Finite Group Databasebeta

← C137×#25853 #46875239

Subgroups — C137×#25853 #46875239

Order 35072.

1094 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501×C137C211yesmaximal, characteristic
2#2502×C137C212yesmaximal, complemented
3#2487×C137C211yesmaximal, characteristic
4#2062×C137C214yesmaximal, complemented
5#1996×C137C212yesmaximal, complemented
6#1867×C137C214yesmaximal
7C4×#99×C137C211yesmaximal, characteristic
8#554×C137—22no—
9#554×C137—24no—
10D4×Q8×C137—22no—
11#544×C137—21no—
12#543×C137—21no—
13#543×C137C2212yes—
14#533×C137—21no—
15C2×C22⋊Q8×C137—21no—
16C2×C22⋊Q8×C137C2212yes—
17#518×C137C2212yes—
18C2×C4×D4×C137C2211yescharacteristic
19C22×C4⋊C4×C137C2211yescharacteristic
20C2×#99×C137C2212yes—
21#400×C137—21no—
22#399×C137—21no—
23#398×C137—24no—
24#391×C137—22no—
25#389×C137—21no—
26#386×C137C2212yes—
27#385×C137C2212yes—
28#385×C137C2218yes—
29#384×C137—22no—
30#380×C137—21no—
31#380×C137C2211yescharacteristic
32C4×C22⋊C4×C137C2212yes—
33#342×C137C2212yes—
34#342×C137C2218yes—
35C22×Q8×C137—21no—
36C22×Q8×C137D412yes—
37C22×D4×C137C2311yescharacteristic
38C23×C4×C137D411yescharacteristic
39C23×C4×C137C2312yes—
40C23×C4×C137C2311yescharacteristic
41C4⋊Q8×C137—21no—
42C4⋊Q8×C137—21no—
43C4⋊Q8×C137—24no—
44#125×C137—22no—
45#125×C137—24no—
46C22⋊Q8×C137—22no—
47C22⋊Q8×C137—22no—
48C22⋊Q8×C137—24no—
49C22⋊Q8×C137—24no—
50C22⋊Q8×C137—22no—
51C22⋊Q8×C137—22no—
52C22⋊Q8×C137—22no—
53C22⋊Q8×C137—22no—
54C22⋊Q8×C137—22no—
55C22⋊Q8×C137—22no—
56C22⋊Q8×C137—24no—
57C22⋊Q8×C137—24no—
58C4×Q8×C137—22no—
59C4×Q8×C137—22no—
60C4×D4×C137—22no—
61C4×D4×C137—22no—
62C4×D4×C137—24no—
63C4×D4×C137—24no—
64#117×C137—22no—
65C2×C4⋊C4×C137—21no—
66C2×C4⋊C4×C137—22no—
67C2×C4⋊C4×C137—21no—
68C2×C4⋊C4×C137—21no—
69C2×C4⋊C4×C137—22no—
70C2×C4⋊C4×C137—22no—
71C2×C4⋊C4×C137—21no—
72C2×C4⋊C4×C137—28no—
73C2×C4⋊C4×C137—22no—
74C2×C4⋊C4×C137D412yes—
75C2×C4⋊C4×C137D414yes—
76C2×C4⋊C4×C137D411yescharacteristic
77C2×C22⋊C4×C137—22no—
78C2×C22⋊C4×C137—22no—
79C2×C22⋊C4×C137D411yescharacteristic
80C2×C22⋊C4×C137C2312yes—
81C2×C22⋊C4×C137C2311yescharacteristic
82C2×C22⋊C4×C137C2314yes—
83C2×C22⋊C4×C137C2314yes—
84C2×C42×C137—22no—
85C2×C42×C137D411yescharacteristic
86#99×C137Q818yescomplemented
87#95×C137—22no—
88#95×C137—22no—
89#95×C137—21no—
90#95×C137—28no—
91#95×C137—22no—
92C24×C137C2×D411yescharacteristic
93C24×C137C2411yescharacteristic
94C2×Q8×C137—22no—
95C2×Q8×C137—22no—
96C2×Q8×C137—22no—
97C2×Q8×C137—22no—
98C2×Q8×C137—44no—
99C2×Q8×C137—82no—
100C2×D4×C137—22no—
101C2×D4×C137—22no—
102C2×D4×C137C2×Q814yes—
103C22×C4×C137—24no—
104C22×C4×C137—22no—
105C22×C4×C137—22no—
106C22×C4×C137—21no—
107C22×C4×C137—21no—
108C22×C4×C137—22no—
109C22×C4×C137—22no—
110C22×C4×C137—24no—
111C22×C4×C137—42no—
112C22×C4×C137—42no—
113C22×C4×C137—82no—
114C22×C4×C137C2×D412yes—
115C22×C4×C137C2×D411yescharacteristic
116C22×C4×C137C2×D411yescharacteristic
117C22×C4×C137C2×D414yes—
118C22×C4×C137C2×D411yescharacteristic
119C22×C4×C137C2×D411yescharacteristic
120C22×C4×C137C2×D414yes—
121C22×C4×C137C2×D411yescharacteristic
122C22×C4×C137C2×D412yes—
123C4⋊C4×C137—22no—
124C4⋊C4×C137—24no—
