Finite Group Databasebeta

← C137×#12093 #46861479

Subgroups — C137×#12093 #46861479

Order 35072.

630 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2812×C137C211yesmaximal, characteristic, complemented
2#2021×C137C211yesmaximal, characteristic, complemented
3#1607×C137C211yesmaximal, characteristic, complemented
4#1595×C137C211yesmaximal, characteristic, complemented
5#1344×C137C211yesmaximal, characteristic, complemented
6#1343×C137C211yesmaximal, characteristic, complemented
7#1316×C137C211yesmaximal, characteristic, complemented
8#534×C137C2211yescharacteristic
9C2×#120×C137—21no—
10C2×#120×C137—21no—
11#480×C137—21no—
12#447×C137—21no—
13C2×#99×C137—21no—
14#397×C137—21no—
15#397×C137C2211yescharacteristic
16#394×C137—21no—
17#394×C137—21no—
18#394×C137—21no—
19#392×C137—21no—
20#392×C137C2211yescharacteristic
21#379×C137C2211yescharacteristic
22#342×C137C2211yescharacteristic
23#327×C137—21no—
24#327×C137—21no—
25#323×C137C2211yescharacteristic
26#323×C137C2211yescharacteristic
27C25×C137D411yescharacteristic
28C22×D4×C137—21no—
29C22×D4×C137—21no—
30C22×D4×C137—21no—
31C22×D4×C137—21no—
32C22×D4×C137—41no—
33C22×D4×C137D411yescharacteristic, complemented
34C2×D8×C137—41no—
35#123×C137—41no—
36#121×C137—21no—
37#121×C137—21no—
38#120×C137—42no—
39#120×C137—41no—
40#120×C137—42no—
41#120×C137—42no—
42#120×C137—42no—
43C2×C4⋊C4×C137—21no—
44C2×C4⋊C4×C137D411yescharacteristic, complemented
45C2×C22⋊C4×C137—21no—
46C2×C22⋊C4×C137—21no—
47C2×C22⋊C4×C137—21no—
48C2×C22⋊C4×C137—21no—
49C2×C22⋊C4×C137—21no—
50C2×C22⋊C4×C137—21no—
51C2×C22⋊C4×C137—21no—
52C2×C22⋊C4×C137—21no—
53C2×C22⋊C4×C137—21no—
54C2×C22⋊C4×C137—41no—
55C2×C22⋊C4×C137D411yescharacteristic
56C2×C22⋊C4×C137C2311yescharacteristic
57#107×C137—41no—
58#102×C137—41no—
59#102×C137—41no—
60#99×C137—42no—
61C22⋊C8×C137—21no—
62C22⋊C8×C137—21no—
63#95×C137—21no—
64#95×C137—21no—
65#95×C137D411yescharacteristic, complemented
66#95×C137D411yescharacteristic, complemented
67C24×C137—21no—
68C24×C137—21no—
69C24×C137—21no—
70C24×C137—21no—
71C24×C137—41no—
72C24×C137—41no—
73C24×C137—81no—
74C24×C137—81no—
75C24×C137D812yescomplemented
76C24×C137C2×D411yescharacteristic
77C2×D4×C137—21no—
78C2×D4×C137—21no—
79C2×D4×C137—41no—
80C2×D4×C137—41no—
81C2×D4×C137—41no—
82C2×D4×C137—42no—
83C2×D4×C137—42no—
84C2×D4×C137—41no—
85C2×D4×C137—44no—
86C2×D4×C137—41no—
87C2×D4×C137—41no—
88C2×D4×C137—41no—
89C2×D4×C137—41no—
90C2×D4×C137—41no—
91C2×D4×C137—81no—
92C2×D4×C137—81no—
93C2×D4×C137—81no—
94C2×D4×C137—81no—
95C2×D4×C137—81no—
96C2×D4×C137—81no—
97C2×D4×C137—81no—
98C2×D4×C137—82no—
99C2×D4×C137—82no—
100C22×C4×C137—21no—
101C22×C4×C137—21no—
102C22×C4×C137—21no—
103C22×C4×C137—21no—
104C22×C4×C137—21no—
105C22×C4×C137—41no—
106C22×C4×C137C2×D411yescharacteristic
107C22×C4×C137C2×D411yescharacteristic
108D8×C137—82no—
109C2×C8×C137—41no—
110C2×C8×C137—41no—
111C4⋊C4×C137—21no—
112C4⋊C4×C137—21no—
113C4⋊C4×C137—41no—
114C4⋊C4×C137—81no—
115C22⋊C4×C137—22no—
116C22⋊C4×C137—41no—
117C22⋊C4×C137—41no—
118C22⋊C4×C137—41no—
119C22⋊C4×C137—42no—
120C22⋊C4×C137—42no—
121C22⋊C4×C137—44no—
122C22⋊C4×C137—41no—
123C22⋊C4×C137—41no—
124C22⋊C4×C137—41no—
125C22⋊C4×C137—41no—
126C22⋊C4×C137—81no—
127C22⋊C4×C137—81no—
128C22⋊C4×C137—81no—
129C22⋊C4×C137—81no—
130C22⋊C4×C137—81no—
131C23×C137—21no—
132C23×C137—21no—
133C23×C137—22no—
134C23×C137—42no—
135C23×C137—41no—
