Finite Group Databasebeta

← C31×#19244 #21476003

Subgroups — C31×#19244 #21476003

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3246C211yesmaximal, characteristic
2C31×#3230C211yesmaximal, characteristic, complemented
3C31×#3141C211yesmaximal, characteristic
4C31×#2962C211yesmaximal, characteristic, complemented
5C31×#1704C212yesmaximal, complemented
6C31×#1698C212yesmaximal, complemented
7C31×#1684C212yesmaximal, complemented
8C31×#1678C212yesmaximal, complemented
9C31×#1565C211yesmaximal, characteristic, complemented
10C2×C31×#333C211yesmaximal, characteristic, complemented
11C31×#1540C211yesmaximal, characteristic
12C31×#533C2211yescharacteristic
13C31×#532C2211yescharacteristic
14C2×C31×C4⋊Q8C2211yescharacteristic
15C31×#500—21no—
16C31×#489—22no—
17C31×#487—21no—
18C31×#487C2212yes—
19C31×#486C2212yes—
20C31×#484—22no—
21C31×#477C2212yes—
22C31×#477C2212yes—
23C31×#476—21no—
24C31×#475C2212yes—
25C31×#465—22no—
26C31×#461—21no—
27C31×#460C2212yes—
28C31×#446—22no—
29C31×#446C2212yes—
30C8×Q8×C31C2212yes—
31C31×#431C2211yescharacteristic
32C31×#424—21no—
33C2×C31×C4⋊C8C2211yescharacteristic
34C2×C31×C4⋊C8C2211yescharacteristic
35C31×#417—21no—
36C31×#417—21no—
37C31×#405—21no—
38C4×M16×C31C2211yescharacteristic
39C31×#335C2212yes—
40C31×#334C2212yes—
41C31×#333C2212yes—
42C31×#333C2212yes—
43C31×#332C2211yescharacteristic
44C31×#332C2211yescharacteristic
45C31×#331C2211yescharacteristic
46C31×#331C2211yescharacteristic
47C2×#41×C31—21no—
48C22×Q8×C31—21no—
49C2×SD16×C31—22no—
50C2×M16×C31D411yescharacteristic
51C2×M16×C31D411yescharacteristic
52C22×C8×C31—21no—
53C22×C8×C31—21no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8D411yescharacteristic
58C31×C4⋊Q8D411yescharacteristic
59C31×C4⋊Q8C2311yescharacteristic
60C31×C4⋊Q8C2312yes—
61C31×#127C2311yescharacteristic
62C31×#125—21no—
63C31×#124—21no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8D412yes—
66C31×#121—21no—
67C31×#121—21no—
68C4×Q8×C31—21no—
69C4×Q8×C31C2312yes—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C31×#117—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C42×C31C2311yescharacteristic
76C31×#107—22no—
77C31×#107—22no—
78C31×#106—22no—
79C31×C4⋊C8—22no—
80C31×C4⋊C8—22no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8C2311yescharacteristic
84C31×C4⋊C8C2311yescharacteristic
85C31×C4⋊C8C2312yes—
86C31×C4⋊C8C2311yescharacteristic
87C31×C4⋊C8C2311yescharacteristic
88C31×#103—22no—
89C31×#103—22no—
90C31×#103—22no—
91C31×#103—22no—
92C31×#103—22no—
93C31×#102—22no—
94C31×#102—22no—
95C31×#102—22no—
96C31×C22⋊C8—21no—
97C31×#97—22no—
98C31×#97D412yes—
99C4×C8×C31—22no—
100C4×C8×C31—21no—
101C4×C8×C31C2312yes—
102#41×C31—42no—
103#41×C31—42no—
104C2×Q8×C31—21no—
105C2×Q8×C31—21no—
106C2×Q8×C31—22no—
107C2×Q8×C31—41no—
108C2×Q8×C31—82no—
109C2×Q8×C31C2×D412yes—
110C2×D4×C31—21no—
111C2×D4×C31—21no—
112C2×D4×C31—21no—
113C2×D4×C31—41no—
114C22×C4×C31—41no—
115C22×C4×C31—41no—
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31#4111yescharacteristic
119SD16×C31—82no—
120M16×C31—22no—
121M16×C31—22no—
122C2×C8×C31—21no—
123C2×C8×C31—21no—
124C2×C8×C31—21no—
125C2×C8×C31—21no—
126C2×C8×C31—22no—
127C2×C8×C31—42no—
128C2×C8×C31—81no—
129C2×C8×C31C2×D412yes—
