Finite Group Databasebeta

← C137×#19512 #46868898

Subgroups — C137×#19512 #46868898

Order 35072.

588 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3235×C137C211yesmaximal, characteristic
2#3229×C137C211yesmaximal, characteristic, complemented
3#3206×C137C211yesmaximal, characteristic
4#2981×C137C211yesmaximal, characteristic
5#1699×C137C212yesmaximal, complemented
6#1691×C137C212yesmaximal, complemented
7#1679×C137C212yesmaximal, complemented
8#1671×C137C212yesmaximal, complemented
9#1572×C137C211yesmaximal, characteristic, complemented
10#1556×C137C211yesmaximal, characteristic, complemented
11#1554×C137C211yesmaximal, characteristic, complemented
12#533×C137C2211yescharacteristic
13#532×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15#501×C137—21no—
16#498×C137C2211yescharacteristic
17#498×C137C2211yescharacteristic
18#492×C137—21no—
19#490×C137—21no—
20#487×C137—21no—
21#487×C137C2211yescharacteristic
22#487×C137C2211yescharacteristic
23#483×C137—22no—
24#482×C137—22no—
25#476×C137C2212yes—
26#475×C137—22no—
27#474×C137C2212yes—
28#461×C137C2212yes—
29#461×C137C2212yes—
30#448×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#332×C137C2212yes—
43#332×C137C2212yes—
44#332×C137C2212yes—
45C2×#41×C137—21no—
46C22×Q8×C137—21no—
47C2×SD16×C137—22no—
48C2×SD16×C137—22no—
49C2×M16×C137—22no—
50C22×C8×C137—22no—
51C4⋊Q8×C137—21no—
52C4⋊Q8×C137—21no—
53C4⋊Q8×C137—21no—
54C4⋊Q8×C137D411yescharacteristic
55C4⋊Q8×C137C2311yescharacteristic
56C4⋊Q8×C137C2311yescharacteristic
57C4⋊Q8×C137C2311yescharacteristic
58C4⋊Q8×C137C2311yescharacteristic
59#127×C137D411yescharacteristic
60#125×C137—21no—
61#124×C137—21no—
62C22⋊Q8×C137—21no—
63C22⋊Q8×C137—21no—
64#121×C137—21no—
65#121×C137D412yes—
66C4×Q8×C137—21no—
67C4×Q8×C137—21no—
68C4×D4×C137—21no—
69C4×D4×C137C2312yes—
70#117×C137—21no—
71C2×C4⋊C4×C137—21no—
72C2×C4⋊C4×C137—21no—
73C2×C42×C137C2311yescharacteristic
74#107×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#103×C137—22no—
84#103×C137—22no—
85#103×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—
100C2×Q8×C137—21no—
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×C137C2×D412yes—
111C22×C4×C137—22no—
112C22×C4×C137—41no—
113C22×C4×C137C2×D411yescharacteristic
114C22×C4×C137C2×D411yescharacteristic
115C22×C4×C137#4111yescharacteristic
116SD16×C137—82no—
117SD16×C137—82no—
118M16×C137—82no—
119C2×C8×C137—22no—
120C2×C8×C137—22no—
121C2×C8×C137—22no—
122C2×C8×C137—82no—
123C2×C8×C137C2×D412yes—
124C2×C8×C137C2×D412yes—
125C4⋊C4×C137—21no—
126C4⋊C4×C137—21no—
127C4⋊C4×C137—21no—
128C4⋊C4×C137—21no—
129C4⋊C4×C137—21no—
130C4⋊C4×C137—21no—
131C4⋊C4×C137—21no—
132C4⋊C4×C137—21no—
133C4⋊C4×C137—21no—
134C4⋊C4×C137—41no—
135C4⋊C4×C137—41no—
136C4⋊C4×C137—41no—
137C4⋊C4×C137—41no—
138C4⋊C4×C137C2×D412yes—
139C22⋊C4×C137—22no—
140C22⋊C4×C137—22no—
141C22⋊C4×C137—41no—
142C42×C137—41no—
143C42×C137C2×D411yescharacteristic
144C42×C137C2×D411yescharacteristic
145C42×C137#4111yescharacteristic
146C42×C137C2411yescharacteristic
