Finite Group Databasebeta

← C107×#18154 #41044118

Subgroups — C107×#18154 #41044118

Order 27392.

960 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C107×#3458C211yesmaximal, characteristic, complemented
2C107×#3110C212yesmaximal, complemented
3C107×#2598C211yesmaximal, characteristic
4C107×#1617C212yesmaximal, complemented
5C107×#1614C212yesmaximal, complemented
6C107×#1599C212yesmaximal, complemented
7C107×#1596C212yesmaximal
8C107×#1515C212yesmaximal, complemented
9C107×#1464C211yesmaximal, characteristic, complemented
10#544×C107—22no—
11#544×C107—22no—
12#544×C107—21no—
13#544×C107C2212yes—
14#543×C107—21no—
15#543×C107C2212yes—
16#533×C107—24no—
17#533×C107—21no—
18#533×C107C2212yes—
19C2×C4×D4×C107C2211yescharacteristic
20C2×#117×C107—21no—
21C2×#117×C107C2211yescharacteristic
22#484×C107C2214yes—
23#474×C107—22no—
24#461×C107—22no—
25#450×C107C2214yes—
26#440×C107—24no—
27#432×C107C2212yes—
28#400×C107C2211yescharacteristic
29#399×C107C2211yescharacteristic
30#394×C107C2211yescharacteristic
31#392×C107C2211yescharacteristic
32#388×C107—21no—
33#386×C107—21no—
34#386×C107—21no—
35#385×C107—21no—
36#383×C107C2211yescharacteristic
37#328×C107C2214yes—
38#328×C107C2214yes—
39#324×C107C2212yes—
40#323×C107C2212yes—
41C22×D4×C107—21no—
42C22×D4×C107C2311yescharacteristic
43C23×C4×C107—21no—
44C23×C4×C107D411yescharacteristic
45C2×SD16×C107—24no—
46C4⋊Q8×C107—24no—
47C4⋊Q8×C107—21no—
48C4⋊Q8×C107—21no—
49C4⋊Q8×C107—24no—
50C4⋊Q8×C107D412yescomplemented
51C4⋊Q8×C107D412yescomplemented
52#125×C107—24no—
53#125×C107—22no—
54#125×C107—24no—
55#125×C107—22no—
56C22⋊Q8×C107—24no—
57C22⋊Q8×C107—24no—
58C22⋊Q8×C107—24no—
59C22⋊Q8×C107—22no—
60C22⋊Q8×C107—22no—
61C22⋊Q8×C107—22no—
62C22⋊Q8×C107—24no—
63C22⋊Q8×C107C2314yes—
64C4×Q8×C107—22no—
65C4×Q8×C107—22no—
66C4×Q8×C107—22no—
67C4×Q8×C107D414yescomplemented
68C4×D4×C107—22no—
69C4×D4×C107—22no—
70#117×C107—22no—
71#117×C107—22no—
72#117×C107—22no—
73#117×C107—22no—
74#117×C107—22no—
75#117×C107—22no—
76C2×C4⋊C4×C107—21no—
77C2×C4⋊C4×C107—21no—
78C2×C4⋊C4×C107—21no—
79C2×C4⋊C4×C107D411yescharacteristic
80C2×C4⋊C4×C107C2311yescharacteristic
81C2×C4⋊C4×C107C2311yescharacteristic
82C2×C22⋊C4×C107—22no—
83C2×C22⋊C4×C107—21no—
84C2×C22⋊C4×C107—21no—
85C2×C22⋊C4×C107—21no—
86C2×C22⋊C4×C107—21no—
87C2×C22⋊C4×C107—21no—
88C2×C22⋊C4×C107D411yescharacteristic
89C2×C22⋊C4×C107C2311yescharacteristic
90C2×C42×C107—22no—
91C2×C42×C107D411yescharacteristic
92C2×C42×C107C2311yescharacteristic
93#107×C107—24no—
94C4⋊C8×C107—22no—
95#103×C107—24no—
96#102×C107—24no—
97C22⋊C8×C107C2314yes—
98#97×C107—22no—
99#95×C107—21no—
100#95×C107—21no—
101#95×C107—21no—
102#95×C107—21no—
103#95×C107C2311yescharacteristic
104#95×C107C2311yescharacteristic
105C24×C107—21no—
106C24×C107C2×D411yescharacteristic
107C2×Q8×C107—22no—
108C2×Q8×C107—22no—
109C2×Q8×C107—22no—
110C2×Q8×C107C2×D414yes—
111C2×D4×C107—22no—
112C2×D4×C107—42no—
113C2×D4×C107—42no—
114C2×D4×C107—42no—
115C2×D4×C107—81no—
116C22×C4×C107—22no—
117C22×C4×C107—21no—
118C22×C4×C107—21no—
119C22×C4×C107—22no—
120C22×C4×C107—22no—
121C22×C4×C107—22no—
122C22×C4×C107—21no—
123C22×C4×C107—21no—
124C22×C4×C107—21no—
125C22×C4×C107—42no—
126C22×C4×C107—81no—
127C22×C4×C107—81no—
128C22×C4×C107C2×D411yescharacteristic
129C22×C4×C107C2×D411yescharacteristic
130C22×C4×C107C2×D411yescharacteristic
131C22×C4×C107C2×D411yescharacteristic
132C22×C4×C107C2×D411yescharacteristic
133C22×C4×C107C2411yescharacteristic
134SD16×C107—48no—
135C2×C8×C107—24no—
136C4⋊C4×C107—22no—
137C4⋊C4×C107—22no—
138C4⋊C4×C107—22no—
139C4⋊C4×C107—24no—
140C4⋊C4×C107—24no—
141C4⋊C4×C107—22no—
142C4⋊C4×C107—22no—
143C4⋊C4×C107—22no—
