Finite Group Databasebeta

← C137×#19513 #46868899

Subgroups — C137×#19513 #46868899

Order 35072.

610 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3227×C137C211yesmaximal, characteristic, complemented
2#3227×C137C211yesmaximal, characteristic, complemented
3#3214×C137C211yesmaximal, characteristic
4#2981×C137C211yesmaximal, characteristic, complemented
5#1697×C137C212yesmaximal, complemented
6#1691×C137C212yesmaximal, complemented
7#1677×C137C212yesmaximal, complemented
8#1671×C137C212yesmaximal, complemented
9#1572×C137C211yesmaximal, characteristic, complemented
10#1558×C137C211yesmaximal, characteristic, complemented
11#1553×C137C211yesmaximal, characteristic, complemented
12#532×C137C2211yescharacteristic
13#532×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15#498×C137—21no—
16#498×C137C2211yescharacteristic
17#498×C137C2211yescharacteristic
18#496×C137—21no—
19#489×C137—21no—
20#486×C137—21no—
21#486×C137C2211yescharacteristic
22#486×C137C2211yescharacteristic
23#481×C137—22no—
24#481×C137—22no—
25#476×C137—22no—
26#476×C137C2212yes—
27#476×C137C2212yes—
28#461×C137C2212yes—
29#459×C137C2212yes—
30#449×C137—22no—
31#436×C137C2212yes—
32#435×C137—22no—
33C8×D4×C137C2212yes—
34#432×C137C2211yescharacteristic
35#432×C137C2211yescharacteristic
36#431×C137C2211yescharacteristic
37#431×C137C2211yescharacteristic
38#408×C137—22no—
39#335×C137C2212yes—
40#334×C137C2212yes—
41#333×C137C2212yes—
42#333×C137C2212yes—
43#332×C137C2212yes—
44#331×C137C2212yes—
45C2×#41×C137—21no—
46C2×#41×C137—21no—
47C22×Q8×C137—21no—
48C2×SD16×C137—22no—
49C2×D8×C137—22no—
50C2×M16×C137—22no—
51C22×C8×C137—22no—
52C4⋊Q8×C137—21no—
53C4⋊Q8×C137—21no—
54C4⋊Q8×C137—21no—
55C4⋊Q8×C137D411yescharacteristic
56C4⋊Q8×C137C2311yescharacteristic
57C4⋊Q8×C137C2311yescharacteristic
58C4⋊Q8×C137C2311yescharacteristic
59#127×C137D411yescharacteristic
60#127×C137C2311yescharacteristic
61#124×C137—21no—
62#124×C137—21no—
63#121×C137—21no—
64#121×C137—21no—
65#121×C137—21no—
66#121×C137D412yes—
67C4×D4×C137—21no—
68C4×D4×C137—21no—
69C4×D4×C137—21no—
70C4×D4×C137C2312yes—
71C2×C4⋊C4×C137—21no—
72C2×C4⋊C4×C137—21no—
73C2×C42×C137C2311yescharacteristic
74#106×C137—22no—
75#106×C137—22no—
76#106×C137—22no—
77#106×C137—22no—
78C4⋊C8×C137D412yes—
79C4⋊C8×C137C2312yes—
80C4⋊C8×C137C2312yes—
81C4⋊C8×C137C2312yes—
82#103×C137—22no—
83#102×C137—22no—
84#102×C137—22no—
85#102×C137—22no—
86#102×C137—22no—
87#102×C137—22no—
88C22⋊C8×C137—21no—
89C22⋊C8×C137—21no—
90C22⋊C8×C137—21no—
91C22⋊C8×C137—22no—
92C22⋊C8×C137—22no—
93C22⋊C8×C137D412yes—
94#97×C137—21no—
95#97×C137—21no—
96C4×C8×C137—21no—
97C4×C8×C137C2311yescharacteristic
98C4×C8×C137C2311yescharacteristic
99#41×C137—82no—
100#41×C137—82no—
101C2×Q8×C137—21no—
102C2×Q8×C137—21no—
103C2×Q8×C137—21no—
104C2×Q8×C137—21no—
105C2×Q8×C137—41no—
106C2×Q8×C137—82no—
107C2×D4×C137—22no—
108C2×D4×C137—21no—
109C2×D4×C137—21no—
110C2×D4×C137—21no—
111C2×D4×C137—21no—
112C2×D4×C137—21no—
113C2×D4×C137—41no—
114C2×D4×C137C2×D412yes—
115C22×C4×C137—22no—
116C22×C4×C137—41no—
117C22×C4×C137—41no—
118C22×C4×C137C2×D411yescharacteristic
119C22×C4×C137C2×D411yescharacteristic
120C22×C4×C137#4111yescharacteristic
121SD16×C137—82no—
122D8×C137—82no—
123M16×C137—82no—
124C2×C8×C137—22no—
125C2×C8×C137—22no—
126C2×C8×C137—22no—
127C2×C8×C137—82no—
128C2×C8×C137C2×D412yes—
