Finite Group Databasebeta

← C31×#9035 #21465794

Subgroups — C31×#9035 #21465794

Order 7936.

612 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
1C31×#2905C211yesmaximal, characteristic, complemented
2C31×#2392C211yesmaximal, characteristic
3C31×#1486C212yesmaximal, complemented
4C31×#1458C211yesmaximal, characteristic, complemented
5C31×#1303C212yesmaximal, complemented
6C22×M16×C31—21no—
7C2×C4×D4×C31—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C2×C31×#117C2211yescharacteristic
10C31×#431C2212yes—
11C31×#424—24no—
12C31×#423—21no—
13C31×#423C2212yes—
14C31×#398C411yescharacteristic
15C31×#397C411yescharacteristic
16C31×#396—21no—
17C31×#394—21no—
18C31×#394C411yescharacteristic
19C31×#392C411yescharacteristic
20C4×C4⋊C4×C31C2211yescharacteristic
21C4×C22⋊C4×C31—21no—
22C31×D4⋊C8—24no—
23C31×#324—21no—
24C31×#323—21no—
25C22×D4×C31—21no—
26C22×D4×C31C2×C411yescharacteristic
27C23×C4×C31—21no—
28C23×C4×C31D411yescharacteristic
29C2×M16×C31—22no—
30C2×M16×C31—22no—
31C2×M16×C31—42no—
32C2×M16×C31—42no—
33C22×C8×C31—22no—
34C4×D4×C31—22no—
35C4×D4×C31—22no—
36C4×D4×C31—42no—
37C4×D4×C31—42no—
38C31×#117—22no—
39C31×#117—22no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31C2×C411yescharacteristic
43C2×C4⋊C4×C31C2×C411yescharacteristic
44C2×C4⋊C4×C31D411yescharacteristic
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31C2×C411yescharacteristic
51C2×C22⋊C4×C31D411yescharacteristic
52C2×C42×C31—21no—
53C2×C42×C31D411yescharacteristic
54C2×C42×C31C2311yescharacteristic
55C31×C4⋊C8—24no—
56C31×C4⋊C8—22no—
57C31×C4⋊C8—22no—
58C31×C4⋊C8D414yescomplemented
59C31×C22⋊C8—22no—
60C4×C8×C31—22no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95C2×C411yescharacteristic
64C31×#95C2×C411yescharacteristic
65C24×C31—21no—
66C24×C31C22⋊C411yescharacteristic
67C2×D4×C31—22no—
68C2×D4×C31—42no—
69C2×D4×C31—42no—
70C2×D4×C31—42no—
71C2×D4×C31—42no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—41no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31—41no—
83C22×C4×C31—41no—
84C22×C4×C31—41no—
85C22×C4×C31C22⋊C411yescharacteristic
86C22×C4×C31C22⋊C411yescharacteristic
87C22×C4×C31C22⋊C411yescharacteristic
88C22×C4×C31D812yes—
89C22×C4×C31SD1612yes—
90C22×C4×C31C22×C411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93M16×C31—44no—
94M16×C31—42no—
95M16×C31—42no—
96C2×C8×C31—24no—
97C2×C8×C31—44no—
98C2×C8×C31—42no—
99C4⋊C4×C31—22no—
100C4⋊C4×C31—22no—
101C4⋊C4×C31—22no—
102C4⋊C4×C31—42no—
103C4⋊C4×C31—42no—
104C22⋊C4×C31—22no—
105C22⋊C4×C31—22no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—42no—
110C22⋊C4×C31—42no—
111C42×C31—21no—
112C42×C31—21no—
113C42×C31—41no—
114C42×C31—41no—
115C42×C31C2×D412yes—
116C42×C31#4112yes—
117#9035C3111yesmaximal, characteristic, direct product factor, Sylow
118C23×C31—21no—
119C23×C31—21no—
120C23×C31—22no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31#10212yes—
130C23×C31C2×C22⋊C411yescharacteristic
131D4×C31—24no—
132D4×C31—42no—
133D4×C31—48no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—22no—
137C2×C4×C31—22no—
138C2×C4×C31—22no—
139C2×C4×C31—22no—
140C2×C4×C31—22no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—81no—
156C2×C4×C31—81no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31#10212yes—
160C2×C4×C31C4×D412yes—
161C2×C4×C31#12011yescharacteristic
162C2×C4×C31#12112yes—
163C2×C4×C31#12311yescharacteristic
164C2×C4×C31C2×D811yescharacteristic
165C2×C4×C31C2×SD1611yescharacteristic
166C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C31#13711yescharacteristic
168C8×C31—44no—
169C8×C31—82no—
170#2905C2×C3111yescharacteristic, complemented
171#2392C2×C3111yescharacteristic
172#1486C2×C3112yescomplemented
173#1458C2×C3111yescharacteristic, complemented
174#1303C2×C3112yescomplemented
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—22no—
178C22×C31—42no—
179C22×C31—42no—
180C22×C31—42no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—42no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—81no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31#38611yescharacteristic
