Finite Group Databasebeta

← C31×#13305 #21470064

Subgroups — C31×#13305 #21470064

Order 7936.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×SD32C211yesmaximal, characteristic, complemented
2C2×C31×#467C211yesmaximal, characteristic, complemented
3C2×C31×#466C211yesmaximal, characteristic
4C2×C31×#358C211yesmaximal, characteristic, complemented
5C2×C31×#357C211yesmaximal, characteristic, complemented
6C2×C31×C22⋊C16C211yesmaximal, characteristic, complemented
7C31×#1380C211yesmaximal, characteristic, complemented
8C22×Q16×C31C2211yescharacteristic
9C22×D8×C31C2211yescharacteristic
10C23×C8×C31C2211yescharacteristic
11C2×C31×C22⋊Q8—21no—
12C2×C31×#121—21no—
13C2×C31×SD32—22no—
14C2×C31×SD32—22no—
15C2×C31×SD32—22no—
16C22×C16×C31C2211yescharacteristic
17C22×C16×C31C2211yescharacteristic
18C31×#467—22no—
19C31×#467—22no—
20C31×#466—22no—
21C31×#466—22no—
22C2×C31×#107C2211yescharacteristic
23C2×C31×#107C2211yescharacteristic
24C2×C31×#103—21no—
25C2×C31×#102—21no—
26C31×#358—22no—
27C31×#357—22no—
28C31×C22⋊C16—22no—
29C22×Q8×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—41no—
32C23×C4×C31D411yescharacteristic
33C2×Q16×C31—21no—
34C2×Q16×C31—21no—
35C2×Q16×C31—21no—
36C2×Q16×C31—21no—
37C2×Q16×C31—21no—
38C2×Q16×C31D412yescomplemented
39C2×D8×C31—21no—
40C2×D8×C31—21no—
41C2×D8×C31—21no—
42C2×D8×C31—21no—
43C2×D8×C31—21no—
44C2×D8×C31D412yes—
45C22×C8×C31—21no—
46C22×C8×C31—21no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C22×C8×C31—21no—
50C22×C8×C31—21no—
51C22×C8×C31D411yescharacteristic
52C22×C8×C31C2311yescharacteristic
53C31×C22⋊Q8—24no—
54C31×C22⋊Q8—42no—
55C31×#121—24no—
56C31×#121—42no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—41no—
60C2×C22⋊C4×C31—41no—
61C2×C22⋊C4×C31—41no—
62C31×SD32—28no—
63C2×C16×C31—21no—
64C2×C16×C31—21no—
65C2×C16×C31—21no—
66C2×C16×C31—21no—
67C2×C16×C31—21no—
68C2×C16×C31D412yescomplemented
69C31×#107—21no—
70C31×#107—21no—
71C31×#107D412yes—
72C31×#107D412yescomplemented
73C31×#103—24no—
74C31×#102—24no—
75C24×C31—41no—
76C24×C31D811yescharacteristic
77C2×Q8×C31—22no—
78C2×Q8×C31—41no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×Q8×C31—41no—
82C2×Q8×C31—41no—
83C2×D4×C31—22no—
84C2×D4×C31—44no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—82no—
91C2×D4×C31—82no—
92C22×C4×C31—21no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—41no—
99C22×C4×C31—41no—
100C22×C4×C31—41no—
101C22×C4×C31D811yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103Q16×C31—22no—
104Q16×C31—22no—
105Q16×C31—41no—
106Q16×C31—41no—
107D8×C31—22no—
108D8×C31—22no—
109D8×C31—41no—
110D8×C31—41no—
111C2×C8×C31—21no—
112C2×C8×C31—21no—
113C2×C8×C31—21no—
114C2×C8×C31—41no—
115C2×C8×C31—41no—
116C2×C8×C31—41no—
117C2×C8×C31—41no—
118C2×C8×C31C2×D411yescharacteristic
119C2×C8×C31C2×D411yescharacteristic
120C2×C8×C31C2×D411yescharacteristic
121C2×C8×C31C2×D411yescharacteristic
122C2×C8×C31C2×D411yescharacteristic
123C2×C8×C31#4111yescharacteristic
124C4⋊C4×C31—22no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—44no—
127C4⋊C4×C31—41no—
128C4⋊C4×C31—41no—
129C22⋊C4×C31—44no—
130C22⋊C4×C31—44no—
131C16×C31—22no—
132C16×C31—41no—
133#13305C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—21no—
