Finite Group Databasebeta

← C31×#10915 #21467674

Subgroups — C31×#10915 #21467674

Order 7936.

974 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2315C211yesmaximal, characteristic
2C31×#2273C211yesmaximal, characteristic
3C2×C31×#342C212yesmaximal
4C2×C31×#342C211yesmaximal, characteristic
5C2×C31×#342C212yesmaximal
6C22×C22⋊C4×C31C2211yescharacteristic
7C22×C22⋊C4×C31C2212yes—
8C22×C22⋊C4×C31C2211yescharacteristic
9C22×C22⋊C4×C31C2211yescharacteristic
10C22×C22⋊C4×C31C2212yes—
11C31×#385—22no—
12C31×#385C414yescomplemented
13C31×#380—22no—
14C4×C22⋊C4×C31—22no—
15C4×C22⋊C4×C31—24no—
16C31×#342—22no—
17C31×#342—24no—
18C31×#342C418yescomplemented
19C25×C31C2311yescharacteristic
20C23×C4×C31—22no—
21C23×C4×C31—21no—
22C23×C4×C31—22no—
23C23×C4×C31C2×C412yes—
24C23×C4×C31D412yes—
25C2×C4⋊C4×C31—24no—
26C2×C22⋊C4×C31—22no—
27C2×C22⋊C4×C31—22no—
28C2×C22⋊C4×C31—21no—
29C2×C22⋊C4×C31—22no—
30C2×C22⋊C4×C31—22no—
31C2×C22⋊C4×C31—22no—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31C2×C414yes—
43C2×C22⋊C4×C31C2×C412yes—
44C2×C22⋊C4×C31C2×C414yes—
45C2×C22⋊C4×C31C2×C412yes—
46C2×C22⋊C4×C31C2×C414yes—
47C2×C22⋊C4×C31D412yes—
48C2×C22⋊C4×C31D414yes—
49C2×C22⋊C4×C31Q814yes—
50C2×C42×C31—24no—
51C31×#95—24no—
52C31×#95—24no—
53C24×C31—82no—
54C24×C31C22×C411yescharacteristic
55C24×C31C22×C412yes—
56C24×C31C2×D411yescharacteristic
57C24×C31C2×D411yescharacteristic
58C24×C31C2×D412yes—
59C24×C31C2×Q812yes—
60C24×C31#4112yes—
61C24×C31#4112yes—
62C24×C31#4112yes—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31—24no—
66C22×C4×C31—24no—
67C22×C4×C31—24no—
68C22×C4×C31—24no—
69C22×C4×C31—22no—
70C22×C4×C31—22no—
71C22×C4×C31—24no—
72C22×C4×C31—24no—
73C22×C4×C31—42no—
74C22×C4×C31—42no—
75C22×C4×C31—82no—
76C22×C4×C31—84no—
77C22×C4×C31—82no—
78C22×C4×C31—82no—
79C22×C4×C31—84no—
80C22×C4×C31C22⋊C414yes—
81C4⋊C4×C31—84no—
82C22⋊C4×C31—22no—
83C22⋊C4×C31—24no—
84C22⋊C4×C31—24no—
85C22⋊C4×C31—24no—
86C22⋊C4×C31—24no—
87C22⋊C4×C31—24no—
88C22⋊C4×C31—24no—
89C22⋊C4×C31—24no—
90C22⋊C4×C31—24no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—41no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31C4214yescomplemented
97C22⋊C4×C31C4⋊C418yescomplemented
98C42×C31—84no—
99#10915C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—44no—
104C23×C31—44no—
105C23×C31—82no—
106C23×C31—82no—
107C23×C31—82no—
108C23×C31—82no—
109C23×C31C2×C4211yescharacteristic
110C23×C31C2×C22⋊C411yescharacteristic
111C23×C31C2×C4⋊C412yes—
112C23×C31#11711yescharacteristic
113C23×C31#11712yes—
114C23×C31C4×D412yes—
115C23×C31C4×D412yes—
116C23×C31C4×D412yes—
117C23×C31C4×D412yes—
118C23×C31C4×Q812yes—
119C23×C31C4×Q812yes—
120C23×C31#12011yescharacteristic
121C23×C31#12112yes—
122C23×C31C22⋊Q812yes—
123C23×C31C22⋊Q812yes—
124C23×C31#12311yescharacteristic
125C23×C31#12312yes—
126C23×C31#12411yescharacteristic
127C23×C31#12512yes—
128C23×C31#12612yes—
129C23×C31C4⋊Q811yescharacteristic
130C2×C4×C31—24no—
131C2×C4×C31—28no—
132C2×C4×C31—24no—
133C2×C4×C31—48no—
134C2×C4×C31—44no—
135C2×C4×C31—44no—
136C2×C4×C31—44no—
137C2×C4×C31—44no—
138C2×C4×C31—42no—
139C2×C4×C31—44no—
140C2×C4×C31—82no—
141C2×C4×C31—84no—
142C2×C4×C31—84no—
143C2×C4×C31—84no—
144C2×C4×C31—84no—
145C2×C4×C31—84no—
146C2×C4×C31—84no—
147C2×C4×C31—84no—
148#2315C2×C3111yescharacteristic
149#2273C2×C3111yescharacteristic
150C2×#342C2×C3112yes—
151C2×#342C2×C3111yescharacteristic
152C2×#342C2×C3112yes—
153C22×C31—22no—
154C22×C31—22no—
155C22×C31—24no—
156C22×C31—24no—
157C22×C31—22no—
158C22×C31—42no—
159C22×C31—42no—
160C22×C31—42no—
161C22×C31—42no—
162C22×C31—42no—
163C22×C31—41no—
164C22×C31—42no—
165C22×C31—41no—
166C22×C31—42no—
167C22×C31—42no—
168C22×C31—42no—
169C22×C31—42no—
170C22×C31—42no—