147C23×C137—22no—
148C23×C137#12111yescharacteristic
149Q8×C137—42no—
150Q8×C137—42no—
151Q8×C137—42no—
152Q8×C137—81no—
153Q8×C137—81no—
154Q8×C137—82no—
155D4×C137—22no—
156D4×C137—22no—
157D4×C137—81no—
158D4×C137—81no—
159D4×C137—82no—
160C2×C4×C137—22no—
161C2×C4×C137—22no—
162C2×C4×C137—22no—
163C2×C4×C137—41no—
164C2×C4×C137—42no—
165C2×C4×C137—42no—
166C2×C4×C137—41no—
167C2×C4×C137—41no—
168C2×C4×C137—41no—
169C2×C4×C137—41no—
170C2×C4×C137—82no—
171C2×C4×C137#12111yescharacteristic
172C2×C4×C137#12111yescharacteristic
173C2×C4×C137#12111yescharacteristic
174C2×C4×C137C22×D411yescharacteristic
175C2×C4×C137C22×D411yescharacteristic
176C2×C4×C137C2×#4111yescharacteristic
177C8×C137—22no—
178C8×C137—22no—
179C8×C137—42no—
180C8×C137—42no—
181C8×C137—42no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—21no—
185C22×C137—42no—
186C22×C137—42no—
187C22×C137—42no—
188C22×C137C2×#12111yescharacteristic
189C4×C137—21no—
190C4×C137—21no—
191C4×C137—21no—
192C4×C137—21no—
193C4×C137—42no—
194C4×C137—81no—
195C4×C137—81no—
196C4×C137—81no—
197C4×C137—81no—
198C4×C137#57712yes—
199C4×C137#57712yes—
200C2×C137—41no—
201C2×C137—82no—
202C2×C137#304211yescharacteristic
203C2×C137#305711yescharacteristic
204C2×C137#306911yescharacteristic
205#19512C13711yesmaximal, characteristic, direct product factor, Sylow
206C137#1951211yescharacteristic, direct product factor, Sylow
207#3235C2×C13711yescharacteristic
208#3229C2×C13711yescharacteristic, complemented
209#3206C2×C13711yescharacteristic
210#2981C2×C13711yescharacteristic
211#1699C2×C13712yescomplemented
212#1691C2×C13712yescomplemented
213#1679C2×C13712yescomplemented
214#1671C2×C13712yescomplemented
215#1572C2×C13711yescharacteristic, complemented
216#1556C2×C13711yescharacteristic, complemented
217#1554C2×C13711yescharacteristic, complemented
218#533C22×C13711yescharacteristic
219#532C22×C13711yescharacteristic
220C2×C4⋊Q8C22×C13711yescharacteristic
221#501—21no—
222#498C22×C13711yescharacteristic
223#498C22×C13711yescharacteristic
224#492—21no—
225#490—21no—
226#487—21no—
227#487C22×C13711yescharacteristic
228#487C22×C13711yescharacteristic
229#483—22no—
230#482—22no—
231#476C22×C13712yes—
232#475—22no—
233#474C22×C13712yes—
234#461C22×C13712yes—
235#461C22×C13712yes—
236#448—22no—
237#436C22×C13712yes—
238#435—22no—
239C8×D4C22×C13712yes—
240#432C22×C13711yescharacteristic
241#432C22×C13711yescharacteristic
242#431C22×C13711yescharacteristic
243#431C22×C13711yescharacteristic
244#408—22no—
245#335C22×C13712yes—
246#334C22×C13712yes—
247#333C22×C13712yes—
248#332C22×C13712yes—
249#332C22×C13712yes—
250#332C22×C13712yes—
251C2×#41—21no—
252C22×Q8—21no—
253C2×SD16—22no—
254C2×SD16—22no—
255C2×M16—22no—
256C22×C8—22no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8D4×C13711yescharacteristic
261C4⋊Q8C23×C13711yescharacteristic
262C4⋊Q8C23×C13711yescharacteristic
263C4⋊Q8C23×C13711yescharacteristic
264C4⋊Q8C23×C13711yescharacteristic
265#127D4×C13711yescharacteristic
266#125—21no—
