Finite Group Databasebeta

← C137×#19474 #46868860

Subgroups — C137×#19474 #46868860

Order 35072.

630 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3227×C137C211yesmaximal, characteristic
2#3112×C137C211yesmaximal, characteristic
3C2×#462×C137C211yesmaximal, characteristic, complemented
4#2984×C137C211yesmaximal, characteristic, complemented
5#1702×C137C212yesmaximal, complemented
6#1692×C137C212yesmaximal, complemented
7#1682×C137C212yesmaximal, complemented
8#1672×C137C212yesmaximal, complemented
9#1573×C137C211yesmaximal, characteristic, complemented
10#1556×C137C211yesmaximal, characteristic, complemented
11#1551×C137C211yesmaximal, characteristic
12C22×Q16×C137—21no—
13#532×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15C2×C4×Q8×C137C2211yescharacteristic
16#498×C137C2212yes—
17#486×C137C2212yes—
18#481×C137—21no—
19#481×C137—21no—
20#480×C137—21no—
21#477×C137C2211yescharacteristic
22#477×C137C2211yescharacteristic
23#476×C137—22no—
24#462×C137—21no—
25#462×C137—21no—
26#462×C137C2212yes—
27#462×C137C2212yes—
28#460×C137C2211yescharacteristic
29#460×C137C2211yescharacteristic
30#448×C137—21no—
31#446×C137—22no—
32#446×C137C2212yes—
33C8×Q8×C137C2212yes—
34#440×C137—21no—
35#440×C137—21no—
36#432×C137—21no—
37#432×C137C2211yescharacteristic
38#432×C137C2211yescharacteristic
39#431×C137—21no—
40#431×C137C2211yescharacteristic
41C2×C4⋊C8×C137C2211yescharacteristic
42C2×#103×C137—21no—
43#335×C137C2212yes—
44#334×C137C2212yes—
45#333×C137C2212yes—
46#332×C137C2212yes—
47#332×C137C2212yes—
48#331×C137C2212yes—
49C2×#41×C137—21no—
50C22×Q8×C137—21no—
51C22×Q8×C137D411yescharacteristic
52C2×Q16×C137—22no—
53C2×Q16×C137—22no—
54C2×Q16×C137—44no—
55C2×SD16×C137—21no—
56C2×SD16×C137—21no—
57C22×C8×C137—21no—
58C4⋊Q8×C137—21no—
59C4⋊Q8×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137C2312yes—
62C4⋊Q8×C137C2311yescharacteristic
63#127×C137C2311yescharacteristic
64#124×C137—21no—
65#121×C137—21no—
66#121×C137—21no—
67C4×Q8×C137—21no—
68C4×Q8×C137—21no—
69C4×Q8×C137D411yescharacteristic
70C4×Q8×C137D411yescharacteristic
71C4×Q8×C137C2311yescharacteristic
72C4×Q8×C137C2311yescharacteristic
73C4×D4×C137—21no—
74C4×D4×C137—21no—
75C2×C4⋊C4×C137—21no—
76C2×C4⋊C4×C137—21no—
77C2×C4⋊C4×C137—21no—
78C2×C4⋊C4×C137D411yescharacteristic
79C2×C42×C137—21no—
80C2×C42×C137C2311yescharacteristic
81#107×C137—21no—
82#107×C137—21no—
83#106×C137—22no—
84#106×C137—22no—
85C4⋊C8×C137—22no—
86C4⋊C8×C137—22no—
87C4⋊C8×C137—21no—
88C4⋊C8×C137D412yes—
89C4⋊C8×C137C2312yes—
90C4⋊C8×C137C2312yes—
91C4⋊C8×C137C2311yescharacteristic
92C4⋊C8×C137C2311yescharacteristic
93#103×C137—21no—
94#103×C137—21no—
95#103×C137—22no—
96#103×C137—22no—
97#102×C137—22no—
98#102×C137—22no—
99#102×C137—21no—
100#102×C137—21no—
101C22⋊C8×C137—21no—
102C22⋊C8×C137—21no—
103C22⋊C8×C137D411yescharacteristic
104C22⋊C8×C137D411yescharacteristic
105#97×C137—22no—
106#97×C137—21no—
107#97×C137—21no—
108C4×C8×C137—22no—
109C4×C8×C137C2312yes—
110#41×C137—82no—
111C2×Q8×C137—21no—
112C2×Q8×C137—22no—
113C2×Q8×C137—22no—
114C2×Q8×C137—21no—
115C2×Q8×C137—41no—
116C2×Q8×C137—42no—
117C2×Q8×C137—42no—
118C2×Q8×C137—41no—
119C2×Q8×C137C2×D411yescharacteristic
120C2×Q8×C137C2×D411yescharacteristic
121C2×D4×C137—21no—
122C2×D4×C137—21no—
123C2×D4×C137—21no—
124C2×D4×C137—41no—
125C22×C4×C137—21no—
126C22×C4×C137—21no—
127C22×C4×C137—41no—
128C22×C4×C137—41no—
129C22×C4×C137C2×D411yescharacteristic
130C22×C4×C137C2×D411yescharacteristic
131C22×C4×C137#4111yescharacteristic
132Q16×C137—44no—
133Q16×C137—44no—
134SD16×C137—42no—
135SD16×C137—42no—
136C2×C8×C137—22no—
137C2×C8×C137—21no—
138C2×C8×C137—21no—
139C2×C8×C137—22no—
140C2×C8×C137—42no—
141C2×C8×C137C2×D412yes—
