Finite Group Databasebeta

← C31×#7560 #21464319

Subgroups — C31×#7560 #21464319

Order 7936.

538 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2293C211yesmaximal, characteristic
2C4×C31×C4⋊C8C211yesmaximal, characteristic, complemented
3C31×#1744C211yesmaximal, characteristic, complemented
4C31×#1271C212yesmaximal, complemented
5C31×#1270C212yesmaximal
6C2×C4×D4×C31C411yescharacteristic
7C2×C31×#117—21no—
8C2×C31×C4⋊C8—21no—
9C2×C4×C8×C31—21no—
10C2×C4×C8×C31—21no—
11C31×#390C411yescharacteristic
12C31×#390C2211yescharacteristic
13C31×#389C411yescharacteristic
14C31×#389C2211yescharacteristic
15C31×#388—21no—
16C31×#388—21no—
17C4×C4⋊C4×C31C411yescharacteristic
18C4×C22⋊C4×C31—21no—
19C43×C31C2211yescharacteristic
20C31×#336—21no—
21C31×#336C2212yes—
22C31×#336C2212yes—
23C22×D4×C31C2×C411yescharacteristic
24C23×C4×C31—21no—
25C22×C8×C31—22no—
26C22×C8×C31—22no—
27C4×D4×C31—22no—
28C4×D4×C31C814yescomplemented
29C31×#117—42no—
30C31×#117—42no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C42×C31—21no—
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31D411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C2×C42×C31D411yescharacteristic
47C2×C42×C31D411yescharacteristic
48C2×C42×C31C2311yescharacteristic
49C31×C4⋊C8—24no—
50C4×C8×C31—24no—
51C4×C8×C31—42no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C24×C31—21no—
56C2×D4×C31—21no—
57C2×D4×C31—41no—
58C2×D4×C31—41no—
59C2×D4×C31C2×C812yes—
60C22×C4×C31—21no—
61C22×C4×C31—24no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—42no—
65C22×C4×C31—42no—
66C22×C4×C31C22⋊C411yescharacteristic
67C22×C4×C31C22⋊C411yescharacteristic
68C22×C4×C31C22⋊C411yescharacteristic
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22×C411yescharacteristic
71C22×C4×C31C2×D411yescharacteristic
72C22×C4×C31C2×D411yescharacteristic
73C2×C8×C31—24no—
74C2×C8×C31—24no—
75C2×C8×C31—42no—
76C2×C8×C31—42no—
77C2×C8×C31—42no—
78C2×C8×C31—42no—
79C4⋊C4×C31—21no—
80C4⋊C4×C31—41no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31C2×C812yes—
86C22⋊C4×C31—24no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—41no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—42no—
92C42×C31—22no—
93C42×C31—22no—
94C42×C31—22no—
95C42×C31—41no—
96C42×C31—41no—
97C42×C31—42no—
98C42×C31C2×C812yes—
99C42×C31M1612yes—
100C42×C31D812yes—
101C42×C31SD1612yes—
102#7560C3111yesmaximal, characteristic, direct product factor, Sylow
103C23×C31—22no—
104C23×C31—41no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31C2×C22⋊C411yescharacteristic
111D4×C31—42no—
112D4×C31—42no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—22no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—48no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31C22⋊C812yes—
136C2×C4×C31C22⋊C812yes—
137C2×C4×C31#10212yes—
138C2×C4×C31#10212yes—
139C2×C4×C31#10412yes—
140C2×C4×C31#10412yes—
141C2×C4×C31C22×C811yescharacteristic
142C2×C4×C31C2×M1611yescharacteristic
143C2×C4×C31C2×D811yescharacteristic
144C2×C4×C31C2×SD1611yescharacteristic
145C2×C4×C31#13512yes—
146C8×C31—44no—
147C8×C31—44no—
148#2293C2×C3111yescharacteristic
149C4×C4⋊C8C2×C3111yescharacteristic, complemented
150#1744C2×C3111yescharacteristic, complemented
151#1271C2×C3112yescomplemented
152#1270C2×C3112yes—
153C22×C31—42no—
154C22×C31—42no—
155C22×C31—42no—
156C22×C31—81no—
157C22×C31—81no—
158C22×C31—81no—
159C22×C31—81no—
160C22×C31C2×C22⋊C811yescharacteristic
161C22×C31#41011yescharacteristic
162C22×C31#41311yescharacteristic
163C22×C31C2×#10211yescharacteristic
164C22×C31#41611yescharacteristic
165C22×C31C2×#10411yescharacteristic
166C22×C31C2×#10411yescharacteristic
167C4×C31—22no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—22no—
171C4×C31—22no—
172C4×C31—21no—
173C4×C31—21no—
174C4×C31—22no—
175C4×C31—42no—
176C4×C31—81no—
177C4×C31—81no—
178C4×C31D4⋊C814yes—
179C2×C4×D4C4×C3111yescharacteristic
