Finite Group Databasebeta

← C137×#34485 #46883871

Subgroups — C137×#34485 #46883871

Order 35072.

2340 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C137C211yesmaximal, characteristic
2#2379×C137C214yesmaximal, complemented
3#2369×C137C211yesmaximal, characteristic, complemented
4#2361×C137C214yesmaximal, complemented
5D4×C4⋊C4×C137C212yesmaximal, complemented
6#2346×C137C214yesmaximal, complemented
7#2326×C137C214yesmaximal, complemented
8#2324×C137C214yesmaximal, complemented
9#2317×C137C214yesmaximal, complemented
10C4×C4⋊Q8×C137C211yesmaximal, characteristic, complemented
11#2293×C137C212yesmaximal, complemented
12#555×C137C4116yescomplemented
13C2×C4⋊Q8×C137C2211yescharacteristic
14C2×#124×C137C2214yes—
15C2×#121×C137C2214yes—
16C2×C4×Q8×C137C2212yes—
17C2×C4×D4×C137C2214yes—
18C2×C4×D4×C137C2214yes—
19C2×C4×D4×C137C2212yes—
20C2×#117×C137C2214yes—
21C2×#117×C137C2214yes—
22C22×C4⋊C4×C137C2214yes—
23#391×C137C2212yes—
24#391×C137C2214yes—
25#390×C137C2212yes—
26#390×C137C2212yes—
27#390×C137C2214yes—
28#389×C137C2212yes—
29#389×C137C2212yes—
30#389×C137C2214yes—
31#389×C137C2214yes—
32#389×C137C2214yes—
33#388×C137C2214yes—
34#388×C137C2214yes—
35#388×C137C2218yes—
36#388×C137C2218yes—
37#386×C137C2218yes—
38#385×C137C2218yes—
39#385×C137C2218yes—
40#385×C137C2218yes—
41#384×C137C2211yescharacteristic
42#384×C137C2214yes—
43#382×C137C2218yes—
44#380×C137C2214yes—
45C4×C4⋊C4×C137C2212yes—
46C4×C4⋊C4×C137C2214yes—
47C4×C4⋊C4×C137C2214yes—
48C4×C22⋊C4×C137C2214yes—
49#376×C137C2214yes—
50C43×C137C2211yescharacteristic
51C22×Q8×C137C2312yes—
52C22×D4×C137C2312yes—
53C22×D4×C137C2314yes—
54C23×C4×C137C2314yes—
55C23×C4×C137C2318yes—
56C4⋊Q8×C137C2×C418yescomplemented
57#124×C137C2×C4132yescomplemented
58#121×C137C2×C4132yescomplemented
59C4×Q8×C137—44no—
60C4×D4×C137—48no—
61C4×D4×C137—48no—
62C4×D4×C137C2×C4116yescomplemented
63#117×C137—48no—
64#117×C137C2×C4132yescomplemented
65C2×C4⋊C4×C137—48no—
66C2×C4⋊C4×C137C2312yes—
67C2×C4⋊C4×C137C2314yes—
68C2×C4⋊C4×C137C2312yes—
69C2×C4⋊C4×C137C2314yes—
70C2×C4⋊C4×C137C2318yes—
71C2×C4⋊C4×C137C2314yes—
72C2×C4⋊C4×C137C2312yes—
73C2×C4⋊C4×C137C2314yes—
74C2×C4⋊C4×C137C2314yes—
75C2×C4⋊C4×C137C2314yes—
76C2×C4⋊C4×C137C2312yes—
77C2×C22⋊C4×C137C2314yes—
78C2×C22⋊C4×C137C2314yes—
79C2×C22⋊C4×C137C2318yes—
80C2×C22⋊C4×C137C2314yes—
81C2×C22⋊C4×C137C2314yes—
82C2×C22⋊C4×C137C2318yes—
83C2×C22⋊C4×C137C2314yes—
84C2×C42×C137C2312yes—
85C2×C42×C137C2311yescharacteristic
86C2×C42×C137C2312yes—
87C2×C42×C137C2314yes—
88C2×C42×C137C2314yes—
89C2×C42×C137C2311yescharacteristic
90C2×C42×C137C2314yes—
91C2×C42×C137C2311yescharacteristic
92C2×C42×C137C2314yes—
93#95×C137C2314yes—
94#95×C137C2318yes—
95#95×C137C2314yes—
96#95×C137C2318yes—
97#95×C137C2318yes—
98#95×C137C2314yes—
99C24×C137C2414yes—
100C2×Q8×C137—42no—
101C2×Q8×C137—42no—
102C2×Q8×C137C22×C418yes—
