Finite Group Databasebeta

← C107×#18151 #41044115

Subgroups — C107×#18151 #41044115

Order 27392.

994 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C107×#3460C211yesmaximal, characteristic
2C107×#3092C212yesmaximal, complemented
3C107×#2487C211yesmaximal, characteristic, complemented
4C107×#1618C214yesmaximal, complemented
5C107×#1617C212yesmaximal, complemented
6C107×#1616C212yesmaximal, complemented
7C107×#1513C212yesmaximal, complemented
8C107×#1465C211yesmaximal, characteristic, complemented
9#544×C107C2211yescharacteristic
10#542×C107—22no—
11#540×C107—22no—
12#538×C107—21no—
13#538×C107C2212yes—
14#537×C107C2211yescharacteristic
15#535×C107—21no—
16#532×C107—22no—
17#529×C107—22no—
18#525×C107—21no—
19#525×C107C2212yes—
20C4×#41×C107—21no—
21C2×C4×D4×C107C2211yescharacteristic
22C22×C4⋊C4×C107C2211yescharacteristic
23#484×C107C2214yes—
24#483×C107—22no—
25#481×C107C2214yes—
26#480×C107—22no—
27#428×C107—22no—
28#417×C107—22no—
29#407×C107C2212yes—
30#400×C107C2212yes—
31#399×C107C2212yes—
32#389×C107—21no—
33#386×C107—21no—
34#385×C107—21no—
35#380×C107—21no—
36#380×C107C2211yescharacteristic
37#328×C107C2214yes—
38#327×C107C2214yes—
39#324×C107C2212yes—
40#323×C107C2212yes—
41C2×#41×C107—21no—
42C2×#41×C107—21no—
43C2×#41×C107D412yes—
44C22×D4×C107D411yescharacteristic
45C23×C4×C107—21no—
46C23×C4×C107D411yescharacteristic
47C23×C4×C107C2311yescharacteristic
48C2×M16×C107—22no—
49C4⋊Q8×C107—22no—
50#127×C107—22no—
51#126×C107—22no—
52#126×C107—22no—
53#125×C107—22no—
54#124×C107—22no—
55#124×C107—22no—
56#124×C107—22no—
57#123×C107—22no—
58#123×C107—22no—
59#123×C107—22no—
60#123×C107—22no—
61C22⋊Q8×C107—21no—
62C22⋊Q8×C107—21no—
63C22⋊Q8×C107—22no—
64C22⋊Q8×C107D412yes—
65C22⋊Q8×C107C2312yes—
66#121×C107—21no—
67#121×C107—22no—
68#121×C107—22no—
69#121×C107—22no—
70#121×C107—22no—
71#121×C107—22no—
72#121×C107—21no—
73#121×C107D412yes—
74#121×C107C2312yes—
75#120×C107—22no—
76#120×C107—22no—
77C4×Q8×C107—21no—
78C4×Q8×C107—21no—
79C4×D4×C107—21no—
80C4×D4×C107—21no—
81C4×D4×C107—22no—
82C4×D4×C107—22no—
83C4×D4×C107—22no—
84C4×D4×C107—22no—
85C4×D4×C107—22no—
86C4×D4×C107—22no—
87#117×C107—22no—
88#117×C107—21no—
89#117×C107—21no—
90#117×C107D412yes—
91C2×C4⋊C4×C107—22no—
92C2×C4⋊C4×C107—22no—
93C2×C4⋊C4×C107—21no—
94C2×C4⋊C4×C107—21no—
95C2×C4⋊C4×C107—22no—
96C2×C4⋊C4×C107C2312yes—
97C2×C4⋊C4×C107C2311yescharacteristic
98C2×C22⋊C4×C107—21no—
99C2×C22⋊C4×C107—22no—
100C2×C22⋊C4×C107D411yescharacteristic
101C2×C22⋊C4×C107C2311yescharacteristic
102C2×C42×C107—22no—
103C2×C42×C107D411yescharacteristic
104#107×C107—24no—
105#106×C107—24no—
106#103×C107—24no—
107#102×C107—24no—
108C22⋊C8×C107—22no—
109C22⋊C8×C107C2314yes—
110#95×C107—22no—
111#95×C107—22no—
112#95×C107C2312yes—
113C24×C107C2×D411yescharacteristic