267#124—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270#121—21no—
271#121D4×C13712yes—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×D4—21no—
275C4×D4C23×C13712yes—
276#117—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C42C23×C13711yescharacteristic
280#107—22no—
281#106—22no—
282#106—22no—
283#106—22no—
284C4⋊C8D4×C13712yes—
285C4⋊C8C23×C13712yes—
286C4⋊C8C23×C13712yes—
287C4⋊C8C23×C13712yes—
288#103—22no—
289#103—22no—
290#103—22no—
291#103—22no—
292#102—22no—
293#102—22no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8D4×C13712yes—
300#97—21no—
301#97—21no—
302C4×C8—21no—
303C4×C8C23×C13711yescharacteristic
304C4×C8C23×C13711yescharacteristic
305#41—82no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—41no—
312C2×Q8—82no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4C2×D4×C13712yes—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4C2×D4×C13711yescharacteristic
320C22×C4C2×D4×C13711yescharacteristic
321C22×C4#41×C13711yescharacteristic
322SD16—82no—
323SD16—82no—
324M16—82no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—82no—
329C2×C8C2×D4×C13712yes—
330C2×C8C2×D4×C13712yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4C2×D4×C13712yes—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—41no—
348C42—41no—
349C42C2×D4×C13711yescharacteristic
350C42C2×D4×C13711yescharacteristic
351C42#41×C13711yescharacteristic
352C42C24×C13711yescharacteristic
353C23—22no—
354C23#121×C13711yescharacteristic
355Q8—42no—
356Q8—42no—
357Q8—42no—
358Q8—81no—
359Q8—81no—
360Q8—82no—
361D4—22no—
362D4—22no—
363D4—81no—
364D4—81no—
365D4—82no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4#121×C13711yescharacteristic
378C2×C4#121×C13711yescharacteristic
379C2×C4#121×C13711yescharacteristic
380C2×C4C22×D4×C13711yescharacteristic
381C2×C4C22×D4×C13711yescharacteristic
382C2×C4C2×#41×C13711yescharacteristic
383C8—22no—
384C8—22no—
385C8—42no—
386C8—42no—
387C8—42no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22C2×#121×C13711yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4#577×C13712yes—
405C4#577×C13712yes—
406C2—41no—
407C2—82no—
408C2#3042×C13711yescharacteristic
409C2#3057×C13711yescharacteristic
410C2#3069×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radical#532×C137C22
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
2. #3229×C137C2semidirect
5. #1699×C137C2semidirect
6. #1691×C137C2semidirect
7. #1679×C137C2semidirect
8. #1671×C137C2semidirect
9. #1572×C137C2semidirect
10. #1556×C137C2semidirect
11. #1554×C137C2semidirect
205. #19512C137direct (N × H)
206. C137#19512direct (N × H)
208. #3229C2×C137semidirect
211. #1699C2×C137semidirect
212. #1691C2×C137semidirect
213. #1679C2×C137semidirect
214. #1671C2×C137semidirect
215. #1572C2×C137semidirect
216. #1556C2×C137semidirect
217. #1554C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3235×C1372
#3229×C1372
#3206×C1372
#2981×C1372
#1699×C1372
#1691×C1372
#1679×C1372
#1671×C1372
#1572×C1372
#1556×C1372
#1554×C1372
#19512137