Finite Group Databasebeta

← C31×#11811 #21468570

Subgroups — C31×#11811 #21468570

Order 7936.

496 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2626C211yesmaximal, characteristic
2C31×#1819C211yesmaximal, characteristic
3C31×#1462C211yesmaximal, characteristic
4C31×#1352C212yesmaximal, complemented
5C31×#1351C212yesmaximal, complemented
6C2×C31×C22⋊Q8C411yescharacteristic, complemented
7C22×C22⋊C4×C31C2211yescharacteristic
8C31×#407—21no—
9C31×#399C411yescharacteristic, complemented
10C31×#399C2211yescharacteristic
11C31×#396—21no—
12C31×#396—21no—
13C31×#393C411yescharacteristic, complemented
14C31×#393C2211yescharacteristic
15C31×#385—21no—
16C31×#380C411yescharacteristic, complemented
17C31×#379—21no—
18C31×#379—21no—
19C31×#342—21no—
20C31×#342C2212yes—
21C31×#323—21no—
22C31×#323C2212yes—
23C25×C31D411yescharacteristic
24C22×Q8×C31C2×C411yescharacteristic
25C23×C4×C31C2×C411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—41no—
28C31×C22⋊Q8—41no—
29C31×C22⋊Q8—41no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C22⋊C4×C31—21no—
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—41no—
43C2×C22⋊C4×C31—41no—
44C2×C22⋊C4×C31—41no—
45C2×C22⋊C4×C31—41no—
46C2×C22⋊C4×C31C2×C411yescharacteristic
47C2×C22⋊C4×C31D411yescharacteristic
48C2×C22⋊C4×C31D411yescharacteristic
49C2×C22⋊C4×C31C2311yescharacteristic
50C31×C22⋊C8—22no—
51C31×C22⋊C8—41no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95C2×C411yescharacteristic
55C31×#95C2×C411yescharacteristic
56C24×C31—21no—
57C24×C31—21no—
58C24×C31—22no—
59C24×C31—41no—
60C24×C31—81no—
61C24×C31—81no—
62C24×C31C22⋊C411yescharacteristic
63C24×C31C22⋊C411yescharacteristic
64C24×C31C2×D411yescharacteristic
65C2×Q8×C31—41no—
66C2×Q8×C31—81no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—42no—
76C22×C4×C31—81no—
77C22×C4×C31—81no—
78C22×C4×C31C22⋊C411yescharacteristic
79C22×C4×C31C22⋊C411yescharacteristic
80C22×C4×C31C22×C411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82M16×C31—82no—
83C2×C8×C31—42no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—81no—
87C4⋊C4×C31—81no—
88C4⋊C4×C31—81no—
89C22⋊C4×C31—41no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—41no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—41no—
95C22⋊C4×C31—41no—
96C22⋊C4×C31—42no—
97C22⋊C4×C31—42no—
98C22⋊C4×C31—81no—
99C22⋊C4×C31—81no—
100C22⋊C4×C31—81no—
101C22⋊C4×C31—81no—
102C22⋊C4×C31—82no—
103C22⋊C4×C31—82no—
104#11811C3111yesmaximal, characteristic, direct product factor, Sylow
105C23×C31—21no—
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—21no—
111C23×C31—21no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—42no—
115C23×C31—42no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—42no—
121C23×C31—42no—
122C23×C31—81no—
123C23×C31—81no—
124C23×C31—84no—
125C23×C31—84no—
126C23×C31#9911yescharacteristic
127C23×C31#9911yescharacteristic
128C23×C31C2×C22⋊C411yescharacteristic
129C23×C31#13712yes—
130Q8×C31—82no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—42no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
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—41no—
152C2×C4×C31—81no—
153C2×C4×C31—81no—
154C2×C4×C31—81no—
155C2×C4×C31—82no—
156C2×C4×C31—82no—
157C2×C4×C31—82no—
158C8×C31—82no—
159#2626C2×C3111yescharacteristic
160#1819C2×C3111yescharacteristic
161#1462C2×C3111yescharacteristic
162#1352C2×C3112yescomplemented
163#1351C2×C3112yescomplemented
164C22×C31—21no—
165C22×C31—21no—
166C22×C31—21no—
167C22×C31—21no—