114C24×C107C2×D411yescharacteristic
115#41×C107—42no—
116#41×C107—44no—
117#41×C107—42no—
118#41×C107—82no—
119C2×Q8×C107—21no—
120C2×Q8×C107—21no—
121C2×Q8×C107C2×D412yes—
122C2×D4×C107—21no—
123C2×D4×C107—22no—
124C2×D4×C107—22no—
125C2×D4×C107—22no—
126C2×D4×C107—22no—
127C2×D4×C107—21no—
128C2×D4×C107—22no—
129C2×D4×C107—22no—
130C2×D4×C107—22no—
131C2×D4×C107—41no—
132C2×D4×C107C2×D412yes—
133C22×C4×C107—22no—
134C22×C4×C107—22no—
135C22×C4×C107—22no—
136C22×C4×C107—21no—
137C22×C4×C107—21no—
138C22×C4×C107—22no—
139C22×C4×C107—22no—
140C22×C4×C107—42no—
141C22×C4×C107—42no—
142C22×C4×C107—42no—
143C22×C4×C107—82no—
144C22×C4×C107C2×D411yescharacteristic
145C22×C4×C107C2×D412yes—
146C22×C4×C107C2×D411yescharacteristic
147C22×C4×C107C2×D411yescharacteristic
148C22×C4×C107C2×D411yescharacteristic
149C22×C4×C107C2×D412yes—
150C22×C4×C107C2411yescharacteristic
151M16×C107—44no—
152C2×C8×C107—24no—
153C4⋊C4×C107—21no—
154C4⋊C4×C107—21no—
155C4⋊C4×C107—22no—
156C4⋊C4×C107—21no—
157C4⋊C4×C107—21no—
158C4⋊C4×C107—22no—
159C4⋊C4×C107—22no—
160C4⋊C4×C107—22no—
161C4⋊C4×C107—22no—
162C4⋊C4×C107—24no—
163C4⋊C4×C107—42no—
164C4⋊C4×C107—41no—
165C4⋊C4×C107—41no—
166C4⋊C4×C107—82no—
167C4⋊C4×C107C2×D412yes—
168C4⋊C4×C107C2×D412yes—
169C22⋊C4×C107—22no—
170C22⋊C4×C107—22no—
171C22⋊C4×C107—22no—
172C22⋊C4×C107—22no—
173C22⋊C4×C107—22no—
174C22⋊C4×C107—22no—
175C22⋊C4×C107—22no—
176C22⋊C4×C107—22no—
177C22⋊C4×C107—22no—
178C22⋊C4×C107—22no—
179C22⋊C4×C107—22no—
180C22⋊C4×C107—42no—
181C22⋊C4×C107—82no—
182C42×C107—22no—
183C42×C107—22no—
184C42×C107—21no—
185C42×C107—22no—
186C42×C107—21no—
187C23×C107—21no—
188C23×C107—21no—
189C23×C107—21no—
190C23×C107—22no—
191C23×C107—21no—
192C23×C107—21no—
193C23×C107—41no—
194C23×C107—41no—
195C23×C107—41no—
196C23×C107—41no—
197C23×C107#12012yes—
198C23×C107C22×D411yescharacteristic
199Q8×C107—42no—
200Q8×C107—42no—
201Q8×C107—81no—
202D4×C107—44no—
203D4×C107—42no—
204D4×C107—42no—
205D4×C107—42no—
206D4×C107—42no—
207D4×C107—82no—
208D4×C107—82no—
209D4×C107—82no—
210D4×C107—81no—
211D4×C107—82no—
212C2×C4×C107—21no—
213C2×C4×C107—21no—
214C2×C4×C107—22no—
215C2×C4×C107—24no—
216C2×C4×C107—21no—
217C2×C4×C107—22no—
218C2×C4×C107—22no—
219C2×C4×C107—22no—
220C2×C4×C107—21no—
221C2×C4×C107—42no—
222C2×C4×C107—41no—
223C2×C4×C107—44no—
224C2×C4×C107—42no—
225C2×C4×C107—44no—
226C2×C4×C107—41no—
227C2×C4×C107—41no—
228C2×C4×C107—42no—
229C2×C4×C107—42no—
230C2×C4×C107—42no—
231C2×C4×C107—42no—
232C2×C4×C107—42no—
233C2×C4×C107—42no—
234C2×C4×C107—81no—
235C2×C4×C107—82no—
236C2×C4×C107—84no—
237C2×C4×C107#12011yescharacteristic
238C2×C4×C107#12011yescharacteristic
239C2×C4×C107#13712yes—
240C2×C4×C107C22×D412yes—
241C8×C107—44no—
242C22×C107—21no—
243C22×C107—21no—
244C22×C107—22no—
245C22×C107—22no—
