Finite Group Databasebeta

← C137×#18899 #46868285

Subgroups — C137×#18899 #46868285

Order 35072.

882 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507×C137C211yesmaximal, characteristic
2#3227×C137C211yesmaximal, characteristic
3#3209×C137C211yesmaximal, characteristic, complemented
4C2×#477×C137C211yesmaximal, characteristic, complemented
5#1638×C137C214yesmaximal, complemented
6#1632×C137C214yesmaximal, complemented
7#1542×C137C211yesmaximal, characteristic, complemented
8#1480×C137C211yesmaximal, characteristic
9#1479×C137C211yesmaximal, characteristic, complemented
10#559×C137—21no—
11#557×C137—21no—
12#556×C137—22no—
13#556×C137C2212yes—
14#554×C137C2212yes—
15#552×C137—22no—
16#539×C137—22no—
17#536×C137—21no—
18#533×C137C2211yescharacteristic
19#532×C137C2211yescharacteristic
20C2×C4⋊Q8×C137C2211yescharacteristic
21#529×C137—22no—
22#520×C137—21no—
23C2×C4×Q8×C137C2211yescharacteristic
24#498×C137—21no—
25#494×C137C2212yes—
26#487×C137C2212yes—
27#486×C137—21no—
28#481×C137—22no—
29#477×C137—22no—
30#477×C137C2214yes—
31#476×C137C2214yes—
32#462×C137—22no—
33#451×C137—22no—
34#431×C137C2211yescharacteristic
35#431×C137C2211yescharacteristic
36C2×#107×C137—21no—
37C2×C4⋊C8×C137C2211yescharacteristic
38C2×#103×C137—21no—
39#333×C137C2212yes—
40#332×C137C2212yes—
41Q8⋊C8×C137C2214yes—
42D4⋊C8×C137C2214yes—
43C2×#41×C137—21no—
44C22×Q8×C137—21no—
45C22×Q8×C137D411yescharacteristic
46C2×Q16×C137—24no—
47C22×C8×C137—21no—
48C4⋊Q8×C137—22no—
49C4⋊Q8×C137—21no—
50C4⋊Q8×C137—21no—
51C4⋊Q8×C137—21no—
52C4⋊Q8×C137D411yescharacteristic
53C4⋊Q8×C137C2312yes—
54C4⋊Q8×C137C2311yescharacteristic
55C4⋊Q8×C137C2311yescharacteristic
56#127×C137D411yescharacteristic
57#126×C137—24no—
58#126×C137—24no—
59#125×C137—24no—
60#125×C137—22no—
61#125×C137—21no—
62#124×C137—22no—
63#124×C137—22no—
64#124×C137—21no—
65#123×C137—24no—
66#123×C137—24no—
67C22⋊Q8×C137—22no—
68C22⋊Q8×C137—22no—
69C22⋊Q8×C137—24no—
70C22⋊Q8×C137—21no—
71C22⋊Q8×C137D412yes—
72#121×C137—21no—
73#121×C137D412yes—
74C4×Q8×C137—22no—
75C4×Q8×C137—22no—
76C4×Q8×C137—21no—
77C4×Q8×C137—21no—
78C4×Q8×C137—21no—
79C4×Q8×C137D412yes—
80C4×Q8×C137D412yes—
81C4×Q8×C137C2312yes—
82C4×D4×C137—24no—
83C4×D4×C137—22no—
84C4×D4×C137—22no—
85C4×D4×C137—21no—
86C4×D4×C137C2312yes—
87#117×C137—24no—
88#117×C137—21no—
89#117×C137—22no—
90C2×C4⋊C4×C137—21no—
91C2×C4⋊C4×C137—22no—
92C2×C4⋊C4×C137—21no—
93C2×C4⋊C4×C137—21no—
94C2×C4⋊C4×C137D411yescharacteristic
95C2×C42×C137—21no—
96C2×C42×C137C2311yescharacteristic
97#107×C137—24no—
98#106×C137—24no—
99C4⋊C8×C137—21no—
100C4⋊C8×C137—21no—
101C4⋊C8×C137C2312yes—
102C4⋊C8×C137C2312yes—
103#103×C137—24no—
104#103×C137—24no—
105#102×C137—24no—
106C22⋊C8×C137—21no—
107C22⋊C8×C137—21no—
108C4×C8×C137—21no—
109C4×C8×C137C2312yes—
110#41×C137—42no—
111#41×C137—42no—
112C2×Q8×C137—21no—
113C2×Q8×C137—21no—
114C2×Q8×C137—21no—
115C2×Q8×C137—22no—
116C2×Q8×C137—22no—
117C2×Q8×C137—22no—
118C2×Q8×C137—41no—
119C2×Q8×C137—82no—
120C2×Q8×C137C2×D412yes—
121C2×Q8×C137C2×D412yes—
122C2×D4×C137—22no—
123C2×D4×C137—21no—
124C2×D4×C137—21no—
125C2×D4×C137C2×D412yes—
126C22×C4×C137—21no—
127C22×C4×C137—24no—
128C22×C4×C137—21no—
129C22×C4×C137—22no—
130C22×C4×C137—41no—
131C22×C4×C137C2×D411yescharacteristic
132C22×C4×C137C2×D411yescharacteristic
133C22×C4×C137C2×D411yescharacteristic
134Q16×C137—84no—
135C2×C8×C137—22no—
136C2×C8×C137—22no—
137C2×C8×C137—22no—
138C2×C8×C137—42no—
139C4⋊C4×C137—24no—
140C4⋊C4×C137—24no—
141C4⋊C4×C137—24no—
142C4⋊C4×C137—22no—
