Finite Group Databasebeta

← C137×#28671 #46878057

Subgroups — C137×#28671 #46878057

Order 35072.

640 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3166×C137C211yesmaximal, characteristic, complemented
2#3140×C137C211yesmaximal, characteristic
3C2×#488×C137C211yesmaximal, characteristic
4#2960×C137C211yesmaximal, characteristic, complemented
5#2064×C137C212yesmaximal, complemented
6#2063×C137C212yesmaximal
7#1998×C137C211yesmaximal, characteristic, complemented
8#1967×C137C211yesmaximal, characteristic
9#1931×C137C212yesmaximal
10#1930×C137C212yesmaximal, complemented
11C4×#108×C137C211yesmaximal, characteristic, complemented
12#533×C137C2211yescharacteristic
13C2×#125×C137C2211yescharacteristic
14C2×#124×C137C2211yescharacteristic
15#501×C137—21no—
16#501×C137—21no—
17#499×C137—22no—
18#489×C137—21no—
19#489×C137—21no—
20#488×C137—22no—
21#488×C137—21no—
22#488×C137—21no—
23#488×C137—21no—
24#488×C137—21no—
25#446×C137—21no—
26#446×C137—21no—
27C8×Q8×C137—21no—
28C8×Q8×C137—21no—
29C2×#108×C137C2212yes—
30#428×C137C2212yes—
31#428×C137C2212yes—
32#424×C137C2212yes—
33#423×C137C2211yescharacteristic
34#419×C137C2212yes—
35#418×C137C2212yes—
36C2×#103×C137C2211yescharacteristic
37C2×#103×C137C2211yescharacteristic
38C2×#102×C137C2211yescharacteristic
39C2×#102×C137C2211yescharacteristic
40C2×#101×C137C2212yes—
41C2×#100×C137C2212yes—
42#405×C137C2212yes—
43C4×M16×C137C2211yescharacteristic
44C4×M16×C137C2211yescharacteristic
45C2×C4×C8×C137C2211yescharacteristic
46#343×C137C2212yes—
47#343×C137C2212yes—
48#341×C137C2212yes—
49C22×Q8×C137C2311yescharacteristic
50C22×D4×C137C2311yescharacteristic
51C2×M16×C137C2312yes—
52C2×M16×C137C2312yes—
53C2×M16×C137C2312yes—
54C22×C8×C137C2311yescharacteristic
55C22×C8×C137C2311yescharacteristic
56C4⋊Q8×C137D411yescharacteristic
57C4⋊Q8×C137D411yescharacteristic
58#125×C137—21no—
59#125×C137—21no—
60#125×C137—21no—
61#125×C137—21no—
62#125×C137D412yes—
63#124×C137—21no—
64#124×C137—21no—
65#124×C137—21no—
66#124×C137—21no—
67C22⋊Q8×C137—21no—
68C22⋊Q8×C137—21no—
69C4×Q8×C137—21no—
70C4×Q8×C137—21no—
71#117×C137C2312yes—
72C2×C4⋊C4×C137—21no—
73C2×C4⋊C4×C137—21no—
74C2×C4⋊C4×C137C2311yescharacteristic
75C2×C4⋊C4×C137C2311yescharacteristic
76C2×C22⋊C4×C137—21no—
77C2×C22⋊C4×C137—21no—
78C2×C42×C137C2311yescharacteristic
79#108×C137—24no—
80#107×C137—22no—
81#107×C137—22no—
82#106×C137—22no—
83#106×C137—22no—
84C4⋊C8×C137—22no—
85C4⋊C8×C137—22no—
86C4⋊C8×C137—21no—
87C4⋊C8×C137—21no—
88#103×C137—21no—
89#103×C137—21no—
90#103×C137—21no—
91#103×C137—21no—
92#103×C137—22no—
93#103×C137—22no—
94#102×C137—21no—
95#102×C137—21no—
96#102×C137—21no—
97#102×C137—21no—
98#102×C137—22no—
99#102×C137—22no—
100#101×C137—24no—
101#100×C137—24no—
102#97×C137—22no—
103#97×C137—21no—
104#97×C137—21no—
105C4×C8×C137—22no—
106C4×C8×C137—21no—
107C4×C8×C137—21no—
108C4×C8×C137D411yescharacteristic
109C4×C8×C137D411yescharacteristic
110C4×C8×C137D411yescharacteristic
111C4×C8×C137D411yescharacteristic
112C24×C137—21no—
113C2×Q8×C137—21no—
114C2×Q8×C137—21no—
115C2×Q8×C137—21no—
116C2×Q8×C137—21no—
117C2×Q8×C137—22no—
118C2×Q8×C137—22no—
119C2×D4×C137—21no—
120C2×D4×C137—21no—
121C2×D4×C137—22no—
122C2×D4×C137—22no—
123C22×C4×C137—21no—
124C22×C4×C137—21no—
125C22×C4×C137—21no—
126C22×C4×C137#4111yescharacteristic
127C22×C4×C137#4111yescharacteristic
128C22×C4×C137C2411yescharacteristic
129M16×C137—22no—
130M16×C137—22no—
131M16×C137—22no—
132M16×C137—22no—
133M16×C137—22no—
134M16×C137#4114yes—
135C2×C8×C137—22no—
136C2×C8×C137—22no—
137C2×C8×C137—22no—
138C2×C8×C137—22no—
139C2×C8×C137—22no—
140C2×C8×C137C2×D411yescharacteristic
