Finite Group Databasebeta

← C137×#19490 #46868876

Subgroups — C137×#19490 #46868876

Order 35072.

624 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3230×C137C211yesmaximal, characteristic, complemented
2#3107×C137C211yesmaximal, characteristic
3#3039×C137C211yesmaximal, characteristic
4#2990×C137C211yesmaximal, characteristic
5#1700×C137C212yesmaximal, complemented
6#1694×C137C212yesmaximal, complemented
7#1679×C137C212yesmaximal, complemented
8#1673×C137C212yesmaximal, complemented
9#1574×C137C211yesmaximal, characteristic, complemented
10#1561×C137C211yesmaximal, characteristic, complemented
11#1549×C137C211yesmaximal, characteristic, complemented
12C2×#137×C137—21no—
13#533×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15C4×#41×C137C2211yescharacteristic
16#500×C137C2212yes—
17#487×C137C2212yes—
18#485×C137—21no—
19#484×C137—22no—
20#477×C137—21no—
21#477×C137—21no—
22#475×C137C2211yescharacteristic
23#474×C137C2211yescharacteristic
24#462×C137—21no—
25#462×C137C2212yes—
26#462×C137C2211yescharacteristic
27#461×C137—21no—
28#461×C137C2212yes—
29#461×C137C2211yescharacteristic
30#452×C137—21no—
31#446×C137C2212yes—
32C4×Q16×C137—21no—
33C4×SD16×C137—21no—
34#435×C137—22no—
35C8×D4×C137C2212yes—
36#432×C137—21no—
37#431×C137C2211yescharacteristic
38#431×C137C2211yescharacteristic
39#423×C137C2211yescharacteristic
40#423×C137C2211yescharacteristic
41C2×C4⋊C8×C137—21no—
42#419×C137—21no—
43#335×C137C2212yes—
44#334×C137C2212yes—
45#332×C137C2212yes—
46#332×C137C2212yes—
47#332×C137C2212yes—
48#332×C137C2212yes—
49C2×#41×C137D411yescharacteristic
50C22×Q8×C137—21no—
51#137×C137—44no—
52C2×Q16×C137—21no—
53C2×Q16×C137—22no—
54C2×SD16×C137—21no—
55C2×SD16×C137—22no—
56C2×M16×C137—21no—
57C2×M16×C137—21no—
58C22×C8×C137—22no—
59C4⋊Q8×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137C2312yes—
63C4⋊Q8×C137C2311yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65#125×C137—21no—
66C22⋊Q8×C137—21no—
67C22⋊Q8×C137—21no—
68C4×Q8×C137—21no—
69C4×Q8×C137—21no—
70C4×Q8×C137D411yescharacteristic
71C4×Q8×C137C2311yescharacteristic
72C4×D4×C137—21no—
73C4×D4×C137—21no—
74C4×D4×C137D411yescharacteristic
75C4×D4×C137C2311yescharacteristic
76#117×C137—21no—
77#117×C137—21no—
78#117×C137D411yescharacteristic
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137—21no—
81C2×C42×C137—21no—
82C2×C42×C137C2311yescharacteristic
83#107×C137—22no—
84#107×C137—22no—
85#107×C137—21no—
86#106×C137—21no—
87C4⋊C8×C137—22no—
88C4⋊C8×C137—21no—
89C4⋊C8×C137—21no—
90C4⋊C8×C137D411yescharacteristic
91C4⋊C8×C137D411yescharacteristic
92C4⋊C8×C137C2312yes—
93C4⋊C8×C137C2312yes—
94C4⋊C8×C137C2311yescharacteristic
95C4⋊C8×C137C2311yescharacteristic
96#103×C137—21no—
97#103×C137—22no—
98#103×C137—22no—
99#103×C137—22no—
100#103×C137—22no—
101#103×C137—21no—
102#103×C137—21no—
103#102×C137—21no—
104C22⋊C8×C137—21no—
105C22⋊C8×C137—22no—
106C22⋊C8×C137D412yes—
107#97×C137—22no—
108C4×C8×C137—21no—
109C4×C8×C137C2312yes—
110#41×C137—22no—
111#41×C137—41no—
112C2×Q8×C137—21no—
113C2×Q8×C137—21no—
114C2×Q8×C137—22no—
115C2×Q8×C137—42no—
116C2×Q8×C137—42no—
117C2×Q8×C137—41no—
118C2×Q8×C137C2×D411yescharacteristic
119C2×D4×C137—21no—
120C2×D4×C137C2×D411yescharacteristic
121C22×C4×C137—21no—
122C22×C4×C137—21no—
123C22×C4×C137—21no—
124C22×C4×C137—41no—
125C22×C4×C137C2×D411yescharacteristic
126C22×C4×C137C2×D411yescharacteristic
127C22×C4×C137#4111yescharacteristic
128Q16×C137—42no—
129Q16×C137—44no—
130SD16×C137—42no—
131SD16×C137—44no—
132M16×C137—42no—
133M16×C137—81no—
134C2×C8×C137—21no—
135C2×C8×C137—21no—
136C2×C8×C137—22no—
137C2×C8×C137—22no—
138C2×C8×C137—82no—
139C2×C8×C137C2×D412yes—
140C2×C8×C137C2×D412yes—
141C4⋊C4×C137—21no—
142C4⋊C4×C137—21no—
