Finite Group Databasebeta

← C31×#10271 #21467030

Subgroups — C31×#10271 #21467030

Order 7936.

594 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×#498C211yesmaximal, characteristic, complemented
2C2×C31×#476C211yesmaximal, characteristic
3C2×C31×#461C211yesmaximal, characteristic
4C31×#1954C211yesmaximal, characteristic
5C2×C31×#333C211yesmaximal, characteristic, complemented
6C2×C31×#332C211yesmaximal, characteristic, complemented
7C31×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C31—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#498—22no—
13C31×#498—22no—
14C31×#476—22no—
15C31×#476—22no—
16C31×#461—22no—
17C31×#461—22no—
18C2×C31×#106—22no—
19C2×C31×#106—21no—
20C2×C31×C4⋊C8C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×C4⋊C8C2211yescharacteristic
23C2×C31×#103—21no—
24C2×C31×#102—21no—
25C2×C31×C22⋊C8—22no—
26C2×C4×C8×C31C2211yescharacteristic
27C31×#333—22no—
28C31×#332—22no—
29C22×Q8×C31—21no—
30C22×Q8×C31—21no—
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C2×SD16×C31—24no—
34C2×SD16×C31—42no—
35C2×SD16×C31—42no—
36C22×C8×C31—21no—
37C22×C8×C31—21no—
38C22×C8×C31—21no—
39C22×C8×C31D412yes—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8D412yes—
47C31×C4⋊Q8D412yes—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—21no—
58C2×C42×C31C2311yescharacteristic
59C31×#106—24no—
60C31×#106—28no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8D412yes—
66C31×C4⋊C8D412yes—
67C31×#103—24no—
68C31×#102—24no—
69C31×C22⋊C8—44no—
70C4×C8×C31—21no—
71C4×C8×C31—21no—
72C24×C31—21no—
73C2×Q8×C31—22no—
74C2×Q8×C31—22no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×Q8×C31—82no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31—21no—
87C2×D4×C31SD1612yes—
88C22×C4×C31—21no—
89C22×C4×C31—41no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—41no—
93C22×C4×C31—42no—
94C22×C4×C31—42no—
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31#4111yescharacteristic
98SD16×C31—48no—
99C2×C8×C31—22no—
100C2×C8×C31—22no—
101C2×C8×C31—22no—
102C2×C8×C31—22no—
103C2×C8×C31—41no—
104C2×C8×C31—41no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—42no—
109C2×C8×C31SD1614yescomplemented
110C4⋊C4×C31—22no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—22no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C42×C31C2×D411yescharacteristic
123C42×C31C2×D411yescharacteristic
124C42×C31C2×D411yescharacteristic
125C42×C31C2×D411yescharacteristic
126#10271C3111yesmaximal, characteristic, direct product factor, Sylow
127C23×C31—22no—
128C23×C31—41no—
129C23×C31—42no—
130C23×C31—41no—
131C23×C31—42no—
132C23×C31#12111yescharacteristic
133Q8×C31—42no—
134Q8×C31—42no—
135Q8×C31—82no—
136Q8×C31—82no—
137Q8×C31—81no—
138Q8×C31—81no—
139D4×C31—22no—
140D4×C31—22no—
141D4×C31—41no—
142D4×C31—41no—
143C2×C4×C31—22no—
144C2×C4×C31—41no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—81no—
149C2×C4×C31—81no—
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#12011yescharacteristic
157C2×C4×C31#12111yescharacteristic
158C2×C4×C31#12111yescharacteristic
159C2×C4×C31#12711yescharacteristic
160C2×C4×C31C2×SD1611yescharacteristic
161C2×C4×C31C2×SD1612yes—
162C2×C4×C31#13511yescharacteristic
163C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C31#13711yescharacteristic
165C2×C4×C31#13712yes—
166C8×C31—24no—
167C8×C31—42no—
168C8×C31—42no—
169C8×C31—81no—
170C2×#498C2×C3111yescharacteristic, complemented
171C2×#476C2×C3111yescharacteristic
172C2×#461C2×C3111yescharacteristic
173#1954C2×C3111yescharacteristic
174C2×#333C2×C3111yescharacteristic, complemented
175C2×#332C2×C3111yescharacteristic, complemented
176#1277C2×C3111yescharacteristic, complemented
177C22×C31—44no—
178C22×C31—42no—
179C22×C31—82no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31#39211yescharacteristic
183C22×C31#46112yes—
184C22×C31#46511yescharacteristic
185C22×C31#46811yescharacteristic
186C22×C31#47112yes—
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—82no—
