Finite Group Databasebeta

← C31×#8536 #21465295

Subgroups — C31×#8536 #21465295

Order 7936.

1074 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic
2C31×#1900C212yesmaximal, complemented
3C31×#1870C211yesmaximal, characteristic, complemented
4C31×#1471C212yesmaximal, complemented
5C31×#1465C211yesmaximal, characteristic, complemented
6C2×C31×H(5,𝔽2)C412yes—
7C22×M16×C31C2211yescharacteristic, complemented
8C31×#547—22no—
9C31×#546—24no—
10D42×C31—22no—
11C31×#537—21no—
12C31×#534—21no—
13C31×#534C412yes—
14C31×#532—21no—
15C2×C31×#121—21no—
16C2×C31×#121C2212yes—
17C2×C4×D4×C31C2211yescharacteristic
18C2×C31×#100—22no—
19C31×#407C2212yescomplemented
20C2×C31×C22⋊C8—21no—
21C2×C31×C22⋊C8C2211yescharacteristic, complemented
22C31×#343—22no—
23C31×#336—21no—
24C31×#324—21no—
25C31×#323—21no—
26C31×#323—24no—
27C31×H(5,𝔽2)—44no—
28C31×H(5,𝔽2)—44no—
29C2×#41×C31—22no—
30C2×#41×C31—24no—
31C22×D4×C31—22no—
32C22×D4×C31—24no—
33C22×D4×C31—21no—
34C22×D4×C31C2×C411yescharacteristic
35C22×D4×C31C2×C414yes—
36C22×D4×C31D412yes—
37C23×C4×C31D411yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C2×M16×C31—22no—
40C2×M16×C31—22no—
41C2×M16×C31—42no—
42C22×C8×C31—21no—
43C22×C8×C31—21no—
44C31×C4⋊Q8—21no—
45C31×#127—21no—
46C31×#124—22no—
47C31×#123—24no—
48C31×#123—24no—
49C31×C22⋊Q8—24no—
50C31×#121—22no—
51C31×#121—22no—
52C31×#121—24no—
53C31×#121—22no—
54C31×#121—22no—
55C31×#121—22no—
56C31×#121—22no—
57C31×#121—22no—
58C31×#121—22no—
59C31×#120—24no—
60C31×#120—24no—
61C4×D4×C31—22no—
62C4×D4×C31—22no—
63C4×D4×C31—22no—
64C4×D4×C31—22no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31D411yescharacteristic
67C2×C4⋊C4×C31D412yes—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31C2×C411yescharacteristic
70C2×C22⋊C4×C31D411yescharacteristic
71C2×C22⋊C4×C31D414yes—
72C2×C42×C31D411yescharacteristic
73C31×#100—44no—
74C31×C22⋊C8—22no—
75C31×C22⋊C8—22no—
76C31×C22⋊C8—22no—
77C31×C22⋊C8—42no—
78C24×C31—22no—
79C24×C31C22⋊C411yescharacteristic
80C24×C31C22⋊C414yes—
81C24×C31C22×C411yescharacteristic
82#41×C31—44no—
83#41×C31—416no—
84#41×C31—44no—
85C2×Q8×C31—22no—
86C2×D4×C31—22no—
87C2×D4×C31—22no—
88C2×D4×C31—24no—
89C2×D4×C31—22no—
90C2×D4×C31—24no—
91C2×D4×C31—24no—
92C2×D4×C31—22no—
93C2×D4×C31—22no—
94C2×D4×C31—22no—
95C2×D4×C31—22no—
96C2×D4×C31—22no—
97C2×D4×C31—42no—
98C2×D4×C31—42no—
99C2×D4×C31—416no—
100C2×D4×C31—44no—
101C2×D4×C31—44no—
102C2×D4×C31—44no—
103C2×D4×C31—44no—
104C2×D4×C31—81no—
105C2×D4×C31—81no—
106C22×C4×C31—22no—
107C22×C4×C31—24no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—41no—
111C22×C4×C31—41no—
112C22×C4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31C22⋊C411yescharacteristic
115C22×C4×C31C22⋊C411yescharacteristic
116C22×C4×C31C22⋊C411yescharacteristic
117C22×C4×C31C22⋊C414yes—
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D412yes—
120C22×C4×C31C2×D412yes—
121C22×C4×C31C2×D411yescharacteristic
122M16×C31—24no—
123M16×C31—42no—
124M16×C31—42no—
125M16×C31—42no—
126C2×C8×C31—22no—
127C2×C8×C31—41no—
128C2×C8×C31—41no—
129C2×C8×C31—41no—
130C2×C8×C31—41no—
131C2×C8×C31—41no—