103C2×D4×C137—24no—
104C2×D4×C137—24no—
105C2×D4×C137—44no—
106C2×D4×C137—44no—
107C2×D4×C137C22×C418yes—
108C2×D4×C137C22×C4116yes—
109C22×C4×C137—28no—
110C22×C4×C137—28no—
111C22×C4×C137—48no—
112C22×C4×C137—48no—
113C22×C4×C137—48no—
114C22×C4×C137C22×C4116yes—
115C22×C4×C137C2414yes—
116C22×C4×C137C2412yes—
117C22×C4×C137C2414yes—
118C22×C4×C137C2414yes—
119C22×C4×C137C2411yescharacteristic
120C22×C4×C137C2411yescharacteristic
121C22×C4×C137C2414yes—
122C22×C4×C137C2414yes—
123C22×C4×C137C2411yescharacteristic
124C22×C4×C137C2412yes—
125C4⋊C4×C137—24no—
126C4⋊C4×C137—28no—
127C4⋊C4×C137—24no—
128C4⋊C4×C137—48no—
129C4⋊C4×C137—44no—
130C4⋊C4×C137—44no—
131C4⋊C4×C137—44no—
132C4⋊C4×C137—44no—
133C4⋊C4×C137—42no—
134C4⋊C4×C137C22×C418yes—
135C4⋊C4×C137C22×C4116yes—
136C22⋊C4×C137—28no—
137C22⋊C4×C137—44no—
138C22⋊C4×C137—44no—
139C22⋊C4×C137—44no—
140C22⋊C4×C137—48no—
141C22⋊C4×C137C22×C4132yes—
142C22⋊C4×C137C22×C4116yes—
143C42×C137—24no—
144C42×C137—28no—
145C42×C137—28no—
146C42×C137—22no—
147C42×C137—24no—
148C42×C137—44no—
149C42×C137—41no—
150C42×C137C22×C4116yes—
151C42×C137C22×C414yes—
152C23×C137—24no—
153C23×C137—24no—
154C23×C137—24no—
155C23×C137—24no—
156C23×C137—24no—
157C23×C137—24no—
158C23×C137C23×C418yes—
159C23×C137C2511yescharacteristic
160Q8×C137—48no—
161D4×C137—216no—
162D4×C137—416no—
163C2×C4×C137—24no—
164C2×C4×C137—21no—
165C2×C4×C137—24no—
166C2×C4×C137—22no—
167C2×C4×C137—24no—
168C2×C4×C137—24no—
169C2×C4×C137—21no—
170C2×C4×C137—232no—
171C2×C4×C137—24no—
172C2×C4×C137—22no—
173C2×C4×C137—22no—
174C2×C4×C137—24no—
175C2×C4×C137—21no—
176C2×C4×C137—24no—
177C2×C4×C137—24no—
178C2×C4×C137—28no—
179C2×C4×C137—24no—
180C2×C4×C137—22no—
181C2×C4×C137—24no—
182C2×C4×C137—24no—
183C2×C4×C137—22no—
184C2×C4×C137—816no—
185C2×C4×C137C23×C418yes—
186C2×C4×C137C23×C412yes—
187C2×C4×C137C23×C418yes—
188C2×C4×C137C23×C414yes—
189C2×C4×C137H(5,𝔽2)12yes—
190C2×C4×C137H(5,𝔽2)14yes—
191C2×C4×C1372−1+414yes—
192C22×C137—28no—
193C22×C137—28no—
194C22×C137—28no—
195C22×C137—44no—
196C22×C137—44no—
197C22×C137—44no—
198C22×C137—44no—
199C22×C137C24×C411yescharacteristic
200C22×C137C2×H(5,𝔽2)12yes—
201C22×C137C2×H(5,𝔽2)11yescharacteristic
202C22×C137C2×2−1+412yes—
203C22×C137#58511yescharacteristic
204C4×C137—28no—
205C4×C137—28no—
206C4×C137—22no—
207C4×C137—22no—
208C4×C137—44no—
209C4×C137—42no—
210C4×C137—41no—
211C4×C137—44no—
212C4×C137—44no—
213C4×C137#52018yes—
214C2×C137—48no—
215C2×C137C2×#52012yes—
216C2×C137C4×H(5,𝔽2)11yescharacteristic
217C2×C137#354411yescharacteristic
218C2×C137#354712yes—
219C2×C137#354911yescharacteristic
220#34485C13711yesmaximal, characteristic, direct product factor, Sylow
221C137#3448511yescharacteristic, direct product factor, Sylow
222C2×#555C2×C13711yescharacteristic
223#2379C2×C13714yescomplemented
224#2369C2×C13711yescharacteristic, complemented
