Finite Group Databasebeta

← C31×#10060 #21466819

Subgroups — C31×#10060 #21466819

Order 7936.

638 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#487C211yesmaximal, characteristic, complemented
2C2×C31×#476C211yesmaximal, characteristic
3C31×#2542C211yesmaximal, characteristic
4C31×#1987C211yesmaximal, characteristic, complemented
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C2×C31×D4⋊C8C211yesmaximal, characteristic
7C31×#1279C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×C22⋊Q8—22no—
11C2×C4×D4×C31C2211yescharacteristic
12C31×#487—22no—
13C31×#487—22no—
14C31×#476—22no—
15C31×#476—22no—
16C2×C31×#106—21no—
17C2×C31×C4⋊C8C2211yescharacteristic
18C2×C31×C4⋊C8C2211yescharacteristic
19C2×C31×#103—22no—
20C2×C31×#102—21no—
21C2×C4×C8×C31C2211yescharacteristic
22C31×#399—22no—
23C31×#390—21no—
24C31×#384—21no—
25C4×C4⋊C4×C31C2211yescharacteristic
26C31×#340—22no—
27C31×#332—22no—
28C31×D4⋊C8—22no—
29C22×Q8×C31—21no—
30C22×Q8×C31—21no—
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C22×C8×C31—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8D412yes—
44C31×C22⋊Q8—44no—
45C31×C22⋊Q8—44no—
46C4×D4×C31—21no—
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31D412yes—
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D412yes—
57C2×C22⋊C4×C31—24no—
58C2×C22⋊C4×C31—21no—
59C2×C42×C31—21no—
60C2×C42×C31C2311yescharacteristic
61C31×#106—24no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8D412yes—
66C31×#103—28no—
67C31×#102—24no—
68C4×C8×C31—21no—
69C4×C8×C31D412yes—
70C31×#95—21no—
71C24×C31—21no—
72C2×Q8×C31—22no—
73C2×Q8×C31—44no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—42no—
78C2×Q8×C31—42no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31Q1612yes—
85C22×C4×C31—22no—
86C22×C4×C31—22no—
87C22×C4×C31—21no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31—41no—
91C22×C4×C31—42no—
92C22×C4×C31—42no—
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96C2×C8×C31—22no—
97C2×C8×C31—22no—
98C2×C8×C31—42no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—41no—
102C2×C8×C31—42no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—48no—
111C4⋊C4×C31—41no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31SD1614yes—
116C22⋊C4×C31—48no—
117C22⋊C4×C31—41no—
118C22⋊C4×C31—41no—
119C42×C31—41no—
120C42×C31—41no—
121C42×C31C2×D411yescharacteristic
122C42×C31C2×D411yescharacteristic
123C42×C31C2×D411yescharacteristic
124C42×C31#4111yescharacteristic
125#10060C3111yesmaximal, characteristic, direct product factor, Sylow
126C23×C31—22no—
127C23×C31—41no—
128C23×C31—42no—
129C23×C31—41no—
130C23×C31—42no—
131C23×C31#12011yescharacteristic
132Q8×C31—44no—
133Q8×C31—82no—
134Q8×C31—82no—
135Q8×C31—81no—
136Q8×C31—81no—
137D4×C31—22no—
138D4×C31—22no—
139D4×C31—41no—
140D4×C31—41no—
141C2×C4×C31—22no—
142C2×C4×C31—24no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—41no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—44no—
149C2×C4×C31—42no—
150C2×C4×C31—82no—
151C2×C4×C31—82no—
152C2×C4×C31—81no—
153C2×C4×C31—81no—
154C2×C4×C31#12011yescharacteristic
155C2×C4×C31#12011yescharacteristic
156C2×C4×C31#12111yescharacteristic
157C2×C4×C31#12111yescharacteristic
158C2×C4×C31#12111yescharacteristic
159C2×C4×C31#12711yescharacteristic
160C2×C4×C31C2×SD1612yes—
161C2×C4×C31C2×Q1611yescharacteristic
162C2×C4×C31#13511yescharacteristic
163C2×C4×C31AGL(1,ℤ/8)12yes—
164C2×C4×C31#13711yescharacteristic
165C2×C4×C31#13711yescharacteristic
166C8×C31—42no—
167C8×C31—42no—
168C8×C31—81no—
169C2×#487C2×C3111yescharacteristic, complemented
170C2×#476C2×C3111yescharacteristic
171#2542C2×C3111yescharacteristic
172#1987C2×C3111yescharacteristic, complemented
173C2×#332C2×C3111yescharacteristic, complemented
174C2×D4⋊C8C2×C3111yescharacteristic
175#1279C2×C3111yescharacteristic, complemented
176C22×C31—44no—
177C22×C31—42no—
178C22×C31—82no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31#39211yescharacteristic
182C22×C31#45012yes—
183C22×C31#45111yescharacteristic