129C2×C8×C137C2×D412yes—
130C4⋊C4×C137—21no—
131C4⋊C4×C137—21no—
132C4⋊C4×C137—21no—
133C4⋊C4×C137—21no—
134C4⋊C4×C137—21no—
135C4⋊C4×C137—21no—
136C4⋊C4×C137—21no—
137C4⋊C4×C137—41no—
138C4⋊C4×C137—41no—
139C4⋊C4×C137C2×D412yes—
140C22⋊C4×C137—22no—
141C22⋊C4×C137—22no—
142C22⋊C4×C137—41no—
143C22⋊C4×C137—41no—
144C42×C137C2×D411yescharacteristic
145C42×C137C2×D411yescharacteristic
146C42×C137#4111yescharacteristic
147C42×C137C2411yescharacteristic
148C23×C137—22no—
149C23×C137—41no—
150C23×C137#12111yescharacteristic
151Q8×C137—42no—
152Q8×C137—81no—
153Q8×C137—81no—
154Q8×C137—81no—
155Q8×C137—82no—
156D4×C137—22no—
157D4×C137—22no—
158D4×C137—42no—
159D4×C137—42no—
160D4×C137—81no—
161D4×C137—81no—
162D4×C137—81no—
163D4×C137—82no—
164D4×C137—82no—
165C2×C4×C137—22no—
166C2×C4×C137—22no—
167C2×C4×C137—22no—
168C2×C4×C137—41no—
169C2×C4×C137—42no—
170C2×C4×C137—42no—
171C2×C4×C137—41no—
172C2×C4×C137—41no—
173C2×C4×C137—41no—
174C2×C4×C137—82no—
175C2×C4×C137—82no—
176C2×C4×C137#12111yescharacteristic
177C2×C4×C137#12111yescharacteristic
178C2×C4×C137#12111yescharacteristic
179C2×C4×C137C22×D411yescharacteristic
180C2×C4×C137C22×D411yescharacteristic
181C2×C4×C137C2×#4111yescharacteristic
182C8×C137—22no—
183C8×C137—22no—
184C8×C137—42no—
185C8×C137—42no—
186C8×C137—42no—
187C22×C137—21no—
188C22×C137—21no—
189C22×C137—21no—
190C22×C137—42no—
191C22×C137—42no—
192C22×C137—42no—
193C22×C137—81no—
194C22×C137—81no—
195C22×C137—81no—
196C22×C137C2×#12111yescharacteristic
197C4×C137—21no—
198C4×C137—21no—
199C4×C137—21no—
200C4×C137—21no—
201C4×C137—42no—
202C4×C137—81no—
203C4×C137—81no—
204C4×C137—81no—
205C4×C137#57712yes—
206C4×C137#57712yes—
207C2×C137—41no—
208C2×C137—82no—
209C2×C137—161no—
210C2×C137#304211yescharacteristic
211C2×C137#305711yescharacteristic
212C2×C137#306811yescharacteristic
213#19513C13711yesmaximal, characteristic, direct product factor, Sylow
214C137#1951311yescharacteristic, direct product factor, Sylow
215#3227C2×C13711yescharacteristic, complemented
216#3227C2×C13711yescharacteristic, complemented
217#3214C2×C13711yescharacteristic
218#2981C2×C13711yescharacteristic, complemented
219#1697C2×C13712yescomplemented
220#1691C2×C13712yescomplemented
221#1677C2×C13712yescomplemented
222#1671C2×C13712yescomplemented
223#1572C2×C13711yescharacteristic, complemented
224#1558C2×C13711yescharacteristic, complemented
225#1553C2×C13711yescharacteristic, complemented
226#532C22×C13711yescharacteristic
227#532C22×C13711yescharacteristic
228C2×C4⋊Q8C22×C13711yescharacteristic
229#498—21no—
230#498C22×C13711yescharacteristic
231#498C22×C13711yescharacteristic
232#496—21no—
233#489—21no—
234#486—21no—
235#486C22×C13711yescharacteristic
236#486C22×C13711yescharacteristic
237#481—22no—
238#481—22no—
239#476—22no—
240#476C22×C13712yes—
241#476C22×C13712yes—
242#461C22×C13712yes—
243#459C22×C13712yes—
244#449—22no—
245#436C22×C13712yes—
246#435—22no—
247C8×D4C22×C13712yes—
248#432C22×C13711yescharacteristic
249#432C22×C13711yescharacteristic
250#431C22×C13711yescharacteristic
251#431C22×C13711yescharacteristic
252#408—22no—
253#335C22×C13712yes—
254#334C22×C13712yes—
255#333C22×C13712yes—
256#333C22×C13712yes—
257#332C22×C13712yes—
258#331C22×C13712yes—
259C2×#41—21no—
260C2×#41—21no—
261C22×Q8—21no—
262C2×SD16—22no—
263C2×D8—22no—
264C2×M16—22no—