132C2×C8×C31—42no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C22⋊C4×C31—24no—
139C22⋊C4×C31—24no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—24no—
144C42×C31—21no—
145C42×C31—21no—
146#8536C3111yesmaximal, characteristic, direct product factor, Sylow
147C23×C31—21no—
148C23×C31—21no—
149C23×C31—21no—
150C23×C31—21no—
151C23×C31—24no—
152C23×C31—41no—
153C23×C31—41no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—44no—
157C23×C31—44no—
158C23×C31—44no—
159C23×C31—44no—
160C23×C31—82no—
161C23×C31—82no—
162C23×C31#10012yescomplemented
163C23×C31C2×C22⋊C412yes—
164C23×C31C2×C22⋊C411yescharacteristic
165Q8×C31—44no—
166D4×C31—28no—
167D4×C31—44no—
168D4×C31—48no—
169D4×C31—48no—
170D4×C31—44no—
171D4×C31—44no—
172D4×C31—42no—
173D4×C31—42no—
174D4×C31—82no—
175D4×C31—82no—
176D4×C31—82no—
177C2×C4×C31—21no—
178C2×C4×C31—21no—
179C2×C4×C31—21no—
180C2×C4×C31—22no—
181C2×C4×C31—21no—
182C2×C4×C31—22no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—24no—
186C2×C4×C31—42no—
187C2×C4×C31—41no—
188C2×C4×C31—48no—
189C2×C4×C31—44no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—44no—
195C2×C4×C31—81no—
196C2×C4×C31#10012yescomplemented
197C2×C4×C31#12011yescharacteristic
198C2×C4×C31#12012yes—
199C2×C4×C31#12011yescharacteristic
200C8×C31—42no—
201C8×C31—81no—
202C8×C31—82no—
203#3496C2×C3111yescharacteristic
204#1900C2×C3112yescomplemented
205#1870C2×C3111yescharacteristic, complemented
206#1471C2×C3112yescomplemented
207#1465C2×C3111yescharacteristic, complemented
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—22no—
211C22×C31—22no—
212C22×C31—22no—
213C22×C31—22no—
214C22×C31—42no—
215C22×C31—41no—
216C22×C31—41no—
217C22×C31—44no—
218C22×C31—44no—
219C22×C31—44no—
220C22×C31—81no—
221C22×C31—81no—
222C22×C31—81no—
223C22×C31—164no—
224C22×C31—164no—
225C22×C31#37911yescharacteristic
226C22×C31C2×#10011yescharacteristic
227C22×C31C2×#10011yescharacteristic
228C4×C31—22no—
229C4×C31—21no—
230C4×C31—22no—
231C4×C31—44no—
232C4×C31—41no—
233C4×C31—41no—
234C4×C31B(4,𝔽2)11yescharacteristic
235C4×C31#45811yescharacteristic
236C2×H(5,𝔽2)C4×C3112yes—
237C22×M16C22×C3111yescharacteristic, complemented
238#547—22no—
239#546—24no—
240D42—22no—
241#537—21no—
242#534—21no—
243#534C4×C3112yes—
244#532—21no—
245C2×#121—21no—
246C2×#121C22×C3112yes—
247C2×C4×D4C22×C3111yescharacteristic
248C2×#100—22no—
249#407C22×C3112yescomplemented
250C2×C22⋊C8—21no—
251C2×C22⋊C8C22×C3111yescharacteristic, complemented
252#343—22no—
253#336—21no—
254#324—21no—
255#323—21no—
256#323—24no—
257C2×C31—21no—
258C2×C31—21no—
259C2×C31—42no—
260C2×C31—42no—
261C2×C31—84no—
262C2×C31#178311yescharacteristic
263C2×C31#179011yescharacteristic
264C2×C31#179111yescharacteristic
265H(5,𝔽2)—44no—
266H(5,𝔽2)—44no—
267C2×#41—22no—
268C2×#41—24no—
269C22×D4—22no—
270C22×D4—24no—
271C22×D4—21no—
272C22×D4C2×C4×C3111yescharacteristic
273C22×D4C2×C4×C3114yes—
274C22×D4D4×C3112yes—
275C23×C4D4×C3111yescharacteristic
276C23×C4C23×C3111yescharacteristic
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—42no—
280C22×C8—21no—
281C22×C8—21no—
282C4⋊Q8—21no—
283#127—21no—
284#124—22no—
285#123—24no—
286#123—24no—
287C22⋊Q8—24no—
288#121—22no—
