Finite Group Databasebeta

← C31×#9952 #21466711

Subgroups — C31×#9952 #21466711

Order 7936.

692 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3186C211yesmaximal, characteristic
2C2×C31×#480C211yesmaximal, characteristic, complemented
3C31×#2501C211yesmaximal, characteristic, complemented
4C31×#1964C211yesmaximal, characteristic, complemented
5C31×#1823C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C2×C31×#123—22no—
9C2×C31×#121C2211yescharacteristic
10C2×C31×#121C2211yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C31×#480—22no—
13C31×#480—21no—
14C31×#480—21no—
15C31×#480—22no—
16C31×#466—22no—
17C31×#447—21no—
18C31×#447—21no—
19C2×C31×#107—21no—
20C2×C31×#102—21no—
21C2×C31×#99—22no—
22C2×C31×C22⋊C8C2211yescharacteristic
23C2×C31×C22⋊C8C2211yescharacteristic
24C31×#399—22no—
25C31×#390—21no—
26C31×#384—21no—
27C31×#380C2211yescharacteristic
28C4×C22⋊C4×C31C2211yescharacteristic
29C31×#340—22no—
30C31×#327—22no—
31C22×D4×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C23×C4×C31C2311yescharacteristic
37C2×D8×C31—22no—
38C22×C8×C31—21no—
39C22×C8×C31D412yes—
40C31×#123—44no—
41C31×#123—44no—
42C31×#121—21no—
43C31×#121—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#121—21no—
48C31×#121—21no—
49C31×#121—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31D412yes—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31D412yes—
65C2×C42×C31—21no—
66C31×#107—24no—
67C31×#107—22no—
68C31×#102—24no—
69C31×#102—22no—
70C31×#102—22no—
71C31×#99—44no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8D412yes—
76C31×#95—21no—
77C31×#95—21no—
78C24×C31—21no—
79C24×C31—21no—
80C24×C31C2×D411yescharacteristic
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—42no—
86C2×D4×C31—42no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—42no—
92C2×D4×C31—42no—
93C2×D4×C31—42no—
94C2×D4×C31—42no—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31—42no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31—42no—
103C22×C4×C31—42no—
104C22×C4×C31—41no—
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31#4111yescharacteristic
111D8×C31—82no—
112C2×C8×C31—22no—
113C2×C8×C31—22no—
114C2×C8×C31—42no—
115C2×C8×C31—42no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—48no—
123C4⋊C4×C31—42no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C22⋊C4×C31—22no—
127C22⋊C4×C31—48no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—41no—
131C22⋊C4×C31—41no—
132C22⋊C4×C31—42no—
133C22⋊C4×C31D814yescomplemented
134C42×C31—42no—
135#9952C3111yesmaximal, characteristic, direct product factor, Sylow
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—42no—
139C23×C31—42no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—42no—
143C23×C31—42no—
144C23×C31—81no—
145C23×C31—81no—
146C23×C31#12011yescharacteristic
147C23×C31#12111yescharacteristic
148C23×C31#12711yescharacteristic
149C23×C31C2×D812yes—
150C23×C31AGL(1,ℤ/8)12yes—
151D4×C31—42no—
152D4×C31—42no—
153D4×C31—42no—
154D4×C31—81no—
155D4×C31—82no—
156D4×C31—82no—
157D4×C31—81no—
158D4×C31—81no—
159D4×C31—81no—
160D4×C31—81no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—24no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—42no—
173C2×C4×C31—44no—
174C2×C4×C31—42no—
175C2×C4×C31—42no—
176C2×C4×C31—82no—
177C2×C4×C31—81no—
178C2×C4×C31—81no—
179C2×C4×C31—81no—
180C2×C4×C31—81no—
181C2×C4×C31—81no—
182C2×C4×C31#12011yescharacteristic
183C2×C4×C31#12011yescharacteristic
184C2×C4×C31#12111yescharacteristic
185C2×C4×C31#12111yescharacteristic
186C8×C31—42no—
187C8×C31—42no—
188#3186C2×C3111yescharacteristic
189C2×#480C2×C3111yescharacteristic, complemented
190#2501C2×C3111yescharacteristic, complemented
191#1964C2×C3111yescharacteristic, complemented
192#1823C2×C3111yescharacteristic, complemented
193#1519C2×C3111yescharacteristic, complemented
194#1282C2×C3111yescharacteristic, complemented
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—42no—
