Finite Group Databasebeta

← C131×#12712 #45698596

Subgroups — C131×#12712 #45698596

Order 33536.

952 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3486×C131C211yesmaximal, characteristic
2#2135×C131C211yesmaximal, characteristic, complemented
3#2129×C131C211yesmaximal, characteristic, complemented
4#2120×C131C212yesmaximal, complemented
5#1796×C131C212yesmaximal, complemented
6C2×2−1+4×C131C2211yescharacteristic
7C23×Q8×C131C411yescharacteristic, complemented
8D4×Q8×C131—22no—
9D4×Q8×C131—22no—
10#548×C131—22no—
11#548×C131—22no—
12#536×C131—22no—
13#536×C131C411yescharacteristic, complemented
14#536×C131C2211yescharacteristic
15#525×C131—22no—
16#525×C131C411yescharacteristic, complemented
17#525×C131C2211yescharacteristic
18#519×C131C411yescharacteristic, complemented
19#421×C131—22no—
20#419×C131—22no—
21#413×C131C2212yes—
22#410×C131C2212yes—
23#356×C131—22no—
24#354×C131—22no—
25#352×C131—22no—
26AΣL(1,𝔽16)×C131—22no—
272−1+4×C131—44no—
28C2×#41×C131—22no—
29C2×#41×C131—21no—
30C2×#41×C131C2×C411yescharacteristic
31C2×#41×C131D412yes—
32C2×#41×C131C2311yescharacteristic
33C22×Q8×C131—22no—
34C22×Q8×C131—22no—
35C22×Q8×C131—21no—
36C22×Q8×C131—21no—
37C22×Q8×C131—44no—
38C22×Q8×C131C2×C411yescharacteristic
39C22×Q8×C131C2×C411yescharacteristic
40C22×Q8×C131D412yes—
41C23×C4×C131—21no—
42C23×C4×C131C2×C411yescharacteristic
43C2×M16×C131—22no—
44C4⋊Q8×C131—24no—
45C4⋊Q8×C131—21no—
46C4⋊Q8×C131D412yes—
47#124×C131—24no—
48#124×C131—21no—
49#124×C131D412yes—
50#123×C131—22no—
51#123×C131—21no—
52#123×C131—22no—
53#123×C131D412yes—
54C22⋊Q8×C131—24no—
55C22⋊Q8×C131—24no—
56C22⋊Q8×C131—24no—
57C22⋊Q8×C131—21no—
58C22⋊Q8×C131—22no—
59C22⋊Q8×C131—21no—
60C22⋊Q8×C131—22no—
61#121×C131—24no—
62#121×C131—21no—
63#121×C131—21no—
64#120×C131—21no—
65#120×C131D412yes—
66C4×Q8×C131—22no—
67C4×Q8×C131—22no—
68C4×D4×C131—24no—
69C4×D4×C131—24no—
70C4×D4×C131—22no—
71C4×D4×C131—22no—
72#117×C131—22no—
73#117×C131—22no—
74#117×C131C2×C411yescharacteristic
75#117×C131C2×C411yescharacteristic
76#104×C131—44no—
77#103×C131—44no—
78#101×C131—22no—
79#100×C131—22no—
80#99×C131—22no—
81#99×C131—22no—
82C24×C131C22⋊C411yescharacteristic
83#41×C131—24no—
84#41×C131—42no—
85#41×C131—42no—
86#41×C131—42no—
87#41×C131—48no—
88#41×C131—44no—
89C2×Q8×C131—24no—
90C2×Q8×C131—24no—
91C2×Q8×C131—21no—
92C2×Q8×C131—21no—
93C2×Q8×C131—22no—
94C2×Q8×C131—24no—
95C2×Q8×C131—42no—
96C2×Q8×C131—42no—
97C2×Q8×C131—44no—
98C2×Q8×C131—48no—
99C2×Q8×C131—44no—
100C2×Q8×C131—44no—
101C2×Q8×C131—44no—
102C2×Q8×C131C22⋊C412yes—
103C2×Q8×C131C22⋊C411yescharacteristic
104C2×Q8×C131C22⋊C412yes—
105C2×Q8×C131C2×D411yescharacteristic
106C2×D4×C131—22no—
107C2×D4×C131—21no—
108C2×D4×C131—22no—
109C2×D4×C131—22no—
110C2×D4×C131C22⋊C411yescharacteristic
111C2×D4×C131C2×D411yescharacteristic
112C2×D4×C131C2×D412yes—
113C22×C4×C131—21no—
114C22×C4×C131—24no—
115C22×C4×C131—21no—
116C22×C4×C131—22no—
117C22×C4×C131—21no—
118C22×C4×C131—21no—
119C22×C4×C131—21no—
120C22×C4×C131—22no—
121C22×C4×C131—44no—
122C22×C4×C131—42no—
123C22×C4×C131C22⋊C411yescharacteristic
124C22×C4×C131C22×C411yescharacteristic
125C22×C4×C131C2×D412yes—
126M16×C131—42no—
