristic
238C4×C8D4×C3111yescharacteristic
239#95—21no—
240#95C2×C4×C3112yes—
241C31#825011yescharacteristic, direct product factor, Sylow
242C2×Q8C42×C3111yescharacteristic
243C2×D4—21no—
244C2×D4C42×C3111yescharacteristic
245C22×C4—21no—
246C22×C4—22no—
247C22×C4C22×C4×C3111yescharacteristic
248C22×C4C2×D4×C3111yescharacteristic
249C22×C4#41×C3111yescharacteristic
250M16—24no—
251M16—42no—
252C2×C8—22no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8C22⋊C4×C3111yescharacteristic
263C2×C8C22⋊C4×C3111yescharacteristic
264C2×C8C22⋊C4×C3111yescharacteristic
265C2×C8C22⋊C4×C3111yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C42×C3111yescharacteristic
271C4⋊C4C42×C3111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C42—41no—
279C42—41no—
280C42C22×C4×C3111yescharacteristic
281C42C22×C4×C3111yescharacteristic
282C42C2×D4×C3111yescharacteristic
283C42#41×C3111yescharacteristic
284C23—21no—
285C23C4×D4×C3111yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4C2×C42×C3111yescharacteristic
302C2×C4C2×C22⋊C4×C3111yescharacteristic
303C2×C4C31×#11711yescharacteristic
304C2×C4C4×D4×C3111yescharacteristic
305C2×C4C4×D4×C3111yescharacteristic
306C2×C4C4×D4×C3111yescharacteristic
307C8—24no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—21no—
312C8—42no—
313C8—42no—
314C8—41no—
315C8—41no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C4×C22⋊C4×C3111yescharacteristic
322C22C31×#44312yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4C31×#42111yescharacteristic
333C4C31×#42111yescharacteristic
334C2—22no—
335C2—81no—
336C2C31×#175711yescharacteristic
337C2C31×#175911yescharacteristic
338C2C31×#176411yescharacteristic
subgroupG/H
centerC2×C4×C31C2×C22⋊C4
commutatorC2×C4C2×C42×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31C4×C22⋊C4
prime radicalC31×#121C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C4×C8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1746C2semidirect
5. C8×M16×C31C2semidirect
6. C31×#1302C2semidirect
7. C31×#1300C2semidirect
9. C31×#446C4semidirect
10. C8×Q8×C31C4semidirect
11. C31×#436C4semidirect
14. C8×D4×C31C4semidirect
25. C31×#330C4semidirect
26. C31×#329C4semidirect
114. #8250C31direct (N × H)
148. #1746C2×C31semidirect
151. C8×M16C2×C31semidirect
152. #1302C2×C31semidirect
153. #1300C2×C31semidirect
173. #446C4×C31semidirect
174. C8×Q8C4×C31semidirect
175. #436C4×C31semidirect
178. C8×D4C4×C31semidirect
189. #330C4×C31semidirect
190. #329C4×C31semidirect
241. C31#8250direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29662
C31×#17462
C31×#15242
C31×#15002
C8×M16×C312
C31×#13022
C31×#13002
#825031