Finite Group Databasebeta

← C31×#8972 #21465731

Subgroups — C31×#8972 #21465731

Order 7936.

476 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×#3107C211yesmaximal, characteristic
2C31×#2991C211yesmaximal, characteristic
3C31×#1994C211yesmaximal, characteristic, complemented
4C4×C31×#104C211yesmaximal, characteristic
5C31×#1485C211yesmaximal, characteristic, complemented
6C31×#1299C211yesmaximal, characteristic, complemented
7C31×#1276C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C4×#41×C31C2211yescharacteristic
10C31×#485—21no—
11C31×#475C411yescharacteristic, complemented
12C31×#474C411yescharacteristic, complemented
13C31×#462C411yescharacteristic, complemented
14C31×#461C411yescharacteristic, complemented
15C31×#452—21no—
16C8×Q8×C31—21no—
17C4×Q16×C31—21no—
18C4×SD16×C31—21no—
19C31×#436—22no—
20C31×#436—21no—
21C31×#431—21no—
22C31×#431C2211yescharacteristic
23C31×#431C2211yescharacteristic
24C31×#423—21no—
25C31×#423C2211yescharacteristic
26C2×C31×#104—21no—
27C4×M16×C31C2211yescharacteristic
28C43×C31C2211yescharacteristic
29C31×#339—22no—
30C31×#339—21no—
31C31×Q8⋊C8—21no—
32C31×Q8⋊C8—21no—
33C2×#41×C31D411yescharacteristic
34C2×Q16×C31—21no—
35C2×SD16×C31—21no—
36C2×M16×C31—22no—
37C2×M16×C31—21no—
38C2×M16×C31D412yes—
39C31×C4⋊Q8C2×C411yescharacteristic
40C31×C4⋊Q8C2×C411yescharacteristic
41C31×#125—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C4×Q8×C31—21no—
45C4×Q8×C31—21no—
46C4×Q8×C31C2×C411yescharacteristic
47C4×Q8×C31D411yescharacteristic
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C4×D4×C31C2×C411yescharacteristic
51C4×D4×C31D411yescharacteristic
52C31×#117—21no—
53C31×#117—21no—
54C31×#117D411yescharacteristic
55C2×C42×C31—21no—
56C2×C42×C31—21no—
57C2×C42×C31—21no—
58C2×C42×C31—21no—
59C2×C42×C31C2311yescharacteristic
60C31×#107—21no—
61C31×#106—21no—
62C31×C4⋊C8—22no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8C2×C411yescharacteristic
66C31×C4⋊C8C2×C411yescharacteristic
67C31×C4⋊C8D411yescharacteristic
68C31×C4⋊C8D411yescharacteristic
69C31×#104—24no—
70C31×#103—21no—
71C31×#103—21no—
72C31×#103—21no—
73C31×#102—21no—
74C31×C22⋊C8—22no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×#97—21no—
78C4×C8×C31—22no—
79C4×C8×C31—21no—
80C4×C8×C31—21no—
81C4×C8×C31—21no—
82#41×C31—22no—
83#41×C31—41no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31C22⋊C411yescharacteristic
87C2×D4×C31—21no—
88C2×D4×C31C22⋊C411yescharacteristic
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—41no—
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31#4111yescharacteristic
96Q16×C31—42no—
97SD16×C31—42no—
98M16×C31—22no—
99M16×C31—41no—
100M16×C31—42no—
101M16×C31—82no—
102C2×C8×C31—21no—
103C2×C8×C31—21no—
104C2×C8×C31—21no—
105C2×C8×C31—22no—
106C2×C8×C31—21no—
107C2×C8×C31—41no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31C22⋊C411yescharacteristic
117C4⋊C4×C31C22⋊C411yescharacteristic
118C22⋊C4×C31—21no—
119C22⋊C4×C31—21no—
120C22⋊C4×C31—21no—
121C22⋊C4×C31—41no—
122C42×C31—21no—
123C42×C31—21no—
124C42×C31—21no—
125C42×C31—22no—
126C42×C31—22no—
127C42×C31—22no—
128C42×C31—41no—
129C42×C31—41no—
130C42×C31—41no—
131C42×C31—41no—
132C42×C31C22×C411yescharacteristic
133C42×C31C2×D411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31#4111yescharacteristic
136#8972C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—21no—
138C23×C31#12111yescharacteristic
139Q8×C31—21no—
140Q8×C31—41no—
141Q8×C31—41no—
142Q8×C31—41no—
143Q8×C31—41no—
144Q8×C31#10412yes—
145D4×C31—21no—
146D4×C31—21no—
147D4×C31—42no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31C2×C22⋊C411yescharacteristic
168C2×C4×C31C4×D411yescharacteristic
169C2×C4×C31C4×D411yescharacteristic
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31#12311yescharacteristic
173C8×C31—42no—
174C8×C31—41no—
175C8×C31—41no—
176C8×C31—42no—
177C8×C31—41no—
178C8×C31—41no—
179#3107C2×C3111yescharacteristic
180#2991C2×C3111yescharacteristic
181#1994C2×C3111yescharacteristic, complemented
182C4×#104C2×C3111yescharacteristic
183#1485C2×C3111yescharacteristic, complemented
184#1299C2×C3111yescharacteristic, complemented
185#1276C2×C3111yescharacteristic, complemented
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—41no—
190C22×C31—41no—
191C22×C31—41no—
192C22×C31#38611yescharacteristic
193C4×C31—21no—
194C4×C31—22no—
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—44no—
198C4×C31—41no—
199C4×C31—41no—
200C4×C31—81no—