265C22×C8—22no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8D4×C13711yescharacteristic
270C4⋊Q8C23×C13711yescharacteristic
271C4⋊Q8C23×C13711yescharacteristic
272C4⋊Q8C23×C13711yescharacteristic
273#127D4×C13711yescharacteristic
274#127C23×C13711yescharacteristic
275#124—21no—
276#124—21no—
277#121—21no—
278#121—21no—
279#121—21no—
280#121D4×C13712yes—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4C23×C13712yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C42C23×C13711yescharacteristic
288#106—22no—
289#106—22no—
290#106—22no—
291#106—22no—
292C4⋊C8D4×C13712yes—
293C4⋊C8C23×C13712yes—
294C4⋊C8C23×C13712yes—
295C4⋊C8C23×C13712yes—
296#103—22no—
297#102—22no—
298#102—22no—
299#102—22no—
300#102—22no—
301#102—22no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8D4×C13712yes—
308#97—21no—
309#97—21no—
310C4×C8—21no—
311C4×C8C23×C13711yescharacteristic
312C4×C8C23×C13711yescharacteristic
313#41—82no—
314#41—82no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×D4—22no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4C2×D4×C13712yes—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4C2×D4×C13711yescharacteristic
333C22×C4C2×D4×C13711yescharacteristic
334C22×C4#41×C13711yescharacteristic
335SD16—82no—
336D8—82no—
337M16—82no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—82no—
342C2×C8C2×D4×C13712yes—
343C2×C8C2×D4×C13712yes—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4C2×D4×C13712yes—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C42C2×D4×C13711yescharacteristic
359C42C2×D4×C13711yescharacteristic
360C42#41×C13711yescharacteristic
361C42C24×C13711yescharacteristic
362C23—22no—
363C23—41no—
364C23#121×C13711yescharacteristic
365Q8—42no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369Q8—82no—
370D4—22no—
371D4—22no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—82no—
378D4—82no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#121×C13711yescharacteristic
391C2×C4#121×C13711yescharacteristic
392C2×C4#121×C13711yescharacteristic
393C2×C4C22×D4×C13711yescharacteristic
394C2×C4C22×D4×C13711yescharacteristic
395C2×C4C2×#41×C13711yescharacteristic
396C8—22no—
397C8—22no—
398C8—42no—
399C8—42no—
400C8—42no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C2×#121×C13711yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4#577×C13712yes—
420C4#577×C13712yes—
421C2—41no—
422C2—82no—
423C2—161no—
424C2#3042×C13711yescharacteristic
425C2#3057×C13711yescharacteristic
426C2#3068×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radical#3214×C137C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3227×C137C2semidirect
2. #3227×C137C2semidirect
4. #2981×C137C2semidirect
5. #1697×C137C2semidirect
6. #1691×C137C2semidirect
7. #1677×C137C2semidirect
8. #1671×C137C2semidirect
9. #1572×C137C2semidirect
10. #1558×C137C2semidirect
11. #1553×C137C2semidirect
213. #19513C137direct (N × H)
214. C137#19513direct (N × H)
215. #3227C2×C137semidirect
216. #3227C2×C137semidirect
218. #2981C2×C137semidirect
219. #1697C2×C137semidirect
220. #1691C2×C137semidirect
221. #1677C2×C137semidirect
222. #1671C2×C137semidirect
223. #1572C2×C137semidirect
224. #1558C2×C137semidirect
225. #1553C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3227×C1372
#3214×C1372
#2981×C1372
#1697×C1372
#1691×C1372
#1677×C1372
#1671×C1372
#1572×C1372
#1558×C1372
#1553×C1372
#19513137