Finite Group Databasebeta

← C31×#11685 #21468444

Subgroups — C31×#11685 #21468444

Order 7936.

448 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3054C211yesmaximal, characteristic
2C2×C31×#349C211yesmaximal, characteristic, complemented
3C2×C31×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
4C31×#1340C212yesmaximal, complemented
5C31×#1339C212yesmaximal, complemented
6C2×C31×#135C411yescharacteristic, complemented
7C22×M16×C31C2211yescharacteristic
8C31×#525—21no—
9C2×C31×#117—21no—
10C2×C31×M32—21no—
11C31×#470C411yescharacteristic, complemented
12C31×#470C2211yescharacteristic
13C31×#469C411yescharacteristic, complemented
14C31×#469C2211yescharacteristic
15C31×#468—21no—
16C31×#468—21no—
17C31×#427C411yescharacteristic, complemented
18C31×#416—21no—
19C31×#349—21no—
20C31×#349C2212yes—
21C31×H(3,ℤ/4)—21no—
22C31×H(3,ℤ/4)C2212yes—
23C2×#41×C31—21no—
24C23×C4×C31D411yescharacteristic
25C31×#135—41no—
26C31×#135—41no—
27C2×Q16×C31C2×C411yescharacteristic, complemented
28C2×SD16×C31—21no—
29C2×D8×C31C2×C411yescharacteristic, complemented
30C2×M16×C31—41no—
31C2×M16×C31—41no—
32C2×M16×C31C2×C411yescharacteristic
33C2×M16×C31D411yescharacteristic
34C2×M16×C31D411yescharacteristic
35C2×M16×C31C2311yescharacteristic
36C22×C8×C31C2×C411yescharacteristic
37C22×C8×C31D411yescharacteristic
38C31×#123—41no—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—21no—
41C31×#121—21no—
42C31×#121—21no—
43C31×#120—41no—
44C4×D4×C31—41no—
45C4×D4×C31—41no—
46C31×#117—22no—
47C31×#117—22no—
48C31×#117—21no—
49C31×#117—41no—
50C31×#117—41no—
51C2×C4⋊C4×C31—41no—
52C2×C22⋊C4×C31—41no—
53C2×C42×C31—22no—
54C31×M32—22no—
55C31×M32—41no—
56C2×C16×C31—41no—
57C31×#107—21no—
58C31×#106C2×C411yescharacteristic, complemented
59C31×#106C2×C411yescharacteristic, complemented
60C31×#103—21no—
61C31×#103—21no—
62C31×#102—21no—
63C31×#102—21no—
64C24×C31D811yescharacteristic
65#41×C31—41no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69C2×Q8×C31—21no—
70C2×D4×C31—21no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—42no—
79C22×C4×C31—41no—
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31C22⋊C411yescharacteristic
82C22×C4×C31D811yescharacteristic
83C22×C4×C31SD1611yescharacteristic
84C22×C4×C31SD1611yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86Q16×C31—41no—
87SD16×C31—42no—
88D8×C31—41no—
89M16×C31—41no—
90M16×C31—41no—
91M16×C31—41no—
92M16×C31—41no—
93C2×C8×C31—21no—
94C2×C8×C31—21no—
95C2×C8×C31—21no—
96C2×C8×C31—21no—
97C2×C8×C31C22⋊C411yescharacteristic
98C2×C8×C31C22⋊C411yescharacteristic
99C2×C8×C31C22×C411yescharacteristic
100C2×C8×C31C2×D411yescharacteristic
101C4⋊C4×C31—21no—
102C4⋊C4×C31—21no—
103C4⋊C4×C31—42no—
104C4⋊C4×C31—42no—
105C4⋊C4×C31—41no—
106C4⋊C4×C31—41no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C42×C31—24no—
115C42×C31—42no—
116C42×C31—41no—
117C16×C31—42no—
118#11685C3111yesmaximal, characteristic, direct product factor, Sylow
119C23×C31—21no—
120C23×C31—21no—
121C23×C31—41no—
122C23×C31—81no—
123C23×C31#10211yescharacteristic
124C23×C31#10211yescharacteristic
125C23×C31C2×D811yescharacteristic
126Q8×C31—41no—
127Q8×C31—41no—
128D4×C31—41no—
129D4×C31—41no—
130D4×C31—82no—
131D4×C31—82no—
132D4×C31—81no—
133D4×C31—81no—
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—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—84no—
149C2×C4×C31—81no—
150C2×C4×C31—81no—
151C2×C4×C31#9911yescharacteristic
152C2×C4×C31#9911yescharacteristic
153C2×C4×C31#10211yescharacteristic
154C2×C4×C31#10211yescharacteristic
155C2×C4×C31#10411yescharacteristic
156C2×C4×C31#10411yescharacteristic
157C2×C4×C31C2×C22⋊C411yescharacteristic
158C2×C4×C31C2×SD1611yescharacteristic
159C8×C31—21no—
160C8×C31—21no—
161C8×C31—21no—
162C8×C31—21no—
163#3054C2×C3111yescharacteristic
164C2×#349C2×C3111yescharacteristic, complemented
165C2×H(3,ℤ/4)C2×C3111yescharacteristic, complemented
166#1340C2×C3112yescomplemented
167#1339C2×C3112yescomplemented
168C22×C31—21no—
169C22×C31—21no—
170C22×C31—41no—
171C22×C31—41no—
172C22×C31—41no—
173C22×C31—42no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31#32711yescharacteristic
179C22×C31#32711yescharacteristic
180C22×C31C2×#10211yescharacteristic
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31—84no—
186C4×C31—81no—
187C4×C31#32711yescharacteristic
188C4×C31#32711yescharacteristic
189C4×C31C2×#9911yescharacteristic
