Finite Group Databasebeta

← C31×#8470 #21465229

Subgroups — C31×#8470 #21465229

Order 7936.

754 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic
2C31×#1997C211yesmaximal, characteristic, complemented
3C31×#1966C211yesmaximal, characteristic, complemented
4C31×#1560C211yesmaximal, characteristic, complemented
5C31×#1525C211yesmaximal, characteristic, complemented
6C31×#1489C211yesmaximal, characteristic, complemented
7C31×#1489C211yesmaximal, characteristic, complemented
8C31×#556—21no—
9C31×#555—21no—
10C31×#554—21no—
11C31×#553—22no—
12C31×#548—22no—
13C31×#547—21no—
14C31×#542—22no—
15C31×#537C412yes—
16C31×#533—21no—
17C31×#532—21no—
18C31×#532C2211yescharacteristic
19C2×C31×C4⋊Q8C2211yescharacteristic
20C31×#520C412yes—
21C4×#41×C31C2211yescharacteristic
22C31×#423C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C2×C31×#101—22no—
25C4×M16×C31C2211yescharacteristic
26C4×M16×C31C2211yescharacteristic
27C31×#343—22no—
28C31×#333—21no—
29C31×#331—21no—
30C31×#329—22no—
31C31×Q8⋊C8—21no—
32C31×Q8⋊C8—21no—
33C31×D4⋊C8—21no—
34C31×D4⋊C8—21no—
35C2×#41×C31—22no—
36C2×#41×C31C2×C411yescharacteristic
37C2×#41×C31C2×C412yes—
38C22×Q8×C31D412yes—
39C2×M16×C31—21no—
40C2×M16×C31—21no—
41C2×M16×C31—21no—
42C2×M16×C31—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8D411yescharacteristic
48C31×#127D411yescharacteristic
49C31×#126—24no—
50C31×#125—22no—
51C31×#124—22no—
52C31×#124—24no—
53C31×#124D412yes—
54C31×#123—24no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—24no—
57C31×C22⋊Q8—22no—
58C31×C22⋊Q8—22no—
59C31×#121—22no—
60C31×#121—22no—
61C31×#121—22no—
62C31×#121—24no—
63C31×#121—21no—
64C31×#121—21no—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31D411yescharacteristic
68C4×Q8×C31D411yescharacteristic
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—24no—
72C4×D4×C31—21no—
73C4×D4×C31—21no—
74C4×D4×C31—21no—
75C4×D4×C31—21no—
76C4×D4×C31D411yescharacteristic
77C4×D4×C31D411yescharacteristic
78C31×#117—22no—
79C31×#117—22no—
80C31×#117—21no—
81C31×#117C2×C411yescharacteristic
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31C2×C412yes—
84C2×C4⋊C4×C31D412yes—
85C2×C42×C31—21no—
86C2×C42×C31C2311yescharacteristic
87C31×C4⋊C8—21no—
88C31×C4⋊C8—21no—
89C31×C4⋊C8—21no—
90C31×C4⋊C8—21no—
91C31×#101—44no—
92C31×#97—21no—
93C31×#97—21no—
94C4×C8×C31—21no—
95C4×C8×C31—21no—
96#41×C31—42no—
97#41×C31—44no—
98#41×C31—82no—
99#41×C31—82no—
100C2×Q8×C31—22no—
101C2×Q8×C31—22no—
102C2×Q8×C31—42no—
103C2×Q8×C31—42no—
104C2×Q8×C31C22⋊C411yescharacteristic
105C2×Q8×C31C22⋊C412yes—
106C2×D4×C31—22no—
107C2×D4×C31—21no—
108C2×D4×C31—24no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31C22⋊C411yescharacteristic
112C2×D4×C31C22⋊C412yes—
113C22×C4×C31—24no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31C22⋊C414yes—
119C22×C4×C31C22×C411yescharacteristic
120C22×C4×C31C2×D411yescharacteristic
121C22×C4×C31C2×D411yescharacteristic
122M16×C31—22no—
123M16×C31—22no—
124M16×C31—41no—
125M16×C31—41no—
126M16×C31—42no—
127M16×C31—42no—
128C2×C8×C31—41no—
129C2×C8×C31—41no—
130C2×C8×C31—41no—
131C2×C8×C31—41no—
132C4⋊C4×C31—24no—
133C4⋊C4×C31—24no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—21no—
139C4⋊C4×C31—21no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31C22⋊C411yescharacteristic