143C4⋊C4×C137—21no—
144C4⋊C4×C137—22no—
145C4⋊C4×C137—21no—
146C4⋊C4×C137—22no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—22no—
149C4⋊C4×C137—21no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—22no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—41no—
154C4⋊C4×C137—41no—
155C4⋊C4×C137C2×D412yes—
156C4⋊C4×C137C2×D412yes—
157C4⋊C4×C137C2×D412yes—
158C22⋊C4×C137—24no—
159C22⋊C4×C137—24no—
160C22⋊C4×C137—24no—
161C22⋊C4×C137—22no—
162C22⋊C4×C137—22no—
163C22⋊C4×C137—22no—
164C42×C137—24no—
165C42×C137—22no—
166C42×C137—22no—
167C42×C137—22no—
168C42×C137C2×D411yescharacteristic
169C42×C137C2×D411yescharacteristic
170C42×C137C2×D411yescharacteristic
171C42×C137C2411yescharacteristic
172C23×C137—22no—
173C23×C137#12011yescharacteristic
174Q8×C137—22no—
175Q8×C137—22no—
176Q8×C137—42no—
177Q8×C137—44no—
178Q8×C137—42no—
179Q8×C137—42no—
180Q8×C137—42no—
181Q8×C137—82no—
182D4×C137—24no—
183D4×C137—44no—
184D4×C137—42no—
185D4×C137—81no—
186C2×C4×C137—22no—
187C2×C4×C137—22no—
188C2×C4×C137—22no—
189C2×C4×C137—22no—
190C2×C4×C137—22no—
191C2×C4×C137—22no—
192C2×C4×C137—41no—
193C2×C4×C137—42no—
194C2×C4×C137—42no—
195C2×C4×C137—42no—
196C2×C4×C137—42no—
197C2×C4×C137—81no—
198C2×C4×C137—84no—
199C2×C4×C137—82no—
200C2×C4×C137#12011yescharacteristic
201C2×C4×C137#12011yescharacteristic
202C2×C4×C137#12011yescharacteristic
203C2×C4×C137#13512yes—
204C2×C4×C137#13712yes—
205C2×C4×C137C22×D411yescharacteristic
206C2×C4×C137C22×D411yescharacteristic
207C2×C4×C137C22×D411yescharacteristic
208C8×C137—42no—
209C8×C137—42no—
210C8×C137—42no—
211C22×C137—21no—
212C22×C137—21no—
213C22×C137—42no—
214C22×C137—42no—
215C22×C137—42no—
216C22×C137#45212yes—
217C22×C137C2×#12011yescharacteristic
218C4×C137—21no—
219C4×C137—21no—
220C4×C137—21no—
221C4×C137—42no—
222C4×C137—42no—
223C4×C137—42no—
224C4×C137—42no—
225C4×C137—42no—
226C4×C137—82no—
227C4×C137#45212yes—
228C4×C137C2×#13511yescharacteristic
229C4×C137C2×#13711yescharacteristic
230C4×C137#57812yes—
231C2×C137—22no—
232C2×C137—82no—
233C2×C137C2×#45211yescharacteristic
234C2×C137#301111yescharacteristic
235C2×C137#302111yescharacteristic
236#18899C13711yesmaximal, characteristic, direct product factor, Sylow
237C137#1889911yescharacteristic, direct product factor, Sylow
238#3507C2×C13711yescharacteristic
239#3227C2×C13711yescharacteristic
240#3209C2×C13711yescharacteristic, complemented
241C2×#477C2×C13711yescharacteristic, complemented
242#1638C2×C13714yescomplemented
243#1632C2×C13714yescomplemented
244#1542C2×C13711yescharacteristic, complemented
245#1480C2×C13711yescharacteristic
246#1479C2×C13711yescharacteristic, complemented
247#559—21no—
248#557—21no—
249#556—22no—
250#556C22×C13712yes—
251#554C22×C13712yes—
252#552—22no—
253#539—22no—
254#536—21no—
255#533C22×C13711yescharacteristic
256#532C22×C13711yescharacteristic
257C2×C4⋊Q8C22×C13711yescharacteristic
258#529—22no—
259#520—21no—
260C2×C4×Q8C22×C13711yescharacteristic
261#498—21no—
262#494C22×C13712yes—
263#487C22×C13712yes—
264#486—21no—
265#481—22no—
266#477—22no—
267#477C22×C13714yes—
268#476C22×C13714yes—
269#462—22no—
270#451—22no—
271#431C22×C13711yescharacteristic
272#431C22×C13711yescharacteristic
273C2×#107—21no—
274C2×C4⋊C8C22×C13711yescharacteristic
275C2×#103—21no—
276#333C22×C13712yes—
277#332C22×C13712yes—
278Q8⋊C8C22×C13714yes—
279D4⋊C8C22×C13714yes—
280C2×#41—21no—
281C22×Q8—21no—
282C22×Q8D4×C13711yescharacteristic
283C2×Q16—24no—
284C22×C8—21no—