136C23×C137—41no—
137C23×C137—41no—
138C23×C137—41no—
139C23×C137—41no—
140C23×C137—41no—
141C23×C137—41no—
142C23×C137—41no—
143C23×C137—41no—
144C23×C137—41no—
145C23×C137—41no—
146C23×C137—41no—
147C23×C137—42no—
148C23×C137—42no—
149C23×C137—41no—
150C23×C137—41no—
151C23×C137—41no—
152C23×C137—41no—
153C23×C137—41no—
154C23×C137—41no—
155C23×C137—81no—
156C23×C137—82no—
157C23×C137—82no—
158C23×C137—81no—
159C23×C137—81no—
160C23×C137—84no—
161C23×C137—81no—
162C23×C137—81no—
163C23×C137#12011yescharacteristic
164C23×C137C2×D811yescharacteristic
165C23×C137AGL(1,ℤ/8)11yescharacteristic
166D4×C137—42no—
167D4×C137—81no—
168D4×C137—81no—
169D4×C137—81no—
170D4×C137—81no—
171D4×C137—81no—
172D4×C137—82no—
173D4×C137—82no—
174D4×C137—84no—
175D4×C137—88no—
176D4×C137—84no—
177D4×C137—81no—
178D4×C137—81no—
179C2×C4×C137—21no—
180C2×C4×C137—21no—
181C2×C4×C137—41no—
182C2×C4×C137—41no—
183C2×C4×C137—42no—
184C2×C4×C137—41no—
185C2×C4×C137—41no—
186C2×C4×C137—41no—
187C2×C4×C137—41no—
188C2×C4×C137—41no—
189C2×C4×C137—41no—
190C2×C4×C137—41no—
191C2×C4×C137—41no—
192C2×C4×C137—41no—
193C2×C4×C137—41no—
194C2×C4×C137—41no—
195C2×C4×C137—41no—
196C2×C4×C137—41no—
197C2×C4×C137—41no—
198C2×C4×C137—41no—
199C2×C4×C137—81no—
200C2×C4×C137—81no—
201C8×C137—81no—
202C8×C137—81no—
203C22×C137—21no—
204C22×C137—21no—
205C22×C137—21no—
206C22×C137—21no—
207C22×C137—41no—
208C22×C137—41no—
209C22×C137—42no—
210C22×C137—42no—
211C22×C137—41no—
212C22×C137—41no—
213C22×C137—44no—
214C22×C137—42no—
215C22×C137—81no—
216C22×C137—81no—
217C22×C137—81no—
218C22×C137—81no—
219C22×C137—81no—
220C22×C137—81no—
221C22×C137—81no—
222C22×C137—82no—
223C22×C137—82no—
224C22×C137—82no—
225C22×C137—82no—
226C22×C137—81no—
227C22×C137—81no—
228C22×C137—81no—
229C22×C137—81no—
230C22×C137—81no—
231C22×C137—81no—
232C22×C137—81no—
233C22×C137—81no—
234C22×C137—161no—
235C22×C137#44711yescharacteristic
236C22×C137#45311yescharacteristic
237C22×C137B(4,𝔽2)11yescharacteristic
238C4×C137—41no—
239C4×C137—41no—
240C4×C137—81no—
241C4×C137—81no—
242C4×C137—81no—
243C4×C137—82no—
244C4×C137—81no—
245C4×C137—81no—
246C2×C137—21no—
247C2×C137—21no—
248C2×C137—41no—
249C2×C137—41no—
250C2×C137—82no—
251C2×C137—161no—
252C2×C137—161no—
253C2×C137#159511yescharacteristic
254C2×C137#219911yescharacteristic
255C2×C137#220011yescharacteristic
256#12093C13711yesmaximal, characteristic, direct product factor, Sylow
257C137#1209311yescharacteristic, direct product factor, Sylow
258#2812C2×C13711yescharacteristic, complemented
259#2021C2×C13711yescharacteristic, complemented
260#1607C2×C13711yescharacteristic, complemented
261#1595C2×C13711yescharacteristic, complemented
262#1344C2×C13711yescharacteristic, complemented
263#1343C2×C13711yescharacteristic, complemented
264#1316C2×C13711yescharacteristic, complemented
265#534C22×C13711yescharacteristic
266C2×#120—21no—
267C2×#120—21no—
268#480—21no—
269#447—21no—
270C2×#99—21no—
271#397—21no—
272#397C22×C13711yescharacteristic
273#394—21no—
274#394—21no—
275#394—21no—
276#392—21no—
277#392C22×C13711yescharacteristic
278#379C22×C13711yescharacteristic
279#342C22×C13711yescharacteristic
280#327—21no—
281#327—21no—
282#323C22×C13711yescharacteristic
283#323C22×C13711yescharacteristic
284C25D4×C13711yescharacteristic
285C22×D4—21no—
286C22×D4—21no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—41no—
