Finite Group Databasebeta

← C31×#21748 #21478507

Subgroups — C31×#21748 #21478507

Order 7936.

1554 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#525C211yesmaximal, characteristic
2C31×#2280C211yesmaximal, characteristic, complemented
3C2×C31×#381C211yesmaximal, characteristic, complemented
4C31×#1899C214yesmaximal, complemented
5C31×#1898C214yesmaximal, complemented
6C31×#1820C212yesmaximal, complemented
7C31×#1782C212yesmaximal, complemented
8C22×#41×C31C2211yescharacteristic
9C24×C4×C31C2211yescharacteristic
10C31×#525—22no—
11C31×#525—22no—
12C31×#525—22no—
13C31×#525—22no—
14C2×C31×#123C2212yes—
15C2×C31×C22⋊Q8C412yescomplemented
16C2×C31×#121C412yescomplemented
17C2×C31×#120C2212yes—
18C2×C4×D4×C31C414yescomplemented
19C2×C31×#117C2212yes—
20C2×C31×#117C2211yescharacteristic
21C22×C22⋊C4×C31C2212yes—
22C2×C31×#99C2218yes—
23C31×#381—22no—
24C31×#381C2212yes—
25C31×#381C2212yes—
26C31×#381C2212yes—
27C31×#380—22no—
28C31×#379C2212yes—
29C4×C22⋊C4×C31—22no—
30C2×C31×#95—22no—
31C31×#342C2218yes—
32C25×C31C2311yescharacteristic
33C2×#41×C31—22no—
34C2×#41×C31—22no—
35C2×#41×C31—44no—
36C22×Q8×C31C2×C411yescharacteristic, complemented
37C22×D4×C31C2×C414yes—
38C22×D4×C31C2×C411yescharacteristic, complemented
39C22×D4×C31C2312yes—
40C23×C4×C31—22no—
41C23×C4×C31—44no—
42C23×C4×C31C2×C412yes—
43C23×C4×C31C2×C412yes—
44C23×C4×C31C2×C412yescomplemented
45C23×C4×C31D412yes—
46C23×C4×C31D412yes—
47C23×C4×C31D412yes—
48C23×C4×C31D412yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C31×#123—24no—
52C31×#123—22no—
53C31×#123—22no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—22no—
58C31×#121—22no—
59C31×#121—22no—
60C31×#121—22no—
61C31×#121—22no—
62C31×#120—22no—
63C31×#120—24no—
64C31×#120—22no—
65C4×D4×C31—24no—
66C4×D4×C31—24no—
67C4×D4×C31—24no—
68C4×D4×C31—24no—
69C31×#117—22no—
70C31×#117—22no—
71C31×#117—44no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31C2×C412yescomplemented
74C2×C4⋊C4×C31C2×C414yescomplemented
75C2×C22⋊C4×C31—22no—
76C2×C22⋊C4×C31—24no—
77C2×C22⋊C4×C31—44no—
78C2×C22⋊C4×C31C2×C414yes—
79C2×C22⋊C4×C31C2×C414yescomplemented
80C2×C22⋊C4×C31C2314yes—
81C2×C22⋊C4×C31C2314yes—
82C2×C22⋊C4×C31C2312yes—
83C2×C42×C31—22no—
84C2×C42×C31C2×C412yescomplemented
85C31×#99—48no—
86C31×#95—24no—
87C31×#95—24no—
88C31×#95—24no—
89C24×C31—24no—
90C24×C31—44no—
91C24×C31C22×C412yes—
92C24×C31C22×C412yes—
93C24×C31C2×D411yescharacteristic
94C24×C31C2×D412yes—
95C24×C31C2×D411yescharacteristic
96C24×C31C2411yescharacteristic
97#41×C31—48no—
98#41×C31—48no—
99C2×Q8×C31—22no—
100C2×Q8×C31—42no—
101C2×D4×C31—24no—
102C2×D4×C31—24no—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—28no—
106C2×D4×C31—22no—
107C2×D4×C31—42no—
