Finite Group Databasebeta

← C137×#18655 #46868041

Subgroups — C137×#18655 #46868041

Order 35072.

884 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#3509×C137C211yesmaximal, characteristic
2#3209×C137C211yesmaximal, characteristic, complemented
3C2×#498×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic
5#1653×C137C214yesmaximal, complemented
6#1652×C137C214yesmaximal, complemented
7#1546×C137C211yesmaximal, characteristic, complemented
8#1493×C137C211yesmaximal, characteristic, complemented
9#1476×C137C211yesmaximal, characteristic
10#557×C137—21no—
11#557×C137C2212yes—
12#556×C137—21no—
13#556×C137C2212yes—
14#552×C137—22no—
15#552×C137—22no—
16#541×C137—22no—
17#536×C137—21no—
18#533×C137—21no—
19#533×C137—21no—
20#533×C137C2211yescharacteristic
21C2×C4⋊Q8×C137C2211yescharacteristic
22C2×C4⋊Q8×C137C2211yescharacteristic
23#519×C137—21no—
24C4×#41×C137C2211yescharacteristic
25#498×C137—22no—
26#498×C137C2214yes—
27#494×C137—21no—
28#487×C137—21no—
29#477×C137—21no—
30#476×C137—21no—
31#475×C137C2212yes—
32#474×C137C2212yes—
33#462×C137C2214yes—
34#451×C137—22no—
35#431×C137C2211yescharacteristic
36#428×C137—21no—
37C2×#106×C137—22no—
38#423×C137C2211yescharacteristic
39#417×C137—21no—
40C2×C4×C8×C137C2211yescharacteristic
41#332×C137C2214yes—
42Q8⋊C8×C137C2214yes—
43Q8⋊C8×C137C2212yes—
44D4⋊C8×C137C2212yes—
45C2×#41×C137D411yescharacteristic
46C22×Q8×C137—21no—
47C22×Q8×C137—21no—
48C2×Q16×C137—24no—
49C2×M16×C137—21no—
50C22×C8×C137—21no—
51C4⋊Q8×C137—22no—
52C4⋊Q8×C137—21no—
53C4⋊Q8×C137—21no—
54C4⋊Q8×C137—21no—
55C4⋊Q8×C137—21no—
56C4⋊Q8×C137—21no—
57C4⋊Q8×C137D411yescharacteristic
58C4⋊Q8×C137D411yescharacteristic
59C4⋊Q8×C137D412yes—
60C4⋊Q8×C137C2312yes—
61C4⋊Q8×C137C2312yes—
62#126×C137—24no—
63#126×C137—24no—
64#125×C137—22no—
65#125×C137—24no—
66#125×C137—22no—
67#125×C137—24no—
68#125×C137—21no—
69#124×C137—22no—
70#123×C137—24no—
71C22⋊Q8×C137—22no—
72C22⋊Q8×C137—22no—
73C22⋊Q8×C137—22no—
74C22⋊Q8×C137—22no—
75C22⋊Q8×C137—21no—
76C22⋊Q8×C137D412yes—
77C4×Q8×C137—22no—
78C4×Q8×C137—22no—
79C4×Q8×C137—22no—
80C4×Q8×C137—22no—
81C4×Q8×C137—22no—
82C4×Q8×C137—22no—
83C4×Q8×C137—21no—
84C4×Q8×C137D411yescharacteristic
85C4×Q8×C137C2312yes—
86C4×Q8×C137C2311yescharacteristic
87C4×D4×C137—21no—
88C4×D4×C137—21no—
89C4×D4×C137D411yescharacteristic
90C4×D4×C137C2311yescharacteristic
91#117×C137—24no—
92#117×C137—22no—
93#117×C137—21no—
94#117×C137—22no—
95#117×C137—21no—
96#117×C137D411yescharacteristic
97C2×C4⋊C4×C137—21no—
98C2×C4⋊C4×C137—21no—
99C2×C4⋊C4×C137—21no—
100C2×C4⋊C4×C137D412yes—
101C2×C42×C137—21no—
102C2×C42×C137C2311yescharacteristic
103#107×C137—22no—
104#106×C137—28no—
105#106×C137—22no—
106C4⋊C8×C137—21no—
107C4⋊C8×C137C2312yes—
108C4⋊C8×C137C2312yes—
109#103×C137—22no—
110#103×C137—24no—
111#102×C137—22no—
112C22⋊C8×C137—21no—
113C4×C8×C137—21no—
114C4×C8×C137—21no—
115C4×C8×C137C2312yes—
116#41×C137—22no—
117#41×C137—22no—
118C2×Q8×C137—21no—
119C2×Q8×C137—22no—
120C2×Q8×C137—22no—
121C2×Q8×C137—22no—
122C2×Q8×C137—41no—
123C2×Q8×C137—42no—
124C2×Q8×C137—42no—
125C2×Q8×C137—82no—
126C2×Q8×C137C2×D411yescharacteristic
127C2×Q8×C137C2×D412yes—
128C2×D4×C137—21no—
129C2×D4×C137C2×D411yescharacteristic
130C22×C4×C137—22no—
131C22×C4×C137—21no—
132C22×C4×C137—21no—
133C22×C4×C137—21no—
134C22×C4×C137—41no—
135C22×C4×C137C2×D411yescharacteristic
136C22×C4×C137C2×D411yescharacteristic
137C22×C4×C137C2×D411yescharacteristic
138Q16×C137—84no—
139M16×C137—42no—
140C2×C8×C137—24no—
141C2×C8×C137—22no—
142C2×C8×C137—22no—
143C2×C8×C137—22no—
144C4⋊C4×C137—24no—
145C4⋊C4×C137—24no—
146C4⋊C4×C137—24no—
147C4⋊C4×C137—24no—
148C4⋊C4×C137—22no—
