Finite Group Databasebeta

← C137×#25313 #46874699

Subgroups — C137×#25313 #46874699

Order 35072.

712 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3139×C137C211yesmaximal, characteristic, complemented
2#3077×C137C211yesmaximal, characteristic
3C2×#461×C137C211yesmaximal, characteristic
4#2962×C137C211yesmaximal, characteristic, complemented
5#2064×C137C212yesmaximal, complemented
6#2046×C137C212yesmaximal
7#1977×C137C212yesmaximal
8#1966×C137C211yesmaximal, characteristic, complemented
9#1950×C137C211yesmaximal, characteristic, complemented
10#1930×C137C212yesmaximal, complemented
11C4×#100×C137C211yesmaximal, characteristic
12C22×SD16×C137C2212yes—
13#533×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15C2×C4×D4×C137C2211yescharacteristic
16#491×C137—22no—
17#487×C137—21no—
18#487×C137—21no—
19#478×C137—22no—
20#476×C137—21no—
21#474×C137—21no—
22#461×C137—21no—
23#461×C137—22no—
24#461×C137—21no—
25#446×C137—22no—
26#446×C137—21no—
27C8×Q8×C137—21no—
28C2×#108×C137C2212yes—
29#428×C137C2212yes—
30#424×C137C2212yes—
31#423×C137C2211yescharacteristic
32C2×C4⋊C8×C137C2211yescharacteristic
33#419×C137C2212yes—
34#419×C137C2212yes—
35#418×C137C2212yes—
36C2×#103×C137C2212yes—
37C2×#101×C137C2212yes—
38C2×#101×C137C2212yes—
39C2×#100×C137C2211yescharacteristic
40C2×#100×C137C2211yescharacteristic
41#405×C137C2212yes—
42C4×M16×C137C2211yescharacteristic
43C4×M16×C137C2211yescharacteristic
44#343×C137C2211yescharacteristic
45#343×C137C2211yescharacteristic
46#341×C137C2212yes—
47C22×Q8×C137C2312yes—
48C22×D4×C137C2311yescharacteristic
49C23×C4×C137—21no—
50C2×SD16×C137—24no—
51C2×SD16×C137—22no—
52C2×SD16×C137—22no—
53C2×SD16×C137—42no—
54C2×M16×C137C2311yescharacteristic
55C2×M16×C137C2311yescharacteristic
56C2×M16×C137C2311yescharacteristic
57C2×M16×C137C2312yes—
58C2×M16×C137C2311yescharacteristic
59C22×C8×C137C2312yes—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137D411yescharacteristic
65C4⋊Q8×C137D411yescharacteristic
66#125×C137D412yes—
67C22⋊Q8×C137—21no—
68C22⋊Q8×C137—21no—
69C4×Q8×C137—21no—
70C4×Q8×C137—21no—
71C4×D4×C137—21no—
72C4×D4×C137—21no—
73C4×D4×C137—21no—
74C4×D4×C137—21no—
75#117×C137C2312yes—
76C2×C4⋊C4×C137—21no—
77C2×C4⋊C4×C137—21no—
78C2×C4⋊C4×C137C2311yescharacteristic
79C2×C22⋊C4×C137—21no—
80C2×C42×C137C2311yescharacteristic
81#108×C137—42no—
82#107×C137—22no—
83#106×C137—22no—
84C4⋊C8×C137—22no—
85C4⋊C8×C137—22no—
86C4⋊C8×C137—21no—
87C4⋊C8×C137—21no—
88C4⋊C8×C137D411yescharacteristic
89C4⋊C8×C137D411yescharacteristic
90C4⋊C8×C137D412yes—
91#103×C137—22no—
92#103×C137—22no—
93#103×C137—22no—
94#103×C137—22no—
95#103×C137—22no—
96#103×C137—22no—
97#102×C137—22no—
98#102×C137—22no—
99#101×C137—24no—
100#101×C137—42no—
101#100×C137—22no—
102#100×C137—22no—
103#97×C137—22no—
104#97×C137—21no—
105#97×C137—21no—
106C4×C8×C137—22no—
107C4×C8×C137—21no—
108C4×C8×C137—21no—
109C24×C137—21no—
110C2×Q8×C137—21no—
111C2×Q8×C137—21no—
112C2×Q8×C137—22no—
113C2×Q8×C137—22no—
114C2×Q8×C137—22no—
115C2×Q8×C137—22no—
116C2×Q8×C137—22no—
117C2×Q8×C137—22no—
118C2×D4×C137—21no—
119C2×D4×C137—21no—
120C2×D4×C137—21no—
121C2×D4×C137—21no—
122C2×D4×C137—21no—
123C2×D4×C137#4112yes—
124C22×C4×C137—22no—
125C22×C4×C137—21no—
126C22×C4×C137—41no—
127C22×C4×C137—41no—
128C22×C4×C137—82no—
129C22×C4×C137#4111yescharacteristic
130C22×C4×C137#4111yescharacteristic
131C22×C4×C137C2411yescharacteristic
132SD16×C137—44no—
133SD16×C137—44no—
134M16×C137—21no—
135M16×C137—21no—
136M16×C137—21no—
137M16×C137—21no—
138M16×C137—22no—
139M16×C137—21no—
140M16×C137—22no—
141M16×C137—21no—
142M16×C137—21no—
143M16×C137#4112yes—
144C2×C8×C137—21no—
145C2×C8×C137—22no—
146C2×C8×C137—22no—
147C2×C8×C137—21no—