246C22×C107—22no—
247C22×C107—42no—
248C22×C107—42no—
249C22×C107—42no—
250C22×C107—42no—
251C22×C107—41no—
252C22×C107—42no—
253C22×C107—41no—
254C22×C107—41no—
255C22×C107—42no—
256C22×C107—81no—
257C22×C107C2×#12011yescharacteristic
258C22×C107C2×#13711yescharacteristic
259C22×C107#57511yescharacteristic
260C4×C107—22no—
261C4×C107—22no—
262C4×C107—21no—
263C4×C107—42no—
264C4×C107—41no—
265C4×C107—42no—
266C4×C107—41no—
267C4×C107—42no—
268C4×C107—82no—
269C4×C107—82no—
270C4×C107#45812yes—
271#18151C10711yesmaximal, characteristic, direct product factor, Sylow
272C2×C107—21no—
273C2×C107—21no—
274C2×C107—42no—
275C2×C107—82no—
276C2×C107—81no—
277C2×C107#300711yescharacteristic
278C2×C107#302111yescharacteristic
279C2×C107C2×#45811yescharacteristic
280#3460C2×C10711yescharacteristic
281#3092C2×C10712yescomplemented
282#2487C2×C10711yescharacteristic, complemented
283#1618C2×C10714yescomplemented
284#1617C2×C10712yescomplemented
285#1616C2×C10712yescomplemented
286#1513C2×C10712yescomplemented
287#1465C2×C10711yescharacteristic, complemented
288C107#1815111yescharacteristic, direct product factor, Sylow
289#544C22×C10711yescharacteristic
290#542—22no—
291#540—22no—
292#538—21no—
293#538C22×C10712yes—
294#537C22×C10711yescharacteristic
295#535—21no—
296#532—22no—
297#529—22no—
298#525—21no—
299#525C22×C10712yes—
300C4×#41—21no—
301C2×C4×D4C22×C10711yescharacteristic
302C22×C4⋊C4C22×C10711yescharacteristic
303#484C22×C10714yes—
304#483—22no—
305#481C22×C10714yes—
306#480—22no—
307#428—22no—
308#417—22no—
309#407C22×C10712yes—
310#400C22×C10712yes—
311#399C22×C10712yes—
312#389—21no—
313#386—21no—
314#385—21no—
315#380—21no—
316#380C22×C10711yescharacteristic
317#328C22×C10714yes—
318#327C22×C10714yes—
319#324C22×C10712yes—
320#323C22×C10712yes—
321C2×#41—21no—
322C2×#41—21no—
323C2×#41D4×C10712yes—
324C22×D4D4×C10711yescharacteristic
325C23×C4—21no—
326C23×C4D4×C10711yescharacteristic
327C23×C4C23×C10711yescharacteristic
328C2×M16—22no—
329C4⋊Q8—22no—
330#127—22no—
331#126—22no—
332#126—22no—
333#125—22no—
334#124—22no—
335#124—22no—
336#124—22no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—22no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—22no—
344C22⋊Q8D4×C10712yes—
345C22⋊Q8C23×C10712yes—
346#121—21no—
347#121—22no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121—22no—
352#121—21no—
353#121D4×C10712yes—
354#121C23×C10712yes—
355#120—22no—
356#120—22no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367#117—22no—
368#117—21no—
369#117—21no—
370#117D4×C10712yes—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4C23×C10712yes—
377C2×C4⋊C4C23×C10711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4D4×C10711yescharacteristic
381C2×C22⋊C4C23×C10711yescharacteristic
382C2×C42—22no—