135C23×C31—21no—
136C23×C31—21no—
137C23×C31—21no—
138C23×C31—21no—
139C23×C31—21no—
140C23×C31—42no—
141C23×C31—81no—
142C23×C31—81no—
143C23×C31—81no—
144C23×C31—81no—
145C23×C31—81no—
146C23×C31—81no—
147C23×C31SD3212yescomplemented
148C23×C31C2×D811yescharacteristic
149Q8×C31—42no—
150Q8×C31—42no—
151Q8×C31—81no—
152Q8×C31—81no—
153D4×C31—42no—
154D4×C31—42no—
155D4×C31—88no—
156D4×C31—81no—
157D4×C31—81no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C2×C4×C31#12011yescharacteristic
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12111yescharacteristic
177C2×C4×C31#13511yescharacteristic
178C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
180C8×C31—41no—
181C8×C31#12011yescharacteristic
182C8×C31#12111yescharacteristic
183C8×C31#12111yescharacteristic
184C8×C31#12711yescharacteristic
185C22×SD32C2×C3111yescharacteristic, complemented
186C2×#467C2×C3111yescharacteristic, complemented
187C2×#466C2×C3111yescharacteristic
188C2×#358C2×C3111yescharacteristic, complemented
189C2×#357C2×C3111yescharacteristic, complemented
190C2×C22⋊C16C2×C3111yescharacteristic, complemented
191#1380C2×C3111yescharacteristic, complemented
192C22×C31—22no—
193C22×C31—22no—
194C22×C31—22no—
195C22×C31—41no—
196C22×C31—41no—
197C22×C31—41no—
198C22×C31—41no—
199C22×C31—82no—
200C22×C31—82no—
201C22×C31—82no—
202C22×C31—161no—
203C22×C31—161no—
204C22×C31—161no—
205C22×C31—161no—
206C22×C31#44711yescharacteristic
207C22×C31#44711yescharacteristic
208C22×C31#46611yescharacteristic
209C22×C31C2×SD3211yescharacteristic
210C22×C31#50811yescharacteristic
211C22×C31#50911yescharacteristic
212C22×C31#51011yescharacteristic
213C4×C31—41no—
214C4×C31—82no—
215C4×C31—82no—
216C4×C31—82no—
217C4×C31#39211yescharacteristic
218C4×C31#44911yescharacteristic
219C4×C31#44911yescharacteristic
220C4×C31#46911yescharacteristic
221C22×Q16C22×C3111yescharacteristic
222C22×D8C22×C3111yescharacteristic
223C23×C8C22×C3111yescharacteristic
224C2×C22⋊Q8—21no—
225C2×#121—21no—
226C2×SD32—22no—
227C2×SD32—22no—
228C2×SD32—22no—
229C22×C16C22×C3111yescharacteristic
230C22×C16C22×C3111yescharacteristic
231#467—22no—
232#467—22no—
233#466—22no—
234#466—22no—
235C2×#107C22×C3111yescharacteristic
236C2×#107C22×C3111yescharacteristic
237C2×#103—21no—
238C2×#102—21no—
239#358—22no—
240#357—22no—
241C22⋊C16—22no—
242C2×C31—42no—
243C2×C31—162no—
244C2×C31#199911yescharacteristic
245C2×C31#218511yescharacteristic
246C2×C31#218611yescharacteristic
247C2×C31#218711yescharacteristic
248C2×C31#218911yescharacteristic
249C2×C31#221711yescharacteristic
250C2×C31#222011yescharacteristic
251C22×Q8—21no—
252C22×D4—21no—
253C22×D4—41no—
254C23×C4D4×C3111yescharacteristic
255C2×Q16—21no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16D4×C3112yescomplemented
261C2×D8—21no—
262C2×D8—21no—
263C2×D8—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8D4×C3112yes—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8D4×C3111yescharacteristic
274C22×C8C23×C3111yescharacteristic
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277#121—24no—
278#121—42no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284SD32—28no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16D4×C3112yescomplemented