108C2×D4×C31—48no—
109C22×C4×C31—24no—
110C22×C4×C31—24no—
111C22×C4×C31—24no—
112C22×C4×C31—24no—
113C22×C4×C31—24no—
114C22×C4×C31—44no—
115C22×C4×C31—44no—
116C22×C4×C31—44no—
117C22×C4×C31—44no—
118C22×C4×C31—44no—
119C22×C4×C31—44no—
120C22×C4×C31—44no—
121C22×C4×C31—44no—
122C22×C4×C31—48no—
123C22×C4×C31C22⋊C412yes—
124C22×C4×C31C22⋊C412yes—
125C22×C4×C31C22⋊C412yes—
126C22×C4×C31C22⋊C412yes—
127C22×C4×C31C22⋊C412yes—
128C22×C4×C31C22⋊C412yes—
129C22×C4×C31C22⋊C412yes—
130C22×C4×C31C22⋊C412yes—
131C22×C4×C31C22×C412yes—
132C22×C4×C31C22×C411yescharacteristic
133C22×C4×C31C22×C412yes—
134C22×C4×C31C22×C411yescharacteristic
135C22×C4×C31C22×C414yes—
136C22×C4×C31C2×D412yes—
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—24no—
144C4⋊C4×C31—24no—
145C4⋊C4×C31—44no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—24no—
148C22⋊C4×C31—22no—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—28no—
153C22⋊C4×C31—28no—
154C22⋊C4×C31—48no—
155C22⋊C4×C31—44no—
156C42×C31—24no—
157C42×C31—44no—
158#21748C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—22no—
160C23×C31—24no—
161C23×C31—24no—
162C23×C31—22no—
163C23×C31—24no—
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—22no—
167C23×C31—24no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—416no—
173C23×C31—44no—
174C23×C31C2×C22⋊C412yes—
175C23×C31C2×C22⋊C412yes—
176C23×C31C23×C411yescharacteristic
177C23×C31C22×D411yescharacteristic
178C23×C31C22×D411yescharacteristic
179C23×C31H(5,𝔽2)14yes—
180Q8×C31—44no—
181D4×C31—44no—
182D4×C31—48no—
183D4×C31—44no—
184D4×C31—44no—
185D4×C31—48no—
186C2×C4×C31—22no—
187C2×C4×C31—24no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—24no—
192C2×C4×C31—24no—
193C2×C4×C31—24no—
194C2×C4×C31—24no—
195C2×C4×C31—22no—
196C2×C4×C31—24no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—48no—
206C2×C4×C31—44no—
207C2×C4×C31—48no—
208C2×C4×C31—48no—
209C2×C4×C31—44no—
210C2×C4×C31—44no—
211C2×C4×C31—88no—
212C2×C4×C31C2×C22⋊C412yes—
213C2×C4×C31C2×C22⋊C412yes—
214C2×C4×C31C2×C22⋊C412yes—
215C2×C4×C31C2×C22⋊C412yes—
216C2×#525C2×C3111yescharacteristic
217#2280C2×C3111yescharacteristic, complemented
218C2×#381C2×C3111yescharacteristic, complemented
219#1899C2×C3114yescomplemented
220#1898C2×C3114yescomplemented
221#1820C2×C3112yescomplemented
222#1782C2×C3112yescomplemented
223C22×C31—28no—
224C22×C31—22no—
225C22×C31—22no—
226C22×C31—22no—
227C22×C31—22no—
228C22×C31—22no—
229C22×C31—22no—
230C22×C31—22no—
231C22×C31—22no—
232C22×C31—44no—
233C22×C31—48no—
234C22×C31—48no—
235C22×C31—44no—
236C22×C31—42no—
237C22×C31—44no—
238C22×C31—84no—
239C22×C31#41014yes—
240C22×C31C22×C22⋊C411yescharacteristic
241C22×C31#51812yes—
242C22×C31#53412yes—
243C22×C31#53412yes—