148C2×C8×C137—21no—
149C2×C8×C137—21no—
150C2×C8×C137—22no—
151C2×C8×C137C2×D412yes—
152C2×C8×C137C2×D412yes—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—22no—
155C4⋊C4×C137—21no—
156C4⋊C4×C137—21no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—21no—
159C4⋊C4×C137—41no—
160C4⋊C4×C137—41no—
161C4⋊C4×C137—41no—
162C4⋊C4×C137—41no—
163C4⋊C4×C137C2×D412yes—
164C4⋊C4×C137C2×D412yes—
165C22⋊C4×C137—22no—
166C22⋊C4×C137—41no—
167C22⋊C4×C137—41no—
168C42×C137—22no—
169C42×C137C2×D411yescharacteristic
170C42×C137C2×D411yescharacteristic
171C42×C137C2×D411yescharacteristic
172C42×C137C2×D411yescharacteristic
173C23×C137—22no—
174C23×C137—41no—
175C23×C137—41no—
176C23×C137—81no—
177C23×C137—81no—
178C23×C137C2×#4111yescharacteristic
179Q8×C137—42no—
180Q8×C137—42no—
181Q8×C137—42no—
182Q8×C137—42no—
183Q8×C137—42no—
184Q8×C137—81no—
185D4×C137—22no—
186D4×C137—22no—
187D4×C137—41no—
188D4×C137—41no—
189C2×C4×C137—22no—
190C2×C4×C137—22no—
191C2×C4×C137—22no—
192C2×C4×C137—41no—
193C2×C4×C137—41no—
194C2×C4×C137—42no—
195C2×C4×C137—42no—
196C2×C4×C137—42no—
197C2×C4×C137—82no—
198C2×C4×C137—82no—
199C2×C4×C137—81no—
200C2×C4×C137#12111yescharacteristic
201C2×C4×C137#12111yescharacteristic
202C2×C4×C137#12111yescharacteristic
203C2×C4×C137#12111yescharacteristic
204C2×C4×C137C22×D411yescharacteristic
205C2×C4×C137C22×D411yescharacteristic
206C2×C4×C137C2×#4111yescharacteristic
207C2×C4×C137C2×#4111yescharacteristic
208C2×C4×C137H(5,𝔽2)11yescharacteristic
209C2×C4×C1372−1+411yescharacteristic
210C8×C137—22no—
211C8×C137—22no—
212C8×C137—22no—
213C8×C137—41no—
214C8×C137—41no—
215C8×C137—41no—
216C8×C137—42no—
217C22×C137—21no—
218C22×C137—21no—
219C22×C137—42no—
220C22×C137—82no—
221C22×C137—82no—
222C22×C137—81no—
223C22×C137—81no—
224C22×C137C2×#12111yescharacteristic
225C22×C137#52911yescharacteristic
226C22×C137#54211yescharacteristic
227C4×C137—21no—
228C4×C137—21no—
229C4×C137—21no—
230C4×C137—42no—
231C4×C137—42no—
232C4×C137—82no—
233C4×C137—81no—
234C4×C137#47112yes—
235C4×C137#54611yescharacteristic
236C4×C137#54811yescharacteristic
237C4×C137#54811yescharacteristic
238C4×C137#55011yescharacteristic
239C2×C137—21no—
240C2×C137—21no—
241C2×C137—82no—
242C2×C137#242611yescharacteristic
243C2×C137C2×#47111yescharacteristic
244C2×C137#306911yescharacteristic
245#25313C13711yesmaximal, characteristic, direct product factor, Sylow
246C137#2531311yescharacteristic, direct product factor, Sylow
247#3139C2×C13711yescharacteristic, complemented
248#3077C2×C13711yescharacteristic
249C2×#461C2×C13711yescharacteristic
250#2962C2×C13711yescharacteristic, complemented
251#2064C2×C13712yescomplemented
252#2046C2×C13712yes—
253#1977C2×C13712yes—
254#1966C2×C13711yescharacteristic, complemented
255#1950C2×C13711yescharacteristic, complemented
256#1930C2×C13712yescomplemented
257C4×#100C2×C13711yescharacteristic
258C22×SD16C22×C13712yes—
259#533C22×C13711yescharacteristic
260C2×C4⋊Q8C22×C13711yescharacteristic
261C2×C4×D4C22×C13711yescharacteristic
262#491—22no—
263#487—21no—
264#487—21no—
265#478—22no—
266#476—21no—
267#474—21no—
268#461—21no—
269#461—22no—
270#461—21no—
271#446—22no—
272#446—21no—
273C8×Q8—21no—
274C2×#108C22×C13712yes—
275#428C22×C13712yes—
276#424C22×C13712yes—
277#423C22×C13711yescharacteristic
278C2×C4⋊C8C22×C13711yescharacteristic
279#419C22×C13712yes—
280#419C22×C13712yes—
281#418C22×C13712yes—
282C2×#103C22×C13712yes—
283C2×#101C22×C13712yes—
284C2×#101C22×C13712yes—
285C2×#100C22×C13711yescharacteristic
286C2×#100C22×C13711yescharacteristic
287#405C22×C13712yes—
288C4×M16C22×C13711yescharacteristic
289C4×M16C22×C13711yescharacteristic
290#343C22×C13711yescharacteristic
291#343C22×C13711yescharacteristic