127M16×C131—42no—
128C2×C8×C131—42no—
129C4⋊C4×C131—24no—
130C4⋊C4×C131—24no—
131C4⋊C4×C131—22no—
132C4⋊C4×C131—21no—
133C4⋊C4×C131—21no—
134C4⋊C4×C131—24no—
135C4⋊C4×C131—22no—
136C4⋊C4×C131—42no—
137C22⋊C4×C131—22no—
138C22⋊C4×C131—24no—
139C22⋊C4×C131—22no—
140C22⋊C4×C131—21no—
141C22⋊C4×C131—24no—
142C22⋊C4×C131—22no—
143C22⋊C4×C131—22no—
144C22⋊C4×C131—42no—
145C22⋊C4×C131C22⋊C412yes—
146C42×C131—24no—
147C42×C131—21no—
148C42×C131—42no—
149C42×C131C22⋊C412yes—
150C23×C131—21no—
151C23×C131—21no—
152C23×C131—41no—
153C23×C131—42no—
154C23×C131C2×C22⋊C411yescharacteristic
155C23×C131#12012yes—
156Q8×C131—24no—
157Q8×C131—22no—
158Q8×C131—22no—
159Q8×C131—41no—
160Q8×C131—44no—
161Q8×C131—42no—
162Q8×C131—48no—
163Q8×C131—42no—
164Q8×C131#9914yes—
165D4×C131—22no—
166D4×C131—42no—
167D4×C131—41no—
168D4×C131—44no—
169D4×C131—44no—
170D4×C131—44no—
171D4×C131—42no—
172C2×C4×C131—24no—
173C2×C4×C131—22no—
174C2×C4×C131—22no—
175C2×C4×C131—22no—
176C2×C4×C131—22no—
177C2×C4×C131—22no—
178C2×C4×C131—22no—
179C2×C4×C131—22no—
180C2×C4×C131—21no—
181C2×C4×C131—21no—
182C2×C4×C131—44no—
183C2×C4×C131—44no—
184C2×C4×C131—48no—
185C2×C4×C131—42no—
186C2×C4×C131—42no—
187C2×C4×C131—42no—
188C2×C4×C131—42no—
189C2×C4×C131—41no—
190C2×C4×C131—82no—
191C2×C4×C131C2×C22⋊C411yescharacteristic
192C2×C4×C131C2×C22⋊C411yescharacteristic
193C2×C4×C131#12012yes—
194C8×C131—82no—
195C22×C131—22no—
196C22×C131—21no—
197C22×C131—22no—
198C22×C131—42no—
199C22×C131—41no—
200C22×C131—41no—
201C22×C131—42no—
202C22×C131—42no—
203C22×C131—42no—
204C22×C131—41no—
205C22×C131—41no—
206C22×C131#37911yescharacteristic
207C4×C131—22no—
208C4×C131—24no—
209C4×C131—21no—
210C4×C131—42no—
211C4×C131—42no—
212C4×C131—41no—
213C4×C131—42no—
214C4×C131—41no—
215C4×C131—162no—
216C4×C131C2×#9912yes—
217C2×C131—21no—
218C2×C131—42no—
219C2×C131—42no—
220C2×C131—41no—
221C2×C131—81no—
222C2×C131#178111yescharacteristic
223#12712C13111yesmaximal, characteristic, direct product factor, Sylow
224C131#1271211yescharacteristic, direct product factor, Sylow
225#3486C2×C13111yescharacteristic
226#2135C2×C13111yescharacteristic, complemented
227#2129C2×C13111yescharacteristic, complemented
228#2120C2×C13112yescomplemented
229#1796C2×C13112yescomplemented
230C2×2−1+4C22×C13111yescharacteristic
231C23×Q8C4×C13111yescharacteristic, complemented
232D4×Q8—22no—
233D4×Q8—22no—
234#548—22no—
235#548—22no—
236#536—22no—
237#536C4×C13111yescharacteristic, complemented
238#536C22×C13111yescharacteristic
239#525—22no—
240#525C4×C13111yescharacteristic, complemented
241#525C22×C13111yescharacteristic
242#519C4×C13111yescharacteristic, complemented
243#421—22no—
244#419—22no—
245#413C22×C13112yes—
246#410C22×C13112yes—
247#356—22no—
248#354—22no—
249#352—22no—
250AΣL(1,𝔽16)—22no—
2512−1+4—44no—
252C2×#41—22no—
253C2×#41—21no—
254C2×#41C2×C4×C13111yescharacteristic
255C2×#41D4×C13112yes—
256C2×#41C23×C13111yescharacteristic
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—44no—
262C22×Q8C2×C4×C13111yescharacteristic
263C22×Q8C2×C4×C13111yescharacteristic
264C22×Q8D4×C13112yes—
265C23×C4—21no—
266C23×C4C2×C4×C13111yescharacteristic
267C2×M16—22no—
268C4⋊Q8—24no—