141C2×C8×C137C2×D411yescharacteristic
142C2×C8×C137C2×D411yescharacteristic
143C2×C8×C137C2×D411yescharacteristic
144C4⋊C4×C137—22no—
145C4⋊C4×C137—22no—
146C4⋊C4×C137—21no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—21no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—41no—
153C4⋊C4×C137—41no—
154C4⋊C4×C137—41no—
155C4⋊C4×C137—41no—
156C4⋊C4×C137C2×D412yes—
157C4⋊C4×C137C2×D412yes—
158C22⋊C4×C137—22no—
159C22⋊C4×C137—41no—
160C22⋊C4×C137—41no—
161C22⋊C4×C137—41no—
162C22⋊C4×C137—41no—
163C42×C137—22no—
164C42×C137C2×D411yescharacteristic
165C42×C137C2×D411yescharacteristic
166C42×C137C2×D411yescharacteristic
167C42×C137C2×D411yescharacteristic
168C23×C137—41no—
169C23×C137—42no—
170C23×C137—81no—
171C23×C137—81no—
172C23×C137C2×#4111yescharacteristic
173Q8×C137—42no—
174Q8×C137—42no—
175Q8×C137—41no—
176Q8×C137—41no—
177Q8×C137—42no—
178D4×C137—41no—
179D4×C137—41no—
180D4×C137—42no—
181C2×C4×C137—22no—
182C2×C4×C137—22no—
183C2×C4×C137—41no—
184C2×C4×C137—41no—
185C2×C4×C137—41no—
186C2×C4×C137—41no—
187C2×C4×C137—41no—
188C2×C4×C137—42no—
189C2×C4×C137—81no—
190C2×C4×C137—81no—
191C2×C4×C137#12111yescharacteristic
192C2×C4×C137#12111yescharacteristic
193C2×C4×C137#12111yescharacteristic
194C2×C4×C137#12111yescharacteristic
195C2×C4×C137C22×D411yescharacteristic
196C2×C4×C137C22×D411yescharacteristic
197C2×C4×C137C2×#4112yes—
198C2×C4×C1372−1+412yes—
199C8×C137—21no—
200C8×C137—21no—
201C8×C137—21no—
202C8×C137—21no—
203C8×C137—24no—
204C8×C137—42no—
205C8×C137—42no—
206C22×C137—21no—
207C22×C137—21no—
208C22×C137—81no—
209C22×C137—81no—
210C22×C137—81no—
211C22×C137—162no—
212C22×C137C2×#12111yescharacteristic
213C22×C137#55612yes—
214C4×C137—21no—
215C4×C137—21no—
216C4×C137—21no—
217C4×C137—21no—
218C4×C137—42no—
219C4×C137—42no—
220C4×C137—81no—
221C4×C137—81no—
222C4×C137—81no—
223C4×C137#53211yescharacteristic
224C4×C137#53611yescharacteristic
225C4×C137#54812yes—
226C2×C137—22no—
227C2×C137—161no—
228C2×C137#254511yescharacteristic
229C2×C137#306711yescharacteristic
230C2×C137#306711yescharacteristic
231#28671C13711yesmaximal, characteristic, direct product factor, Sylow
232C137#2867111yescharacteristic, direct product factor, Sylow
233#3166C2×C13711yescharacteristic, complemented
234#3140C2×C13711yescharacteristic
235C2×#488C2×C13711yescharacteristic
236#2960C2×C13711yescharacteristic, complemented
237#2064C2×C13712yescomplemented
238#2063C2×C13712yes—
239#1998C2×C13711yescharacteristic, complemented
240#1967C2×C13711yescharacteristic
241#1931C2×C13712yes—
242#1930C2×C13712yescomplemented
243C4×#108C2×C13711yescharacteristic, complemented
244#533C22×C13711yescharacteristic
245C2×#125C22×C13711yescharacteristic
246C2×#124C22×C13711yescharacteristic
247#501—21no—
248#501—21no—
249#499—22no—
250#489—21no—
251#489—21no—
252#488—22no—
253#488—21no—
254#488—21no—
255#488—21no—
256#488—21no—
257#446—21no—
258#446—21no—
259C8×Q8—21no—
260C8×Q8—21no—
261C2×#108C22×C13712yes—
262#428C22×C13712yes—
263#428C22×C13712yes—
264#424C22×C13712yes—
265#423C22×C13711yescharacteristic
266#419C22×C13712yes—
267#418C22×C13712yes—
268C2×#103C22×C13711yescharacteristic
269C2×#103C22×C13711yescharacteristic
270C2×#102C22×C13711yescharacteristic
271C2×#102C22×C13711yescharacteristic
272C2×#101C22×C13712yes—
273C2×#100C22×C13712yes—
274#405C22×C13712yes—
275C4×M16C22×C13711yescharacteristic
276C4×M16C22×C13711yescharacteristic
277C2×C4×C8C22×C13711yescharacteristic
278#343C22×C13712yes—
279#343C22×C13712yes—
280#341C22×C13712yes—
281C22×Q8C23×C13711yescharacteristic
282C22×D4C23×C13711yescharacteristic
283C2×M16C23×C13712yes—