180C2×#117—21no—
181C2×C4⋊C8—21no—
182C2×C4×C8—21no—
183C2×C4×C8—21no—
184#390C4×C3111yescharacteristic
185#390C22×C3111yescharacteristic
186#389C4×C3111yescharacteristic
187#389C22×C3111yescharacteristic
188#388—21no—
189#388—21no—
190C4×C4⋊C4C4×C3111yescharacteristic
191C4×C22⋊C4—21no—
192C43C22×C3111yescharacteristic
193#336—21no—
194#336C22×C3112yes—
195#336C22×C3112yes—
196C2×C31—82no—
197C2×C31#145811yescharacteristic
198C2×C31C2×D4⋊C811yescharacteristic
199C2×C31#147611yescharacteristic
200C2×C31#150011yescharacteristic
201C2×C31#150211yescharacteristic
202C2×C31#152411yescharacteristic
203C2×C31#154011yescharacteristic
204C22×D4C2×C4×C3111yescharacteristic
205C23×C4—21no—
206C22×C8—22no—
207C22×C8—22no—
208C4×D4—22no—
209C4×D4C8×C3114yescomplemented
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×C4×C3111yescharacteristic
215C2×C4⋊C4C2×C4×C3111yescharacteristic
216C2×C4⋊C4C2×C4×C3111yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42C2×C4×C3111yescharacteristic
224C2×C42C2×C4×C3111yescharacteristic
225C2×C42D4×C3111yescharacteristic
226C2×C42D4×C3111yescharacteristic
227C2×C42D4×C3111yescharacteristic
228C2×C42D4×C3111yescharacteristic
229C2×C42C23×C3111yescharacteristic
230C4⋊C8—24no—
231C4×C8—24no—
232C4×C8—42no—
233#95—21no—
234#95—21no—
235#95—21no—
236C31#756011yescharacteristic, direct product factor, Sylow
237C24—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4C2×C8×C3112yes—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C22⋊C4×C3111yescharacteristic
249C22×C4C22⋊C4×C3111yescharacteristic
250C22×C4C22⋊C4×C3111yescharacteristic
251C22×C4C22⋊C4×C3111yescharacteristic
252C22×C4C22×C4×C3111yescharacteristic
253C22×C4C2×D4×C3111yescharacteristic
254C22×C4C2×D4×C3111yescharacteristic
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4C2×C8×C3112yes—
268C22⋊C4—24no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—41no—
278C42—41no—
279C42—42no—
280C42C2×C8×C3112yes—
281C42M16×C3112yes—
282C42D8×C3112yes—
283C42SD16×C3112yes—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊C4×C3111yescharacteristic
292D4—42no—
293D4—42no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—48no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4C31×C22⋊C812yes—
317C2×C4C31×C22⋊C812yes—
318C2×C4C31×#10212yes—
319C2×C4C31×#10212yes—
320C2×C4C31×#10412yes—
321C2×C4C31×#10412yes—
322C2×C4C22×C8×C3111yescharacteristic
323C2×C4C2×M16×C3111yescharacteristic
324C2×C4C2×D8×C3111yescharacteristic
325C2×C4C2×SD16×C3111yescharacteristic
326C2×C4C31×#13512yes—
327C8—44no—
328C8—44no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×C31×C22⋊C811yescharacteristic
337C22C31×#41011yescharacteristic
338C22C31×#41311yescharacteristic
339C22C2×C31×#10211yescharacteristic
340C22C31×#41611yescharacteristic
341C22C2×C31×#10411yescharacteristic
342C22C2×C31×#10411yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—21no—
349C4—21no—
350C4—22no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4C31×D4⋊C814yes—
355C2—82no—
356C2C31×#145811yescharacteristic
357C2C2×C31×D4⋊C811yescharacteristic
358C2C31×#147611yescharacteristic
359C2C31×#150011yescharacteristic
360C2C31×#150211yescharacteristic
361C2C31×#152411yescharacteristic
362C2C31×#154011yescharacteristic
subgroupG/H
centerC22×C4×C31C22⋊C4
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
socleC23×C31C2×C22⋊C4
prime radicalC22×D4×C31C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C43×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C31×C4⋊C8C2semidirect
3. C31×#1744C2semidirect
4. C31×#1271C2semidirect
28. C4×D4×C31C8semidirect
102. #7560C31direct (N × H)
149. C4×C4⋊C8C2×C31semidirect
150. #1744C2×C31semidirect
151. #1271C2×C31semidirect
209. C4×D4C8×C31semidirect
236. C31#7560direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#22932
C4×C31×C4⋊C82
C31×#17442
C31×#12712
C31×#12702
#756031