Finite Group Databasebeta

← C31×#9754 #21466513

Subgroups — C31×#9754 #21466513

Order 7936.

416 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3131C211yesmaximal, characteristic
2C31×#1770C211yesmaximal, characteristic, complemented
3C31×#1564C211yesmaximal, characteristic, complemented
4C31×#1503C211yesmaximal, characteristic, complemented
5C31×#1502C211yesmaximal, characteristic
6C31×#1481C211yesmaximal, characteristic, complemented
7C31×#1480C211yesmaximal, characteristic
8C31×#533C2211yescharacteristic
9C31×#532C2211yescharacteristic
10C31×#488—21no—
11C31×#488—21no—
12C31×#487C411yescharacteristic, complemented
13C31×#487C411yescharacteristic, complemented
14C31×#486C411yescharacteristic, complemented
15C31×#486C411yescharacteristic, complemented
16C31×#431C2211yescharacteristic
17C31×#431C2211yescharacteristic
18C2×C31×C4⋊C8C2211yescharacteristic
19C31×#416—21no—
20C31×#416—21no—
21C2×C31×#97—21no—
22C2×C4×C8×C31C2211yescharacteristic
23C4×C4⋊C4×C31C2211yescharacteristic
24C31×#336—21no—
25C31×#335—21no—
26C31×#334—21no—
27C31×#328—22no—
28C31×#327—22no—
29C31×Q8⋊C8—22no—
30C31×D4⋊C8—22no—
31C2×#41×C31—21no—
32C22×C8×C31—21no—
33C22×C8×C31D411yescharacteristic
34C22×C8×C31D411yescharacteristic
35C31×C4⋊Q8C2×C411yescharacteristic
36C31×C4⋊Q8C2×C411yescharacteristic
37C31×C4⋊Q8C2×C411yescharacteristic
38C31×#127C2×C411yescharacteristic
39C31×#125—21no—
40C31×#124—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×#121—21no—
44C31×#121—21no—
45C4×Q8×C31—21no—
46C4×Q8×C31—21no—
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C31×#117—21no—
50C2×C4⋊C4×C31—21no—
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×C4⋊C8—21no—
54C31×C4⋊C8—21no—
55C31×C4⋊C8—21no—
56C31×C4⋊C8D412yes—
57C31×#103—21no—
58C31×#103—21no—
59C31×#103—21no—
60C31×#103—21no—
61C31×#102—21no—
62C31×#102—21no—
63C31×#102—21no—
64C31×#102—21no—
65C31×C22⋊C8—21no—
66C31×C22⋊C8—21no—
67C31×#97—42no—
68C4×C8×C31—21no—
69C4×C8×C31—21no—
70C4×C8×C31C2×C411yescharacteristic
71C4×C8×C31C2×C411yescharacteristic
72C4×C8×C31D411yescharacteristic
73C4×C8×C31D411yescharacteristic
74C31×#95D412yes—
75#41×C31—41no—
76#41×C31—41no—
77#41×C31—41no—
78#41×C31—41no—
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×Q8×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—41no—
86C22×C4×C31—22no—
87C22×C4×C31—41no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31#4111yescharacteristic
91C2×C8×C31—21no—
92C2×C8×C31—21no—
93C2×C8×C31—21no—
94C2×C8×C31—21no—
95C2×C8×C31—22no—
96C2×C8×C31—41no—
97C2×C8×C31—41no—
98C2×C8×C31—41no—
99C2×C8×C31—41no—
100C2×C8×C31C22⋊C411yescharacteristic
101C2×C8×C31C22⋊C411yescharacteristic
102C2×C8×C31C22⋊C411yescharacteristic
103C2×C8×C31C22⋊C411yescharacteristic
104C4⋊C4×C31—21no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—42no—
111C4⋊C4×C31—41no—
112C4⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—41no—
116C42×C31—41no—
117C42×C31—41no—
118C42×C31—41no—
119C42×C31C22×C411yescharacteristic
120C42×C31C2×D411yescharacteristic
121C42×C31C2×D411yescharacteristic
122C42×C31#4111yescharacteristic
123#9754C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—41no—
125C23×C31#12111yescharacteristic
126Q8×C31—41no—
127Q8×C31—41no—
128Q8×C31—42no—
129Q8×C31—81no—
130D4×C31—41no—
131D4×C31—41no—
132D4×C31—42no—
133D4×C31—81no—
134D4×C31—81no—
135D4×C31—81no—
136C2×C4×C31—22no—
137C2×C4×C31—41no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—42no—
144C2×C4×C31—81no—
145C2×C4×C31—81no—
146C2×C4×C31C2×C22⋊C411yescharacteristic
147C2×C4×C31C4×D411yescharacteristic
148C2×C4×C31C4×D411yescharacteristic
149C2×C4×C31#12111yescharacteristic
150C2×C4×C31#12411yescharacteristic
151C2×C4×C31#12711yescharacteristic
152C2×C4×C31#13511yescharacteristic
153C2×C4×C31#13511yescharacteristic
154C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C31#13711yescharacteristic
156C8×C31—21no—
157C8×C31—21no—
158C8×C31—21no—
159C8×C31—21no—
160C8×C31—42no—
161C8×C31—81no—
162C8×C31—81no—
163#3131C2×C3111yescharacteristic
164#1770C2×C3111yescharacteristic, complemented
165#1564C2×C3111yescharacteristic, complemented
166#1503C2×C3111yescharacteristic, complemented
167#1502C2×C3111yescharacteristic
168#1481C2×C3111yescharacteristic, complemented
169#1480C2×C3111yescharacteristic
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31#39011yescharacteristic