130C2×C8×C31C2×D412yes—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—21no—
140C4⋊C4×C31—41no—
141C4⋊C4×C31—41no—
142C4⋊C4×C31C2×D412yes—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—41no—
145C22⋊C4×C31—41no—
146C42×C31—22no—
147C42×C31C2×D411yescharacteristic
148C42×C31C2×D411yescharacteristic
149C42×C31#4111yescharacteristic
150C42×C31C2411yescharacteristic
151#19244C3111yesmaximal, characteristic, direct product factor, Sylow
152C23×C31—41no—
153C23×C31#12111yescharacteristic
154Q8×C31—22no—
155Q8×C31—22no—
156Q8×C31—41no—
157Q8×C31—41no—
158Q8×C31—81no—
159Q8×C31—82no—
160Q8×C31—82no—
161D4×C31—41no—
162D4×C31—41no—
163D4×C31—41no—
164D4×C31—41no—
165D4×C31—81no—
166D4×C31—82no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—22no—
170C2×C4×C31—22no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—42no—
174C2×C4×C31—82no—
175C2×C4×C31—82no—
176C2×C4×C31#12111yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C2×C4×C31#12111yescharacteristic
179C2×C4×C31#13512yes—
180C2×C4×C31#13712yes—
181C2×C4×C31C22×D411yescharacteristic
182C2×C4×C31C22×D411yescharacteristic
183C2×C4×C31C2×#4111yescharacteristic
184C8×C31—22no—
185C8×C31—22no—
186C8×C31—41no—
187C8×C31—41no—
188C8×C31—41no—
189C8×C31—42no—
190C8×C31—41no—
191#3246C2×C3111yescharacteristic
192#3230C2×C3111yescharacteristic, complemented
193#3141C2×C3111yescharacteristic
194#2962C2×C3111yescharacteristic, complemented
195#1704C2×C3112yescomplemented
196#1698C2×C3112yescomplemented
197#1684C2×C3112yescomplemented
198#1678C2×C3112yescomplemented
199#1565C2×C3111yescharacteristic, complemented
200C2×#333C2×C3111yescharacteristic, complemented
201#1540C2×C3111yescharacteristic
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31#46412yes—
208C22×C31C2×#12111yescharacteristic
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—21no—
212C4×C31—42no—
213C4×C31—42no—
214C4×C31—81no—
215C4×C31—82no—
216C4×C31#46412yes—
217C4×C31C2×#13511yescharacteristic
218C4×C31C2×#13711yescharacteristic
219C4×C31#57711yescharacteristic
220C4×C31#57811yescharacteristic
221#533C22×C3111yescharacteristic
222#532C22×C3111yescharacteristic
223C2×C4⋊Q8C22×C3111yescharacteristic
224#500—21no—
225#489—22no—
226#487—21no—
227#487C22×C3112yes—
228#486C22×C3112yes—
229#484—22no—
230#477C22×C3112yes—
231#477C22×C3112yes—
232#476—21no—
233#475C22×C3112yes—
234#465—22no—
235#461—21no—
236#460C22×C3112yes—
237#446—22no—
238#446C22×C3112yes—
239C8×Q8C22×C3112yes—
240#431C22×C3111yescharacteristic
241#424—21no—
242C2×C4⋊C8C22×C3111yescharacteristic
243C2×C4⋊C8C22×C3111yescharacteristic
244#417—21no—
245#417—21no—
246#405—21no—
247C4×M16C22×C3111yescharacteristic
248#335C22×C3112yes—
249#334C22×C3112yes—
250#333C22×C3112yes—
251#333C22×C3112yes—
252#332C22×C3111yescharacteristic
253#332C22×C3111yescharacteristic
254#331C22×C3111yescharacteristic
255#331C22×C3111yescharacteristic
256C2×C31—22no—
257C2×C31—161no—
258C2×C31C2×#46411yescharacteristic
259C2×C31#306311yescharacteristic
260C2×C31#306811yescharacteristic
261C2×#41—21no—
262C22×Q8—21no—
263C2×SD16—22no—
264C2×M16D4×C3111yescharacteristic
265C2×M16D4×C3111yescharacteristic
266C22×C8—21no—
267C22×C8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8D4×C3111yescharacteristic
272C4⋊Q8D4×C3111yescharacteristic
273C4⋊Q8C23×C3112yes—