285C4⋊Q8—22no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8D4×C13711yescharacteristic
290C4⋊Q8C23×C13712yes—
291C4⋊Q8C23×C13711yescharacteristic
292C4⋊Q8C23×C13711yescharacteristic
293#127D4×C13711yescharacteristic
294#126—24no—
295#126—24no—
296#125—24no—
297#125—22no—
298#125—21no—
299#124—22no—
300#124—22no—
301#124—21no—
302#123—24no—
303#123—24no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—24no—
307C22⋊Q8—21no—
308C22⋊Q8D4×C13712yes—
309#121—21no—
310#121D4×C13712yes—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8D4×C13712yes—
317C4×Q8D4×C13712yes—
318C4×Q8C23×C13712yes—
319C4×D4—24no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—21no—
323C4×D4C23×C13712yes—
324#117—24no—
325#117—21no—
326#117—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4D4×C13711yescharacteristic
332C2×C42—21no—
333C2×C42C23×C13711yescharacteristic
334#107—24no—
335#106—24no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8C23×C13712yes—
339C4⋊C8C23×C13712yes—
340#103—24no—
341#103—24no—
342#102—24no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C4×C8—21no—
346C4×C8C23×C13712yes—
347#41—42no—
348#41—42no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—41no—
356C2×Q8—82no—
357C2×Q8C2×D4×C13712yes—
358C2×Q8C2×D4×C13712yes—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4C2×D4×C13712yes—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4C2×D4×C13711yescharacteristic
369C22×C4C2×D4×C13711yescharacteristic
370C22×C4C2×D4×C13711yescharacteristic
371Q16—84no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4C2×D4×C13712yes—
393C4⋊C4C2×D4×C13712yes—
394C4⋊C4C2×D4×C13712yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C42—24no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42C2×D4×C13711yescharacteristic
406C42C2×D4×C13711yescharacteristic
407C42C2×D4×C13711yescharacteristic
408C42C24×C13711yescharacteristic
409C23—22no—
410C23#120×C13711yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—42no—
414Q8—44no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—82no—
419D4—24no—
420D4—44no—
421D4—42no—
422D4—81no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—81no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4#120×C13711yescharacteristic
438C2×C4#120×C13711yescharacteristic
439C2×C4#120×C13711yescharacteristic
440C2×C4#135×C13712yes—
441C2×C4#137×C13712yes—
442C2×C4C22×D4×C13711yescharacteristic
443C2×C4C22×D4×C13711yescharacteristic
444C2×C4C22×D4×C13711yescharacteristic
445C8—42no—
446C8—42no—
447C8—42no—
448C22—21no—
449C22—21no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22#452×C13712yes—
454C22C2×#120×C13711yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4#452×C13712yes—
465C4C2×#135×C13711yescharacteristic
466C4C2×#137×C13711yescharacteristic
467C4#578×C13712yes—
468C2—22no—
469C2—82no—
470C2C2×#452×C13711yescharacteristic
471C2#3011×C13711yescharacteristic
472C2#3021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
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. #3209×C137C2semidirect
4. C2×#477×C137C2semidirect
5. #1638×C137C2semidirect
6. #1632×C137C2semidirect
7. #1542×C137C2semidirect
9. #1479×C137C2semidirect
236. #18899C137direct (N × H)
237. C137#18899direct (N × H)
240. #3209C2×C137semidirect
241. C2×#477C2×C137semidirect
242. #1638C2×C137semidirect
243. #1632C2×C137semidirect
244. #1542C2×C137semidirect
246. #1479C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3507×C1372
#3227×C1372
#3209×C1372
C2×#477×C1372
#1638×C1372
#1632×C1372
#1542×C1372
#1480×C1372
#1479×C1372
#18899137