Finite Group Databasebeta

← C31×#9180 #21465939

Subgroups — C31×#9180 #21465939

Order 7936.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3037C211yesmaximal, characteristic
2C31×#2351C211yesmaximal, characteristic, complemented
3C31×#1517C211yesmaximal, characteristic
4C31×#1502C211yesmaximal, characteristic
5C31×#1479C211yesmaximal, characteristic, complemented
6C31×#1303C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#532C2211yescharacteristic
10C2×C4×Q8×C31—21no—
11C2×C4×Q8×C31C2211yescharacteristic
12C31×#464—22no—
13C31×#462C412yescomplemented
14C31×#460C412yescomplemented
15C31×#431C2211yescharacteristic
16C31×#431C2211yescharacteristic
17C31×#423C2211yescharacteristic
18C31×#417—21no—
19C31×#389—21no—
20C31×#389—22no—
21C31×#382—21no—
22C31×#382C2211yescharacteristic
23C4×C4⋊C4×C31—21no—
24C4×C4⋊C4×C31C2211yescharacteristic
25C31×#327—21no—
26C31×#327—21no—
27C31×#327—22no—
28C31×Q8⋊C8—22no—
29C2×#41×C31—21no—
30C22×Q8×C31D411yescharacteristic
31C31×#137—42no—
32C31×#137—42no—
33C2×Q16×C31—22no—
34C2×SD16×C31—22no—
35C2×M16×C31—21no—
36C31×C4⋊Q8C2×C411yescharacteristic
37C31×#127C2×C411yescharacteristic
38C31×#124—21no—
39C31×#121—21no—
40C31×#121—21no—
41C4×Q8×C31—21no—
42C4×Q8×C31—21no—
43C4×Q8×C31—42no—
44C4×Q8×C31—42no—
45C4×Q8×C31C2×C412yes—
46C4×Q8×C31D412yes—
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—22no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31D411yescharacteristic
55C2×C42×C31—21no—
56C2×C42×C31—22no—
57C2×C42×C31—21no—
58C2×C42×C31C2311yescharacteristic
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8C2×C412yes—
63C31×#103—22no—
64C31×#102—22no—
65C31×C22⋊C8—21no—
66C31×C22⋊C8—21no—
67C4×C8×C31—21no—
68C4×C8×C31—21no—
69C31×#95—22no—
70C31×#95—21no—
71C31×#95D411yescharacteristic
72C31×#95D411yescharacteristic
73C31×#95D412yes—
74#41×C31—41no—
75#41×C31—41no—
76#41×C31—41no—
77#41×C31—41no—
78C2×Q8×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×Q8×C31—21no—
82C2×Q8×C31C22⋊C412yes—
83C2×Q8×C31D812yes—
84C2×Q8×C31SD1612yes—
85C2×D4×C31—21no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31—41no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—22no—
94C22×C4×C31—41no—
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98Q16×C31—44no—
99SD16×C31—44no—
100M16×C31—41no—
101M16×C31—41no—
102C2×C8×C31—22no—
103C2×C8×C31—41no—
104C2×C8×C31—41no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—42no—
111C4⋊C4×C31—44no—
112C4⋊C4×C31—44no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31C22⋊C412yes—
116C22⋊C4×C31—41no—
117C22⋊C4×C31—41no—
118C42×C31—22no—
119C42×C31—22no—
120C42×C31—44no—
121C42×C31—41no—
122C42×C31—41no—
123C42×C31C22×C411yescharacteristic
124C42×C31C2×D411yescharacteristic
125C42×C31#4111yescharacteristic
126C42×C31#4111yescharacteristic
127#9180C3111yesmaximal, characteristic, direct product factor, Sylow
128C23×C31—41no—
129C23×C31#12011yescharacteristic
130Q8×C31—22no—
131Q8×C31—22no—
132Q8×C31—22no—
133Q8×C31—41no—
134Q8×C31—41no—
135Q8×C31#10214yes—
136D4×C31—41no—
137D4×C31—41no—
138D4×C31—81no—
139D4×C31—81no—
140D4×C31—81no—
141D4×C31—81no—
142C2×C4×C31—22no—
143C2×C4×C31—22no—
144C2×C4×C31—22no—
145C2×C4×C31—22no—
146C2×C4×C31—22no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—42no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—41no—
157C2×C4×C31—42no—
158C2×C4×C31—41no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31C2×C22⋊C411yescharacteristic
162C2×C4×C31C4×D411yescharacteristic
163C2×C4×C31C4×D411yescharacteristic
164C2×C4×C31#12111yescharacteristic
165C2×C4×C31#12111yescharacteristic
166C2×C4×C31#12311yescharacteristic
167C2×C4×C31C2×D811yescharacteristic
168C2×C4×C31C2×SD1611yescharacteristic
169C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C31#13711yescharacteristic
171C8×C31—42no—
172C8×C31—81no—
173C8×C31—81no—
174#3037C2×C3111yescharacteristic
175#2351C2×C3111yescharacteristic, complemented
176#1517C2×C3111yescharacteristic
177#1502C2×C3111yescharacteristic
178#1479C2×C3111yescharacteristic, complemented
179#1303C2×C3111yescharacteristic, complemented
180#1286C2×C3111yescharacteristic, complemented