274C4⋊Q8C23×C3111yescharacteristic
275#127C23×C3111yescharacteristic
276#125—21no—
277#124—21no—
278C22⋊Q8—21no—
279C22⋊Q8D4×C3112yes—
280#121—21no—
281#121—21no—
282C4×Q8—21no—
283C4×Q8C23×C3112yes—
284C4×D4—21no—
285C4×D4—21no—
286#117—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C42C23×C3111yescharacteristic
290#107—22no—
291#107—22no—
292#106—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8C23×C3111yescharacteristic
298C4⋊C8C23×C3111yescharacteristic
299C4⋊C8C23×C3111yescharacteristic
300C4⋊C8C23×C3111yescharacteristic
301C4⋊C8C23×C3112yes—
302#103—22no—
303#103—22no—
304#103—22no—
305#103—22no—
306#103—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310C22⋊C8—21no—
311#97—22no—
312#97D4×C3112yes—
313C4×C8—22no—
314C4×C8—21no—
315C4×C8C23×C3112yes—
316C31#1924411yescharacteristic, direct product factor, Sylow
317#41—42no—
318#41—42no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—82no—
324C2×Q8C2×D4×C3112yes—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×D4×C3111yescharacteristic
332C22×C4C2×D4×C3111yescharacteristic
333C22×C4#41×C3111yescharacteristic
334SD16—82no—
335M16—22no—
336M16—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—42no—
343C2×C8—81no—
344C2×C8C2×D4×C3112yes—
345C2×C8C2×D4×C3112yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4C2×D4×C3112yes—
358C22⋊C4—22no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—22no—
362C42C2×D4×C3111yescharacteristic
363C42C2×D4×C3111yescharacteristic
364C42#41×C3111yescharacteristic
365C42C24×C3111yescharacteristic
366C23—41no—
367C23C31×#12111yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372Q8—81no—
373Q8—82no—
374Q8—82no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—82no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4C31×#12111yescharacteristic
391C2×C4C31×#12111yescharacteristic
392C2×C4C31×#12111yescharacteristic
393C2×C4C31×#13512yes—
394C2×C4C31×#13712yes—
395C2×C4C22×D4×C3111yescharacteristic
396C2×C4C22×D4×C3111yescharacteristic
397C2×C4C2×#41×C3111yescharacteristic
398C8—22no—
399C8—22no—
400C8—41no—
401C8—41no—
402C8—41no—
403C8—42no—
404C8—41no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C31×#46412yes—
411C22C2×C31×#12111yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—42no—
417C4—81no—
418C4—82no—
419C4C31×#46412yes—
420C4C2×C31×#13511yescharacteristic
421C4C2×C31×#13711yescharacteristic
422C4C31×#57711yescharacteristic
423C4C31×#57811yescharacteristic
424C2—22no—
425C2—161no—
426C2C2×C31×#46411yescharacteristic
427C2C31×#306311yescharacteristic
428C2C31×#306811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3230C2semidirect
4. C31×#2962C2semidirect
5. C31×#1704C2semidirect
6. C31×#1698C2semidirect
7. C31×#1684C2semidirect
8. C31×#1678C2semidirect
9. C31×#1565C2semidirect
10. C2×C31×#333C2semidirect
151. #19244C31direct (N × H)
192. #3230C2×C31semidirect
194. #2962C2×C31semidirect
195. #1704C2×C31semidirect
196. #1698C2×C31semidirect
197. #1684C2×C31semidirect
198. #1678C2×C31semidirect
199. #1565C2×C31semidirect
200. C2×#333C2×C31semidirect
316. C31#19244direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32462
C31×#32302
C31×#31412
C31×#29622
C31×#17042
C31×#16982
C31×#16842
C31×#16782
C31×#15652
C2×C31×#3332
C31×#15402
#1924431