Finite Group Databasebeta

← C107×#18165 #41044129

Subgroups — C107×#18165 #41044129

Order 27392.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C107×#3492C211yesmaximal, characteristic
2C107×#3004C212yesmaximal, complemented
3C2×#395×C107C211yesmaximal, characteristic, complemented
4C107×#1605C212yesmaximal, complemented
5C107×#1604C212yesmaximal, complemented
6C107×#1603C214yesmaximal, complemented
7C107×#1504C212yesmaximal, complemented
8C107×#1463C211yesmaximal, characteristic, complemented
9C23×Q8×C107C2211yescharacteristic
10C2×#137×C107—22no—
11#554×C107—22no—
12#553×C107—22no—
13#552×C107—22no—
14#547×C107—22no—
15#544×C107C2211yescharacteristic
16#539×C107—21no—
17#539×C107C2212yes—
18#537×C107C2211yescharacteristic
19#525×C107—21no—
20#525×C107—21no—
21#525×C107C2212yes—
22#520×C107—21no—
23C22×C4⋊C4×C107—21no—
24C22×C4⋊C4×C107C2211yescharacteristic
25#451×C107—22no—
26#451×C107C2214yes—
27#448×C107—22no—
28#448×C107C2214yes—
29#419×C107—22no—
30#407×C107C2212yes—
31#395×C107—21no—
32#395×C107—21no—
33#395×C107C2212yes—
34#395×C107C2212yes—
35C2×#95×C107—21no—
36C2×#95×C107C2211yescharacteristic
37#328×C107C2214yes—
38#327×C107C2214yes—
39#324×C107C2212yes—
40#324×C107C2212yes—
41C2×#41×C107—21no—
42C2×#41×C107—21no—
43C2×#41×C107D412yes—
44C22×Q8×C107—21no—
45C22×Q8×C107—22no—
46C22×Q8×C107—22no—
47C22×Q8×C107—44no—
48C22×Q8×C107C2312yes—
49C23×C4×C107—21no—
50C23×C4×C107—21no—
51C23×C4×C107D411yescharacteristic
52C23×C4×C107D411yescharacteristic
53C23×C4×C107C2311yescharacteristic
54#137×C107—48no—
55C2×Q16×C107—24no—
56C2×SD16×C107—24no—
57C2×M16×C107—22no—
58C4⋊Q8×C107—22no—
59#126×C107—22no—
60#126×C107—22no—
61#125×C107—22no—
62#125×C107—22no—
63#125×C107—22no—
64#123×C107—22no—
65#123×C107—22no—
66#123×C107—22no—
67#123×C107—22no—
68#123×C107—22no—
69C22⋊Q8×C107—21no—
70C22⋊Q8×C107—21no—
71C22⋊Q8×C107—21no—
72C22⋊Q8×C107—22no—
73C22⋊Q8×C107—22no—
74C22⋊Q8×C107—21no—
75C22⋊Q8×C107—22no—
76C22⋊Q8×C107D412yes—
77C22⋊Q8×C107C2312yes—
78#121×C107—21no—
79#121×C107—21no—
80#121×C107—21no—
81#121×C107—22no—
82#121×C107—21no—
83#121×C107D412yes—
84#121×C107C2312yes—
85#120×C107—22no—
86C4×Q8×C107—22no—
87C4×D4×C107—22no—
88C4×D4×C107—22no—
89C4×D4×C107—22no—
90C4×D4×C107—22no—
91C4×D4×C107—22no—
92C4×D4×C107—22no—
93C4×D4×C107—22no—
94#117×C107—22no—
95#117×C107—21no—
96#117×C107—21no—
97#117×C107D412yes—
98C2×C4⋊C4×C107—22no—
99C2×C4⋊C4×C107—22no—
100C2×C4⋊C4×C107—22no—
101C2×C4⋊C4×C107—22no—
102C2×C4⋊C4×C107—22no—
103C2×C4⋊C4×C107—44no—
104C2×C4⋊C4×C107D411yescharacteristic
105C2×C4⋊C4×C107D411yescharacteristic
106C2×C4⋊C4×C107C2311yescharacteristic
107C2×C4⋊C4×C107C2311yescharacteristic
108#103×C107—24no—
109#103×C107—24no—
110C22⋊C8×C107—22no—
111C22⋊C8×C107C2314yes—
112#95×C107—22no—
113#95×C107—22no—
114#95×C107—21no—
115#95×C107C2312yes—
116C24×C107C2×D411yescharacteristic