149C4⋊C4×C137—22no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—21no—
153C4⋊C4×C137—21no—
154C4⋊C4×C137—22no—
155C4⋊C4×C137—22no—
156C4⋊C4×C137—21no—
157C4⋊C4×C137—22no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—41no—
161C4⋊C4×C137—41no—
162C4⋊C4×C137C2×D412yes—
163C4⋊C4×C137C2×D414yes—
164C4⋊C4×C137C2×D411yescharacteristic
165C4⋊C4×C137C2×D411yescharacteristic
166C22⋊C4×C137—24no—
167C22⋊C4×C137—24no—
168C22⋊C4×C137—21no—
169C22⋊C4×C137—22no—
170C22⋊C4×C137—21no—
171C22⋊C4×C137—21no—
172C42×C137—24no—
173C42×C137—24no—
174C42×C137—21no—
175C42×C137—21no—
176C42×C137—21no—
177C42×C137—22no—
178C42×C137C2×D411yescharacteristic
179C42×C137C2×D411yescharacteristic
180C42×C137C2×D411yescharacteristic
181C42×C137C2411yescharacteristic
182C23×C137—21no—
183C23×C137#12011yescharacteristic
184Q8×C137—21no—
185Q8×C137—21no—
186Q8×C137—24no—
187Q8×C137—42no—
188Q8×C137—42no—
189Q8×C137—44no—
190Q8×C137—42no—
191Q8×C137—44no—
192Q8×C137—82no—
193D4×C137—21no—
194D4×C137—21no—
195D4×C137—42no—
196C2×C4×C137—22no—
197C2×C4×C137—22no—
198C2×C4×C137—21no—
199C2×C4×C137—21no—
200C2×C4×C137—22no—
201C2×C4×C137—22no—
202C2×C4×C137—24no—
203C2×C4×C137—21no—
204C2×C4×C137—44no—
205C2×C4×C137—42no—
206C2×C4×C137—42no—
207C2×C4×C137—42no—
208C2×C4×C137—42no—
209C2×C4×C137—82no—
210C2×C4×C137#12011yescharacteristic
211C2×C4×C137#12011yescharacteristic
212C2×C4×C137#12011yescharacteristic
213C2×C4×C137#13512yes—
214C2×C4×C137#13712yes—
215C2×C4×C137C22×D411yescharacteristic
216C2×C4×C137C22×D411yescharacteristic
217C2×C4×C137C22×D411yescharacteristic
218C8×C137—24no—
219C8×C137—42no—
220C8×C137—42no—
221C22×C137—21no—
222C22×C137—21no—
223C22×C137—41no—
224C22×C137—41no—
225C22×C137—41no—
226C22×C137#45212yes—
227C22×C137C2×#12011yescharacteristic
228C4×C137—21no—
229C4×C137—21no—
230C4×C137—41no—
231C4×C137—44no—
232C4×C137—41no—
233C4×C137—42no—
234C4×C137—41no—
235C4×C137—42no—
236C4×C137—82no—
237C4×C137#45212yes—
238C4×C137#45612yes—
239C4×C137C2×#13511yescharacteristic
240C4×C137C2×#13711yescharacteristic
241C4×C137#57712yes—
242C2×C137—22no—
243C2×C137—81no—
244C2×C137C2×#45211yescharacteristic
245C2×C137#301011yescharacteristic
246C2×C137C2×#45611yescharacteristic
247#18655C13711yesmaximal, characteristic, direct product factor, Sylow
248C137#1865511yescharacteristic, direct product factor, Sylow
249#3509C2×C13711yescharacteristic
250#3209C2×C13711yescharacteristic, complemented
251C2×#498C2×C13711yescharacteristic, complemented
252#3079C2×C13711yescharacteristic
253#1653C2×C13714yescomplemented
254#1652C2×C13714yescomplemented
255#1546C2×C13711yescharacteristic, complemented
256#1493C2×C13711yescharacteristic, complemented
257#1476C2×C13711yescharacteristic
258#557—21no—
259#557C22×C13712yes—
260#556—21no—
261#556C22×C13712yes—
262#552—22no—
263#552—22no—
264#541—22no—
265#536—21no—
266#533—21no—
267#533—21no—
268#533C22×C13711yescharacteristic
269C2×C4⋊Q8C22×C13711yescharacteristic
270C2×C4⋊Q8C22×C13711yescharacteristic
271#519—21no—
272C4×#41C22×C13711yescharacteristic
273#498—22no—
274#498C22×C13714yes—
275#494—21no—
276#487—21no—
277#477—21no—
278#476—21no—
279#475C22×C13712yes—
280#474C22×C13712yes—
281#462C22×C13714yes—
282#451—22no—
283#431C22×C13711yescharacteristic
284#428—21no—
285C2×#106—22no—
286#423C22×C13711yescharacteristic
287#417—21no—
288C2×C4×C8C22×C13711yescharacteristic
289#332C22×C13714yes—
290Q8⋊C8C22×C13714yes—
291Q8⋊C8C22×C13712yes—
292D4⋊C8C22×C13712yes—
293C2×#41D4×C13711yescharacteristic
294C22×Q8—21no—
295C22×Q8—21no—
296C2×Q16—24no—
297C2×M16—21no—
298C22×C8—21no—
299C4⋊Q8—22no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8D4×C13711yescharacteristic
