Finite Group Databasebeta

← C31×#7710 #21464469

Subgroups — C31×#7710 #21464469

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2850C211yesmaximal, characteristic
2C31×#1797C211yesmaximal, characteristic, complemented
3C31×#1728C211yesmaximal, characteristic, complemented
4C31×#1272C212yesmaximal, complemented
5C31×#1270C212yesmaximal, complemented
6C22×M16×C31—21no—
7C2×C31×#124C411yescharacteristic
8C2×C31×C22⋊Q8C411yescharacteristic
9C2×C31×#121C411yescharacteristic
10C22×C42×C31C2211yescharacteristic
11C2×C31×C4⋊C8—21no—
12C2×C31×C22⋊C8—21no—
13C31×#391C2211yescharacteristic
14C31×#390C2211yescharacteristic
15C31×#388—21no—
16C31×#388—21no—
17C31×#386—21no—
18C31×#386—21no—
19C31×#382C411yescharacteristic
20C31×#336—21no—
21C31×#336C2212yes—
22C31×#336C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C22×D4×C31—21no—
25C22×D4×C31C2×C411yescharacteristic
26C23×C4×C31C2×C411yescharacteristic
27C23×C4×C31D411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C2×M16×C31—24no—
30C2×M16×C31—42no—
31C2×M16×C31—42no—
32C22×C8×C31—22no—
33C22×C8×C31—22no—
34C31×#124—42no—
35C31×C22⋊Q8—42no—
36C31×#121—42no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C4⋊C4×C31C2×C411yescharacteristic
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C42×C31—42no—
47C2×C42×C31C2×C411yescharacteristic
48C2×C42×C31D411yescharacteristic
49C2×C42×C31D411yescharacteristic
50C2×C42×C31C2311yescharacteristic
51C31×C4⋊C8—42no—
52C31×C22⋊C8—42no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C24×C31—21no—
58C24×C31C22⋊C411yescharacteristic
59C2×Q8×C31—22no—
60C2×Q8×C31—41no—
61C2×Q8×C31—41no—
62C2×D4×C31—22no—
63C2×D4×C31—41no—
64C2×D4×C31—41no—
65C2×D4×C31—42no—
66C2×D4×C31—42no—
67C2×D4×C31—42no—
68C22×C4×C31—22no—
69C22×C4×C31—22no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—22no—
73C22×C4×C31—22no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—41no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22⋊C411yescharacteristic
85C22×C4×C31C22×C411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88M16×C31—48no—
89C2×C8×C31—24no—
90C2×C8×C31—42no—
91C2×C8×C31—42no—
92C2×C8×C31—42no—
93C2×C8×C31—42no—
94C2×C8×C31—42no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—42no—
98C22⋊C4×C31—42no—
99C22⋊C4×C31—42no—
100C22⋊C4×C31—42no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C42×C31—22no—
107C42×C31—22no—
108C42×C31—41no—
109C42×C31—41no—
110#7710C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31#10412yes—
125C23×C31C2×C22⋊C411yescharacteristic
126Q8×C31—82no—
127D4×C31—82no—
128D4×C31—84no—
129C2×C4×C31—24no—
130C2×C4×C31—24no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—44no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31#9912yes—
157C2×C4×C31#10012yes—
158C2×C4×C31#13112yes—
159C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C31#13711yescharacteristic
162C2×C4×C31#13711yescharacteristic
163C8×C31—44no—
164C8×C31—82no—
165#2850C2×C3111yescharacteristic
166#1797C2×C3111yescharacteristic, complemented
167#1728C2×C3111yescharacteristic, complemented
168#1272C2×C3112yescomplemented
169#1270C2×C3112yescomplemented
170C22×C31—22no—
171C22×C31—22no—
172C22×C31—22no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31#40811yescharacteristic
185C22×C31C2×#9911yescharacteristic
186C22×C31C2×#10011yescharacteristic
187C22×C31#41711yescharacteristic
188C22×C31#41711yescharacteristic
189C22×C31C2×#10411yescharacteristic
190C22×C31#42111yescharacteristic