225#2361C2×C13714yescomplemented
226D4×C4⋊C4C2×C13712yescomplemented
227#2346C2×C13714yescomplemented
228#2326C2×C13714yescomplemented
229#2324C2×C13714yescomplemented
230#2317C2×C13714yescomplemented
231C4×C4⋊Q8C2×C13711yescharacteristic, complemented
232#2293C2×C13712yescomplemented
233#555C4×C137116yescomplemented
234C2×C4⋊Q8C22×C13711yescharacteristic
235C2×#124C22×C13714yes—
236C2×#121C22×C13714yes—
237C2×C4×Q8C22×C13712yes—
238C2×C4×D4C22×C13714yes—
239C2×C4×D4C22×C13714yes—
240C2×C4×D4C22×C13712yes—
241C2×#117C22×C13714yes—
242C2×#117C22×C13714yes—
243C22×C4⋊C4C22×C13714yes—
244#391C22×C13712yes—
245#391C22×C13714yes—
246#390C22×C13714yes—
247#390C22×C13712yes—
248#390C22×C13712yes—
249#389C22×C13714yes—
250#389C22×C13712yes—
251#389C22×C13714yes—
252#389C22×C13714yes—
253#389C22×C13712yes—
254#388C22×C13718yes—
255#388C22×C13714yes—
256#388C22×C13718yes—
257#388C22×C13714yes—
258#386C22×C13718yes—
259#385C22×C13718yes—
260#385C22×C13718yes—
261#385C22×C13718yes—
262#384C22×C13711yescharacteristic
263#384C22×C13714yes—
264#382C22×C13718yes—
265#380C22×C13714yes—
266C4×C4⋊C4C22×C13714yes—
267C4×C4⋊C4C22×C13714yes—
268C4×C4⋊C4C22×C13712yes—
269C4×C22⋊C4C22×C13714yes—
270#376C22×C13714yes—
271C43C22×C13711yescharacteristic
272C22×Q8C23×C13712yes—
273C22×D4C23×C13712yes—
274C22×D4C23×C13714yes—
275C23×C4C23×C13714yes—
276C23×C4C23×C13718yes—
277C4⋊Q8C2×C4×C13718yescomplemented
278#124C2×C4×C137132yescomplemented
279#121C2×C4×C137132yescomplemented
280C4×Q8—44no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4C2×C4×C137116yescomplemented
284#117—48no—
285#117C2×C4×C137132yescomplemented
286C2×C4⋊C4—48no—
287C2×C4⋊C4C23×C13712yes—
288C2×C4⋊C4C23×C13714yes—
289C2×C4⋊C4C23×C13712yes—
290C2×C4⋊C4C23×C13718yes—
291C2×C4⋊C4C23×C13714yes—
292C2×C4⋊C4C23×C13714yes—
293C2×C4⋊C4C23×C13714yes—
294C2×C4⋊C4C23×C13712yes—
295C2×C4⋊C4C23×C13714yes—
296C2×C4⋊C4C23×C13712yes—
297C2×C4⋊C4C23×C13714yes—
298C2×C22⋊C4C23×C13714yes—
299C2×C22⋊C4C23×C13714yes—
300C2×C22⋊C4C23×C13714yes—
301C2×C22⋊C4C23×C13718yes—
302C2×C22⋊C4C23×C13714yes—
303C2×C22⋊C4C23×C13718yes—
304C2×C22⋊C4C23×C13714yes—
305C2×C42C23×C13712yes—
306C2×C42C23×C13714yes—
307C2×C42C23×C13714yes—
308C2×C42C23×C13711yescharacteristic
309C2×C42C23×C13712yes—
310C2×C42C23×C13711yescharacteristic
311C2×C42C23×C13714yes—
312C2×C42C23×C13714yes—
313C2×C42C23×C13711yescharacteristic
314#95C23×C13714yes—
315#95C23×C13718yes—
316#95C23×C13714yes—
317#95C23×C13718yes—
318#95C23×C13718yes—
319#95C23×C13714yes—
320C24C24×C13714yes—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8C22×C4×C13718yes—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4C22×C4×C13718yes—
329C2×D4C22×C4×C137116yes—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4C22×C4×C137116yes—
336C22×C4C24×C13714yes—
337C22×C4C24×C13712yes—
338C22×C4C24×C13714yes—
339C22×C4C24×C13714yes—