143C4⋊C4×C137—21no—
144C4⋊C4×C137—22no—
145C4⋊C4×C137—21no—
146C4⋊C4×C137—22no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—41no—
150C4⋊C4×C137—41no—
151C4⋊C4×C137—41no—
152C4⋊C4×C137—41no—
153C4⋊C4×C137C2×D411yescharacteristic
154C4⋊C4×C137C2×D411yescharacteristic
155C22⋊C4×C137—21no—
156C22⋊C4×C137—21no—
157C22⋊C4×C137—21no—
158C22⋊C4×C137—41no—
159C42×C137—21no—
160C42×C137—21no—
161C42×C137—21no—
162C42×C137—41no—
163C42×C137C2×D411yescharacteristic
164C42×C137C2×D411yescharacteristic
165C42×C137#4111yescharacteristic
166C42×C137C2411yescharacteristic
167C23×C137—21no—
168C23×C137#12111yescharacteristic
169Q8×C137—21no—
170Q8×C137—21no—
171Q8×C137—42no—
172Q8×C137—41no—
173Q8×C137—41no—
174Q8×C137—42no—
175Q8×C137—41no—
176Q8×C137—41no—
177Q8×C137—82no—
178D4×C137—21no—
179D4×C137—42no—
180D4×C137AGL(1,ℤ/8)12yes—
181C2×C4×C137—21no—
182C2×C4×C137—21no—
183C2×C4×C137—21no—
184C2×C4×C137—21no—
185C2×C4×C137—22no—
186C2×C4×C137—21no—
187C2×C4×C137—21no—
188C2×C4×C137—21no—
189C2×C4×C137—42no—
190C2×C4×C137—41no—
191C2×C4×C137—41no—
192C2×C4×C137—42no—
193C2×C4×C137—41no—
194C2×C4×C137—41no—
195C2×C4×C137—42no—
196C2×C4×C137—82no—
197C2×C4×C137#12111yescharacteristic
198C2×C4×C137#12111yescharacteristic
199C2×C4×C137#12111yescharacteristic
200C2×C4×C137C22×D411yescharacteristic
201C2×C4×C137C22×D411yescharacteristic
202C2×C4×C137C2×#4111yescharacteristic
203C8×C137—22no—
204C8×C137—22no—
205C8×C137—42no—
206C8×C137—41no—
207C8×C137—41no—
208C8×C137—42no—
209C22×C137—21no—
210C22×C137—21no—
211C22×C137—22no—
212C22×C137—21no—
213C22×C137—41no—
214C22×C137—41no—
215C22×C137C2×#12111yescharacteristic
216C4×C137—21no—
217C4×C137—21no—
218C4×C137—21no—
219C4×C137—41no—
220C4×C137—41no—
221C4×C137—41no—
222C4×C137—81no—
223C4×C137—82no—
224C4×C137—81no—
225C4×C137#47312yes—
226C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C137#57511yescharacteristic
228C4×C137#57711yescharacteristic
229C4×C137#57811yescharacteristic
230C2×C137—41no—
231C2×C137—42no—
232C2×C137#304611yescharacteristic
233C2×C137#305511yescharacteristic
234C2×C137C2×#47311yescharacteristic
235#19490C13711yesmaximal, characteristic, direct product factor, Sylow
236C137#1949011yescharacteristic, direct product factor, Sylow
237#3230C2×C13711yescharacteristic, complemented
238#3107C2×C13711yescharacteristic
239#3039C2×C13711yescharacteristic
240#2990C2×C13711yescharacteristic
241#1700C2×C13712yescomplemented
242#1694C2×C13712yescomplemented
243#1679C2×C13712yescomplemented
244#1673C2×C13712yescomplemented
245#1574C2×C13711yescharacteristic, complemented
246#1561C2×C13711yescharacteristic, complemented
247#1549C2×C13711yescharacteristic, complemented
248C2×#137—21no—
249#533C22×C13711yescharacteristic
250C2×C4⋊Q8C22×C13711yescharacteristic
251C4×#41C22×C13711yescharacteristic
252#500C22×C13712yes—
253#487C22×C13712yes—
254#485—21no—
255#484—22no—
256#477—21no—
257#477—21no—
258#475C22×C13711yescharacteristic
259#474C22×C13711yescharacteristic
260#462—21no—
261#462C22×C13711yescharacteristic
262#462C22×C13712yes—
263#461—21no—
264#461C22×C13711yescharacteristic
265#461C22×C13712yes—
266#452—21no—
267#446C22×C13712yes—
268C4×Q16—21no—
269C4×SD16—21no—
270#435—22no—
271C8×D4C22×C13712yes—
272#432—21no—
273#431C22×C13711yescharacteristic
274#431C22×C13711yescharacteristic
275#423C22×C13711yescharacteristic
276#423C22×C13711yescharacteristic
277C2×C4⋊C8—21no—
278#419—21no—
279#335C22×C13712yes—
280#334C22×C13712yes—
281#332C22×C13712yes—
282#332C22×C13712yes—
283#332C22×C13712yes—
284#332C22×C13712yes—
285C2×#41D4×C13711yescharacteristic
286C22×Q8—21no—
287#137—44no—
288C2×Q16—22no—
289C2×Q16—21no—
290C2×SD16—22no—
291C2×SD16—21no—
292C2×M16—21no—
293C2×M16—21no—
