Finite Group Databasebeta

← C137×#32974 #46882360

Subgroups — C137×#32974 #46882360

Order 35072.

1186 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C137C211yesmaximal, characteristic
2#3080×C137C212yesmaximal, complemented
3#2565×C137C211yesmaximal, characteristic, complemented
4#1860×C137C212yesmaximal, complemented
5#1859×C137C212yesmaximal, complemented
6#1857×C137C211yesmaximal, characteristic, complemented
7#1857×C137C211yesmaximal, characteristic, complemented
8#1806×C137C212yesmaximal, complemented
9#1801×C137C211yesmaximal, characteristic, complemented
10#1793×C137C211yesmaximal, characteristic
11#1793×C137C211yesmaximal, characteristic
12C2×2−1+4×C137C2211yescharacteristic
13C2×#125×C137C2211yescharacteristic
14#520×C137—22no—
15#520×C137—22no—
16#520×C137C2212yes—
17#519×C137—22no—
18#519×C137C2211yescharacteristic
19C4×#41×C137—22no—
20C4×#41×C137—22no—
21C4×#41×C137C2212yes—
22C2×C4×Q8×C137—22no—
23C2×C4×Q8×C137C2211yescharacteristic
24#479×C137—22no—
25#479×C137—22no—
26#478×C137—22no—
27#478×C137—22no—
28#428×C137C2212yes—
29#428×C137C2212yes—
30#423×C137C2212yes—
31C2×#104×C137C2212yes—
32C2×#104×C137C2212yes—
33#417×C137C2212yes—
34#417×C137C2212yes—
35C2×#101×C137C2211yescharacteristic
36C2×#101×C137C2211yescharacteristic
37#398×C137—22no—
38#391×C137C2211yescharacteristic
39#391×C137C2211yescharacteristic
40#387×C137—22no—
41#384×C137C2211yescharacteristic
42#382×C137C2211yescharacteristic
43C4×C4⋊C4×C137C2211yescharacteristic
44#343×C137C2212yes—
45#339×C137C2212yes—
46#339×C137C2212yes—
472−1+4×C137—24no—
482−1+4×C137—22no—
492−1+4×C137Q814yes—
50C2×#41×C137—22no—
51C2×#41×C137—22no—
52C2×#41×C137C2312yes—
53C22×Q8×C137—22no—
54C22×Q8×C137C2311yescharacteristic
55C2×M16×C137C2312yes—
56C2×M16×C137C2312yes—
57#125×C137—22no—
58#125×C137—22no—
59C4×Q8×C137—22no—
60C4×Q8×C137—24no—
61C4×Q8×C137—22no—
62C4×Q8×C137—22no—
63C4×Q8×C137—22no—
64C4×Q8×C137—22no—
65C4×Q8×C137—22no—
66C4×Q8×C137—22no—
67C4×Q8×C137D412yes—
68C4×Q8×C137D412yes—
69C4×D4×C137—22no—
70C4×D4×C137—24no—
71C4×D4×C137—22no—
72C4×D4×C137—22no—
73C4×D4×C137—22no—
74C4×D4×C137—24no—
75C4×D4×C137—22no—
76C4×D4×C137D412yes—
77C4×D4×C137D412yes—
78#117×C137—24no—
79#117×C137—24no—
80#117×C137—22no—
81#117×C137—22no—
82#117×C137—22no—
83#117×C137C2312yes—
84C2×C4⋊C4×C137—24no—
85C2×C4⋊C4×C137—22no—
86C2×C4⋊C4×C137—21no—
87C2×C4⋊C4×C137—22no—
88C2×C4⋊C4×C137—21no—
89C2×C4⋊C4×C137—21no—
90C2×C4⋊C4×C137D412yes—
91C2×C4⋊C4×C137C2311yescharacteristic
92C2×C4⋊C4×C137C2311yescharacteristic
93C2×C4⋊C4×C137C2311yescharacteristic
94C2×C42×C137—24no—
95C2×C42×C137—22no—
96C2×C42×C137D412yes—
97C2×C42×C137C2311yescharacteristic
98C2×C42×C137C2311yescharacteristic
99C2×C42×C137C2311yescharacteristic
100#107×C137—22no—
101#107×C137—22no—
102#106×C137—22no—
103#106×C137—22no—
104C4⋊C8×C137—22no—