184C22×C31#45211yescharacteristic
185C22×C31#45412yes—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—44no—
189C4×C31—82no—
190C4×C31—81no—
191C4×C31—82no—
192C4×C31#45111yescharacteristic
193C4×C31#45211yescharacteristic
194C4×C31#46012yes—
195C4×C31#46711yescharacteristic
196C4×C31#47011yescharacteristic
197C4×C31#49211yescharacteristic
198C4×C31#49611yescharacteristic
199C2×C4⋊Q8C22×C3111yescharacteristic
200C2×C4⋊Q8C22×C3111yescharacteristic
201C2×C22⋊Q8—22no—
202C2×C4×D4C22×C3111yescharacteristic
203#487—22no—
204#487—22no—
205#476—22no—
206#476—22no—
207C2×#106—21no—
208C2×C4⋊C8C22×C3111yescharacteristic
209C2×C4⋊C8C22×C3111yescharacteristic
210C2×#103—22no—
211C2×#102—21no—
212C2×C4×C8C22×C3111yescharacteristic
213#399—22no—
214#390—21no—
215#384—21no—
216C4×C4⋊C4C22×C3111yescharacteristic
217#340—22no—
218#332—22no—
219D4⋊C8—22no—
220C2×C31—82no—
221C2×C31#163512yes—
222C2×C31#163712yes—
223C2×C31#200111yescharacteristic
224C2×C31#200311yescharacteristic
225C2×C31#201411yescharacteristic
226C22×Q8—21no—
227C22×Q8—21no—
228C22×D4D4×C3111yescharacteristic
229C23×C4—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8D4×C3112yes—
241C22⋊Q8—44no—
242C22⋊Q8—44no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4D4×C3111yescharacteristic
252C2×C4⋊C4D4×C3112yes—
253C2×C4⋊C4D4×C3112yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42C23×C3111yescharacteristic
258#106—24no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8D4×C3112yes—
263#103—28no—
264#102—24no—
265C4×C8—21no—
266C4×C8D4×C3112yes—
267#95—21no—
268C31#1006011yescharacteristic, direct product factor, Sylow
269C24—21no—
270C2×Q8—22no—
271C2×Q8—44no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4Q16×C3112yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C2×D4×C3111yescharacteristic
292C22×C4C2×D4×C3111yescharacteristic
293C22×C4C2×D4×C3111yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—48no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4SD16×C3114yes—
314C22⋊C4—48no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—41no—
318C42—41no—
319C42C2×D4×C3111yescharacteristic
320C42C2×D4×C3111yescharacteristic
321C42C2×D4×C3111yescharacteristic
322C42#41×C3111yescharacteristic
323C23—22no—
324C23—42no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23C31×#12011yescharacteristic
329Q8—44no—
330Q8—82no—
331Q8—82no—
332Q8—81no—
333Q8—81no—
334D4—22no—
335D4—22no—
336D4—41no—
337D4—41no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C31×#12011yescharacteristic
352C2×C4C31×#12011yescharacteristic
353C2×C4C31×#12111yescharacteristic
354C2×C4C31×#12111yescharacteristic
355C2×C4C31×#12111yescharacteristic
356C2×C4C31×#12711yescharacteristic
357C2×C4C2×SD16×C3112yes—
358C2×C4C2×Q16×C3111yescharacteristic
359C2×C4C31×#13511yescharacteristic
360C2×C4C31×AGL(1,ℤ/8)12yes—
361C2×C4C31×#13711yescharacteristic
362C2×C4C31×#13711yescharacteristic
363C8—42no—
364C8—42no—
365C8—81no—
366C22—44no—
367C22—42no—
368C22—82no—
369C22—81no—
370C22—81no—
371C22C31×#39211yescharacteristic
372C22C31×#45012yes—
373C22C31×#45111yescharacteristic
374C22C31×#45211yescharacteristic
375C22C31×#45412yes—
376C4—21no—
377C4—21no—
378C4—44no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4C31×#45111yescharacteristic
383C4C31×#45211yescharacteristic
384C4C31×#46012yes—
385C4C31×#46711yescharacteristic
386C4C31×#47011yescharacteristic
387C4C31×#49211yescharacteristic
388C4C31×#49611yescharacteristic
389C2—82no—
390C2C31×#163512yes—
391C2C31×#163712yes—
392C2C31×#200111yescharacteristic
393C2C31×#200311yescharacteristic
394C2C31×#201411yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#487C2semidirect
4. C31×#1987C2semidirect
5. C2×C31×#332C2semidirect
7. C31×#1279C2semidirect
125. #10060C31direct (N × H)
169. C2×#487C2×C31semidirect
172. #1987C2×C31semidirect
173. C2×#332C2×C31semidirect
175. #1279C2×C31semidirect
268. C31#10060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4872
C2×C31×#4762
C31×#25422
C31×#19872
C2×C31×#3322
C2×C31×D4⋊C82
C31×#12792
#1006031