383C2×C42D4×C10711yescharacteristic
384#107—24no—
385#106—24no—
386#103—24no—
387#102—24no—
388C22⋊C8—22no—
389C22⋊C8C23×C10714yes—
390#95—22no—
391#95—22no—
392#95C23×C10712yes—
393C24C2×D4×C10711yescharacteristic
394C24C2×D4×C10711yescharacteristic
395#41—42no—
396#41—44no—
397#41—42no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C2×D4×C10712yes—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—41no—
412C2×D4C2×D4×C10712yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4C2×D4×C10711yescharacteristic
425C22×C4C2×D4×C10712yes—
426C22×C4C2×D4×C10711yescharacteristic
427C22×C4C2×D4×C10711yescharacteristic
428C22×C4C2×D4×C10711yescharacteristic
429C22×C4C2×D4×C10712yes—
430C22×C4C24×C10711yescharacteristic
431M16—44no—
432C2×C8—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—82no—
447C4⋊C4C2×D4×C10712yes—
448C4⋊C4C2×D4×C10712yes—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—42no—
461C22⋊C4—82no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—22no—
466C42—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#120×C10712yes—
478C23C22×D4×C10711yescharacteristic
479Q8—42no—
480Q8—42no—
481Q8—81no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—82no—
488D4—82no—
489D4—81no—
490D4—82no—
491D4—82no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4#120×C10711yescharacteristic
518C2×C4#120×C10711yescharacteristic
519C2×C4#137×C10712yes—
520C2×C4C22×D4×C10712yes—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—81no—
537C22C2×#120×C10711yescharacteristic
538C22C2×#137×C10711yescharacteristic
539C22#575×C10711yescharacteristic
540C4—22no—
541C4—22no—
542C4—21no—
543C4—42no—
544C4—41no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4#458×C10712yes—
551C2—21no—
552C2—21no—
553C2—42no—
554C2—81no—
555C2—82no—
556C2C107×#300711yescharacteristic
557C2C107×#302111yescharacteristic
558C2C2×#458×C10711yescharacteristic
subgroupG/H
center, socleC22×C107C2×#120
commutator, Frattini, cosocleC22×C4C24×C107
prime radicalC107×#3460C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C107 ◁ C23×C4×C107 ◁ G

Complements

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

NH ≅ G/Ntype
2. C107×#3092C2semidirect
3. C107×#2487C2semidirect
4. C107×#1618C2semidirect
5. C107×#1617C2semidirect
6. C107×#1616C2semidirect
7. C107×#1513C2semidirect
8. C107×#1465C2semidirect
271. #18151C107direct (N × H)
281. #3092C2×C107semidirect
282. #2487C2×C107semidirect
283. #1618C2×C107semidirect
284. #1617C2×C107semidirect
285. #1616C2×C107semidirect
286. #1513C2×C107semidirect
287. #1465C2×C107semidirect
288. C107#18151direct (N × H)

Maximal subgroups

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

subgroupindex
C107×#34602
C107×#30922
C107×#24872
C107×#16182
C107×#16172
C107×#16162
C107×#15132
C107×#14652
#18151107