117#41×C107—42no—
118#41×C107—44no—
119#41×C107—42no—
120#41×C107—82no—
121C2×Q8×C107—21no—
122C2×Q8×C107—21no—
123C2×Q8×C107—28no—
124C2×Q8×C107—24no—
125C2×Q8×C107—44no—
126C2×Q8×C107—44no—
127C2×Q8×C107—44no—
128C2×Q8×C107—48no—
129C2×Q8×C107—42no—
130C2×Q8×C107C2×D412yes—
131C2×D4×C107—21no—
132C2×D4×C107—22no—
133C2×D4×C107—22no—
134C2×D4×C107—22no—
135C2×D4×C107—21no—
136C2×D4×C107C2×D412yes—
137C22×C4×C107—22no—
138C22×C4×C107—24no—
139C22×C4×C107—24no—
140C22×C4×C107—21no—
141C22×C4×C107—21no—
142C22×C4×C107—21no—
143C22×C4×C107—22no—
144C22×C4×C107—22no—
145C22×C4×C107—22no—
146C22×C4×C107—21no—
147C22×C4×C107—22no—
148C22×C4×C107—22no—
149C22×C4×C107—41no—
150C22×C4×C107—42no—
151C22×C4×C107—42no—
152C22×C4×C107—42no—
153C22×C4×C107—42no—
154C22×C4×C107—41no—
155C22×C4×C107—41no—
156C22×C4×C107—42no—
157C22×C4×C107C2×D411yescharacteristic
158C22×C4×C107C2×D412yes—
159C22×C4×C107C2×D411yescharacteristic
160C22×C4×C107C2×D411yescharacteristic
161C22×C4×C107C2×D411yescharacteristic
162C22×C4×C107C2×D411yescharacteristic
163C22×C4×C107C2×D412yes—
164C22×C4×C107C2411yescharacteristic
165Q16×C107—48no—
166SD16×C107—48no—
167M16×C107—44no—
168C2×C8×C107—24no—
169C4⋊C4×C107—21no—
170C4⋊C4×C107—22no—
171C4⋊C4×C107—21no—
172C4⋊C4×C107—21no—
173C4⋊C4×C107—21no—
174C4⋊C4×C107—22no—
175C4⋊C4×C107—22no—
176C4⋊C4×C107—22no—
177C4⋊C4×C107—22no—
178C4⋊C4×C107—22no—
179C4⋊C4×C107—22no—
180C4⋊C4×C107—22no—
181C4⋊C4×C107—22no—
182C4⋊C4×C107—22no—
183C4⋊C4×C107—44no—
184C4⋊C4×C107—44no—
185C4⋊C4×C107C2×D412yes—
186C4⋊C4×C107C2×D412yes—
187C22⋊C4×C107—22no—
188C22⋊C4×C107—22no—
189C22⋊C4×C107—22no—
190C22⋊C4×C107—22no—
191C22⋊C4×C107—22no—
192C22⋊C4×C107—22no—
193C22⋊C4×C107—22no—
194C22⋊C4×C107—22no—
195C22⋊C4×C107—22no—
196C42×C107—22no—
197C42×C107—22no—
198C42×C107—22no—
199C23×C107—21no—
200C23×C107—21no—
201C23×C107—22no—
202C23×C107—22no—
203C23×C107—22no—
204C23×C107#12011yescharacteristic
205C23×C107#12011yescharacteristic
206C23×C107C22×D411yescharacteristic
207Q8×C107—28no—
208Q8×C107—42no—
209Q8×C107—44no—
210Q8×C107—48no—
211Q8×C107—42no—
212Q8×C107—81no—
213D4×C107—42no—
214D4×C107—42no—
215D4×C107—42no—
216D4×C107—42no—
217D4×C107—44no—
218D4×C107—44no—
219D4×C107—81no—
220C2×C4×C107—21no—
221C2×C4×C107—21no—
222C2×C4×C107—24no—
223C2×C4×C107—24no—
224C2×C4×C107—24no—
225C2×C4×C107—22no—
226C2×C4×C107—22no—
227C2×C4×C107—21no—
228C2×C4×C107—22no—
229C2×C4×C107—21no—
230C2×C4×C107—21no—
231C2×C4×C107—21no—
232C2×C4×C107—21no—
233C2×C4×C107—42no—
234C2×C4×C107—41no—
235C2×C4×C107—41no—
236C2×C4×C107—42no—
237C2×C4×C107—42no—
238C2×C4×C107—44no—
239C2×C4×C107—42no—
240C2×C4×C107—48no—
241C2×C4×C107—42no—
242C2×C4×C107—44no—
243C2×C4×C107—42no—
244C2×C4×C107—42no—
245C2×C4×C107—44no—
246C2×C4×C107—42no—
247C2×C4×C107—41no—
248C2×C4×C107—41no—
249C2×C4×C107—82no—
