Finite Group Databasebeta

← C31×#10539 #21467298

Subgroups — C31×#10539 #21467298

Order 7936.

490 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#486C211yesmaximal, characteristic
2C31×#1987C211yesmaximal, characteristic, complemented
3C31×#1982C211yesmaximal, characteristic
4C31×#1939C211yesmaximal, characteristic, complemented
5C2×C31×#333C211yesmaximal, characteristic, complemented
6C2×C31×#331C211yesmaximal, characteristic
7C31×#1277C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#127C2211yescharacteristic
10C31×#486—22no—
11C31×#486—22no—
12C2×C31×C4⋊C8—21no—
13C2×C31×C4⋊C8C2211yescharacteristic
14C2×C31×C4⋊C8C2211yescharacteristic
15C2×C31×C4⋊C8C2211yescharacteristic
16C2×C31×#102—21no—
17C2×C31×#102—21no—
18C2×C31×#102—21no—
19C2×C31×#102—21no—
20C2×C4×C8×C31C2211yescharacteristic
21C4×C4⋊C4×C31C2211yescharacteristic
22C31×#340—21no—
23C31×#340—21no—
24C31×#336—21no—
25C31×#333—22no—
26C31×#331—22no—
27C22×Q8×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×C8×C31—21no—
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31D411yescharacteristic
35C22×C8×C31D411yescharacteristic
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8D412yes—
40C31×#127—21no—
41C31×#127—21no—
42C31×#127—21no—
43C31×#127D412yes—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31D411yescharacteristic
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C42×C31—21no—
49C2×C42×C31C2311yescharacteristic
50C31×C4⋊C8—21no—
51C31×C4⋊C8—21no—
52C31×C4⋊C8—21no—
53C31×C4⋊C8—21no—
54C31×C4⋊C8—21no—
55C31×C4⋊C8—21no—
56C31×C4⋊C8—42no—
57C31×#102—24no—
58C31×#102—24no—
59C31×#102—42no—
60C31×#102—42no—
61C4×C8×C31—21no—
62C4×C8×C31—21no—
63C31×#95—21no—
64C24×C31—41no—
65C2×Q8×C31—22no—
66C2×Q8×C31—41no—
67C2×Q8×C31—81no—
68C2×Q8×C31—81no—
69C2×D4×C31—22no—
70C2×D4×C31—22no—
71C2×D4×C31—22no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—81no—
84C2×D4×C31—81no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—41no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31#4111yescharacteristic
91C2×C8×C31—21no—
92C2×C8×C31—21no—
93C2×C8×C31—21no—
94C2×C8×C31—21no—
95C2×C8×C31—21no—
96C2×C8×C31—21no—
97C2×C8×C31—41no—
98C2×C8×C31—41no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—41no—
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C2×C8×C31—41no—
105C2×C8×C31—41no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—22no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—41no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31SD1612yes—
114C42×C31—41no—
115C42×C31—41no—
116C42×C31C2×D411yescharacteristic
117C42×C31C2×D411yescharacteristic
118C42×C31#4111yescharacteristic
119C42×C31#4111yescharacteristic
120#10539C3111yesmaximal, characteristic, direct product factor, Sylow
121C23×C31—42no—
122C23×C31—81no—
123C23×C31—81no—
124C23×C31—81no—
125C23×C31—81no—
126C23×C31—81no—
127C23×C31—81no—
128C23×C31#12111yescharacteristic
129Q8×C31—82no—
130Q8×C31—82no—
131D4×C31—42no—
132D4×C31—42no—
133D4×C31—42no—
134D4×C31—42no—
135D4×C31—82no—
136D4×C31—82no—
137D4×C31—81no—
138D4×C31—81no—
139D4×C31—81no—
140D4×C31—81no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—22no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—42no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—81no—
151C2×C4×C31—81no—
152C2×C4×C31#12011yescharacteristic
153C2×C4×C31#12111yescharacteristic
154C2×C4×C31#12111yescharacteristic
155C2×C4×C31#12311yescharacteristic
156C2×C4×C31#12311yescharacteristic
157C2×C4×C31#12411yescharacteristic
158C2×C4×C31C2×SD1611yescharacteristic
159C2×C4×C31#13511yescharacteristic
160C2×C4×C31#13511yescharacteristic
161C2×C4×C31#13511yescharacteristic
162C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C31#13711yescharacteristic
165C2×C4×C31#13711yescharacteristic
166C8×C31—41no—
167C8×C31—41no—
168C8×C31—81no—
169C8×C31—81no—
170C8×C31—81no—
171C2×#486C2×C3111yescharacteristic
172#1987C2×C3111yescharacteristic, complemented
173#1982C2×C3111yescharacteristic
174#1939C2×C3111yescharacteristic, complemented
175C2×#333C2×C3111yescharacteristic, complemented
176C2×#331C2×C3111yescharacteristic
177#1277C2×C3111yescharacteristic, complemented
178C22×C31—82no—
179C22×C31—82no—
180C22×C31—82no—
181C22×C31—161no—
182C22×C31—161no—
183C22×C31—161no—
184C22×C31—161no—
185C22×C31#39411yescharacteristic
186C22×C31#46311yescharacteristic
187C22×C31#46411yescharacteristic
188C22×C31#46511yescharacteristic
189C22×C31#46811yescharacteristic
190C22×C31#47211yescharacteristic