190C4×C31C2×#10411yescharacteristic
191C2×#135C4×C3111yescharacteristic, complemented
192C22×M16C22×C3111yescharacteristic
193#525—21no—
194C2×#117—21no—
195C2×M32—21no—
196#470C4×C3111yescharacteristic, complemented
197#470C22×C3111yescharacteristic
198#469C4×C3111yescharacteristic, complemented
199#469C22×C3111yescharacteristic
200#468—21no—
201#468—21no—
202#427C4×C3111yescharacteristic, complemented
203#416—21no—
204#349—21no—
205#349C22×C3112yes—
206H(3,ℤ/4)—21no—
207H(3,ℤ/4)C22×C3112yes—
208C2×C31—21no—
209C2×C31—21no—
210C2×C31—21no—
211C2×C31—41no—
212C2×C31—41no—
213C2×C31—161no—
214C2×C31C2×#32711yescharacteristic
215C2×#41—21no—
216C23×C4D4×C3111yescharacteristic
217#135—41no—
218#135—41no—
219C2×Q16C2×C4×C3111yescharacteristic, complemented
220C2×SD16—21no—
221C2×D8C2×C4×C3111yescharacteristic, complemented
222C2×M16—41no—
223C2×M16—41no—
224C2×M16C2×C4×C3111yescharacteristic
225C2×M16D4×C3111yescharacteristic
226C2×M16D4×C3111yescharacteristic
227C2×M16C23×C3111yescharacteristic
228C22×C8C2×C4×C3111yescharacteristic
229C22×C8D4×C3111yescharacteristic
230#123—41no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233#121—21no—
234#121—21no—
235#120—41no—
236C4×D4—41no—
237C4×D4—41no—
238#117—22no—
239#117—22no—
240#117—21no—
241#117—41no—
242#117—41no—
243C2×C4⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C42—22no—
246M32—22no—
247M32—41no—
248C2×C16—41no—
249#107—21no—
250#106C2×C4×C3111yescharacteristic, complemented
251#106C2×C4×C3111yescharacteristic, complemented
252#103—21no—
253#103—21no—
254#102—21no—
255#102—21no—
256C31#1168511yescharacteristic, direct product factor, Sylow
257C24D8×C3111yescharacteristic
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4C22⋊C4×C3111yescharacteristic
274C22×C4C22⋊C4×C3111yescharacteristic
275C22×C4D8×C3111yescharacteristic
276C22×C4SD16×C3111yescharacteristic
277C22×C4SD16×C3111yescharacteristic
278C22×C4C2×D4×C3111yescharacteristic
279Q16—41no—
280SD16—42no—
281D8—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C22⋊C4×C3111yescharacteristic
291C2×C8C22⋊C4×C3111yescharacteristic
292C2×C8C22×C4×C3111yescharacteristic
293C2×C8C2×D4×C3111yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—24no—
308C42—42no—
309C42—41no—
310C16—42no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—81no—
315C23C31×#10211yescharacteristic
316C23C31×#10211yescharacteristic
317C23C2×D8×C3111yescharacteristic
318Q8—41no—
319Q8—41no—
320D4—41no—
321D4—41no—
322D4—82no—
323D4—82no—
324D4—81no—
325D4—81no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—84no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4C31×#9911yescharacteristic
344C2×C4C31×#9911yescharacteristic
345C2×C4C31×#10211yescharacteristic
346C2×C4C31×#10211yescharacteristic
347C2×C4C31×#10411yescharacteristic
348C2×C4C31×#10411yescharacteristic
349C2×C4C2×C22⋊C4×C3111yescharacteristic
350C2×C4C2×SD16×C3111yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C31×#32711yescharacteristic
366C22C31×#32711yescharacteristic
367C22C2×C31×#10211yescharacteristic
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—84no—
373C4—81no—
374C4C31×#32711yescharacteristic
375C4C31×#32711yescharacteristic
376C4C2×C31×#9911yescharacteristic
377C4C2×C31×#10411yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2C2×C31×#32711yescharacteristic
subgroupG/H
centerC4×C31#327
commutatorC2×C8C22×C4×C31
Frattini, cosocleC2×M16C23×C31
socleC2×C31C2×#327
prime radicalC31×#3054C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C22×C4×C31 ◁ C22×M16×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#349C2semidirect
3. C2×C31×H(3,ℤ/4)C2semidirect
4. C31×#1340C2semidirect
5. C31×#1339C2semidirect
6. C2×C31×#135C4semidirect
11. C31×#470C4semidirect
13. C31×#469C4semidirect
17. C31×#427C4semidirect
27. C2×Q16×C31C2×C4semidirect
29. C2×D8×C31C2×C4semidirect
58. C31×#106C2×C4semidirect
59. C31×#106C2×C4semidirect
118. #11685C31direct (N × H)
164. C2×#349C2×C31semidirect
165. C2×H(3,ℤ/4)C2×C31semidirect
166. #1340C2×C31semidirect
167. #1339C2×C31semidirect
191. C2×#135C4×C31semidirect
196. #470C4×C31semidirect
198. #469C4×C31semidirect
202. #427C4×C31semidirect
219. C2×Q16C2×C4×C31semidirect
221. C2×D8C2×C4×C31semidirect
250. #106C2×C4×C31semidirect
251. #106C2×C4×C31semidirect
256. C31#11685direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30542
C2×C31×#3492
C2×C31×H(3,ℤ/4)2
C31×#13402
C31×#13392
#1168531