Finite Group Databasebeta

← C137×#25852 #46875238

Subgroups — C137×#25852 #46875238

Order 35072.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501×C137C211yesmaximal, characteristic
2#2434×C137C212yesmaximal, complemented
3#2409×C137C211yesmaximal, characteristic
4#2062×C137C214yesmaximal, complemented
5#1996×C137C212yesmaximal, complemented
6#1867×C137C214yesmaximal
7C4×#99×C137C211yesmaximal, characteristic
8C23×Q8×C137C2211yescharacteristic
9#554×C137—22no—
10#554×C137—24no—
11D4×Q8×C137—22no—
12#544×C137—21no—
13#543×C137—21no—
14#543×C137C2212yes—
15#533×C137—21no—
16C2×C4⋊Q8×C137—22no—
17C2×C22⋊Q8×C137—21no—
18C2×C22⋊Q8×C137—21no—
19C2×C22⋊Q8×C137C2212yes—
20#518×C137C2212yes—
21C2×C4×D4×C137C2211yescharacteristic
22C2×#99×C137C2212yes—
23#395×C137—21no—
24#395×C137—24no—
25#393×C137—21no—
26#391×C137—22no—
27#391×C137—21no—
28#386×C137C2212yes—
29#385×C137C2212yes—
30#385×C137C2218yes—
31#380×C137C2211yescharacteristic
32C4×C22⋊C4×C137C2212yes—
33#342×C137C2212yes—
34#342×C137C2218yes—
35C22×Q8×C137—21no—
36C22×Q8×C137—21no—
37C22×Q8×C137—22no—
38C22×Q8×C137—21no—
39C22×Q8×C137—22no—
40C22×Q8×C137—44no—
41C22×Q8×C137D412yes—
42C22×D4×C137C2311yescharacteristic
43C23×C4×C137D411yescharacteristic
44C23×C4×C137C2312yes—
45C23×C4×C137C2311yescharacteristic
46C4⋊Q8×C137—21no—
47C4⋊Q8×C137—21no—
48C4⋊Q8×C137—24no—
49C4⋊Q8×C137—48no—
50#125×C137—22no—
51#125×C137—24no—
52C22⋊Q8×C137—22no—
53C22⋊Q8×C137—22no—
54C22⋊Q8×C137—24no—
55C22⋊Q8×C137—24no—
56C22⋊Q8×C137—22no—
57C22⋊Q8×C137—22no—
58C22⋊Q8×C137—22no—
59C22⋊Q8×C137—22no—
60C22⋊Q8×C137—22no—
61C22⋊Q8×C137—22no—
62C22⋊Q8×C137—24no—
63C22⋊Q8×C137—24no—
64C22⋊Q8×C137—44no—
65C4×Q8×C137—22no—
66C4×Q8×C137—22no—
67C4×D4×C137—22no—
68C4×D4×C137—22no—
69C4×D4×C137—24no—
70C4×D4×C137—24no—
71#117×C137—22no—
72C2×C4⋊C4×C137—21no—
73C2×C4⋊C4×C137—22no—
74C2×C4⋊C4×C137—21no—
75C2×C4⋊C4×C137—22no—
76C2×C4⋊C4×C137—28no—
77C2×C4⋊C4×C137D412yes—
78C2×C4⋊C4×C137D411yescharacteristic
79C2×C4⋊C4×C137D414yes—
80C2×C22⋊C4×C137—22no—
81C2×C22⋊C4×C137—22no—
82C2×C22⋊C4×C137D411yescharacteristic
83C2×C22⋊C4×C137C2312yes—
84C2×C22⋊C4×C137C2311yescharacteristic
85C2×C22⋊C4×C137C2314yes—
86C2×C22⋊C4×C137C2314yes—
87C2×C42×C137—22no—
88C2×C42×C137D411yescharacteristic
89#99×C137Q818yescomplemented
90#95×C137—22no—
91#95×C137—21no—
92#95×C137—22no—
93#95×C137—28no—
94#95×C137—22no—
95C24×C137C2×D411yescharacteristic
96C24×C137C2411yescharacteristic
97C2×Q8×C137—24no—
98C2×Q8×C137—28no—
99C2×Q8×C137—22no—
100C2×Q8×C137—22no—
101C2×Q8×C137—22no—
102C2×Q8×C137—22no—
103C2×Q8×C137—44no—
104C2×Q8×C137—42no—
105C2×Q8×C137—44no—
106C2×Q8×C137—48no—
107C2×Q8×C137—44no—
108C2×Q8×C137—41no—
109C2×Q8×C137—42no—
110C2×Q8×C137—41no—
111C2×Q8×C137—82no—
112C2×D4×C137—22no—
113C2×D4×C137—22no—
114C2×D4×C137C2×Q814yes—
115C22×C4×C137—24no—
116C22×C4×C137—22no—
117C22×C4×C137—24no—
118C22×C4×C137—22no—
119C22×C4×C137—22no—
120C22×C4×C137—21no—
121C22×C4×C137—21no—
122C22×C4×C137—22no—
123C22×C4×C137—44no—
124C22×C4×C137—42no—
125C22×C4×C137—42no—
126C22×C4×C137C2×D412yes—
127C22×C4×C137C2×D411yescharacteristic
128C22×C4×C137C2×D411yescharacteristic
129C22×C4×C137C2×D411yescharacteristic
130C22×C4×C137C2×D414yes—
131C22×C4×C137C2×D414yes—
132C22×C4×C137C2×D411yescharacteristic
133C22×C4×C137C2×D411yescharacteristic
134C22×C4×C137C2×D412yes—
135C4⋊C4×C137—22no—
136C4⋊C4×C137—24no—
137C4⋊C4×C137—22no—
