Finite Group Databasebeta

← C31×#18327 #21475086

Subgroups — C31×#18327 #21475086

Order 7936.

950 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#544C211yesmaximal, characteristic
2C31×#3195C212yesmaximal
3C31×#2583C211yesmaximal, characteristic
4C31×#1611C214yesmaximal, complemented
5C31×#1605C214yesmaximal, complemented
6C31×#1508C212yesmaximal
7C31×#1463C211yesmaximal, characteristic
8C31×#544—24no—
9C31×#544—22no—
10C31×#544C2214yes—
11C2×C31×C4⋊Q8—22no—
12C2×C31×#125—22no—
13C2×C31×C22⋊Q8—21no—
14C2×C31×C22⋊Q8C2212yes—
15C2×C31×C22⋊Q8C2211yescharacteristic
16C2×C31×#117—21no—
17C22×C4⋊C4×C31C2211yescharacteristic
18C31×#483—22no—
19C31×#483C2214yes—
20C31×#470—24no—
21C31×#451—22no—
22C31×#451C2214yes—
23C31×#407C2212yes—
24C31×#398C2212yes—
25C31×#395C2212yes—
26C31×#389—22no—
27C31×#385—21no—
28C31×#385—21no—
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#328C2218yes—
31C31×#324C2214yes—
32C22×Q8×C31—21no—
33C22×Q8×C31D412yes—
34C22×Q8×C31C2311yescharacteristic
35C23×C4×C31D411yescharacteristic
36C23×C4×C31D411yescharacteristic
37C23×C4×C31C2311yescharacteristic
38C2×Q16×C31—24no—
39C2×M16×C31—22no—
40C31×C4⋊Q8—28no—
41C31×C4⋊Q8—28no—
42C31×#125—28no—
43C31×#125—28no—
44C31×C22⋊Q8—24no—
45C31×C22⋊Q8—24no—
46C31×C22⋊Q8—22no—
47C31×C22⋊Q8—22no—
48C31×C22⋊Q8—22no—
49C31×C22⋊Q8—22no—
50C31×C22⋊Q8D414yes—
51C31×C22⋊Q8C2314yes—
52C31×#117—24no—
53C31×#117—24no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31D412yes—
65C2×C4⋊C4×C31D412yes—
66C2×C4⋊C4×C31C2311yescharacteristic
67C2×C4⋊C4×C31C2312yes—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C42×C31—22no—
72C2×C42×C31—21no—
73C31×#106—24no—
74C31×#103—24no—
75C31×#103—24no—
76C31×C22⋊C8—22no—
77C31×C22⋊C8C2314yes—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95C2312yes—
81C24×C31C2×D411yescharacteristic
82C2×Q8×C31—24no—
83C2×Q8×C31—22no—
84C2×Q8×C31—22no—
85C2×Q8×C31—81no—
86C2×Q8×C31—82no—
87C2×Q8×C31—82no—
88C2×Q8×C31C2×D414yes—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—24no—
92C22×C4×C31—22no—
93C22×C4×C31—22no—
94C22×C4×C31—21no—
95C22×C4×C31—22no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31—81no—
100C22×C4×C31—81no—
101C22×C4×C31C2×D412yes—
102C22×C4×C31C2×D412yes—
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D412yes—
105C22×C4×C31C2×D412yes—
106C22×C4×C31C2411yescharacteristic
107Q16×C31—84no—
108M16×C31—82no—
109C2×C8×C31—24no—
110C4⋊C4×C31—24no—
111C4⋊C4×C31—24no—
112C4⋊C4×C31—24no—
113C4⋊C4×C31—28no—
114C4⋊C4×C31—24no—
115C4⋊C4×C31—24no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—42no—
121C4⋊C4×C31—44no—
122C4⋊C4×C31—82no—
123C4⋊C4×C31C2×D414yes—
124C22⋊C4×C31—24no—
125C22⋊C4×C31—24no—
126C22⋊C4×C31—24no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C42×C31—28no—
130C42×C31—81no—
131#18327C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—22no—
133C23×C31—22no—
134C23×C31—41no—
135C23×C31#12011yescharacteristic
136C23×C31#12011yescharacteristic
137C23×C31C22×D411yescharacteristic
138Q8×C31—44no—
139Q8×C31—44no—
140Q8×C31—82no—
141Q8×C31—84no—
142C2×C4×C31—24no—
143C2×C4×C31—24no—
144C2×C4×C31—22no—
145C2×C4×C31—22no—
146C2×C4×C31—24no—
147C2×C4×C31—24no—
148C2×C4×C31—24no—
149C2×C4×C31—22no—
150C2×C4×C31—41no—
151C2×C4×C31—42no—
152C2×C4×C31—42no—
153C2×C4×C31—41no—
154C2×C4×C31—42no—
155C2×C4×C31—42no—
156C2×C4×C31—44no—
157C2×C4×C31—42no—
158C2×C4×C31—44no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—81no—
165C2×C4×C31#12012yes—
166C2×C4×C31#13714yes—
167C2×C4×C31C22×D412yes—
168C8×C31—44no—