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31#38611yescharacteristic
187C22×C31#44711yescharacteristic
188C22×C31#44811yescharacteristic
189C4×C31—24no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—21no—
193C4×C31—21no—
194C4×C31—42no—
195C4×C31—42no—
196C4×C31—82no—
197C4×C31—81no—
198C4×C31—81no—
199C4×C31C2×#10211yescharacteristic
200C4×C31#41711yescharacteristic
201C4×C31#48011yescharacteristic
202C4×C31#48111yescharacteristic
203C2×#137—21no—
204#532C22×C3111yescharacteristic
205C2×C4×Q8—21no—
206C2×C4×Q8C22×C3111yescharacteristic
207#464—22no—
208#462C4×C3112yescomplemented
209#460C4×C3112yescomplemented
210#431C22×C3111yescharacteristic
211#431C22×C3111yescharacteristic
212#423C22×C3111yescharacteristic
213#417—21no—
214#389—21no—
215#389—22no—
216#382—21no—
217#382C22×C3111yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4C22×C3111yescharacteristic
220#327—21no—
221#327—21no—
222#327—22no—
223Q8⋊C8—22no—
224C2×C31—21no—
225C2×C31—21no—
226C2×C31—161no—
227C2×C31#187411yescharacteristic
228C2×C31#190511yescharacteristic
229C2×C31#190611yescharacteristic
230C2×#41—21no—
231C22×Q8D4×C3111yescharacteristic
232#137—42no—
233#137—42no—
234C2×Q16—22no—
235C2×SD16—22no—
236C2×M16—21no—
237C4⋊Q8C2×C4×C3111yescharacteristic
238#127C2×C4×C3111yescharacteristic
239#124—21no—
240#121—21no—
241#121—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8C2×C4×C3112yes—
247C4×Q8D4×C3112yes—
248C4×D4—21no—
249C4×D4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4D4×C3111yescharacteristic
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42C23×C3111yescharacteristic
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8C2×C4×C3112yes—
264#103—22no—
265#102—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270#95—22no—
271#95—21no—
272#95D4×C3111yescharacteristic
273#95D4×C3111yescharacteristic
274#95D4×C3112yes—
275C31#918011yescharacteristic, direct product factor, Sylow
276#41—41no—
277#41—41no—
278#41—41no—
279#41—41no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8C22⋊C4×C3112yes—
285C2×Q8D8×C3112yes—
286C2×Q8SD16×C3112yes—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4C2×D4×C3111yescharacteristic
298C22×C4C2×D4×C3111yescharacteristic
299C22×C4C2×D4×C3111yescharacteristic
300Q16—44no—
301SD16—44no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C22⋊C4×C3112yes—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—41no—
324C42—41no—
325C42C22×C4×C3111yescharacteristic
326C42C2×D4×C3111yescharacteristic
327C42#41×C3111yescharacteristic
328C42#41×C3111yescharacteristic
329C23—41no—
330C23C31×#12011yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8C31×#10214yes—
337D4—41no—
338D4—41no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4C2×C22⋊C4×C3111yescharacteristic
363C2×C4C4×D4×C3111yescharacteristic
364C2×C4C4×D4×C3111yescharacteristic
365C2×C4C31×#12111yescharacteristic
366C2×C4C31×#12111yescharacteristic
367C2×C4C31×#12311yescharacteristic
368C2×C4C2×D8×C3111yescharacteristic
369C2×C4C2×SD16×C3111yescharacteristic
370C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C31×#13711yescharacteristic
372C8—42no—
373C8—81no—
374C8—81no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C31×#38611yescharacteristic
381C22C31×#44711yescharacteristic
382C22C31×#44811yescharacteristic
383C4—24no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—81no—
392C4—81no—
393C4C2×C31×#10211yescharacteristic
394C4C31×#41711yescharacteristic
395C4C31×#48011yescharacteristic
396C4C31×#48111yescharacteristic
397C2—21no—
398C2—21no—
399C2—161no—
400C2C31×#187411yescharacteristic
401C2C31×#190511yescharacteristic
402C2C31×#190611yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2351C2semidirect
5. C31×#1479C2semidirect
6. C31×#1303C2semidirect
7. C31×#1286C2semidirect
13. C31×#462C4semidirect
14. C31×#460C4semidirect
127. #9180C31direct (N × H)
175. #2351C2×C31semidirect
178. #1479C2×C31semidirect
179. #1303C2×C31semidirect
180. #1286C2×C31semidirect
208. #462C4×C31semidirect
209. #460C4×C31semidirect
275. C31#9180direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30372
C31×#23512
C31×#15172
C31×#15022
C31×#14792
C31×#13032
C31×#12862
#918031