269C4⋊Q8—21no—
270C4⋊Q8D4×C13112yes—
271#124—24no—
272#124—21no—
273#124D4×C13112yes—
274#123—22no—
275#123—21no—
276#123—22no—
277#123D4×C13112yes—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—21no—
282C22⋊Q8—22no—
283C22⋊Q8—21no—
284C22⋊Q8—22no—
285#121—24no—
286#121—21no—
287#121—21no—
288#120—21no—
289#120D4×C13112yes—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296#117—22no—
297#117—22no—
298#117C2×C4×C13111yescharacteristic
299#117C2×C4×C13111yescharacteristic
300#104—44no—
301#103—44no—
302#101—22no—
303#100—22no—
304#99—22no—
305#99—22no—
306C24C22⋊C4×C13111yescharacteristic
307#41—24no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—48no—
312#41—44no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8C22⋊C4×C13111yescharacteristic
327C2×Q8C22⋊C4×C13112yes—
328C2×Q8C22⋊C4×C13112yes—
329C2×Q8C2×D4×C13111yescharacteristic
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4C22⋊C4×C13111yescharacteristic
335C2×D4C2×D4×C13111yescharacteristic
336C2×D4C2×D4×C13112yes—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4C22⋊C4×C13111yescharacteristic
348C22×C4C22×C4×C13111yescharacteristic
349C22×C4C2×D4×C13112yes—
350M16—42no—
351M16—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C22⋊C4—22no—
362C22⋊C4—24no—
363C22⋊C4—22no—
364C22⋊C4—21no—
365C22⋊C4—24no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—42no—
369C22⋊C4C22⋊C4×C13112yes—
370C42—24no—
371C42—21no—
372C42—42no—
373C42C22⋊C4×C13112yes—
374C23—21no—
375C23—21no—
376C23—41no—
377C23—42no—
378C23C2×C22⋊C4×C13111yescharacteristic
379C23#120×C13112yes—
380Q8—24no—
381Q8—22no—
382Q8—22no—
383Q8—41no—
384Q8—44no—
385Q8—42no—
386Q8—48no—
387Q8—42no—
388Q8#99×C13114yes—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—44no—
393D4—42no—
394D4—44no—
395D4—44no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4C2×C22⋊C4×C13111yescharacteristic
416C2×C4C2×C22⋊C4×C13111yescharacteristic
417C2×C4#120×C13112yes—
418C8—82no—
419C22—22no—
420C22—21no—
421C22—22no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22#379×C13111yescharacteristic
431C4—22no—
432C4—24no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—162no—
440C4C2×#99×C13112yes—
441C2—21no—
442C2—42no—
443C2—42no—
444C2—41no—
445C2—81no—
446C2#1781×C13111yescharacteristic
subgroupG/H
center, socleC2×C131#1781
commutatorC22×C4C22×C4×C131
Frattini, cosocleC2×#41C23×C131
prime radical#3486×C131C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C131 ◁ C2×C4×C131 ◁ C2×2−1+4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2135×C131C2semidirect
3. #2129×C131C2semidirect
4. #2120×C131C2semidirect
5. #1796×C131C2semidirect
7. C23×Q8×C131C4semidirect
13. #536×C131C4semidirect
16. #525×C131C4semidirect
18. #519×C131C4semidirect
223. #12712C131direct (N × H)
224. C131#12712direct (N × H)
226. #2135C2×C131semidirect
227. #2129C2×C131semidirect
228. #2120C2×C131semidirect
229. #1796C2×C131semidirect
231. C23×Q8C4×C131semidirect
237. #536C4×C131semidirect
240. #525C4×C131semidirect
242. #519C4×C131semidirect

Maximal subgroups

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

subgroupindex
#3486×C1312
#2135×C1312
#2129×C1312
#2120×C1312
#1796×C1312
#12712131