142C2×C8×C137C2×D412yes—
143C4⋊C4×C137—21no—
144C4⋊C4×C137—21no—
145C4⋊C4×C137—21no—
146C4⋊C4×C137—21no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—22no—
149C4⋊C4×C137—21no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—22no—
152C4⋊C4×C137—41no—
153C4⋊C4×C137—41no—
154C4⋊C4×C137C2×D411yescharacteristic
155C4⋊C4×C137C2×D411yescharacteristic
156C22⋊C4×C137—41no—
157C22⋊C4×C137—41no—
158C42×C137—21no—
159C42×C137—21no—
160C42×C137—21no—
161C42×C137—21no—
162C42×C137C2×D411yescharacteristic
163C42×C137C2×D411yescharacteristic
164C42×C137#4111yescharacteristic
165C42×C137C2411yescharacteristic
166C23×C137—41no—
167C23×C137#12111yescharacteristic
168Q8×C137—21no—
169Q8×C137—21no—
170Q8×C137—21no—
171Q8×C137—42no—
172Q8×C137—42no—
173Q8×C137—42no—
174Q8×C137—81no—
175Q8×C137—82no—
176Q8×C137AGL(1,ℤ/8)12yes—
177D4×C137—41no—
178D4×C137—41no—
179D4×C137—42no—
180D4×C137—81no—
181D4×C137—82no—
182C2×C4×C137—21no—
183C2×C4×C137—22no—
184C2×C4×C137—21no—
185C2×C4×C137—21no—
186C2×C4×C137—22no—
187C2×C4×C137—21no—
188C2×C4×C137—21no—
189C2×C4×C137—21no—
190C2×C4×C137—42no—
191C2×C4×C137—42no—
192C2×C4×C137—41no—
193C2×C4×C137—42no—
194C2×C4×C137—82no—
195C2×C4×C137—82no—
196C2×C4×C137#12111yescharacteristic
197C2×C4×C137#12111yescharacteristic
198C2×C4×C137#12111yescharacteristic
199C2×C4×C137C22×D411yescharacteristic
200C2×C4×C137C22×D411yescharacteristic
201C2×C4×C137C2×#4111yescharacteristic
202C8×C137—22no—
203C8×C137—22no—
204C8×C137—42no—
205C8×C137—41no—
206C8×C137—41no—
207C8×C137—42no—
208C22×C137—21no—
209C22×C137—21no—
210C22×C137—81no—
211C22×C137—81no—
212C22×C137—81no—
213C22×C137#47112yes—
214C22×C137C2×#12111yescharacteristic
215C4×C137—21no—
216C4×C137—21no—
217C4×C137—22no—
218C4×C137—21no—
219C4×C137—21no—
220C4×C137—41no—
221C4×C137—41no—
222C4×C137—41no—
223C4×C137—82no—
224C4×C137—81no—
225C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
226C4×C137#57511yescharacteristic
227C4×C137#57812yes—
228C2×C137—22no—
229C2×C137—161no—
230C2×C137#304411yescharacteristic
231C2×C137#305511yescharacteristic
232C2×C137C2×#47111yescharacteristic
233#19474C13711yesmaximal, characteristic, direct product factor, Sylow
234C137#1947411yescharacteristic, direct product factor, Sylow
235#3227C2×C13711yescharacteristic
236#3112C2×C13711yescharacteristic
237C2×#462C2×C13711yescharacteristic, complemented
238#2984C2×C13711yescharacteristic, complemented
239#1702C2×C13712yescomplemented
240#1692C2×C13712yescomplemented
241#1682C2×C13712yescomplemented
242#1672C2×C13712yescomplemented
243#1573C2×C13711yescharacteristic, complemented
244#1556C2×C13711yescharacteristic, complemented
245#1551C2×C13711yescharacteristic
246C22×Q16—21no—
247#532C22×C13711yescharacteristic
248C2×C4⋊Q8C22×C13711yescharacteristic
249C2×C4×Q8C22×C13711yescharacteristic
250#498C22×C13712yes—
251#486C22×C13712yes—
252#481—21no—
253#481—21no—
254#480—21no—
255#477C22×C13711yescharacteristic
256#477C22×C13711yescharacteristic
257#476—22no—
258#462—21no—
259#462—21no—
260#462C22×C13712yes—
261#462C22×C13712yes—
262#460C22×C13711yescharacteristic
263#460C22×C13711yescharacteristic
264#448—21no—
265#446—22no—
266#446C22×C13712yes—
267C8×Q8C22×C13712yes—
268#440—21no—
269#440—21no—
270#432—21no—
271#432C22×C13711yescharacteristic
272#432C22×C13711yescharacteristic
273#431—21no—
274#431C22×C13711yescharacteristic
275C2×C4⋊C8C22×C13711yescharacteristic
276C2×#103—21no—
277#335C22×C13712yes—
278#334C22×C13712yes—
279#333C22×C13712yes—
280#332C22×C13712yes—
281#332C22×C13712yes—
282#331C22×C13712yes—
283C2×#41—21no—
284C22×Q8—21no—
285C22×Q8D4×C13711yescharacteristic
286C2×Q16—22no—
287C2×Q16—22no—
288C2×Q16—44no—
289C2×SD16—21no—
290C2×SD16—21no—