202C22×C31—42no—
203C22×C31—84no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31—162no—
211C22×C31#39211yescharacteristic
212C22×C31#44712yes—
213C22×C31#45912yes—
214C22×C31#49311yescharacteristic
215C22×C31#49611yescharacteristic
216C4×C31—21no—
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—44no—
220C4×C31—82no—
221C4×C31—81no—
222C4×C31—81no—
223C4×C31—161no—
224C4×C31#45812yes—
225C2×#123—22no—
226C2×#121C22×C3111yescharacteristic
227C2×#121C22×C3111yescharacteristic
228C22×C4⋊C4C22×C3111yescharacteristic
229#480—22no—
230#480—21no—
231#480—21no—
232#480—22no—
233#466—22no—
234#447—21no—
235#447—21no—
236C2×#107—21no—
237C2×#102—21no—
238C2×#99—22no—
239C2×C22⋊C8C22×C3111yescharacteristic
240C2×C22⋊C8C22×C3111yescharacteristic
241#399—22no—
242#390—21no—
243#384—21no—
244#380C22×C3111yescharacteristic
245C4×C22⋊C4C22×C3111yescharacteristic
246#340—22no—
247#327—22no—
248C2×C31—22no—
249C2×C31—21no—
250C2×C31—22no—
251C2×C31—21no—
252C2×C31—161no—
253C2×C31—161no—
254C2×C31#200911yescharacteristic
255C2×C31#201111yescharacteristic
256C2×C31#202211yescharacteristic
257C22×D4—21no—
258C22×D4—21no—
259C22×D4D4×C3111yescharacteristic
260C22×D4D4×C3111yescharacteristic
261C23×C4—21no—
262C23×C4C23×C3111yescharacteristic
263C2×D8—22no—
264C22×C8—21no—
265C22×C8D4×C3112yes—
266#123—44no—
267#123—44no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275#121—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4D4×C3112yes—
284C2×C4⋊C4D4×C3111yescharacteristic
285C2×C4⋊C4D4×C3111yescharacteristic
286C2×C22⋊C4—24no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4D4×C3112yes—
291C2×C42—21no—
292#107—24no—
293#107—22no—
294#102—24no—
295#102—22no—
296#102—22no—
297#99—44no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8D4×C3112yes—
302#95—21no—
303#95—21no—
304C31#995211yescharacteristic, direct product factor, Sylow
305C24—21no—
306C24—21no—
307C24C2×D4×C3111yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4C2×D4×C3111yescharacteristic
333C22×C4C2×D4×C3111yescharacteristic
334C22×C4C2×D4×C3111yescharacteristic
335C22×C4C2×D4×C3111yescharacteristic
336C22×C4C2×D4×C3111yescharacteristic
337C22×C4#41×C3111yescharacteristic
338D8—82no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C22⋊C4—22no—
354C22⋊C4—48no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C22⋊C4D8×C3114yescomplemented
361C42—42no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23C31×#12011yescharacteristic
373C23C31×#12111yescharacteristic
374C23C31×#12711yescharacteristic
375C23C2×D8×C3112yes—
376C23C31×AGL(1,ℤ/8)12yes—
377D4—42no—
378D4—42no—
379D4—42no—
380D4—81no—
381D4—82no—
382D4—82no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4C31×#12011yescharacteristic
409C2×C4C31×#12011yescharacteristic
410C2×C4C31×#12111yescharacteristic
411C2×C4C31×#12111yescharacteristic
412C8—42no—
413C8—42no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—162no—
430C22C31×#39211yescharacteristic
431C22C31×#44712yes—
432C22C31×#45912yes—
433C22C31×#49311yescharacteristic
434C22C31×#49611yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—82no—
440C4—81no—
441C4—81no—
442C4—161no—
443C4C31×#45812yes—
444C2—22no—
445C2—21no—
446C2—22no—
447C2—21no—
448C2—161no—
449C2—161no—
450C2C31×#200911yescharacteristic
451C2C31×#201111yescharacteristic
452C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#480C2semidirect
3. C31×#2501C2semidirect
4. C31×#1964C2semidirect
5. C31×#1823C2semidirect
6. C31×#1519C2semidirect
7. C31×#1282C2semidirect
133. C22⋊C4×C31D8semidirect
135. #9952C31direct (N × H)
189. C2×#480C2×C31semidirect
190. #2501C2×C31semidirect
191. #1964C2×C31semidirect
192. #1823C2×C31semidirect
193. #1519C2×C31semidirect
194. #1282C2×C31semidirect
304. C31#9952direct (N × H)
360. C22⋊C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31862
C2×C31×#4802
C31×#25012
C31×#19642
C31×#18232
C31×#15192
C31×#12822
#995231