105C4⋊C8×C137—22no—
106#104×C137—24no—
107#104×C137—24no—
108#103×C137—22no—
109#103×C137—22no—
110#102×C137—22no—
111#102×C137—22no—
112#101×C137—22no—
113#101×C137—22no—
114#95×C137—21no—
115#95×C137—22no—
116#95×C137—22no—
117#95×C137—22no—
118#41×C137—28no—
119#41×C137—28no—
120#41×C137—28no—
121#41×C137—28no—
122#41×C137—22no—
123#41×C137—22no—
124#41×C137C2×Q814yes—
125#41×C137#4114yes—
126C2×Q8×C137—22no—
127C2×Q8×C137—28no—
128C2×Q8×C137—28no—
129C2×Q8×C137—22no—
130C2×Q8×C137—22no—
131C2×Q8×C137C2×D412yes—
132C2×Q8×C137C2×Q812yes—
133C2×Q8×C137#4112yes—
134C2×D4×C137—22no—
135C2×D4×C137—22no—
136C2×D4×C137C2×D412yes—
137C22×C4×C137—24no—
138C22×C4×C137—22no—
139C22×C4×C137—21no—
140C22×C4×C137—24no—
141C22×C4×C137—22no—
142C22×C4×C137—22no—
143C22×C4×C137—21no—
144C22×C4×C137—21no—
145C22×C4×C137—21no—
146C22×C4×C137C2×D412yes—
147C22×C4×C137C2×D412yes—
148C22×C4×C137C2×D411yescharacteristic
149C22×C4×C137C2×D411yescharacteristic
150C22×C4×C137C2411yescharacteristic
151M16×C137—22no—
152M16×C137—22no—
153M16×C137—22no—
154M16×C137—22no—
155C2×C8×C137—22no—
156C2×C8×C137—22no—
157C4⋊C4×C137—24no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—24no—
162C4⋊C4×C137—22no—
163C4⋊C4×C137—22no—
164C4⋊C4×C137—22no—
165C4⋊C4×C137—22no—
166C4⋊C4×C137—42no—
167C4⋊C4×C137—44no—
168C4⋊C4×C137—41no—
169C4⋊C4×C137—41no—
170C4⋊C4×C137—42no—
171C4⋊C4×C137C2×D412yes—
172C4⋊C4×C137C2×D412yes—
173C22⋊C4×C137—24no—
174C22⋊C4×C137—22no—
175C22⋊C4×C137—22no—
176C22⋊C4×C137—22no—
177C42×C137—24no—
178C42×C137—24no—
179C42×C137—22no—
180C42×C137—22no—
181C42×C137—22no—
182C42×C137—22no—
183C42×C137—22no—
184C42×C137—22no—
185C42×C137—22no—
186C42×C137C2×D412yes—
187C42×C137C2×D412yes—
188C23×C137—22no—
189C23×C137C22×D411yescharacteristic
190Q8×C137—28no—
191Q8×C137—24no—
192Q8×C137—22no—
193Q8×C137—24no—
194Q8×C137C22⋊Q814yes—
195D4×C137—28no—
196D4×C137—24no—
197D4×C137—24no—
198D4×C137—22no—
199D4×C137C22⋊Q814yes—
200C2×C4×C137—216no—
201C2×C4×C137—22no—
202C2×C4×C137—22no—
203C2×C4×C137—22no—
204C2×C4×C137—24no—
205C2×C4×C137—24no—
206C2×C4×C137—22no—
207C2×C4×C137—22no—
208C2×C4×C137—22no—
209C2×C4×C137—22no—
210C2×C4×C137—41no—
211C2×C4×C137—41no—
212C2×C4×C137—44no—
213C2×C4×C137—41no—
214C2×C4×C137—41no—
215C2×C4×C137—42no—
216C2×C4×C137—41no—
217C2×C4×C137—42no—
218C2×C4×C137—41no—
219C2×C4×C137—42no—
220C2×C4×C137—41no—
221C2×C4×C137—41no—
222C2×C4×C137—41no—
223C2×C4×C137—82no—
224C2×C4×C137—84no—
225C2×C4×C137#12011yescharacteristic
226C2×C4×C137#12011yescharacteristic
227C2×C4×C137#12012yes—
228C2×C4×C137C22⋊Q814yes—
229C2×C4×C137C22×D412yes—
230C2×C4×C137C22×Q811yescharacteristic
231C2×C4×C137C2×#4112yes—
232C2×C4×C137C2×#4111yescharacteristic
233C8×C137—42no—
234C8×C137—42no—
235C22×C137—21no—
236C22×C137—21no—
237C22×C137—24no—
238C22×C137—42no—
239C22×C137—42no—