168C22×C31—21no—
169C22×C31—21no—
170C22×C31—22no—
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—42no—
174C22×C31—42no—
175C22×C31—42no—
176C22×C31—42no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—84no—
185C22×C31—84no—
186C22×C31C2×#9911yescharacteristic
187C22×C31#41911yescharacteristic
188C22×C31#42111yescharacteristic
189C4×C31—41no—
190C4×C31—41no—
191C4×C31—41no—
192C4×C31—41no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—162no—
198C2×C22⋊Q8C4×C3111yescharacteristic, complemented
199C22×C22⋊C4C22×C3111yescharacteristic
200#407—21no—
201#399C4×C3111yescharacteristic, complemented
202#399C22×C3111yescharacteristic
203#396—21no—
204#396—21no—
205#393C4×C3111yescharacteristic, complemented
206#393C22×C3111yescharacteristic
207#385—21no—
208#380C4×C3111yescharacteristic, complemented
209#379—21no—
210#379—21no—
211#342—21no—
212#342C22×C3112yes—
213#323—21no—
214#323C22×C3112yes—
215C2×C31—21no—
216C2×C31—21no—
217C2×C31—42no—
218C2×C31—41no—
219C2×C31—41no—
220C2×C31—81no—
221C2×C31#151111yescharacteristic
222C2×C31#212111yescharacteristic
223C2×C31#212711yescharacteristic
224C25D4×C3111yescharacteristic
225C22×Q8C2×C4×C3111yescharacteristic
226C23×C4C2×C4×C3111yescharacteristic
227C23×C4D4×C3111yescharacteristic
228C2×M16—41no—
229C22⋊Q8—41no—
230C22⋊Q8—41no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C2×C4×C3111yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4C2×C4×C3111yescharacteristic
248C2×C22⋊C4D4×C3111yescharacteristic
249C2×C22⋊C4D4×C3111yescharacteristic
250C2×C22⋊C4C23×C3111yescharacteristic
251C22⋊C8—22no—
252C22⋊C8—41no—
253#95—21no—
254#95—21no—
255#95C2×C4×C3111yescharacteristic
256#95C2×C4×C3111yescharacteristic
257C31#1181111yescharacteristic, direct product factor, Sylow
258C24—21no—
259C24—21no—
260C24—22no—
261C24—41no—
262C24—81no—
263C24—81no—
264C24C22⋊C4×C3111yescharacteristic
265C24C22⋊C4×C3111yescharacteristic
266C24C2×D4×C3111yescharacteristic
267C2×Q8—41no—
268C2×Q8—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4C22⋊C4×C3111yescharacteristic
281C22×C4C22⋊C4×C3111yescharacteristic
282C22×C4C22×C4×C3111yescharacteristic
283C22×C4C2×D4×C3111yescharacteristic
284M16—82no—
285C2×C8—42no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—84no—
326C23—84no—
327C23C31×#9911yescharacteristic
328C23C31×#9911yescharacteristic
329C23C2×C22⋊C4×C3111yescharacteristic
330C23C31×#13712yes—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
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—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C8—82no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×C31×#9911yescharacteristic
383C22C31×#41911yescharacteristic
384C22C31×#42111yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—162no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2C31×#151111yescharacteristic
401C2C31×#212111yescharacteristic
402C2C31×#212711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C22×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1352C2semidirect
5. C31×#1351C2semidirect
6. C2×C31×C22⋊Q8C4semidirect
9. C31×#399C4semidirect
13. C31×#393C4semidirect
16. C31×#380C4semidirect
104. #11811C31direct (N × H)
162. #1352C2×C31semidirect
163. #1351C2×C31semidirect
198. C2×C22⋊Q8C4×C31semidirect
201. #399C4×C31semidirect
205. #393C4×C31semidirect
208. #380C4×C31semidirect
257. C31#11811direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26262
C31×#18192
C31×#14622
C31×#13522
C31×#13512
#1181131