144C4⋊C4×C107—22no—
145C4⋊C4×C107—22no—
146C4⋊C4×C107—24no—
147C4⋊C4×C107—22no—
148C4⋊C4×C107—24no—
149C4⋊C4×C107—22no—
150C4⋊C4×C107—81no—
151C4⋊C4×C107C2×D414yes—
152C22⋊C4×C107—24no—
153C22⋊C4×C107—24no—
154C22⋊C4×C107—22no—
155C22⋊C4×C107—22no—
156C22⋊C4×C107—42no—
157C22⋊C4×C107—41no—
158C22⋊C4×C107—41no—
159C22⋊C4×C107—42no—
160C22⋊C4×C107—81no—
161C22⋊C4×C107—81no—
162C42×C107—24no—
163C42×C107—24no—
164C42×C107—21no—
165C42×C107—21no—
166C42×C107C2×D412yes—
167C42×C107C2×D412yes—
168C23×C107—21no—
169C23×C107—21no—
170C23×C107—22no—
171C23×C107—41no—
172C23×C107—41no—
173C23×C107—41no—
174C23×C107—41no—
175C23×C107—41no—
176C23×C107—41no—
177C23×C107—41no—
178C23×C107—41no—
179C23×C107#13712yes—
180C23×C107C22×D411yescharacteristic
181Q8×C107—44no—
182Q8×C107—82no—
183Q8×C107—82no—
184Q8×C107—82no—
185D4×C107—24no—
186D4×C107—42no—
187D4×C107—84no—
188C2×C4×C107—24no—
189C2×C4×C107—24no—
190C2×C4×C107—21no—
191C2×C4×C107—21no—
192C2×C4×C107—21no—
193C2×C4×C107—21no—
194C2×C4×C107—24no—
195C2×C4×C107—21no—
196C2×C4×C107—21no—
197C2×C4×C107—24no—
198C2×C4×C107—41no—
199C2×C4×C107—41no—
200C2×C4×C107—41no—
201C2×C4×C107—41no—
202C2×C4×C107—41no—
203C2×C4×C107—41no—
204C2×C4×C107—41no—
205C2×C4×C107—41no—
206C2×C4×C107—41no—
207C2×C4×C107—41no—
208C2×C4×C107—41no—
209C2×C4×C107—41no—
210C2×C4×C107—41no—
211C2×C4×C107—84no—
212C2×C4×C107—82no—
213C2×C4×C107—82no—
214C2×C4×C107#12011yescharacteristic
215C2×C4×C107#12011yescharacteristic
216C2×C4×C107#12012yes—
217C2×C4×C107#13712yes—
218C2×C4×C107C22×D412yes—
219C8×C107—44no—
220C22×C107—21no—
221C22×C107—21no—
222C22×C107—22no—
223C22×C107—42no—
224C22×C107—42no—
225C22×C107—42no—
226C22×C107—41no—
227C22×C107—41no—
228C22×C107—42no—
229C22×C107—81no—
230C22×C107—81no—
231C22×C107—81no—
232C22×C107—81no—
233C22×C107—81no—
234C22×C107C2×#12011yescharacteristic
235C22×C107C2×#13711yescharacteristic
236C22×C107C2×#13711yescharacteristic
237C4×C107—21no—
238C4×C107—21no—
239C4×C107—22no—
240C4×C107—22no—
241C4×C107—22no—
242C4×C107—41no—
243C4×C107—44no—
244C4×C107—41no—
245C4×C107—44no—
246C4×C107—81no—
247C4×C107—81no—
248C4×C107—81no—
249#18154C10711yesmaximal, characteristic, direct product factor, Sylow
250C2×C107—21no—
251C2×C107—21no—
252C2×C107—42no—
253C2×C107—82no—
254C2×C107#300411yescharacteristic
255C2×C107#302111yescharacteristic
256C2×C107#302511yescharacteristic
257#3458C2×C10711yescharacteristic, complemented
258#3110C2×C10712yescomplemented
259#2598C2×C10711yescharacteristic
260#1617C2×C10712yescomplemented
261#1614C2×C10712yescomplemented
262#1599C2×C10712yescomplemented
263#1596C2×C10712yes—
264#1515C2×C10712yescomplemented
265#1464C2×C10711yescharacteristic, complemented
266C107#1815411yescharacteristic, direct product factor, Sylow
267#544—22no—
268#544—22no—
269#544—21no—
270#544C22×C10712yes—
271#543—21no—
272#543C22×C10712yes—
273#533—24no—
274#533—21no—
275#533C22×C10712yes—
276C2×C4×D4C22×C10711yescharacteristic
277C2×#117—21no—
278C2×#117C22×C10711yescharacteristic
279#484C22×C10714yes—
280#474—22no—
281#461—22no—
282#450C22×C10714yes—
283#440—24no—
284#432C22×C10712yes—
285#400C22×C10711yescharacteristic
286#399C22×C10711yescharacteristic
287#394C22×C10711yescharacteristic
288#392C22×C10711yescharacteristic
289#388—21no—
290#386—21no—
291#386—21no—
292#385—21no—
293#383C22×C10711yescharacteristic
294#328C22×C10714yes—
295#328C22×C10714yes—
296#324C22×C10712yes—
297#323C22×C10712yes—
298C22×D4—21no—
299C22×D4C23×C10711yescharacteristic
300C23×C4—21no—
301C23×C4D4×C10711yescharacteristic
302C2×SD16—24no—
303C4⋊Q8—24no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—24no—
307C4⋊Q8D4×C10712yescomplemented