190C22×C31C2×#10211yescharacteristic
191C22×C31#41711yescharacteristic
192C4×C31—22no—
193C4×C31—22no—
194C4×C31—22no—
195C4×C31—42no—
196C4×C31—41no—
197C4×C31—41no—
198C4×C31—81no—
199C4×C31—81no—
200C4×C31#44711yescharacteristic
201C4×C31#44811yescharacteristic
202C4×C31#48011yescharacteristic
203C4×C31#48111yescharacteristic
204C22×M16—21no—
205C2×C4×D4—21no—
206C2×C4×D4C22×C3111yescharacteristic
207C2×#117C22×C3111yescharacteristic
208#431C22×C3112yes—
209#424—24no—
210#423—21no—
211#423C22×C3112yes—
212#398C4×C3111yescharacteristic
213#397C4×C3111yescharacteristic
214#396—21no—
215#394—21no—
216#394C4×C3111yescharacteristic
217#392C4×C3111yescharacteristic
218C4×C4⋊C4C22×C3111yescharacteristic
219C4×C22⋊C4—21no—
220D4⋊C8—24no—
221#324—21no—
222#323—21no—
223C2×C31—21no—
224C2×C31—21no—
225C2×C31—42no—
226C2×C31—82no—
227C2×C31#187411yescharacteristic
228C2×C31#188411yescharacteristic
229C2×C31#189111yescharacteristic
230C22×D4—21no—
231C22×D4C2×C4×C3111yescharacteristic
232C23×C4—21no—
233C23×C4D4×C3111yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—42no—
237C2×M16—42no—
238C22×C8—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—42no—
242C4×D4—42no—
243#117—22no—
244#117—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C2×C4×C3111yescharacteristic
248C2×C4⋊C4C2×C4×C3111yescharacteristic
249C2×C4⋊C4D4×C3111yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4C2×C4×C3111yescharacteristic
256C2×C22⋊C4D4×C3111yescharacteristic
257C2×C42—21no—
258C2×C42D4×C3111yescharacteristic
259C2×C42C23×C3111yescharacteristic
260C4⋊C8—24no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8D4×C3114yescomplemented
264C22⋊C8—22no—
265C4×C8—22no—
266#95—21no—
267#95—21no—
268#95C2×C4×C3111yescharacteristic
269#95C2×C4×C3111yescharacteristic
270C31#903511yescharacteristic, direct product factor, Sylow
271C24—21no—
272C24C22⋊C4×C3111yescharacteristic
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4C22⋊C4×C3111yescharacteristic
292C22×C4C22⋊C4×C3111yescharacteristic
293C22×C4C22⋊C4×C3111yescharacteristic
294C22×C4D8×C3112yes—
295C22×C4SD16×C3112yes—
296C22×C4C22×C4×C3111yescharacteristic
297C22×C4C2×D4×C3111yescharacteristic
298C22×C4C2×D4×C3111yescharacteristic
299M16—44no—
300M16—42no—
301M16—42no—
302C2×C8—24no—
303C2×C8—44no—
304C2×C8—42no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—41no—
321C42C2×D4×C3112yes—
322C42#41×C3112yes—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23C31×#10212yes—
335C23C2×C22⋊C4×C3111yescharacteristic
336D4—24no—
337D4—42no—
338D4—48no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4C31×#10212yes—
365C2×C4C4×D4×C3112yes—
366C2×C4C31×#12011yescharacteristic
367C2×C4C31×#12112yes—
368C2×C4C31×#12311yescharacteristic
369C2×C4C2×D8×C3111yescharacteristic
370C2×C4C2×SD16×C3111yescharacteristic
371C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C31×#13711yescharacteristic
373C8—44no—
374C8—82no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C31×#38611yescharacteristic
390C22C2×C31×#10211yescharacteristic
391C22C31×#41711yescharacteristic
392C4—22no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4C31×#44711yescharacteristic
401C4C31×#44811yescharacteristic
402C4C31×#48011yescharacteristic
403C4C31×#48111yescharacteristic
404C2—21no—
405C2—21no—
406C2—42no—
407C2—82no—
408C2C31×#187411yescharacteristic
409C2C31×#188411yescharacteristic
410C2C31×#189111yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#386
prime radicalC31×#392C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2905C2semidirect
3. C31×#1486C2semidirect
4. C31×#1458C2semidirect
5. C31×#1303C2semidirect
58. C31×C4⋊C8D4semidirect
117. #9035C31direct (N × H)
170. #2905C2×C31semidirect
172. #1486C2×C31semidirect
173. #1458C2×C31semidirect
174. #1303C2×C31semidirect
263. C4⋊C8D4×C31semidirect
270. C31#9035direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29052
C31×#23922
C31×#14862
C31×#14582
C31×#13032
#903531