143C4⋊C4×C31C22⋊C411yescharacteristic
144C22⋊C4×C31—24no—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—21no—
148C22⋊C4×C31—21no—
149C22⋊C4×C31—21no—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—22no—
152C42×C31—24no—
153C42×C31—21no—
154C42×C31—21no—
155C42×C31—21no—
156C42×C31C2×D411yescharacteristic
157C42×C31C2×D411yescharacteristic
158C42×C31C2×D411yescharacteristic
159C42×C31C2×D411yescharacteristic
160#8470C3111yesmaximal, characteristic, direct product factor, Sylow
161C23×C31—21no—
162C23×C31—22no—
163C23×C31C2×C22⋊C411yescharacteristic
164Q8×C31—22no—
165Q8×C31—42no—
166Q8×C31—42no—
167Q8×C31—42no—
168Q8×C31—82no—
169D4×C31—22no—
170D4×C31—42no—
171D4×C31—42no—
172D4×C31—44no—
173D4×C31—41no—
174D4×C31—41no—
175D4×C31—41no—
176D4×C31—41no—
177D4×C31—84no—
178C2×C4×C31—22no—
179C2×C4×C31—21no—
180C2×C4×C31—21no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—22no—
186C2×C4×C31—21no—
187C2×C4×C31—24no—
188C2×C4×C31—42no—
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—84no—
196C2×C4×C31C2×C22⋊C411yescharacteristic
197C2×C4×C31C2×C22⋊C411yescharacteristic
198C2×C4×C31#12011yescharacteristic
199C2×C4×C31#12011yescharacteristic
200C2×C4×C31#12011yescharacteristic
201C2×C4×C31#12011yescharacteristic
202C8×C31—42no—
203C8×C31—42no—
204C8×C31—81no—
205C8×C31—81no—
206#3511C2×C3111yescharacteristic
207#1997C2×C3111yescharacteristic, complemented
208#1966C2×C3111yescharacteristic, complemented
209#1560C2×C3111yescharacteristic, complemented
210#1525C2×C3111yescharacteristic, complemented
211#1489C2×C3111yescharacteristic, complemented
212#1489C2×C3111yescharacteristic, complemented
213C22×C31—21no—
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—41no—
217C22×C31—42no—
218C22×C31—41no—
219C22×C31—42no—
220C22×C31—41no—
221C22×C31—42no—
222C22×C31#37911yescharacteristic
223C4×C31—21no—
224C4×C31—21no—
225C4×C31—21no—
226C4×C31—41no—
227C4×C31—44no—
228C4×C31—41no—
229C4×C31—41no—
230C4×C31—42no—
231C4×C31#41312yes—
232C4×C31#45311yescharacteristic
233C4×C31#45411yescharacteristic
234C4×C31#45511yescharacteristic
235C4×C31#45611yescharacteristic
236#556—21no—
237#555—21no—
238#554—21no—
239#553—22no—
240#548—22no—
241#547—21no—
242#542—22no—
243#537C4×C3112yes—
244#533—21no—
245#532—21no—
246#532C22×C3111yescharacteristic
247C2×C4⋊Q8C22×C3111yescharacteristic
248#520C4×C3112yes—
249C4×#41C22×C3111yescharacteristic
250#423C22×C3111yescharacteristic
251#423C22×C3111yescharacteristic
252C2×#101—22no—
253C4×M16C22×C3111yescharacteristic
254C4×M16C22×C3111yescharacteristic
255#343—22no—
256#333—21no—
257#331—21no—
258#329—22no—
259Q8⋊C8—21no—
260Q8⋊C8—21no—
261D4⋊C8—21no—
262D4⋊C8—21no—
263C2×C31—41no—
264C2×C31—81no—
265C2×C31—82no—
266C2×C31#178511yescharacteristic
267C2×C31#179211yescharacteristic
268C2×C31#179311yescharacteristic
269C2×#41—22no—
270C2×#41C2×C4×C3111yescharacteristic
271C2×#41C2×C4×C3112yes—
272C22×Q8D4×C3112yes—
273C2×M16—21no—
274C2×M16—21no—
275C2×M16—21no—
276C2×M16—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8D4×C3111yescharacteristic
282#127D4×C3111yescharacteristic
283#126—24no—
284#125—22no—
285#124—22no—
286#124—24no—
287#124D4×C3112yes—
288#123—24no—
289C22⋊Q8—22no—
290C22⋊Q8—24no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293#121—22no—