308C4⋊Q8D4×C10712yescomplemented
309#125—24no—
310#125—22no—
311#125—24no—
312#125—22no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—24no—
320C22⋊Q8C23×C10714yes—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8D4×C10714yescomplemented
325C4×D4—22no—
326C4×D4—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4D4×C10711yescharacteristic
337C2×C4⋊C4C23×C10711yescharacteristic
338C2×C4⋊C4C23×C10711yescharacteristic
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4D4×C10711yescharacteristic
346C2×C22⋊C4C23×C10711yescharacteristic
347C2×C42—22no—
348C2×C42D4×C10711yescharacteristic
349C2×C42C23×C10711yescharacteristic
350#107—24no—
351C4⋊C8—22no—
352#103—24no—
353#102—24no—
354C22⋊C8C23×C10714yes—
355#97—22no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95C23×C10711yescharacteristic
361#95C23×C10711yescharacteristic
362C24—21no—
363C24C2×D4×C10711yescharacteristic
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8C2×D4×C10714yes—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—81no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4C2×D4×C10711yescharacteristic
386C22×C4C2×D4×C10711yescharacteristic
387C22×C4C2×D4×C10711yescharacteristic
388C22×C4C2×D4×C10711yescharacteristic
389C22×C4C2×D4×C10711yescharacteristic
390C22×C4C24×C10711yescharacteristic
391SD16—48no—
392C2×C8—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—81no—
408C4⋊C4C2×D4×C10714yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—42no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—42no—
417C22⋊C4—81no—
418C22⋊C4—81no—
419C42—24no—
420C42—24no—
421C42—21no—
422C42—21no—
423C42C2×D4×C10712yes—
424C42C2×D4×C10712yes—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23#137×C10712yes—
437C23C22×D4×C10711yescharacteristic
438Q8—44no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442D4—24no—
443D4—42no—
444D4—84no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—84no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#120×C10711yescharacteristic
472C2×C4#120×C10711yescharacteristic
473C2×C4#120×C10712yes—
474C2×C4#137×C10712yes—
475C2×C4C22×D4×C10712yes—
476C8—44no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22C2×#120×C10711yescharacteristic
492C22C2×#137×C10711yescharacteristic
493C22C2×#137×C10711yescharacteristic
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—41no—
500C4—44no—
501C4—41no—
502C4—44no—
503C4—81no—
504C4—81no—
505C4—81no—
506C2—21no—
507C2—21no—
508C2—42no—
509C2—82no—
510C2C107×#300411yescharacteristic
511C2C107×#302111yescharacteristic
512C2C107×#302511yescharacteristic
subgroupG/H
center, socleC22×C107C2×#120
commutator, Frattini, cosocleC22×C4C24×C107
prime radical#392×C107C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C107 ◁ C2×C42×C107 ◁ G

Complements

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

NH ≅ G/Ntype
1. C107×#3458C2semidirect
2. C107×#3110C2semidirect
4. C107×#1617C2semidirect
5. C107×#1614C2semidirect
6. C107×#1599C2semidirect
8. C107×#1515C2semidirect
9. C107×#1464C2semidirect
50. C4⋊Q8×C107D4semidirect
51. C4⋊Q8×C107D4semidirect
67. C4×Q8×C107D4semidirect
249. #18154C107direct (N × H)
257. #3458C2×C107semidirect
258. #3110C2×C107semidirect
260. #1617C2×C107semidirect
261. #1614C2×C107semidirect
262. #1599C2×C107semidirect
264. #1515C2×C107semidirect
265. #1464C2×C107semidirect
266. C107#18154direct (N × H)
307. C4⋊Q8D4×C107semidirect
308. C4⋊Q8D4×C107semidirect
324. C4×Q8D4×C107semidirect

Maximal subgroups

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

subgroupindex
C107×#34582
C107×#31102
C107×#25982
C107×#16172
C107×#16142
C107×#15992
C107×#15962
C107×#15152
C107×#14642
#18154107