125C4⋊C4×C137—22no—
126C4⋊C4×C137—22no—
127C4⋊C4×C137—24no—
128C4⋊C4×C137—22no—
129C4⋊C4×C137—24no—
130C4⋊C4×C137—22no—
131C4⋊C4×C137—22no—
132C4⋊C4×C137—22no—
133C4⋊C4×C137—24no—
134C4⋊C4×C137—42no—
135C4⋊C4×C137—41no—
136C4⋊C4×C137—42no—
137C4⋊C4×C137—41no—
138C4⋊C4×C137—88no—
139C4⋊C4×C137—82no—
140C22⋊C4×C137—24no—
141C22⋊C4×C137—24no—
142C22⋊C4×C137—22no—
143C22⋊C4×C137—24no—
144C22⋊C4×C137—24no—
145C22⋊C4×C137—22no—
146C22⋊C4×C137—24no—
147C22⋊C4×C137—82no—
148C22⋊C4×C137C2×Q818yescomplemented
149C42×C137—24no—
150C42×C137—21no—
151C42×C137—21no—
152C42×C137—44no—
153C23×C137—21no—
154C23×C137—21no—
155C23×C137—41no—
156C23×C137—41no—
157C23×C137—41no—
158C23×C137—41no—
159C23×C137C4⋊Q812yes—
160C23×C137C22×D411yescharacteristic
161C23×C137C22×D412yes—
162C23×C137C22×Q812yes—
163C23×C1372−1+412yes—
164Q8×C137—44no—
165Q8×C137—82no—
166Q8×C137—82no—
167Q8×C137—82no—
168D4×C137—28no—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137—24no—
172C2×C4×C137—22no—
173C2×C4×C137—21no—
174C2×C4×C137—24no—
175C2×C4×C137—24no—
176C2×C4×C137—21no—
177C2×C4×C137—28no—
178C2×C4×C137—42no—
179C2×C4×C137—41no—
180C2×C4×C137—44no—
181C2×C4×C137—44no—
182C2×C4×C137—42no—
183C2×C4×C137—44no—
184C2×C4×C137—41no—
185C2×C4×C137—41no—
186C2×C4×C137—44no—
187C2×C4×C137—42no—
188C2×C4×C137—42no—
189C2×C4×C137—42no—
190C2×C4×C137—48no—
191C2×C4×C137—42no—
192C2×C4×C137—42no—
193C2×C4×C137—42no—
194C2×C4×C137—81no—
195C2×C4×C137—82no—
196C2×C4×C137—82no—
197C2×C4×C137#12011yescharacteristic
198C2×C4×C137#12012yes—
199C2×C4×C137#12011yescharacteristic
200C2×C4×C137C4⋊Q812yes—
201C22×C137—22no—
202C22×C137—21no—
203C22×C137—22no—
204C22×C137—21no—
205C22×C137—22no—
206C22×C137—42no—
207C22×C137—42no—
208C22×C137—42no—
209C22×C137—41no—
210C22×C137—41no—
211C22×C137—81no—
212C22×C137C2×#12011yescharacteristic
213C22×C137C2×C4⋊Q811yescharacteristic
214C22×C137#53611yescharacteristic
215C22×C137D4×Q814yes—
216C4×C137—22no—
217C4×C137—21no—
218C4×C137—22no—
219C4×C137—48no—
220C4×C137—41no—
221C4×C137—44no—
222C4×C137—44no—
223C4×C137—41no—
224C4×C137—82no—
225C4×C137—82no—
226C4×C137#45812yes—
227C2×C137—24no—
228C2×C137—21no—
229C2×C137—21no—
230C2×C137—42no—
231C2×C137#243411yescharacteristic
232C2×C137C2×#45811yescharacteristic
233C2×C137#302711yescharacteristic
234#25853C13711yesmaximal, characteristic, direct product factor, Sylow
235C137#2585311yescharacteristic, direct product factor, Sylow
236#3501C2×C13711yescharacteristic
237#2502C2×C13712yescomplemented
238#2487C2×C13711yescharacteristic
239#2062C2×C13714yescomplemented
240#1996C2×C13712yescomplemented
241#1867C2×C13714yes—
242C4×#99C2×C13711yescharacteristic
243#554—22no—
244#554—24no—
245D4×Q8—22no—
246#544—21no—
247#543—21no—
248#543C22×C13712yes—
249#533—21no—
250C2×C22⋊Q8—21no—
251C2×C22⋊Q8C22×C13712yes—
252#518C22×C13712yes—
253C2×C4×D4C22×C13711yescharacteristic
254C22×C4⋊C4C22×C13711yescharacteristic
255C2×#99C22×C13712yes—
256#400—21no—
257#399—21no—
258#398—24no—
259#391—22no—
260#389—21no—
261#386C22×C13712yes—
262#385C22×C13712yes—
263#385C22×C13718yes—
264#384—22no—
265#380—21no—
266#380C22×C13711yescharacteristic