171C22×C31—41no—
172C22×C31—42no—
173C22×C31—41no—
174C22×C31C4×C22⋊C411yescharacteristic
175C22×C31C4×C4⋊C412yes—
176C22×C31#38512yes—
177C22×C31#38611yescharacteristic
178C22×C31#38712yes—
179C22×C31#38712yes—
180C22×C31#38812yes—
181C22×C31#38912yes—
182C22×C31#39111yescharacteristic
183C4×C31—44no—
184C4×C31—48no—
185C4×C31—44no—
186C4×C31—84no—
187C4×C31—82no—
188C22×C22⋊C4C22×C3111yescharacteristic
189C22×C22⋊C4C22×C3112yes—
190C22×C22⋊C4C22×C3111yescharacteristic
191C22×C22⋊C4C22×C3111yescharacteristic
192C22×C22⋊C4C22×C3112yes—
193#385—22no—
194#385C4×C3114yescomplemented
195#380—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—24no—
198#342—22no—
199#342—24no—
200#342C4×C3118yescomplemented
201C2×C31—21no—
202C2×C31—22no—
203C2×C31—22no—
204C2×C31—22no—
205C2×C31—22no—
206C2×C31—22no—
207C2×C31—22no—
208C2×C31—21no—
209C2×C31#143611yescharacteristic
210C2×C31#191312yes—
211C25C23×C3111yescharacteristic
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4C2×C4×C3112yes—
216C23×C4D4×C3112yes—
217C2×C4⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4C2×C4×C3114yes—
235C2×C22⋊C4C2×C4×C3112yes—
236C2×C22⋊C4C2×C4×C3114yes—
237C2×C22⋊C4C2×C4×C3112yes—
238C2×C22⋊C4C2×C4×C3114yes—
239C2×C22⋊C4D4×C3112yes—
240C2×C22⋊C4D4×C3114yes—
241C2×C22⋊C4Q8×C3114yes—
242C2×C42—24no—
243#95—24no—
244#95—24no—
245C31#1091511yescharacteristic, direct product factor, Sylow
246C24—82no—
247C24C22×C4×C3111yescharacteristic
248C24C22×C4×C3112yes—
249C24C2×D4×C3111yescharacteristic
250C24C2×D4×C3111yescharacteristic
251C24C2×D4×C3112yes—
252C24C2×Q8×C3112yes—
253C24#41×C3112yes—
254C24#41×C3112yes—
255C24#41×C3112yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—82no—
269C22×C4—84no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4C22⋊C4×C3114yes—
274C4⋊C4—84no—
275C22⋊C4—22no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4C42×C3114yescomplemented
290C22⋊C4C4⋊C4×C3118yescomplemented
291C42—84no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—44no—
296C23—44no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23C2×C42×C3111yescharacteristic
302C23C2×C22⋊C4×C3111yescharacteristic
303C23C2×C4⋊C4×C3112yes—
304C23C31×#11711yescharacteristic
305C23C31×#11712yes—
306C23C4×D4×C3112yes—
307C23C4×D4×C3112yes—
308C23C4×D4×C3112yes—
309C23C4×D4×C3112yes—
310C23C4×Q8×C3112yes—
311C23C4×Q8×C3112yes—
312C23C31×#12011yescharacteristic
313C23C31×#12112yes—
314C23C31×C22⋊Q812yes—
315C23C31×C22⋊Q812yes—
316C23C31×#12311yescharacteristic
317C23C31×#12312yes—
318C23C31×#12411yescharacteristic
319C23C31×#12512yes—
320C23C31×#12612yes—
321C23C31×C4⋊Q811yescharacteristic
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C22—22no—
341C22—22no—
342C22—24no—
343C22—24no—
344C22—22no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22C4×C22⋊C4×C3111yescharacteristic
362C22C4×C4⋊C4×C3112yes—
363C22C31×#38512yes—
364C22C31×#38611yescharacteristic
365C22C31×#38712yes—
366C22C31×#38712yes—
367C22C31×#38812yes—
368C22C31×#38912yes—
369C22C31×#39111yescharacteristic
370C4—44no—
371C4—48no—
372C4—44no—
373C4—84no—
374C4—82no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—22no—
381C2—22no—
382C2—21no—
383C2C31×#143611yescharacteristic
384C2C31×#191312yes—
subgroupG/H
center, socleC22×C31#386
commutatorC23C2×C42×C31
Frattini, cosocleC25C23×C31
prime radicalC25×C31C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
12. C31×#385C4semidirect
18. C31×#342C4semidirect
96. C22⋊C4×C31C42semidirect
97. C22⋊C4×C31C4⋊C4semidirect
99. #10915C31direct (N × H)
194. #385C4×C31semidirect
200. #342C4×C31semidirect
245. C31#10915direct (N × H)
289. C22⋊C4C42×C31semidirect
290. C22⋊C4C4⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#23152
C31×#22732
C2×C31×#3422
#1091531