250C2×C4×C107—81no—
251C2×C4×C107—82no—
252C2×C4×C107#12012yes—
253C2×C4×C107#13712yescomplemented
254C2×C4×C107C22×D412yes—
255C8×C107—44no—
256C22×C107—24no—
257C22×C107—24no—
258C22×C107—21no—
259C22×C107—21no—
260C22×C107—21no—
261C22×C107—21no—
262C22×C107—21no—
263C22×C107—21no—
264C22×C107—42no—
265C22×C107—42no—
266C22×C107—42no—
267C22×C107#45612yes—
268C22×C107B(4,𝔽2)12yes—
269C22×C107C2×#12011yescharacteristic
270C22×C107C2×#13711yescharacteristic
271C22×C107#57511yescharacteristic
272C4×C107—22no—
273C4×C107—22no—
274C4×C107—22no—
275C4×C107—44no—
276C4×C107—41no—
277C4×C107—42no—
278C4×C107—42no—
279C4×C107—44no—
280C4×C107—41no—
281C4×C107—41no—
282C4×C107—42no—
283#18165C10711yesmaximal, characteristic, direct product factor, Sylow
284C2×C107—21no—
285C2×C107—21no—
286C2×C107—22no—
287C2×C107—22no—
288C2×C107—82no—
289C2×C107#300711yescharacteristic
290C2×C107C2×#45611yescharacteristic
291C2×C107C2×B(4,𝔽2)11yescharacteristic
292#3492C2×C10711yescharacteristic
293#3004C2×C10712yescomplemented
294C2×#395C2×C10711yescharacteristic, complemented
295#1605C2×C10712yescomplemented
296#1604C2×C10712yescomplemented
297#1603C2×C10714yescomplemented
298#1504C2×C10712yescomplemented
299#1463C2×C10711yescharacteristic, complemented
300C107#1816511yescharacteristic, direct product factor, Sylow
301C23×Q8C22×C10711yescharacteristic
302C2×#137—22no—
303#554—22no—
304#553—22no—
305#552—22no—
306#547—22no—
307#544C22×C10711yescharacteristic
308#539—21no—
309#539C22×C10712yes—
310#537C22×C10711yescharacteristic
311#525—21no—
312#525—21no—
313#525C22×C10712yes—
314#520—21no—
315C22×C4⋊C4—21no—
316C22×C4⋊C4C22×C10711yescharacteristic
317#451—22no—
318#451C22×C10714yes—
319#448—22no—
320#448C22×C10714yes—
321#419—22no—
322#407C22×C10712yes—
323#395—21no—
324#395—21no—
325#395C22×C10712yes—
326#395C22×C10712yes—
327C2×#95—21no—
328C2×#95C22×C10711yescharacteristic
329#328C22×C10714yes—
330#327C22×C10714yes—
331#324C22×C10712yes—
332#324C22×C10712yes—
333C2×#41—21no—
334C2×#41—21no—
335C2×#41D4×C10712yes—
336C22×Q8—21no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—44no—
340C22×Q8C23×C10712yes—
341C23×C4—21no—
342C23×C4—21no—
343C23×C4D4×C10711yescharacteristic
344C23×C4D4×C10711yescharacteristic
345C23×C4C23×C10711yescharacteristic
346#137—48no—
347C2×Q16—24no—
348C2×SD16—24no—
349C2×M16—22no—
350C4⋊Q8—22no—
351#126—22no—
352#126—22no—
353#125—22no—
354#125—22no—
355#125—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8D4×C10712yes—
369C22⋊Q8C23×C10712yes—
370#121—21no—
371#121—21no—
372#121—21no—
373#121—22no—
374#121—21no—
375#121D4×C10712yes—
376#121C23×C10712yes—
377#120—22no—
378C4×Q8—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—22no—
386#117—22no—
387#117—21no—
388#117—21no—
389#117D4×C10712yes—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—44no—
396C2×C4⋊C4D4×C10711yescharacteristic