294#121—22no—
295#121—22no—
296#121—24no—
297#121—21no—
298#121—21no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8D4×C3111yescharacteristic
302C4×Q8D4×C3111yescharacteristic
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—24no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4D4×C3111yescharacteristic
311C4×D4D4×C3111yescharacteristic
312#117—22no—
313#117—22no—
314#117—21no—
315#117C2×C4×C3111yescharacteristic
316C2×C4⋊C4—22no—
317C2×C4⋊C4C2×C4×C3112yes—
318C2×C4⋊C4D4×C3112yes—
319C2×C42—21no—
320C2×C42C23×C3111yescharacteristic
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325#101—44no—
326#97—21no—
327#97—21no—
328C4×C8—21no—
329C4×C8—21no—
330C31#847011yescharacteristic, direct product factor, Sylow
331#41—42no—
332#41—44no—
333#41—82no—
334#41—82no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8C22⋊C4×C3111yescharacteristic
340C2×Q8C22⋊C4×C3112yes—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—24no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C22⋊C4×C3111yescharacteristic
347C2×D4C22⋊C4×C3112yes—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4C22⋊C4×C3114yes—
354C22×C4C22×C4×C3111yescharacteristic
355C22×C4C2×D4×C3111yescharacteristic
356C22×C4C2×D4×C3111yescharacteristic
357M16—22no—
358M16—22no—
359M16—41no—
360M16—41no—
361M16—42no—
362M16—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4C22⋊C4×C3111yescharacteristic
378C4⋊C4C22⋊C4×C3111yescharacteristic
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—24no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42C2×D4×C3111yescharacteristic
392C42C2×D4×C3111yescharacteristic
393C42C2×D4×C3111yescharacteristic
394C42C2×D4×C3111yescharacteristic
395C23—21no—
396C23—22no—
397C23C2×C22⋊C4×C3111yescharacteristic
398Q8—22no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—82no—
403D4—22no—
404D4—42no—
405D4—42no—
406D4—44no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—84no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—84no—
430C2×C4C2×C22⋊C4×C3111yescharacteristic
431C2×C4C2×C22⋊C4×C3111yescharacteristic
432C2×C4C31×#12011yescharacteristic
433C2×C4C31×#12011yescharacteristic
434C2×C4C31×#12011yescharacteristic
435C2×C4C31×#12011yescharacteristic
436C8—42no—
437C8—42no—
438C8—81no—
439C8—81no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—41no—
448C22—42no—
449C22C31×#37911yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4C31×#41312yes—
459C4C31×#45311yescharacteristic
460C4C31×#45411yescharacteristic
461C4C31×#45511yescharacteristic
462C4C31×#45611yescharacteristic
463C2—41no—
464C2—81no—
465C2—82no—
466C2C31×#178511yescharacteristic
467C2C31×#179211yescharacteristic
468C2C31×#179311yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3511C2
LCSG ▷ C22×C4 ▷ 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×#1997C2semidirect
3. C31×#1966C2semidirect
4. C31×#1560C2semidirect
5. C31×#1525C2semidirect
6. C31×#1489C2semidirect
7. C31×#1489C2semidirect
160. #8470C31direct (N × H)
207. #1997C2×C31semidirect
208. #1966C2×C31semidirect
209. #1560C2×C31semidirect
210. #1525C2×C31semidirect
211. #1489C2×C31semidirect
212. #1489C2×C31semidirect
330. C31#8470direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#19972
C31×#19662
C31×#15602
C31×#15252
C31×#14892
#847031