190C4×C31—81no—
191C4×C31—82no—
192C4×C31#44811yescharacteristic
193C4×C31#44812yes—
194C4×C31#44911yescharacteristic
195C4×C31#45011yescharacteristic
196C4×C31#45211yescharacteristic
197C4×C31#49211yescharacteristic
198C4×C31#49711yescharacteristic
199C22×SD16—21no—
200C2×C4⋊Q8C22×C3111yescharacteristic
201C2×C4⋊Q8C22×C3111yescharacteristic
202C2×C4×D4C22×C3111yescharacteristic
203#498—22no—
204#498—22no—
205#476—22no—
206#476—22no—
207#461—22no—
208#461—22no—
209C2×#106—22no—
210C2×#106—21no—
211C2×C4⋊C8C22×C3111yescharacteristic
212C2×C4⋊C8C22×C3111yescharacteristic
213C2×C4⋊C8C22×C3111yescharacteristic
214C2×#103—21no—
215C2×#102—21no—
216C2×C22⋊C8—22no—
217C2×C4×C8C22×C3111yescharacteristic
218#333—22no—
219#332—22no—
220C2×C31—82no—
221C2×C31#167112yes—
222C2×C31#169112yes—
223C2×C31#200011yescharacteristic
224C2×C31#201311yescharacteristic
225C2×C31#201911yescharacteristic
226C22×Q8—21no—
227C22×Q8—21no—
228C22×D4D4×C3111yescharacteristic
229C23×C4—21no—
230C2×SD16—24no—
231C2×SD16—42no—
232C2×SD16—42no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8D4×C3112yes—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8D4×C3112yes—
244C4⋊Q8D4×C3112yes—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4D4×C3111yescharacteristic
254C2×C22⋊C4—21no—
255C2×C42C23×C3111yescharacteristic
256#106—24no—
257#106—28no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8D4×C3112yes—
263C4⋊C8D4×C3112yes—
264#103—24no—
265#102—24no—
266C22⋊C8—44no—
267C4×C8—21no—
268C4×C8—21no—
269C31#1027111yescharacteristic, direct product factor, Sylow
270C24—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4SD16×C3112yes—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C2×D4×C3111yescharacteristic
294C22×C4C2×D4×C3111yescharacteristic
295C22×C4#41×C3111yescharacteristic
296SD16—48no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8SD16×C3114yescomplemented
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42C2×D4×C3111yescharacteristic
321C42C2×D4×C3111yescharacteristic
322C42C2×D4×C3111yescharacteristic
323C42C2×D4×C3111yescharacteristic
324C23—22no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—41no—
329C23C31×#12111yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336D4—22no—
337D4—22no—
338D4—41no—
339D4—41no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C31×#12011yescharacteristic
352C2×C4C31×#12011yescharacteristic
353C2×C4C31×#12011yescharacteristic
354C2×C4C31×#12111yescharacteristic
355C2×C4C31×#12111yescharacteristic
356C2×C4C31×#12711yescharacteristic
357C2×C4C2×SD16×C3111yescharacteristic
358C2×C4C2×SD16×C3112yes—
359C2×C4C31×#13511yescharacteristic
360C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
361C2×C4C31×#13711yescharacteristic
362C2×C4C31×#13712yes—
363C8—24no—
364C8—42no—
365C8—42no—
366C8—81no—
367C22—44no—
368C22—42no—
369C22—82no—
370C22—81no—
371C22—81no—
372C22C31×#39211yescharacteristic
373C22C31×#46112yes—
374C22C31×#46511yescharacteristic
375C22C31×#46811yescharacteristic
376C22C31×#47112yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4C31×#44811yescharacteristic
383C4C31×#44812yes—
384C4C31×#44911yescharacteristic
385C4C31×#45011yescharacteristic
386C4C31×#45211yescharacteristic
387C4C31×#49211yescharacteristic
388C4C31×#49711yescharacteristic
389C2—82no—
390C2C31×#167112yes—
391C2C31×#169112yes—
392C2C31×#200011yescharacteristic
393C2C31×#201311yescharacteristic
394C2C31×#201911yescharacteristic
subgroupG/H
center, socleC23×C31#121
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×#498C2semidirect
5. C2×C31×#333C2semidirect
6. C2×C31×#332C2semidirect
7. C31×#1277C2semidirect
109. C2×C8×C31SD16semidirect
126. #10271C31direct (N × H)
170. C2×#498C2×C31semidirect
174. C2×#333C2×C31semidirect
175. C2×#332C2×C31semidirect
176. #1277C2×C31semidirect
269. C31#10271direct (N × H)
307. C2×C8SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4982
C2×C31×#4762
C2×C31×#4612
C31×#19542
C2×C31×#3332
C2×C31×#3322
C31×#12772
#1027131