176C22×C31#46311yescharacteristic
177C22×C31#46411yescharacteristic
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—21no—
181C4×C31—82no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31#41611yescharacteristic
186C4×C31#41711yescharacteristic
187C4×C31#44412yes—
188C4×C31#46311yescharacteristic
189C4×C31#46411yescharacteristic
190#533C22×C3111yescharacteristic
191#532C22×C3111yescharacteristic
192#488—21no—
193#488—21no—
194#487C4×C3111yescharacteristic, complemented
195#487C4×C3111yescharacteristic, complemented
196#486C4×C3111yescharacteristic, complemented
197#486C4×C3111yescharacteristic, complemented
198#431C22×C3111yescharacteristic
199#431C22×C3111yescharacteristic
200C2×C4⋊C8C22×C3111yescharacteristic
201#416—21no—
202#416—21no—
203C2×#97—21no—
204C2×C4×C8C22×C3111yescharacteristic
205C4×C4⋊C4C22×C3111yescharacteristic
206#336—21no—
207#335—21no—
208#334—21no—
209#328—22no—
210#327—22no—
211Q8⋊C8—22no—
212D4⋊C8—22no—
213C2×C31—21no—
214C2×C31—21no—
215C2×C31—161no—
216C2×C31#197511yescharacteristic
217C2×C31#197611yescharacteristic
218C2×C31#198311yescharacteristic
219C2×#41—21no—
220C22×C8—21no—
221C22×C8D4×C3111yescharacteristic
222C22×C8D4×C3111yescharacteristic
223C4⋊Q8C2×C4×C3111yescharacteristic
224C4⋊Q8C2×C4×C3111yescharacteristic
225C4⋊Q8C2×C4×C3111yescharacteristic
226#127C2×C4×C3111yescharacteristic
227#125—21no—
228#124—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4—21no—
237#117—21no—
238C2×C4⋊C4—21no—
239C2×C42—21no—
240C2×C42C23×C3111yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8D4×C3112yes—
245#103—21no—
246#103—21no—
247#103—21no—
248#103—21no—
249#102—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255#97—42no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8C2×C4×C3111yescharacteristic
259C4×C8C2×C4×C3111yescharacteristic
260C4×C8D4×C3111yescharacteristic
261C4×C8D4×C3111yescharacteristic
262#95D4×C3112yes—
263C31#975411yescharacteristic, direct product factor, Sylow
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4C2×D4×C3111yescharacteristic
278C22×C4C2×D4×C3111yescharacteristic
279C22×C4#41×C3111yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8C22⋊C4×C3111yescharacteristic
290C2×C8C22⋊C4×C3111yescharacteristic
291C2×C8C22⋊C4×C3111yescharacteristic
292C2×C8C22⋊C4×C3111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42C22×C4×C3111yescharacteristic
309C42C2×D4×C3111yescharacteristic
310C42C2×D4×C3111yescharacteristic
311C42#41×C3111yescharacteristic
312C23—41no—
313C23C31×#12111yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—42no—
317Q8—81no—
318D4—41no—
319D4—41no—
320D4—42no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4C2×C22⋊C4×C3111yescharacteristic
335C2×C4C4×D4×C3111yescharacteristic
336C2×C4C4×D4×C3111yescharacteristic
337C2×C4C31×#12111yescharacteristic
338C2×C4C31×#12411yescharacteristic
339C2×C4C31×#12711yescharacteristic
340C2×C4C31×#13511yescharacteristic
341C2×C4C31×#13511yescharacteristic
342C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
343C2×C4C31×#13711yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—42no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C31×#39011yescharacteristic
357C22C31×#46311yescharacteristic
358C22C31×#46411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4C31×#41611yescharacteristic
367C4C31×#41711yescharacteristic
368C4C31×#44412yes—
369C4C31×#46311yescharacteristic
370C4C31×#46411yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2C31×#197511yescharacteristic
375C2C31×#197611yescharacteristic
376C2C31×#198311yescharacteristic
subgroupG/H
centerC2×C4×C31#121
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#390
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1770C2semidirect
3. C31×#1564C2semidirect
4. C31×#1503C2semidirect
6. C31×#1481C2semidirect
12. C31×#487C4semidirect
13. C31×#487C4semidirect
14. C31×#486C4semidirect
15. C31×#486C4semidirect
123. #9754C31direct (N × H)
164. #1770C2×C31semidirect
165. #1564C2×C31semidirect
166. #1503C2×C31semidirect
168. #1481C2×C31semidirect
194. #487C4×C31semidirect
195. #487C4×C31semidirect
196. #486C4×C31semidirect
197. #486C4×C31semidirect
263. C31#9754direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31312
C31×#17702
C31×#15642
C31×#15032
C31×#15022
C31×#14812
C31×#14802
#975431