289#121—22no—
290#121—24no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—22no—
295#121—22no—
296#121—22no—
297#120—24no—
298#120—24no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4D4×C3111yescharacteristic
305C2×C4⋊C4D4×C3112yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C2×C4×C3111yescharacteristic
308C2×C22⋊C4D4×C3111yescharacteristic
309C2×C22⋊C4D4×C3114yes—
310C2×C42D4×C3111yescharacteristic
311#100—44no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—42no—
316C31#853611yescharacteristic, direct product factor, Sylow
317C24—22no—
318C24C22⋊C4×C3111yescharacteristic
319C24C22⋊C4×C3114yes—
320C24C22×C4×C3111yescharacteristic
321#41—44no—
322#41—416no—
323#41—44no—
324C2×Q8—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—416no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—81no—
344C2×D4—81no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C22⋊C4×C3111yescharacteristic
354C22×C4C22⋊C4×C3111yescharacteristic
355C22×C4C22⋊C4×C3111yescharacteristic
356C22×C4C22⋊C4×C3114yes—
357C22×C4C2×D4×C3111yescharacteristic
358C22×C4C2×D4×C3112yes—
359C22×C4C2×D4×C3112yes—
360C22×C4C2×D4×C3111yescharacteristic
361M16—24no—
362M16—42no—
363M16—42no—
364M16—42no—
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C42—21no—
384C42—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—24no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—82no—
399C23—82no—
400C23C31×#10012yescomplemented
401C23C2×C22⋊C4×C3112yes—
402C23C2×C22⋊C4×C3111yescharacteristic
403Q8—44no—
404D4—28no—
405D4—44no—
406D4—48no—
407D4—48no—
408D4—44no—
409D4—44no—
410D4—42no—
411D4—42no—
412D4—82no—
413D4—82no—
414D4—82no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—81no—
434C2×C4C31×#10012yescomplemented
435C2×C4C31×#12011yescharacteristic
436C2×C4C31×#12012yes—
437C2×C4C31×#12011yescharacteristic
438C8—42no—
439C8—81no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—164no—
457C22—164no—
458C22C31×#37911yescharacteristic
459C22C2×C31×#10011yescharacteristic
460C22C2×C31×#10011yescharacteristic
461C4—22no—
462C4—21no—
463C4—22no—
464C4—44no—
465C4—41no—
466C4—41no—
467C4C31×B(4,𝔽2)11yescharacteristic
468C4C31×#45811yescharacteristic
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—84no—
474C2C31×#178311yescharacteristic
475C2C31×#179011yescharacteristic
476C2C31×#179111yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3496C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1900C2semidirect
3. C31×#1870C2semidirect
4. C31×#1471C2semidirect
5. C31×#1465C2semidirect
7. C22×M16×C31C22semidirect
19. C31×#407C22semidirect
21. C2×C31×C22⋊C8C22semidirect
146. #8536C31direct (N × H)
162. C23×C31#100semidirect
196. C2×C4×C31#100semidirect
204. #1900C2×C31semidirect
205. #1870C2×C31semidirect
206. #1471C2×C31semidirect
207. #1465C2×C31semidirect
237. C22×M16C22×C31semidirect
249. #407C22×C31semidirect
251. C2×C22⋊C8C22×C31semidirect
316. C31#8536direct (N × H)
400. C23C31×#100semidirect
434. C2×C4C31×#100semidirect

Maximal subgroups

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

subgroupindex
C31×#34962
C31×#19002
C31×#18702
C31×#14712
C31×#14652
#853631