138C4⋊C4×C137—22no—
139C4⋊C4×C137—24no—
140C4⋊C4×C137—22no—
141C4⋊C4×C137—24no—
142C4⋊C4×C137—22no—
143C4⋊C4×C137—22no—
144C4⋊C4×C137—22no—
145C4⋊C4×C137—24no—
146C4⋊C4×C137—42no—
147C4⋊C4×C137—44no—
148C4⋊C4×C137—416no—
149C22⋊C4×C137—24no—
150C22⋊C4×C137—24no—
151C22⋊C4×C137—22no—
152C22⋊C4×C137—24no—
153C22⋊C4×C137—24no—
154C22⋊C4×C137—22no—
155C22⋊C4×C137—24no—
156C22⋊C4×C137—44no—
157C22⋊C4×C137C2×Q818yescomplemented
158C42×C137—24no—
159C42×C137—21no—
160C42×C137—21no—
161C42×C137—44no—
162C23×C137—21no—
163C23×C137—21no—
164C23×C137—41no—
165C23×C137—41no—
166C23×C137—41no—
167C23×C137—41no—
168C23×C137C4⋊Q812yescomplemented
169C23×C137C22×D411yescharacteristic
170C23×C137C22×D412yes—
171C23×C137C22×Q812yes—
172C23×C1372−1+412yes—
173Q8×C137—44no—
174Q8×C137—48no—
175Q8×C137—44no—
176Q8×C137—44no—
177Q8×C137—82no—
178Q8×C137—82no—
179Q8×C137—82no—
180D4×C137—28no—
181C2×C4×C137—21no—
182C2×C4×C137—21no—
183C2×C4×C137—24no—
184C2×C4×C137—22no—
185C2×C4×C137—24no—
186C2×C4×C137—28no—
187C2×C4×C137—21no—
188C2×C4×C137—24no—
189C2×C4×C137—24no—
190C2×C4×C137—21no—
191C2×C4×C137—28no—
192C2×C4×C137—42no—
193C2×C4×C137—41no—
194C2×C4×C137—44no—
195C2×C4×C137—44no—
196C2×C4×C137—44no—
197C2×C4×C137—42no—
198C2×C4×C137—42no—
199C2×C4×C137—42no—
200C2×C4×C137—41no—
201C2×C4×C137—44no—
202C2×C4×C137—44no—
203C2×C4×C137—41no—
204C2×C4×C137—42no—
205C2×C4×C137—42no—
206C2×C4×C137—42no—
207C2×C4×C137—81no—
208C2×C4×C137—82no—
209C2×C4×C137—82no—
210C2×C4×C137#12011yescharacteristic
211C2×C4×C137#12012yes—
212C2×C4×C137#12011yescharacteristic
213C2×C4×C137C4⋊Q812yescomplemented
214C22×C137—22no—
215C22×C137—21no—
216C22×C137—22no—
217C22×C137—21no—
218C22×C137—22no—
219C22×C137—42no—
220C22×C137—42no—
221C22×C137—42no—
222C22×C137—41no—
223C22×C137—41no—
224C22×C137—81no—
225C22×C137C2×#12011yescharacteristic
226C22×C137C2×C4⋊Q811yescharacteristic
227C22×C137#53611yescharacteristic
228C22×C137D4×Q814yes—
229C4×C137—22no—
230C4×C137—21no—
231C4×C137—22no—
232C4×C137—48no—
233C4×C137—44no—
234C4×C137—41no—
235C4×C137—44no—
236C4×C137—44no—
237C4×C137—41no—
238C4×C137—82no—
239C4×C137B(4,𝔽2)12yes—
240C2×C137—24no—
241C2×C137—21no—
242C2×C137—21no—
243C2×C137—42no—
244C2×C137#243411yescharacteristic
245C2×C137C2×B(4,𝔽2)11yescharacteristic
246C2×C137#302811yescharacteristic
247#25852C13711yesmaximal, characteristic, direct product factor, Sylow
248C137#2585211yescharacteristic, direct product factor, Sylow
249#3501C2×C13711yescharacteristic
250#2434C2×C13712yescomplemented
251#2409C2×C13711yescharacteristic
252#2062C2×C13714yescomplemented
253#1996C2×C13712yescomplemented
254#1867C2×C13714yes—
255C4×#99C2×C13711yescharacteristic
256C23×Q8C22×C13711yescharacteristic
257#554—22no—
258#554—24no—
259D4×Q8—22no—
260#544—21no—
261#543—21no—
262#543C22×C13712yes—
263#533—21no—
264C2×C4⋊Q8—22no—
265C2×C22⋊Q8—21no—
266C2×C22⋊Q8—21no—
267C2×C22⋊Q8C22×C13712yes—
268#518C22×C13712yes—
269C2×C4×D4C22×C13711yescharacteristic
270C2×#99C22×C13712yes—
271#395—21no—
272#395—24no—
273#393—21no—
274#391—22no—
275#391—21no—
276#386C22×C13712yes—
277#385C22×C13712yes—
278#385C22×C13718yes—
279#380C22×C13711yescharacteristic
280C4×C22⋊C4C22×C13712yes—
281#342C22×C13712yes—
282#342C22×C13718yes—
283C22×Q8—21no—
284C22×Q8—21no—
285C22×Q8—22no—
286C22×Q8—21no—
287C22×Q8—22no—
288C22×Q8—44no—
289C22×Q8D4×C13712yes—
290C22×D4C23×C13711yescharacteristic
