Finite Group Databasebeta

← C31×#11730 #21468489

Subgroups — C31×#11730 #21468489

Order 7936.

500 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2631C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic, complemented
3C31×#1459C211yesmaximal, characteristic, complemented
4C31×#1351C212yesmaximal, complemented
5C31×#1343C212yesmaximal, complemented
6C2×C31×#124C411yescharacteristic, complemented
7C2×C4×D4×C31C2211yescharacteristic
8C2×C31×#99—21no—
9C2×C31×C22⋊C8—21no—
10C31×#397—21no—
11C31×#393C411yescharacteristic, complemented
12C31×#393C2211yescharacteristic
13C31×#392—21no—
14C31×#392C411yescharacteristic, complemented
15C31×#392C2211yescharacteristic
16C31×#386—21no—
17C31×#386—21no—
18C31×#383C411yescharacteristic, complemented
19C4×C22⋊C4×C31—21no—
20C31×#342C2212yes—
21C31×#324—21no—
22C31×#323C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C22×D4×C31—21no—
25C22×D4×C31—21no—
26C22×D4×C31C2×C411yescharacteristic
27C22×D4×C31D411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C23×C4×C31D411yescharacteristic
30C22×C8×C31—41no—
31C31×#124—41no—
32C31×#124—41no—
33C4×D4×C31—41no—
34C4×D4×C31—41no—
35C2×C4⋊C4×C31D411yescharacteristic
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31C2×C411yescharacteristic
44C2×C22⋊C4×C31C2311yescharacteristic
45C2×C42×C31—41no—
46C2×C42×C31C2×C411yescharacteristic
47C31×#99—24no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8—41no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—41no—
55C31×#95C2×C411yescharacteristic
56C31×#95C2×C411yescharacteristic
57C24×C31—21no—
58C24×C31C22⋊C411yescharacteristic
59C24×C31C2×D411yescharacteristic
60C2×Q8×C31—41no—
61C2×Q8×C31—81no—
62C2×D4×C31—22no—
63C2×D4×C31—22no—
64C2×D4×C31—41no—
65C2×D4×C31—41no—
66C2×D4×C31—42no—
67C2×D4×C31—42no—
68C2×D4×C31—81no—
69C2×D4×C31—81no—
70C2×D4×C31—81no—
71C2×D4×C31—81no—
72C2×D4×C31—81no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—42no—
81C22×C4×C31—81no—
82C22×C4×C31—81no—
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22⋊C411yescharacteristic
85C22×C4×C31C22⋊C411yescharacteristic
86C22×C4×C31C22×C411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88C2×C8×C31—42no—
89C2×C8×C31—82no—
90C4⋊C4×C31—41no—
91C22⋊C4×C31—24no—
92C22⋊C4×C31—41no—
93C22⋊C4×C31—41no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—81no—
99C22⋊C4×C31—81no—
100C22⋊C4×C31—81no—
101C22⋊C4×C31—81no—
102C42×C31—21no—
103C42×C31—21no—
104C42×C31—82no—
105#11730C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31#10412yes—
122C23×C31C2×C22⋊C411yescharacteristic
123C23×C31#13512yes—
124Q8×C31—82no—
125D4×C31—44no—
126D4×C31—82no—
127D4×C31—84no—
128D4×C31—84no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—41no—
136C2×C4×C31—42no—
137C2×C4×C31—42no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—44no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—81no—
152C2×C4×C31—81no—
153C2×C4×C31—82no—
154C2×C4×C31—82no—
155C2×C4×C31#9911yescharacteristic
156C2×C4×C31#9911yescharacteristic
157C8×C31—82no—
158#2631C2×C3111yescharacteristic
159C4×#99C2×C3111yescharacteristic, complemented
160#1459C2×C3111yescharacteristic, complemented
161#1351C2×C3112yescomplemented
162#1343C2×C3112yescomplemented
163C22×C31—21no—
164C22×C31—21no—
165C22×C31—22no—
166C22×C31—22no—
167C22×C31—42no—
168C22×C31—42no—
169C22×C31—41no—
170C22×C31—41no—
171C22×C31—41no—
172C22×C31—41no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31C2×#9911yescharacteristic