244C4×C31—22no—
245C4×C31—22no—
246C4×C31—22no—
247C4×C31—22no—
248C4×C31—42no—
249C4×C31—44no—
250C4×C31—84no—
251C4×C31—84no—
252C22×#41C22×C3111yescharacteristic
253C24×C4C22×C3111yescharacteristic
254#525—22no—
255#525—22no—
256#525—22no—
257#525—22no—
258C2×#123C22×C3112yes—
259C2×C22⋊Q8C4×C3112yescomplemented
260C2×#121C4×C3112yescomplemented
261C2×#120C22×C3112yes—
262C2×C4×D4C4×C3114yescomplemented
263C2×#117C22×C3112yes—
264C2×#117C22×C3111yescharacteristic
265C22×C22⋊C4C22×C3112yes—
266C2×#99C22×C3118yes—
267#381—22no—
268#381C22×C3112yes—
269#381C22×C3112yes—
270#381C22×C3112yes—
271#380—22no—
272#379C22×C3112yes—
273C4×C22⋊C4—22no—
274C2×#95—22no—
275#342C22×C3118yes—
276C2×C31—22no—
277C2×C31—22no—
278C2×C31—24no—
279C2×C31—22no—
280C2×C31—42no—
281C2×C31—82no—
282C2×C31#230811yescharacteristic
283C2×C31C2×#41012yes—
284C25C23×C3111yescharacteristic
285C2×#41—22no—
286C2×#41—22no—
287C2×#41—44no—
288C22×Q8C2×C4×C3111yescharacteristic, complemented
289C22×D4C2×C4×C3114yes—
290C22×D4C2×C4×C3111yescharacteristic, complemented
291C22×D4C23×C3112yes—
292C23×C4—22no—
293C23×C4—44no—
294C23×C4C2×C4×C3112yes—
295C23×C4C2×C4×C3112yes—
296C23×C4C2×C4×C3112yescomplemented
297C23×C4D4×C3112yes—
298C23×C4D4×C3112yes—
299C23×C4D4×C3112yes—
300C23×C4D4×C3112yes—
301C23×C4C23×C3111yescharacteristic
302C23×C4C23×C3111yescharacteristic
303#123—24no—
304#123—22no—
305#123—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310#121—22no—
311#121—22no—
312#121—22no—
313#121—22no—
314#120—22no—
315#120—24no—
316#120—22no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321#117—22no—
322#117—22no—
323#117—44no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4C2×C4×C3112yescomplemented
326C2×C4⋊C4C2×C4×C3114yescomplemented
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×C4×C3114yes—
331C2×C22⋊C4C2×C4×C3114yescomplemented
332C2×C22⋊C4C23×C3114yes—
333C2×C22⋊C4C23×C3114yes—
334C2×C22⋊C4C23×C3112yes—
335C2×C42—22no—
336C2×C42C2×C4×C3112yescomplemented
337#99—48no—
338#95—24no—
339#95—24no—
340#95—24no—
341C31#2174811yescharacteristic, direct product factor, Sylow
342C24—24no—
343C24—44no—
344C24C22×C4×C3112yes—
345C24C22×C4×C3112yes—
346C24C2×D4×C3111yescharacteristic
347C24C2×D4×C3112yes—
348C24C2×D4×C3111yescharacteristic
349C24C24×C3111yescharacteristic
350#41—48no—
351#41—48no—
352C2×Q8—22no—
353C2×Q8—42no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—42no—
361C2×D4—48no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4C22⋊C4×C3112yes—
377C22×C4C22⋊C4×C3112yes—
378C22×C4C22⋊C4×C3112yes—
379C22×C4C22⋊C4×C3112yes—
380C22×C4C22⋊C4×C3112yes—
381C22×C4C22⋊C4×C3112yes—
382C22×C4C22⋊C4×C3112yes—
383C22×C4C22⋊C4×C3112yes—
384C22×C4C22×C4×C3112yes—