284C2×M16C23×C13712yes—
285C2×M16C23×C13712yes—
286C22×C8C23×C13711yescharacteristic
287C22×C8C23×C13711yescharacteristic
288C4⋊Q8D4×C13711yescharacteristic
289C4⋊Q8D4×C13711yescharacteristic
290#125—21no—
291#125—21no—
292#125—21no—
293#125—21no—
294#125D4×C13712yes—
295#124—21no—
296#124—21no—
297#124—21no—
298#124—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303#117C23×C13712yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C23×C13711yescharacteristic
307C2×C4⋊C4C23×C13711yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C42C23×C13711yescharacteristic
311#108—24no—
312#107—22no—
313#107—22no—
314#106—22no—
315#106—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320#103—21no—
321#103—21no—
322#103—21no—
323#103—21no—
324#103—22no—
325#103—22no—
326#102—21no—
327#102—21no—
328#102—21no—
329#102—21no—
330#102—22no—
331#102—22no—
332#101—24no—
333#100—24no—
334#97—22no—
335#97—21no—
336#97—21no—
337C4×C8—22no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8D4×C13711yescharacteristic
341C4×C8D4×C13711yescharacteristic
342C4×C8D4×C13711yescharacteristic
343C4×C8D4×C13711yescharacteristic
344C24—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4#41×C13711yescharacteristic
359C22×C4#41×C13711yescharacteristic
360C22×C4C24×C13711yescharacteristic
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16#41×C13714yes—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8C2×D4×C13711yescharacteristic
373C2×C8C2×D4×C13711yescharacteristic
374C2×C8C2×D4×C13711yescharacteristic
375C2×C8C2×D4×C13711yescharacteristic
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C2×D4×C13712yes—
389C4⋊C4C2×D4×C13712yes—
390C22⋊C4—22no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—22no—
396C42C2×D4×C13711yescharacteristic
397C42C2×D4×C13711yescharacteristic
398C42C2×D4×C13711yescharacteristic
399C42C2×D4×C13711yescharacteristic
400C23—41no—
401C23—42no—
402C23—81no—
403C23—81no—
404C23C2×#41×C13711yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—41no—
408Q8—41no—
409Q8—42no—
410D4—41no—
411D4—41no—
412D4—42no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4#121×C13711yescharacteristic
424C2×C4#121×C13711yescharacteristic
425C2×C4#121×C13711yescharacteristic
426C2×C4#121×C13711yescharacteristic
427C2×C4C22×D4×C13711yescharacteristic
428C2×C4C22×D4×C13711yescharacteristic
429C2×C4C2×#41×C13712yes—
430C2×C42−1+4×C13712yes—
431C8—21no—
432C8—21no—
433C8—21no—
434C8—21no—
435C8—24no—
436C8—42no—
437C8—42no—
438C22—21no—
439C22—21no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—162no—
444C22C2×#121×C13711yescharacteristic
445C22#556×C13712yes—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—42no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4#532×C13711yescharacteristic
456C4#536×C13711yescharacteristic
457C4#548×C13712yes—
458C2—22no—
459C2—161no—
460C2#2545×C13711yescharacteristic
461C2#3067×C13711yescharacteristic
462C2#3067×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC22×D4×C137C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3166×C137C2semidirect
4. #2960×C137C2semidirect
5. #2064×C137C2semidirect
7. #1998×C137C2semidirect
10. #1930×C137C2semidirect
11. C4×#108×C137C2semidirect
231. #28671C137direct (N × H)
232. C137#28671direct (N × H)
233. #3166C2×C137semidirect
236. #2960C2×C137semidirect
237. #2064C2×C137semidirect
239. #1998C2×C137semidirect
242. #1930C2×C137semidirect
243. C4×#108C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3166×C1372
#3140×C1372
C2×#488×C1372
#2960×C1372
#2064×C1372
#2063×C1372
#1998×C1372
#1967×C1372
#1931×C1372
#1930×C1372
C4×#108×C1372
#28671137