240C22×C137C2×#12011yescharacteristic
241C22×C137C2×C22⋊Q811yescharacteristic
242C22×C137#52511yescharacteristic
243C4×C137—24no—
244C4×C137—24no—
245C4×C137—22no—
246C4×C137—21no—
247C4×C137—21no—
248C4×C137—42no—
249C4×C137—42no—
250C4×C137—42no—
251C4×C137—42no—
252C4×C137—81no—
253C4×C137—81no—
254C4×C137—81no—
255C4×C137C2×C22⋊Q812yes—
256C4×C137#52512yes—
257C2×C137—21no—
258C2×C137—21no—
259C2×C137—44no—
260C2×C137#284811yescharacteristic
261C2×C137#301811yescharacteristic
262C2×C137#302011yescharacteristic
263#32974C13711yesmaximal, characteristic, direct product factor, Sylow
264C137#3297411yescharacteristic, direct product factor, Sylow
265C4×2−1+4C2×C13711yescharacteristic
266#3080C2×C13712yescomplemented
267#2565C2×C13711yescharacteristic, complemented
268#1860C2×C13712yescomplemented
269#1859C2×C13712yescomplemented
270#1857C2×C13711yescharacteristic, complemented
271#1857C2×C13711yescharacteristic, complemented
272#1806C2×C13712yescomplemented
273#1801C2×C13711yescharacteristic, complemented
274#1793C2×C13711yescharacteristic
275#1793C2×C13711yescharacteristic
276C2×2−1+4C22×C13711yescharacteristic
277C2×#125C22×C13711yescharacteristic
278#520—22no—
279#520—22no—
280#520C22×C13712yes—
281#519—22no—
282#519C22×C13711yescharacteristic
283C4×#41—22no—
284C4×#41—22no—
285C4×#41C22×C13712yes—
286C2×C4×Q8—22no—
287C2×C4×Q8C22×C13711yescharacteristic
288#479—22no—
289#479—22no—
290#478—22no—
291#478—22no—
292#428C22×C13712yes—
293#428C22×C13712yes—
294#423C22×C13712yes—
295C2×#104C22×C13712yes—
296C2×#104C22×C13712yes—
297#417C22×C13712yes—
298#417C22×C13712yes—
299C2×#101C22×C13711yescharacteristic
300C2×#101C22×C13711yescharacteristic
301#398—22no—
302#391C22×C13711yescharacteristic
303#391C22×C13711yescharacteristic
304#387—22no—
305#384C22×C13711yescharacteristic
306#382C22×C13711yescharacteristic
307C4×C4⋊C4C22×C13711yescharacteristic
308#343C22×C13712yes—
309#339C22×C13712yes—
310#339C22×C13712yes—
3112−1+4—24no—
3122−1+4—22no—
3132−1+4Q8×C13714yes—
314C2×#41—22no—
315C2×#41—22no—
316C2×#41C23×C13712yes—
317C22×Q8—22no—
318C22×Q8C23×C13711yescharacteristic
319C2×M16C23×C13712yes—
320C2×M16C23×C13712yes—
321#125—22no—
322#125—22no—
323C4×Q8—22no—
324C4×Q8—24no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8D4×C13712yes—
332C4×Q8D4×C13712yes—
333C4×D4—22no—
334C4×D4—24no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—24no—
339C4×D4—22no—
340C4×D4D4×C13712yes—
341C4×D4D4×C13712yes—
342#117—24no—
343#117—24no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117C23×C13712yes—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4D4×C13712yes—
355C2×C4⋊C4C23×C13711yescharacteristic
356C2×C4⋊C4C23×C13711yescharacteristic
357C2×C4⋊C4C23×C13711yescharacteristic
358C2×C42—24no—
359C2×C42—22no—
360C2×C42D4×C13712yes—
361C2×C42C23×C13711yescharacteristic