201C4×C31—81no—
202C4×C31C2×#10411yescharacteristic
203C4×C31#42111yescharacteristic
204C4×C31#44311yescharacteristic
205C4×C31#44411yescharacteristic
206C4×C31#45511yescharacteristic
207C4×C31#45611yescharacteristic
208#533C22×C3111yescharacteristic
209C4×#41C22×C3111yescharacteristic
210#485—21no—
211#475C4×C3111yescharacteristic, complemented
212#474C4×C3111yescharacteristic, complemented
213#462C4×C3111yescharacteristic, complemented
214#461C4×C3111yescharacteristic, complemented
215#452—21no—
216C8×Q8—21no—
217C4×Q16—21no—
218C4×SD16—21no—
219#436—22no—
220#436—21no—
221#431—21no—
222#431C22×C3111yescharacteristic
223#431C22×C3111yescharacteristic
224#423—21no—
225#423C22×C3111yescharacteristic
226C2×#104—21no—
227C4×M16C22×C3111yescharacteristic
228C43C22×C3111yescharacteristic
229#339—22no—
230#339—21no—
231Q8⋊C8—21no—
232Q8⋊C8—21no—
233C2×C31—41no—
234C2×C31—81no—
235C2×C31#187111yescharacteristic
236C2×C31#187611yescharacteristic
237C2×C31#188211yescharacteristic
238C2×#41D4×C3111yescharacteristic
239C2×Q16—21no—
240C2×SD16—21no—
241C2×M16—22no—
242C2×M16—21no—
243C2×M16D4×C3112yes—
244C4⋊Q8C2×C4×C3111yescharacteristic
245C4⋊Q8C2×C4×C3111yescharacteristic
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8C2×C4×C3111yescharacteristic
252C4×Q8D4×C3111yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C4×D4C2×C4×C3111yescharacteristic
256C4×D4D4×C3111yescharacteristic
257#117—21no—
258#117—21no—
259#117D4×C3111yescharacteristic
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C23×C3111yescharacteristic
265#107—21no—
266#106—21no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8C2×C4×C3111yescharacteristic
271C4⋊C8C2×C4×C3111yescharacteristic
272C4⋊C8D4×C3111yescharacteristic
273C4⋊C8D4×C3111yescharacteristic
274#104—24no—
275#103—21no—
276#103—21no—
277#103—21no—
278#102—21no—
279C22⋊C8—22no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282#97—21no—
283C4×C8—22no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C31#897211yescharacteristic, direct product factor, Sylow
288#41—22no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C22⋊C4×C3111yescharacteristic
293C2×D4—21no—
294C2×D4C22⋊C4×C3111yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4C2×D4×C3111yescharacteristic
300C22×C4C2×D4×C3111yescharacteristic
301C22×C4#41×C3111yescharacteristic
302Q16—42no—
303SD16—42no—
304M16—22no—
305M16—41no—
306M16—42no—
307M16—82no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—41no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4C22⋊C4×C3111yescharacteristic
323C4⋊C4C22⋊C4×C3111yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42C22×C4×C3111yescharacteristic
339C42C2×D4×C3111yescharacteristic
340C42C2×D4×C3111yescharacteristic
341C42#41×C3111yescharacteristic
342C23—21no—
343C23C31×#12111yescharacteristic
344Q8—21no—
345Q8—41no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8C31×#10412yes—
350D4—21no—
351D4—21no—
352D4—42no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4C2×C22⋊C4×C3111yescharacteristic
373C2×C4C4×D4×C3111yescharacteristic
374C2×C4C4×D4×C3111yescharacteristic
375C2×C4C31×#12011yescharacteristic
376C2×C4C31×#12111yescharacteristic
377C2×C4C31×#12311yescharacteristic
378C8—42no—
379C8—41no—
380C8—41no—
381C8—42no—
382C8—41no—
383C8—41no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22C31×#38611yescharacteristic
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4C2×C31×#10411yescharacteristic
401C4C31×#42111yescharacteristic
402C4C31×#44311yescharacteristic
403C4C31×#44411yescharacteristic
404C4C31×#45511yescharacteristic
405C4C31×#45611yescharacteristic
406C2—41no—
407C2—81no—
408C2C31×#187111yescharacteristic
409C2C31×#187611yescharacteristic
410C2C31×#188211yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#386
prime radicalC2×#41×C31D4
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
3. C31×#1994C2semidirect
5. C31×#1485C2semidirect
6. C31×#1299C2semidirect
7. C31×#1276C2semidirect
11. C31×#475C4semidirect
12. C31×#474C4semidirect
13. C31×#462C4semidirect
14. C31×#461C4semidirect
136. #8972C31direct (N × H)
181. #1994C2×C31semidirect
183. #1485C2×C31semidirect
184. #1299C2×C31semidirect
185. #1276C2×C31semidirect
211. #475C4×C31semidirect
212. #474C4×C31semidirect
213. #462C4×C31semidirect
214. #461C4×C31semidirect
287. C31#8972direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31072
C31×#29912
C31×#19942
C4×C31×#1042
C31×#14852
C31×#12992
C31×#12762
#897231