183C22×C31#41611yescharacteristic
184C22×C31C2×#10411yescharacteristic
185C4×C31—21no—
186C4×C31—21no—
187C4×C31—41no—
188C4×C31—41no—
189C4×C31—41no—
190C4×C31—41no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31—84no—
194C4×C31—81no—
195C2×#124C4×C3111yescharacteristic, complemented
196C2×C4×D4C22×C3111yescharacteristic
197C2×#99—21no—
198C2×C22⋊C8—21no—
199#397—21no—
200#393C4×C3111yescharacteristic, complemented
201#393C22×C3111yescharacteristic
202#392—21no—
203#392C4×C3111yescharacteristic, complemented
204#392C22×C3111yescharacteristic
205#386—21no—
206#386—21no—
207#383C4×C3111yescharacteristic, complemented
208C4×C22⋊C4—21no—
209#342C22×C3112yes—
210#324—21no—
211#323C22×C3112yes—
212C2×C31—21no—
213C2×C31—21no—
214C2×C31—42no—
215C2×C31—42no—
216C2×C31—161no—
217C2×C31#150111yescharacteristic
218C2×C31#212211yescharacteristic
219C2×C31#212711yescharacteristic
220C22×Q8C2×C4×C3111yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2×C4×C3111yescharacteristic
224C22×D4D4×C3111yescharacteristic
225C23×C4D4×C3111yescharacteristic
226C23×C4D4×C3111yescharacteristic
227C22×C8—41no—
228#124—41no—
229#124—41no—
230C4×D4—41no—
231C4×D4—41no—
232C2×C4⋊C4D4×C3111yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4C2×C4×C3111yescharacteristic
241C2×C22⋊C4C23×C3111yescharacteristic
242C2×C42—41no—
243C2×C42C2×C4×C3111yescharacteristic
244#99—24no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—41no—
252#95C2×C4×C3111yescharacteristic
253#95C2×C4×C3111yescharacteristic
254C31#1173011yescharacteristic, direct product factor, Sylow
255C24—21no—
256C24C22⋊C4×C3111yescharacteristic
257C24C2×D4×C3111yescharacteristic
258C2×Q8—41no—
259C2×Q8—81no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—81no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4C22⋊C4×C3111yescharacteristic
282C22×C4C22⋊C4×C3111yescharacteristic
283C22×C4C22⋊C4×C3111yescharacteristic
284C22×C4C22×C4×C3111yescharacteristic
285C22×C4C2×D4×C3111yescharacteristic
286C2×C8—42no—
287C2×C8—82no—
288C4⋊C4—41no—
289C22⋊C4—24no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—21no—
301C42—21no—
302C42—82no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23C31×#10412yes—
319C23C2×C22⋊C4×C3111yescharacteristic
320C23C31×#13512yes—
321Q8—82no—
322D4—44no—
323D4—82no—
324D4—84no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4C31×#9911yescharacteristic
353C2×C4C31×#9911yescharacteristic
354C8—82no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22C2×C31×#9911yescharacteristic
375C22C31×#41611yescharacteristic
376C22C2×C31×#10411yescharacteristic
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—84no—
386C4—81no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—161no—
392C2C31×#150111yescharacteristic
393C2C31×#212211yescharacteristic
394C2C31×#212711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2631C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C31×#99C2semidirect
3. C31×#1459C2semidirect
4. C31×#1351C2semidirect
5. C31×#1343C2semidirect
6. C2×C31×#124C4semidirect
11. C31×#393C4semidirect
14. C31×#392C4semidirect
18. C31×#383C4semidirect
105. #11730C31direct (N × H)
159. C4×#99C2×C31semidirect
160. #1459C2×C31semidirect
161. #1351C2×C31semidirect
162. #1343C2×C31semidirect
195. C2×#124C4×C31semidirect
200. #393C4×C31semidirect
203. #392C4×C31semidirect
207. #383C4×C31semidirect
254. C31#11730direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26312
C4×C31×#992
C31×#14592
C31×#13512
C31×#13432
#1173031