362C2×C42C23×C13711yescharacteristic
363C2×C42C23×C13711yescharacteristic
364#107—22no—
365#107—22no—
366#106—22no—
367#106—22no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370#104—24no—
371#104—24no—
372#103—22no—
373#103—22no—
374#102—22no—
375#102—22no—
376#101—22no—
377#101—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#41—28no—
383#41—28no—
384#41—28no—
385#41—28no—
386#41—22no—
387#41—22no—
388#41C2×Q8×C13714yes—
389#41#41×C13714yes—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C2×D4×C13712yes—
396C2×Q8C2×Q8×C13712yes—
397C2×Q8#41×C13712yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4C2×D4×C13712yes—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4C2×D4×C13712yes—
411C22×C4C2×D4×C13712yes—
412C22×C4C2×D4×C13711yescharacteristic
413C22×C4C2×D4×C13711yescharacteristic
414C22×C4C24×C13711yescharacteristic
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419C2×C8—22no—
420C2×C8—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4C2×D4×C13712yes—
436C4⋊C4C2×D4×C13712yes—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42C2×D4×C13712yes—
451C42C2×D4×C13712yes—
452C23—22no—
453C23C22×D4×C13711yescharacteristic
454Q8—28no—
455Q8—24no—
456Q8—22no—
457Q8—24no—
458Q8C22⋊Q8×C13714yes—
459D4—28no—
460D4—24no—
461D4—24no—
462D4—22no—
463D4C22⋊Q8×C13714yes—
464C2×C4—216no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4#120×C13711yescharacteristic
490C2×C4#120×C13711yescharacteristic
491C2×C4#120×C13712yes—
492C2×C4C22⋊Q8×C13714yes—
493C2×C4C22×D4×C13712yes—
494C2×C4C22×Q8×C13711yescharacteristic
495C2×C4C2×#41×C13712yes—
496C2×C4C2×#41×C13711yescharacteristic
497C8—42no—
498C8—42no—
499C22—21no—
500C22—21no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22C2×#120×C13711yescharacteristic
505C22C2×C22⋊Q8×C13711yescharacteristic
506C22#525×C13711yescharacteristic
507C4—24no—
508C4—24no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4C2×C22⋊Q8×C13712yes—
520C4#525×C13712yes—
521C2—21no—
522C2—21no—
523C2—44no—
524C2#2848×C13711yescharacteristic
525C2#3018×C13711yescharacteristic
526C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×2−1+4×C137C22
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
2. #3080×C137C2semidirect
3. #2565×C137C2semidirect
4. #1860×C137C2semidirect
5. #1859×C137C2semidirect
6. #1857×C137C2semidirect
7. #1857×C137C2semidirect
8. #1806×C137C2semidirect
9. #1801×C137C2semidirect
263. #32974C137direct (N × H)
264. C137#32974direct (N × H)
266. #3080C2×C137semidirect
267. #2565C2×C137semidirect
268. #1860C2×C137semidirect
269. #1859C2×C137semidirect
270. #1857C2×C137semidirect
271. #1857C2×C137semidirect
272. #1806C2×C137semidirect
273. #1801C2×C137semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4×C1372
#3080×C1372
#2565×C1372
#1860×C1372
#1859×C1372
#1857×C1372
#1806×C1372
#1801×C1372
#1793×C1372
#32974137