291#107—21no—
292#107—21no—
293#107D4×C3112yes—
294#107D4×C3112yescomplemented
295#103—24no—
296#102—24no—
297C31#1330511yescharacteristic, direct product factor, Sylow
298C24—41no—
299C24D8×C3111yescharacteristic
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—22no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C2×D4—82no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4D8×C3111yescharacteristic
325C22×C4C2×D4×C3111yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330D8—22no—
331D8—22no—
332D8—41no—
333D8—41no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8C2×D4×C3111yescharacteristic
342C2×C8C2×D4×C3111yescharacteristic
343C2×C8C2×D4×C3111yescharacteristic
344C2×C8C2×D4×C3111yescharacteristic
345C2×C8C2×D4×C3111yescharacteristic
346C2×C8#41×C3111yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C16—22no—
355C16—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23C31×SD3212yescomplemented
370C23C2×D8×C3111yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—42no—
376D4—42no—
377D4—88no—
378D4—81no—
379D4—81no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4C31×#12011yescharacteristic
397C2×C4C31×#12011yescharacteristic
398C2×C4C31×#12111yescharacteristic
399C2×C4C31×#13511yescharacteristic
400C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
402C8—41no—
403C8C31×#12011yescharacteristic
404C8C31×#12111yescharacteristic
405C8C31×#12111yescharacteristic
406C8C31×#12711yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22C31×#44711yescharacteristic
422C22C31×#44711yescharacteristic
423C22C31×#46611yescharacteristic
424C22C2×C31×SD3211yescharacteristic
425C22C31×#50811yescharacteristic
426C22C31×#50911yescharacteristic
427C22C31×#51011yescharacteristic
428C4—41no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4C31×#39211yescharacteristic
433C4C31×#44911yescharacteristic
434C4C31×#44911yescharacteristic
435C4C31×#46911yescharacteristic
436C2—42no—
437C2—162no—
438C2C31×#199911yescharacteristic
439C2C31×#218511yescharacteristic
440C2C31×#218611yescharacteristic
441C2C31×#218711yescharacteristic
442C2C31×#218911yescharacteristic
443C2C31×#221711yescharacteristic
444C2C31×#222011yescharacteristic
subgroupG/H
center, socleC23×C31C2×D8
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC2×C31×#466C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ C23×C8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×SD32C2semidirect
2. C2×C31×#467C2semidirect
4. C2×C31×#358C2semidirect
5. C2×C31×#357C2semidirect
6. C2×C31×C22⋊C16C2semidirect
7. C31×#1380C2semidirect
38. C2×Q16×C31D4semidirect
68. C2×C16×C31D4semidirect
72. C31×#107D4semidirect
133. #13305C31direct (N × H)
147. C23×C31SD32semidirect
185. C22×SD32C2×C31semidirect
186. C2×#467C2×C31semidirect
188. C2×#358C2×C31semidirect
189. C2×#357C2×C31semidirect
190. C2×C22⋊C16C2×C31semidirect
191. #1380C2×C31semidirect
260. C2×Q16D4×C31semidirect
290. C2×C16D4×C31semidirect
294. #107D4×C31semidirect
297. C31#13305direct (N × H)
369. C23C31×SD32semidirect

Maximal subgroups

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

subgroupindex
C22×C31×SD322
C2×C31×#4672
C2×C31×#4662
C2×C31×#3582
C2×C31×#3572
C2×C31×C22⋊C162
C31×#13802
#1330531