385C22×C4C22×C4×C3111yescharacteristic
386C22×C4C22×C4×C3112yes—
387C22×C4C22×C4×C3111yescharacteristic
388C22×C4C22×C4×C3114yes—
389C22×C4C2×D4×C3112yes—
390C22×C4C2×D4×C3112yes—
391C22×C4C2×D4×C3112yes—
392C22×C4C2×D4×C3111yescharacteristic
393C22×C4C2×D4×C3111yescharacteristic
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—44no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—48no—
408C22⋊C4—44no—
409C42—24no—
410C42—44no—
411C23—24no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—416no—
425C23—44no—
426C23C2×C22⋊C4×C3112yes—
427C23C2×C22⋊C4×C3112yes—
428C23C23×C4×C3111yescharacteristic
429C23C22×D4×C3111yescharacteristic
430C23C22×D4×C3111yescharacteristic
431C23C31×H(5,𝔽2)14yes—
432Q8—44no—
433D4—44no—
434D4—48no—
435D4—48no—
436D4—44no—
437D4—44no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—88no—
464C2×C4C2×C22⋊C4×C3112yes—
465C2×C4C2×C22⋊C4×C3112yes—
466C2×C4C2×C22⋊C4×C3112yes—
467C2×C4C2×C22⋊C4×C3112yes—
468C22—28no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—44no—
478C22—48no—
479C22—48no—
480C22—44no—
481C22—42no—
482C22—44no—
483C22—84no—
484C22C31×#41014yes—
485C22C22×C22⋊C4×C3111yescharacteristic
486C22C31×#51812yes—
487C22C31×#53412yes—
488C22C31×#53412yes—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—44no—
495C4—84no—
496C4—84no—
497C2—22no—
498C2—22no—
499C2—24no—
500C2—22no—
501C2—42no—
502C2—82no—
503C2C31×#230811yescharacteristic
504C2C2×C31×#41012yes—
subgroupG/H
center, socleC22×C31C22×C22⋊C4
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC2×C31×#525C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2280C2semidirect
3. C2×C31×#381C2semidirect
4. C31×#1899C2semidirect
5. C31×#1898C2semidirect
6. C31×#1820C2semidirect
7. C31×#1782C2semidirect
15. C2×C31×C22⋊Q8C4semidirect
16. C2×C31×#121C4semidirect
18. C2×C4×D4×C31C4semidirect
36. C22×Q8×C31C2×C4semidirect
38. C22×D4×C31C2×C4semidirect
44. C23×C4×C31C2×C4semidirect
73. C2×C4⋊C4×C31C2×C4semidirect
74. C2×C4⋊C4×C31C2×C4semidirect
79. C2×C22⋊C4×C31C2×C4semidirect
84. C2×C42×C31C2×C4semidirect
158. #21748C31direct (N × H)
217. #2280C2×C31semidirect
218. C2×#381C2×C31semidirect
219. #1899C2×C31semidirect
220. #1898C2×C31semidirect
221. #1820C2×C31semidirect
222. #1782C2×C31semidirect
259. C2×C22⋊Q8C4×C31semidirect
260. C2×#121C4×C31semidirect
262. C2×C4×D4C4×C31semidirect
288. C22×Q8C2×C4×C31semidirect
290. C22×D4C2×C4×C31semidirect
296. C23×C4C2×C4×C31semidirect
325. C2×C4⋊C4C2×C4×C31semidirect
326. C2×C4⋊C4C2×C4×C31semidirect
331. C2×C22⋊C4C2×C4×C31semidirect
336. C2×C42C2×C4×C31semidirect
341. C31#21748direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5252
C31×#22802
C2×C31×#3812
C31×#18992
C31×#18982
C31×#18202
C31×#17822
#2174831