397C2×C4⋊C4D4×C10711yescharacteristic
398C2×C4⋊C4C23×C10711yescharacteristic
399C2×C4⋊C4C23×C10711yescharacteristic
400#103—24no—
401#103—24no—
402C22⋊C8—22no—
403C22⋊C8C23×C10714yes—
404#95—22no—
405#95—22no—
406#95—21no—
407#95C23×C10712yes—
408C24C2×D4×C10711yescharacteristic
409#41—42no—
410#41—44no—
411#41—42no—
412#41—82no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—28no—
416C2×Q8—24no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—42no—
422C2×Q8C2×D4×C10712yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4C2×D4×C10712yes—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4C2×D4×C10711yescharacteristic
450C22×C4C2×D4×C10712yes—
451C22×C4C2×D4×C10711yescharacteristic
452C22×C4C2×D4×C10711yescharacteristic
453C22×C4C2×D4×C10711yescharacteristic
454C22×C4C2×D4×C10711yescharacteristic
455C22×C4C2×D4×C10712yes—
456C22×C4C24×C10711yescharacteristic
457Q16—48no—
458SD16—48no—
459M16—44no—
460C2×C8—24no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4C2×D4×C10712yes—
478C4⋊C4C2×D4×C10712yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23#120×C10711yescharacteristic
497C23#120×C10711yescharacteristic
498C23C22×D4×C10711yescharacteristic
499Q8—28no—
500Q8—42no—
501Q8—44no—
502Q8—48no—
503Q8—42no—
504Q8—81no—
505D4—42no—
506D4—42no—
507D4—44no—
508D4—44no—
509D4—42no—
510D4—42no—
511D4—81no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4#120×C10712yes—
545C2×C4#137×C10712yescomplemented
546C2×C4C22×D4×C10712yes—
547C8—44no—
548C22—24no—
549C22—24no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22#456×C10712yes—
560C22B(4,𝔽2)×C10712yes—
561C22C2×#120×C10711yescharacteristic
562C22C2×#137×C10711yescharacteristic
563C22#575×C10711yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—44no—
572C4—41no—
573C4—41no—
574C4—42no—
575C2—21no—
576C2—22no—
577C2—22no—
578C2—21no—
579C2—82no—
580C2C107×#300711yescharacteristic
581C2C2×#456×C10711yescharacteristic
582C2C2×B(4,𝔽2)×C10711yescharacteristic
subgroupG/H
center, socleC22×C107C2×#120
commutator, Frattini, cosocleC22×C4C24×C107
prime radicalC107×#3492C2
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×#3004C2semidirect
3. C2×#395×C107C2semidirect
4. C107×#1605C2semidirect
5. C107×#1604C2semidirect
6. C107×#1603C2semidirect
7. C107×#1504C2semidirect
8. C107×#1463C2semidirect
253. C2×C4×C107#137semidirect
283. #18165C107direct (N × H)
293. #3004C2×C107semidirect
294. C2×#395C2×C107semidirect
295. #1605C2×C107semidirect
296. #1604C2×C107semidirect
297. #1603C2×C107semidirect
298. #1504C2×C107semidirect
299. #1463C2×C107semidirect
300. C107#18165direct (N × H)
545. C2×C4#137×C107semidirect

Maximal subgroups

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

subgroupindex
C107×#34922
C107×#30042
C2×#395×C1072
C107×#16052
C107×#16042
C107×#16032
C107×#15042
C107×#14632
#18165107