169C2×#544C2×C3111yescharacteristic
170#3195C2×C3112yes—
171#2583C2×C3111yescharacteristic
172#1611C2×C3114yescomplemented
173#1605C2×C3114yescomplemented
174#1508C2×C3112yes—
175#1463C2×C3111yescharacteristic
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—44no—
184C22×C31#45612yes—
185C22×C31C2×#12011yescharacteristic
186C22×C31C2×#13712yes—
187C4×C31—24no—
188C4×C31—22no—
189C4×C31—22no—
190C4×C31—44no—
191C4×C31—44no—
192C4×C31—42no—
193C4×C31—44no—
194C4×C31—81no—
195C4×C31—81no—
196#544—24no—
197#544—22no—
198#544C22×C3114yes—
199C2×C4⋊Q8—22no—
200C2×#125—22no—
201C2×C22⋊Q8—21no—
202C2×C22⋊Q8C22×C3112yes—
203C2×C22⋊Q8C22×C3111yescharacteristic
204C2×#117—21no—
205C22×C4⋊C4C22×C3111yescharacteristic
206#483—22no—
207#483C22×C3114yes—
208#470—24no—
209#451—22no—
210#451C22×C3114yes—
211#407C22×C3112yes—
212#398C22×C3112yes—
213#395C22×C3112yes—
214#389—22no—
215#385—21no—
216#385—21no—
217C4×C22⋊C4C22×C3111yescharacteristic
218#328C22×C3118yes—
219#324C22×C3114yes—
220C2×C31—22no—
221C2×C31—22no—
222C2×C31—41no—
223C2×C31#300411yescharacteristic
224C2×C31C2×#45611yescharacteristic
225C2×C31#302611yescharacteristic
226C22×Q8—21no—
227C22×Q8D4×C3112yes—
228C22×Q8C23×C3111yescharacteristic
229C23×C4D4×C3111yescharacteristic
230C23×C4D4×C3111yescharacteristic
231C23×C4C23×C3111yescharacteristic
232C2×Q16—24no—
233C2×M16—22no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236#125—28no—
237#125—28no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8D4×C3114yes—
245C22⋊Q8C23×C3114yes—
246#117—24no—
247#117—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4D4×C3112yes—
259C2×C4⋊C4D4×C3112yes—
260C2×C4⋊C4C23×C3111yescharacteristic
261C2×C4⋊C4C23×C3112yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C42—22no—
266C2×C42—21no—
267#106—24no—
268#103—24no—
269#103—24no—
270C22⋊C8—22no—
271C22⋊C8C23×C3114yes—
272#95—22no—
273#95—22no—
274#95C23×C3112yes—
275C31#1832711yescharacteristic, direct product factor, Sylow
276C24C2×D4×C3111yescharacteristic
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—81no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8C2×D4×C3114yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C2×D4×C3112yes—
297C22×C4C2×D4×C3112yes—
298C22×C4C2×D4×C3111yescharacteristic
299C22×C4C2×D4×C3112yes—
300C22×C4C2×D4×C3112yes—
301C22×C4C24×C3111yescharacteristic
302Q16—84no—
303M16—82no—
304C2×C8—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—82no—
318C4⋊C4C2×D4×C3114yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—28no—
325C42—81no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23C31×#12011yescharacteristic
330C23C31×#12011yescharacteristic
331C23C22×D4×C3111yescharacteristic
332Q8—44no—
333Q8—44no—
334Q8—82no—
335Q8—84no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4C31×#12012yes—
360C2×C4C31×#13714yes—
361C2×C4C22×D4×C3112yes—
362C8—44no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22C31×#45612yes—
372C22C2×C31×#12011yescharacteristic
373C22C2×C31×#13712yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—44no—
379C4—42no—
380C4—44no—
381C4—81no—
382C4—81no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2C31×#300411yescharacteristic
387C2C2×C31×#45611yescharacteristic
388C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1611C2semidirect
5. C31×#1605C2semidirect
131. #18327C31direct (N × H)
172. #1611C2×C31semidirect
173. #1605C2×C31semidirect
275. C31#18327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5442
C31×#31952
C31×#25832
C31×#16112
C31×#16052
C31×#15082
C31×#14632
#1832731