294C22×C8—22no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8C23×C13712yes—
299C4⋊Q8C23×C13711yescharacteristic
300C4⋊Q8C23×C13711yescharacteristic
301#125—21no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8D4×C13711yescharacteristic
307C4×Q8C23×C13711yescharacteristic
308C4×D4—21no—
309C4×D4—21no—
310C4×D4D4×C13711yescharacteristic
311C4×D4C23×C13711yescharacteristic
312#117—21no—
313#117—21no—
314#117D4×C13711yescharacteristic
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C42—21no—
318C2×C42C23×C13711yescharacteristic
319#107—22no—
320#107—22no—
321#107—21no—
322#106—21no—
323C4⋊C8—22no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8D4×C13711yescharacteristic
327C4⋊C8D4×C13711yescharacteristic
328C4⋊C8C23×C13712yes—
329C4⋊C8C23×C13711yescharacteristic
330C4⋊C8C23×C13712yes—
331C4⋊C8C23×C13711yescharacteristic
332#103—21no—
333#103—22no—
334#103—22no—
335#103—22no—
336#103—22no—
337#103—21no—
338#103—21no—
339#102—21no—
340C22⋊C8—21no—
341C22⋊C8—22no—
342C22⋊C8D4×C13712yes—
343#97—22no—
344C4×C8—21no—
345C4×C8C23×C13712yes—
346#41—22no—
347#41—41no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—41no—
354C2×Q8C2×D4×C13711yescharacteristic
355C2×D4—21no—
356C2×D4C2×D4×C13711yescharacteristic
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4C2×D4×C13711yescharacteristic
362C22×C4C2×D4×C13711yescharacteristic
363C22×C4#41×C13711yescharacteristic
364Q16—42no—
365Q16—44no—
366SD16—42no—
367SD16—44no—
368M16—42no—
369M16—81no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—82no—
375C2×C8C2×D4×C13712yes—
376C2×C8C2×D4×C13712yes—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4C2×D4×C13711yescharacteristic
390C4⋊C4C2×D4×C13711yescharacteristic
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—41no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42C2×D4×C13711yescharacteristic
400C42C2×D4×C13711yescharacteristic
401C42#41×C13711yescharacteristic
402C42C24×C13711yescharacteristic
403C23—21no—
404C23#121×C13711yescharacteristic
405Q8—21no—
406Q8—21no—
407Q8—42no—
408Q8—41no—
409Q8—41no—
410Q8—42no—
411Q8—41no—
412Q8—41no—
413Q8—82no—
414D4—21no—
415D4—42no—
416D4AGL(1,ℤ/8)×C13712yes—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4#121×C13711yescharacteristic
434C2×C4#121×C13711yescharacteristic
435C2×C4#121×C13711yescharacteristic
436C2×C4C22×D4×C13711yescharacteristic
437C2×C4C22×D4×C13711yescharacteristic
438C2×C4C2×#41×C13711yescharacteristic
439C8—22no—
440C8—22no—
441C8—42no—
442C8—41no—
443C8—41no—
444C8—42no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22C2×#121×C13711yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—81no—
459C4—82no—
460C4—81no—
461C4#473×C13712yes—
462C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
463C4#575×C13711yescharacteristic
464C4#577×C13711yescharacteristic
465C4#578×C13711yescharacteristic
466C2—41no—
467C2—42no—
468C2#3046×C13711yescharacteristic
469C2#3055×C13711yescharacteristic
470C2C2×#473×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radicalC2×#41×C137D4
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
1. #3230×C137C2semidirect
5. #1700×C137C2semidirect
6. #1694×C137C2semidirect
7. #1679×C137C2semidirect
8. #1673×C137C2semidirect
9. #1574×C137C2semidirect
10. #1561×C137C2semidirect
11. #1549×C137C2semidirect
235. #19490C137direct (N × H)
236. C137#19490direct (N × H)
237. #3230C2×C137semidirect
241. #1700C2×C137semidirect
242. #1694C2×C137semidirect
243. #1679C2×C137semidirect
244. #1673C2×C137semidirect
245. #1574C2×C137semidirect
246. #1561C2×C137semidirect
247. #1549C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3230×C1372
#3107×C1372
#3039×C1372
#2990×C1372
#1700×C1372
#1694×C1372
#1679×C1372
#1673×C1372
#1574×C1372
#1561×C1372
#1549×C1372
#19490137