191C22×C31#47211yescharacteristic
192C4×C31—21no—
193C4×C31—21no—
194C4×C31—42no—
195C4×C31—81no—
196C4×C31—82no—
197C4×C31#44911yescharacteristic
198C4×C31#45211yescharacteristic
199C4×C31#46011yescharacteristic
200C4×C31#46111yescharacteristic
201C4×C31#46311yescharacteristic
202C4×C31#46411yescharacteristic
203C4×C31#48811yescharacteristic
204C4×C31#48911yescharacteristic
205C2×C4⋊Q8C22×C3111yescharacteristic
206C2×#127C22×C3111yescharacteristic
207#486—22no—
208#486—22no—
209C2×C4⋊C8—21no—
210C2×C4⋊C8C22×C3111yescharacteristic
211C2×C4⋊C8C22×C3111yescharacteristic
212C2×C4⋊C8C22×C3111yescharacteristic
213C2×#102—21no—
214C2×#102—21no—
215C2×#102—21no—
216C2×#102—21no—
217C2×C4×C8C22×C3111yescharacteristic
218C4×C4⋊C4C22×C3111yescharacteristic
219#340—21no—
220#340—21no—
221#336—21no—
222#333—22no—
223#331—22no—
224C2×C31—162no—
225C2×C31#167711yescharacteristic
226C2×C31#167811yescharacteristic
227C2×C31#169711yescharacteristic
228C2×C31#169811yescharacteristic
229C2×C31#204211yescharacteristic
230C2×C31#204511yescharacteristic
231C2×C31#205511yescharacteristic
232C22×Q8—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8D4×C3111yescharacteristic
240C22×C8D4×C3111yescharacteristic
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C3112yes—
245#127—21no—
246#127—21no—
247#127—21no—
248#127D4×C3112yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4D4×C3111yescharacteristic
252C2×C4⋊C4D4×C3111yescharacteristic
253C2×C42—21no—
254C2×C42C23×C3111yescharacteristic
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—42no—
262#102—24no—
263#102—24no—
264#102—42no—
265#102—42no—
266C4×C8—21no—
267C4×C8—21no—
268#95—21no—
269C31#1053911yescharacteristic, direct product factor, Sylow
270C24—41no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—81no—
274C2×Q8—81no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—81no—
290C2×D4—81no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4C2×D4×C3111yescharacteristic
295C22×C4C2×D4×C3111yescharacteristic
296C22×C4#41×C3111yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4SD16×C3112yes—
320C42—41no—
321C42—41no—
322C42C2×D4×C3111yescharacteristic
323C42C2×D4×C3111yescharacteristic
324C42#41×C3111yescharacteristic
325C42#41×C3111yescharacteristic
326C23—42no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23C31×#12111yescharacteristic
334Q8—82no—
335Q8—82no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—42no—
340D4—82no—
341D4—82no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4C31×#12011yescharacteristic
358C2×C4C31×#12111yescharacteristic
359C2×C4C31×#12111yescharacteristic
360C2×C4C31×#12311yescharacteristic
361C2×C4C31×#12311yescharacteristic
362C2×C4C31×#12411yescharacteristic
363C2×C4C2×SD16×C3111yescharacteristic
364C2×C4C31×#13511yescharacteristic
365C2×C4C31×#13511yescharacteristic
366C2×C4C31×#13511yescharacteristic
367C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C31×#13711yescharacteristic
370C2×C4C31×#13711yescharacteristic
371C8—41no—
372C8—41no—
373C8—81no—
374C8—81no—
375C8—81no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—161no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22C31×#39411yescharacteristic
384C22C31×#46311yescharacteristic
385C22C31×#46411yescharacteristic
386C22C31×#46511yescharacteristic
387C22C31×#46811yescharacteristic
388C22C31×#47211yescharacteristic
389C22C31×#47211yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—82no—
395C4C31×#44911yescharacteristic
396C4C31×#45211yescharacteristic
397C4C31×#46011yescharacteristic
398C4C31×#46111yescharacteristic
399C4C31×#46311yescharacteristic
400C4C31×#46411yescharacteristic
401C4C31×#48811yescharacteristic
402C4C31×#48911yescharacteristic
403C2—162no—
404C2C31×#167711yescharacteristic
405C2C31×#167811yescharacteristic
406C2C31×#169711yescharacteristic
407C2C31×#169811yescharacteristic
408C2C31×#204211yescharacteristic
409C2C31×#204511yescharacteristic
410C2C31×#205511yescharacteristic
subgroupG/H
center, socleC23×C31#121
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
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
2. C31×#1987C2semidirect
4. C31×#1939C2semidirect
5. C2×C31×#333C2semidirect
7. C31×#1277C2semidirect
120. #10539C31direct (N × H)
172. #1987C2×C31semidirect
174. #1939C2×C31semidirect
175. C2×#333C2×C31semidirect
177. #1277C2×C31semidirect
269. C31#10539direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4862
C31×#19872
C31×#19822
C31×#19392
C2×C31×#3332
C2×C31×#3312
C31×#12772
#1053931