267C4×C22⋊C4C22×C13712yes—
268#342C22×C13712yes—
269#342C22×C13718yes—
270C22×Q8—21no—
271C22×Q8D4×C13712yes—
272C22×D4C23×C13711yescharacteristic
273C23×C4D4×C13711yescharacteristic
274C23×C4C23×C13712yes—
275C23×C4C23×C13711yescharacteristic
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—24no—
279#125—22no—
280#125—24no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—24no—
299#117—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4D4×C13711yescharacteristic
310C2×C4⋊C4D4×C13712yes—
311C2×C4⋊C4D4×C13714yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4D4×C13711yescharacteristic
315C2×C22⋊C4C23×C13712yes—
316C2×C22⋊C4C23×C13711yescharacteristic
317C2×C22⋊C4C23×C13714yes—
318C2×C22⋊C4C23×C13714yes—
319C2×C42—22no—
320C2×C42D4×C13711yescharacteristic
321#99Q8×C13718yescomplemented
322#95—22no—
323#95—22no—
324#95—21no—
325#95—28no—
326#95—22no—
327C24C2×D4×C13711yescharacteristic
328C24C24×C13711yescharacteristic
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—82no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4C2×Q8×C13714yes—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4C2×D4×C13712yes—
350C22×C4C2×D4×C13711yescharacteristic
351C22×C4C2×D4×C13711yescharacteristic
352C22×C4C2×D4×C13714yes—
353C22×C4C2×D4×C13711yescharacteristic
354C22×C4C2×D4×C13711yescharacteristic
355C22×C4C2×D4×C13714yes—
356C22×C4C2×D4×C13711yescharacteristic
357C22×C4C2×D4×C13712yes—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—88no—
374C4⋊C4—82no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—82no—
383C22⋊C4C2×Q8×C13718yescomplemented
384C42—24no—
385C42—21no—
386C42—21no—
387C42—44no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C4⋊Q8×C13712yes—
395C23C22×D4×C13711yescharacteristic
396C23C22×D4×C13712yes—
397C23C22×Q8×C13712yes—
398C232−1+4×C13712yes—
399Q8—44no—
400Q8—82no—
401Q8—82no—
402Q8—82no—
403D4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—28no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#120×C13711yescharacteristic
433C2×C4#120×C13712yes—
434C2×C4#120×C13711yescharacteristic
435C2×C4C4⋊Q8×C13712yes—
436C22—22no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#120×C13711yescharacteristic
448C22C2×C4⋊Q8×C13711yescharacteristic
449C22#536×C13711yescharacteristic
450C22D4×Q8×C13714yes—
451C4—22no—
452C4—21no—
453C4—22no—
454C4—48no—
455C4—41no—
456C4—44no—
457C4—44no—
458C4—41no—
459C4—82no—
460C4—82no—
461C4#458×C13712yes—
462C2—24no—
463C2—21no—
464C2—21no—
465C2—42no—
466C2#2434×C13711yescharacteristic
467C2C2×#458×C13711yescharacteristic
468C2#3027×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radicalC22×D4×C137C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2502×C137C2semidirect
4. #2062×C137C2semidirect
5. #1996×C137C2semidirect
86. #99×C137Q8semidirect
148. C22⋊C4×C137C2×Q8semidirect
234. #25853C137direct (N × H)
235. C137#25853direct (N × H)
237. #2502C2×C137semidirect
239. #2062C2×C137semidirect
240. #1996C2×C137semidirect
321. #99Q8×C137semidirect
383. C22⋊C4C2×Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#3501×C1372
#2502×C1372
#2487×C1372
#2062×C1372
#1996×C1372
#1867×C1372
C4×#99×C1372
#25853137