291C22×C8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8C23×C13712yes—
296C4⋊Q8C23×C13711yescharacteristic
297#127C23×C13711yescharacteristic
298#124—21no—
299#121—21no—
300#121—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8D4×C13711yescharacteristic
304C4×Q8D4×C13711yescharacteristic
305C4×Q8C23×C13711yescharacteristic
306C4×Q8C23×C13711yescharacteristic
307C4×D4—21no—
308C4×D4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4D4×C13711yescharacteristic
313C2×C42—21no—
314C2×C42C23×C13711yescharacteristic
315#107—21no—
316#107—21no—
317#106—22no—
318#106—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—21no—
322C4⋊C8D4×C13712yes—
323C4⋊C8C23×C13712yes—
324C4⋊C8C23×C13712yes—
325C4⋊C8C23×C13711yescharacteristic
326C4⋊C8C23×C13711yescharacteristic
327#103—21no—
328#103—21no—
329#103—22no—
330#103—22no—
331#102—22no—
332#102—22no—
333#102—21no—
334#102—21no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8D4×C13711yescharacteristic
338C22⋊C8D4×C13711yescharacteristic
339#97—22no—
340#97—21no—
341#97—21no—
342C4×C8—22no—
343C4×C8C23×C13712yes—
344#41—82no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—41no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—41no—
353C2×Q8C2×D4×C13711yescharacteristic
354C2×Q8C2×D4×C13711yescharacteristic
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C2×D4×C13711yescharacteristic
364C22×C4C2×D4×C13711yescharacteristic
365C22×C4#41×C13711yescharacteristic
366Q16—44no—
367Q16—44no—
368SD16—42no—
369SD16—42no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—42no—
375C2×C8C2×D4×C13712yes—
376C2×C8C2×D4×C13712yes—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C2×D4×C13711yescharacteristic
389C4⋊C4C2×D4×C13711yescharacteristic
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42C2×D4×C13711yescharacteristic
397C42C2×D4×C13711yescharacteristic
398C42#41×C13711yescharacteristic
399C42C24×C13711yescharacteristic
400C23—41no—
401C23#121×C13711yescharacteristic
402Q8—21no—
403Q8—21no—
404Q8—21no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408Q8—81no—
409Q8—82no—
410Q8AGL(1,ℤ/8)×C13712yes—
411D4—41no—
412D4—41no—
413D4—42no—
414D4—81no—
415D4—82no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#121×C13711yescharacteristic
431C2×C4#121×C13711yescharacteristic
432C2×C4#121×C13711yescharacteristic
433C2×C4C22×D4×C13711yescharacteristic
434C2×C4C22×D4×C13711yescharacteristic
435C2×C4C2×#41×C13711yescharacteristic
436C8—22no—
437C8—22no—
438C8—42no—
439C8—41no—
440C8—41no—
441C8—42no—
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22#471×C13712yes—
448C22C2×#121×C13711yescharacteristic
449C4—21no—
450C4—21no—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—81no—
459C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
460C4#575×C13711yescharacteristic
461C4#578×C13712yes—
462C2—22no—
463C2—161no—
464C2#3044×C13711yescharacteristic
465C2#3055×C13711yescharacteristic
466C2C2×#471×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radical#532×C137C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#462×C137C2semidirect
4. #2984×C137C2semidirect
5. #1702×C137C2semidirect
6. #1692×C137C2semidirect
7. #1682×C137C2semidirect
8. #1672×C137C2semidirect
9. #1573×C137C2semidirect
10. #1556×C137C2semidirect
233. #19474C137direct (N × H)
234. C137#19474direct (N × H)
237. C2×#462C2×C137semidirect
238. #2984C2×C137semidirect
239. #1702C2×C137semidirect
240. #1692C2×C137semidirect
241. #1682C2×C137semidirect
242. #1672C2×C137semidirect
243. #1573C2×C137semidirect
244. #1556C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3227×C1372
#3112×C1372
C2×#462×C1372
#2984×C1372
#1702×C1372
#1692×C1372
#1682×C1372
#1672×C1372
#1573×C1372
#1556×C1372
#1551×C1372
#19474137