292#341C22×C13712yes—
293C22×Q8C23×C13712yes—
294C22×D4C23×C13711yescharacteristic
295C23×C4—21no—
296C2×SD16—24no—
297C2×SD16—22no—
298C2×SD16—22no—
299C2×SD16—42no—
300C2×M16C23×C13711yescharacteristic
301C2×M16C23×C13712yes—
302C2×M16C23×C13711yescharacteristic
303C2×M16C23×C13711yescharacteristic
304C2×M16C23×C13711yescharacteristic
305C22×C8C23×C13712yes—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8D4×C13711yescharacteristic
311C4⋊Q8D4×C13711yescharacteristic
312#125D4×C13712yes—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321#117C23×C13712yes—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C23×C13711yescharacteristic
325C2×C22⋊C4—21no—
326C2×C42C23×C13711yescharacteristic
327#108—42no—
328#107—22no—
329#106—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8D4×C13711yescharacteristic
335C4⋊C8D4×C13711yescharacteristic
336C4⋊C8D4×C13712yes—
337#103—22no—
338#103—22no—
339#103—22no—
340#103—22no—
341#103—22no—
342#103—22no—
343#102—22no—
344#102—22no—
345#101—24no—
346#101—42no—
347#100—22no—
348#100—22no—
349#97—22no—
350#97—21no—
351#97—21no—
352C4×C8—22no—
353C4×C8—21no—
354C4×C8—21no—
355C24—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4#41×C13712yes—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—82no—
375C22×C4#41×C13711yescharacteristic
376C22×C4#41×C13711yescharacteristic
377C22×C4C24×C13711yescharacteristic
378SD16—44no—
379SD16—44no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—22no—
385M16—21no—
386M16—22no—
387M16—21no—
388M16—21no—
389M16#41×C13712yes—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8C2×D4×C13712yes—
398C2×C8C2×D4×C13712yes—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4C2×D4×C13712yes—
410C4⋊C4C2×D4×C13712yes—
411C22⋊C4—22no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C42—22no—
415C42C2×D4×C13711yescharacteristic
416C42C2×D4×C13711yescharacteristic
417C42C2×D4×C13711yescharacteristic
418C42C2×D4×C13711yescharacteristic
419C23—22no—
420C23—41no—
421C23—41no—
422C23—81no—
423C23—81no—
424C23C2×#41×C13711yescharacteristic
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—81no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4#121×C13711yescharacteristic
447C2×C4#121×C13711yescharacteristic
448C2×C4#121×C13711yescharacteristic
449C2×C4#121×C13711yescharacteristic
450C2×C4C22×D4×C13711yescharacteristic
451C2×C4C22×D4×C13711yescharacteristic
452C2×C4C2×#41×C13711yescharacteristic
453C2×C4C2×#41×C13711yescharacteristic
454C2×C4H(5,𝔽2)×C13711yescharacteristic
455C2×C42−1+4×C13711yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—42no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22C2×#121×C13711yescharacteristic
471C22#529×C13711yescharacteristic
472C22#542×C13711yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4#471×C13712yes—
481C4#546×C13711yescharacteristic
482C4#548×C13711yescharacteristic
483C4#548×C13711yescharacteristic
484C4#550×C13711yescharacteristic
485C2—21no—
486C2—21no—
487C2—82no—
488C2#2426×C13711yescharacteristic
489C2C2×#471×C13711yescharacteristic
490C2#3069×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. #3139×C137C2semidirect
4. #2962×C137C2semidirect
5. #2064×C137C2semidirect
8. #1966×C137C2semidirect
9. #1950×C137C2semidirect
10. #1930×C137C2semidirect
245. #25313C137direct (N × H)
246. C137#25313direct (N × H)
247. #3139C2×C137semidirect
250. #2962C2×C137semidirect
251. #2064C2×C137semidirect
254. #1966C2×C137semidirect
255. #1950C2×C137semidirect
256. #1930C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3139×C1372
#3077×C1372
C2×#461×C1372
#2962×C1372
#2064×C1372
#2046×C1372
#1977×C1372
#1966×C1372
#1950×C1372
#1930×C1372
C4×#100×C1372
#25313137