306C4⋊Q8D4×C13712yes—
307C4⋊Q8D4×C13711yescharacteristic
308C4⋊Q8C23×C13712yes—
309C4⋊Q8C23×C13712yes—
310#126—24no—
311#126—24no—
312#125—22no—
313#125—24no—
314#125—22no—
315#125—24no—
316#125—21no—
317#124—22no—
318#123—24no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8D4×C13712yes—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—21no—
332C4×Q8D4×C13711yescharacteristic
333C4×Q8C23×C13711yescharacteristic
334C4×Q8C23×C13712yes—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4D4×C13711yescharacteristic
338C4×D4C23×C13711yescharacteristic
339#117—24no—
340#117—22no—
341#117—21no—
342#117—22no—
343#117—21no—
344#117D4×C13711yescharacteristic
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4D4×C13712yes—
349C2×C42—21no—
350C2×C42C23×C13711yescharacteristic
351#107—22no—
352#106—28no—
353#106—22no—
354C4⋊C8—21no—
355C4⋊C8C23×C13712yes—
356C4⋊C8C23×C13712yes—
357#103—22no—
358#103—24no—
359#102—22no—
360C22⋊C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8C23×C13712yes—
364#41—22no—
365#41—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—82no—
374C2×Q8C2×D4×C13711yescharacteristic
375C2×Q8C2×D4×C13712yes—
376C2×D4—21no—
377C2×D4C2×D4×C13711yescharacteristic
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4C2×D4×C13711yescharacteristic
384C22×C4C2×D4×C13711yescharacteristic
385C22×C4C2×D4×C13711yescharacteristic
386Q16—84no—
387M16—42no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4C2×D4×C13712yes—
411C4⋊C4C2×D4×C13714yes—
412C4⋊C4C2×D4×C13711yescharacteristic
413C4⋊C4C2×D4×C13711yescharacteristic
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—21no—
417C22⋊C4—22no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C42—24no—
421C42—24no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42C2×D4×C13711yescharacteristic
427C42C2×D4×C13711yescharacteristic
428C42C2×D4×C13711yescharacteristic
429C42C24×C13711yescharacteristic
430C23—21no—
431C23#120×C13711yescharacteristic
432Q8—21no—
433Q8—21no—
434Q8—24no—
435Q8—44no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—44no—
440Q8—82no—
441D4—21no—
442D4—21no—
443D4—42no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—82no—
458C2×C4#120×C13711yescharacteristic
459C2×C4#120×C13711yescharacteristic
460C2×C4#120×C13711yescharacteristic
461C2×C4#135×C13712yes—
462C2×C4#137×C13712yes—
463C2×C4C22×D4×C13711yescharacteristic
464C2×C4C22×D4×C13711yescharacteristic
465C2×C4C22×D4×C13711yescharacteristic
466C8—24no—
467C8—42no—
468C8—42no—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22#452×C13712yes—
475C22C2×#120×C13711yescharacteristic
476C4—21no—
477C4—21no—
478C4—41no—
479C4—44no—
480C4—41no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—82no—
485C4#452×C13712yes—
486C4#456×C13712yes—
487C4C2×#135×C13711yescharacteristic
488C4C2×#137×C13711yescharacteristic
489C4#577×C13712yes—
490C2—22no—
491C2—81no—
492C2C2×#452×C13711yescharacteristic
493C2#3010×C13711yescharacteristic
494C2C2×#456×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radicalC2×#41×C137D4
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. #3209×C137C2semidirect
3. C2×#498×C137C2semidirect
5. #1653×C137C2semidirect
6. #1652×C137C2semidirect
7. #1546×C137C2semidirect
8. #1493×C137C2semidirect
247. #18655C137direct (N × H)
248. C137#18655direct (N × H)
250. #3209C2×C137semidirect
251. C2×#498C2×C137semidirect
253. #1653C2×C137semidirect
254. #1652C2×C137semidirect
255. #1546C2×C137semidirect
256. #1493C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3509×C1372
#3209×C1372
C2×#498×C1372
#3079×C1372
#1653×C1372
#1652×C1372
#1546×C1372
#1493×C1372
#1476×C1372
#18655137