291C23×C4D4×C13711yescharacteristic
292C23×C4C23×C13712yes—
293C23×C4C23×C13711yescharacteristic
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—24no—
297C4⋊Q8—48no—
298#125—22no—
299#125—24no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—24no—
311C22⋊Q8—24no—
312C22⋊Q8—44no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—24no—
318C4×D4—24no—
319#117—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4D4×C13711yescharacteristic
326C2×C4⋊C4D4×C13712yes—
327C2×C4⋊C4D4×C13714yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4D4×C13711yescharacteristic
331C2×C22⋊C4C23×C13712yes—
332C2×C22⋊C4C23×C13711yescharacteristic
333C2×C22⋊C4C23×C13714yes—
334C2×C22⋊C4C23×C13714yes—
335C2×C42—22no—
336C2×C42D4×C13711yescharacteristic
337#99Q8×C13718yescomplemented
338#95—22no—
339#95—21no—
340#95—22no—
341#95—28no—
342#95—22no—
343C24C2×D4×C13711yescharacteristic
344C24C24×C13711yescharacteristic
345C2×Q8—24no—
346C2×Q8—28no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—44no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—41no—
357C2×Q8—42no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4C2×Q8×C13714yes—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4C2×D4×C13712yes—
375C22×C4C2×D4×C13711yescharacteristic
376C22×C4C2×D4×C13711yescharacteristic
377C22×C4C2×D4×C13711yescharacteristic
378C22×C4C2×D4×C13714yes—
379C22×C4C2×D4×C13714yes—
380C22×C4C2×D4×C13711yescharacteristic
381C22×C4C2×D4×C13711yescharacteristic
382C22×C4C2×D4×C13712yes—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—416no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—44no—
405C22⋊C4C2×Q8×C13718yescomplemented
406C42—24no—
407C42—21no—
408C42—21no—
409C42—44no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C4⋊Q8×C13712yescomplemented
417C23C22×D4×C13711yescharacteristic
418C23C22×D4×C13712yes—
419C23C22×Q8×C13712yes—
420C232−1+4×C13712yes—
421Q8—44no—
422Q8—48no—
423Q8—44no—
424Q8—44no—
425Q8—82no—
426Q8—82no—
427Q8—82no—
428D4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—28no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#120×C13711yescharacteristic
459C2×C4#120×C13712yes—
460C2×C4#120×C13711yescharacteristic
461C2×C4C4⋊Q8×C13712yescomplemented
462C22—22no—
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22C2×#120×C13711yescharacteristic
474C22C2×C4⋊Q8×C13711yescharacteristic
475C22#536×C13711yescharacteristic
476C22D4×Q8×C13714yes—
477C4—22no—
478C4—21no—
479C4—22no—
480C4—48no—
481C4—44no—
482C4—41no—
483C4—44no—
484C4—44no—
485C4—41no—
486C4—82no—
487C4B(4,𝔽2)×C13712yes—
488C2—24no—
489C2—21no—
490C2—21no—
491C2—42no—
492C2#2434×C13711yescharacteristic
493C2C2×B(4,𝔽2)×C13711yescharacteristic
494C2#3028×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radicalC22×D4×C137C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2434×C137C2semidirect
4. #2062×C137C2semidirect
5. #1996×C137C2semidirect
89. #99×C137Q8semidirect
157. C22⋊C4×C137C2×Q8semidirect
168. C23×C137C4⋊Q8semidirect
213. C2×C4×C137C4⋊Q8semidirect
247. #25852C137direct (N × H)
248. C137#25852direct (N × H)
250. #2434C2×C137semidirect
252. #2062C2×C137semidirect
253. #1996C2×C137semidirect
337. #99Q8×C137semidirect
405. C22⋊C4C2×Q8×C137semidirect
416. C23C4⋊Q8×C137semidirect
461. C2×C4C4⋊Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#3501×C1372
#2434×C1372
#2409×C1372
#2062×C1372
#1996×C1372
#1867×C1372
C4×#99×C1372
#25852137