340C22×C4C24×C13711yescharacteristic
341C22×C4C24×C13711yescharacteristic
342C22×C4C24×C13714yes—
343C22×C4C24×C13714yes—
344C22×C4C24×C13711yescharacteristic
345C22×C4C24×C13712yes—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4C22×C4×C13718yes—
356C4⋊C4C22×C4×C137116yes—
357C22⋊C4—28no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—48no—
362C22⋊C4C22×C4×C137132yes—
363C22⋊C4C22×C4×C137116yes—
364C42—24no—
365C42—28no—
366C42—28no—
367C42—22no—
368C42—24no—
369C42—44no—
370C42—41no—
371C42C22×C4×C137116yes—
372C42C22×C4×C13714yes—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23C23×C4×C13718yes—
380C23C25×C13711yescharacteristic
381Q8—48no—
382D4—216no—
383D4—416no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—232no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—816no—
406C2×C4C23×C4×C13718yes—
407C2×C4C23×C4×C13712yes—
408C2×C4C23×C4×C13718yes—
409C2×C4C23×C4×C13714yes—
410C2×C4H(5,𝔽2)×C13712yes—
411C2×C4H(5,𝔽2)×C13714yes—
412C2×C42−1+4×C13714yes—
413C22—28no—
414C22—28no—
415C22—28no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22C24×C4×C13711yescharacteristic
421C22C2×H(5,𝔽2)×C13712yes—
422C22C2×H(5,𝔽2)×C13711yescharacteristic
423C22C2×2−1+4×C13712yes—
424C22#585×C13711yescharacteristic
425C4—28no—
426C4—28no—
427C4—22no—
428C4—22no—
429C4—44no—
430C4—44no—
431C4—41no—
432C4—44no—
433C4—42no—
434C4#520×C13718yes—
435C2—48no—
436C2C2×#520×C13712yes—
437C2C4×H(5,𝔽2)×C13711yescharacteristic
438C2#3544×C13711yescharacteristic
439C2#3547×C13712yes—
440C2#3549×C13711yescharacteristic
subgroupG/H
center, socleC23×C137C25
commutatorC22C24×C4×C137
Frattini, cosocleC23C25×C137
prime radicalC2×#555×C137C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2379×C137C2semidirect
3. #2369×C137C2semidirect
4. #2361×C137C2semidirect
5. D4×C4⋊C4×C137C2semidirect
6. #2346×C137C2semidirect
7. #2326×C137C2semidirect
8. #2324×C137C2semidirect
9. #2317×C137C2semidirect
10. C4×C4⋊Q8×C137C2semidirect
11. #2293×C137C2semidirect
12. #555×C137C4semidirect
56. C4⋊Q8×C137C2×C4semidirect
57. #124×C137C2×C4semidirect
58. #121×C137C2×C4semidirect
62. C4×D4×C137C2×C4semidirect
64. #117×C137C2×C4semidirect
220. #34485C137direct (N × H)
221. C137#34485direct (N × H)
223. #2379C2×C137semidirect
224. #2369C2×C137semidirect
225. #2361C2×C137semidirect
226. D4×C4⋊C4C2×C137semidirect
227. #2346C2×C137semidirect
228. #2326C2×C137semidirect
229. #2324C2×C137semidirect
230. #2317C2×C137semidirect
231. C4×C4⋊Q8C2×C137semidirect
232. #2293C2×C137semidirect
233. #555C4×C137semidirect
277. C4⋊Q8C2×C4×C137semidirect
278. #124C2×C4×C137semidirect
279. #121C2×C4×C137semidirect
283. C4×D4C2×C4×C137semidirect
285. #117C2×C4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#555×C1372
#2379×C1372
#2369×C1372
#2361×C1372
D4×C4⋊C4×C1372
#2346×C1372
#2326×C1372
#2324×C1372
#2317×C1372
C4×C4⋊Q8×C1372
#2293×C1372
#34485137