290C22×D4D4×C13711yescharacteristic, complemented
291C2×D8—41no—
292#123—41no—
293#121—21no—
294#121—21no—
295#120—42no—
296#120—41no—
297#120—42no—
298#120—42no—
299#120—42no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4D4×C13711yescharacteristic, complemented
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4D4×C13711yescharacteristic
313C2×C22⋊C4C23×C13711yescharacteristic
314#107—41no—
315#102—41no—
316#102—41no—
317#99—42no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320#95—21no—
321#95—21no—
322#95D4×C13711yescharacteristic, complemented
323#95D4×C13711yescharacteristic, complemented
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—41no—
329C24—41no—
330C24—81no—
331C24—81no—
332C24D8×C13712yescomplemented
333C24C2×D4×C13711yescharacteristic
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—81no—
349C2×D4—81no—
350C2×D4—81no—
351C2×D4—81no—
352C2×D4—81no—
353C2×D4—81no—
354C2×D4—81no—
355C2×D4—82no—
356C2×D4—82no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4C2×D4×C13711yescharacteristic
364C22×C4C2×D4×C13711yescharacteristic
365D8—82no—
366C2×C8—41no—
367C2×C8—41no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—81no—
372C22⋊C4—22no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C22⋊C4—81no—
387C22⋊C4—81no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23—81no—
416C23—81no—
417C23—84no—
418C23—81no—
419C23—81no—
420C23#120×C13711yescharacteristic
421C23C2×D8×C13711yescharacteristic
422C23AGL(1,ℤ/8)×C13711yescharacteristic
423D4—42no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—82no—
430D4—82no—
431D4—84no—
432D4—88no—
433D4—84no—
434D4—81no—
435D4—81no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—81no—
457C2×C4—81no—
458C8—81no—
459C8—81no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—41no—
465C22—41no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—41no—
470C22—44no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—161no—
492C22#447×C13711yescharacteristic
493C22#453×C13711yescharacteristic
494C22B(4,𝔽2)×C13711yescharacteristic
495C4—41no—
496C4—41no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—82no—
501C4—81no—
502C4—81no—
503C2—21no—
504C2—21no—
505C2—41no—
506C2—41no—
507C2—82no—
508C2—161no—
509C2—161no—
510C2#1595×C13711yescharacteristic
511C2#2199×C13711yescharacteristic
512C2#2200×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C137
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C2×C22⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2812×C137C2semidirect
2. #2021×C137C2semidirect
3. #1607×C137C2semidirect
4. #1595×C137C2semidirect
5. #1344×C137C2semidirect
6. #1343×C137C2semidirect
7. #1316×C137C2semidirect
33. C22×D4×C137D4semidirect
44. C2×C4⋊C4×C137D4semidirect
65. #95×C137D4semidirect
66. #95×C137D4semidirect
75. C24×C137D8semidirect
256. #12093C137direct (N × H)
257. C137#12093direct (N × H)
258. #2812C2×C137semidirect
259. #2021C2×C137semidirect
260. #1607C2×C137semidirect
261. #1595C2×C137semidirect
262. #1344C2×C137semidirect
263. #1343C2×C137semidirect
264. #1316C2×C137semidirect
290. C22×D4D4×C137semidirect
301. C2×C4⋊C4D4×C137semidirect
322. #95D4×C137semidirect
323. #95D4×C137semidirect
332. C24D8×C137semidirect

Maximal subgroups

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

subgroupindex
#2812×C1372
#2021×C1372
#1607×C1372
#1595×C1372
#1344×C1372
#1343×C1372
#1316×C1372
#12093137