191C4×C31—22no—
192C4×C31—22no—
193C4×C31—21no—
194C4×C31—22no—
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—41no—
199C4×C31—81no—
200C4×C31—81no—
201C4×C31—81no—
202C22×M16—21no—
203C2×#124C4×C3111yescharacteristic
204C2×C22⋊Q8C4×C3111yescharacteristic
205C2×#121C4×C3111yescharacteristic
206C22×C42C22×C3111yescharacteristic
207C2×C4⋊C8—21no—
208C2×C22⋊C8—21no—
209#391C22×C3111yescharacteristic
210#390C22×C3111yescharacteristic
211#388—21no—
212#388—21no—
213#386—21no—
214#386—21no—
215#382C4×C3111yescharacteristic
216#336—21no—
217#336C22×C3112yes—
218#336C22×C3112yes—
219C2×C31—42no—
220C2×C31—161no—
221C2×C31#147111yescharacteristic
222C2×C31#148311yescharacteristic
223C2×C31#149511yescharacteristic
224C2×C31#150411yescharacteristic
225C2×C31#151311yescharacteristic
226C2×C31#152711yescharacteristic
227C2×C31#156011yescharacteristic
228C22×Q8C2×C4×C3111yescharacteristic
229C22×D4—21no—
230C22×D4C2×C4×C3111yescharacteristic
231C23×C4C2×C4×C3111yescharacteristic
232C23×C4D4×C3111yescharacteristic
233C23×C4D4×C3111yescharacteristic
234C2×M16—24no—
235C2×M16—42no—
236C2×M16—42no—
237C22×C8—22no—
238C22×C8—22no—
239#124—42no—
240C22⋊Q8—42no—
241#121—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C2×C4×C3111yescharacteristic
244C2×C4⋊C4C2×C4×C3111yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—42no—
252C2×C42C2×C4×C3111yescharacteristic
253C2×C42D4×C3111yescharacteristic
254C2×C42D4×C3111yescharacteristic
255C2×C42C23×C3111yescharacteristic
256C4⋊C8—42no—
257C22⋊C8—42no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262C31#771011yescharacteristic, direct product factor, Sylow
263C24—21no—
264C24C22⋊C4×C3111yescharacteristic
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C22⋊C4×C3111yescharacteristic
289C22×C4C22⋊C4×C3111yescharacteristic
290C22×C4C22⋊C4×C3111yescharacteristic
291C22×C4C22×C4×C3111yescharacteristic
292C22×C4C2×D4×C3111yescharacteristic
293C22×C4C2×D4×C3111yescharacteristic
294M16—48no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C31×#10412yes—
330C23C2×C22⋊C4×C3111yescharacteristic
331Q8—82no—
332D4—82no—
333D4—84no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4C31×#9912yes—
362C2×C4C31×#10012yes—
363C2×C4C31×#13112yes—
364C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
365C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
366C2×C4C31×#13711yescharacteristic
367C2×C4C31×#13711yescharacteristic
368C8—44no—
369C8—82no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C31×#40811yescharacteristic
385C22C2×C31×#9911yescharacteristic
386C22C2×C31×#10011yescharacteristic
387C22C31×#41711yescharacteristic
388C22C31×#41711yescharacteristic
389C22C2×C31×#10411yescharacteristic
390C22C31×#42111yescharacteristic
391C4—22no—
392C4—22no—
393C4—21no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—81no—
402C2—42no—
403C2—161no—
404C2C31×#147111yescharacteristic
405C2C31×#148311yescharacteristic
406C2C31×#149511yescharacteristic
407C2C31×#150411yescharacteristic
408C2C31×#151311yescharacteristic
409C2C31×#152711yescharacteristic
410C2C31×#156011yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1797C2semidirect
3. C31×#1728C2semidirect
4. C31×#1272C2semidirect
5. C31×#1270C2semidirect
110. #7710C31direct (N × H)
166. #1797C2×C31semidirect
167. #1728C2×C31semidirect
168. #1272C2×C31semidirect
169. #1270C2×C31semidirect
262. C31#7710direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28502
C31×#17972
C31×#17282
C31×#12722
C31×#12702
#771031