Finite Group Databasebeta

← C107×#18196 #41044160

Subgroups — C107×#18196 #41044160

Order 27392.

1370 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C107×#3501C211yesmaximal, characteristic, complemented
2C2×#450×C107C212yesmaximal, complemented
3C107×#2408C211yesmaximal, characteristic
4C107×#1599C214yesmaximal, complemented
5C107×#1596C214yesmaximal, complemented
6C107×#1520C212yesmaximal, complemented
7C107×#1459C211yesmaximal, characteristic, complemented
8C23×D4×C107C2211yescharacteristic
9C22×SD16×C107—22no—
10#554×C107—24no—
11#554×C107—22no—
12D4×Q8×C107—22no—
13#544×C107C2212yescomplemented
14#543×C107—21no—
15#543×C107C2212yescomplemented
16#533×C107—21no—
17C2×C22⋊Q8×C107—21no—
18C2×C22⋊Q8×C107C2212yes—
19C2×#121×C107—21no—
20C2×#120×C107—21no—
21#518×C107—21no—
22C2×C4×D4×C107C2211yescharacteristic
23#450×C107—24no—
24#450×C107C2218yes—
25C2×#102×C107—22no—
26C2×C22⋊C8×C107C2212yes—
27#394×C107C2212yes—
28#392×C107C2212yes—
29#390×C107—21no—
30#380×C107C2211yescharacteristic
31#379×C107—21no—
32#328×C107C2218yescomplemented
33#324×C107C2212yescomplemented
34#323×C107C2212yescomplemented
35C25×C107—21no—
36C22×Q8×C107—21no—
37C22×Q8×C107D412yescomplemented
38C22×D4×C107—22no—
39C22×D4×C107—22no—
40C22×D4×C107—21no—
41C22×D4×C107—22no—
42C22×D4×C107—44no—
43C22×D4×C107D411yescharacteristic
44C22×D4×C107C2312yes—
45C23×C4×C107D411yescharacteristic
46C23×C4×C107C2311yescharacteristic
47C2×SD16×C107—28no—
48C2×SD16×C107—48no—
49C22×C8×C107—22no—
50C4⋊Q8×C107—21no—
51C4⋊Q8×C107—21no—
52C4⋊Q8×C107—24no—
53#125×C107—22no—
54#125×C107—24no—
55C22⋊Q8×C107—22no—
56C22⋊Q8×C107—22no—
57C22⋊Q8×C107—24no—
58C22⋊Q8×C107—24no—
59C22⋊Q8×C107—24no—
60C22⋊Q8×C107—22no—
61C22⋊Q8×C107—22no—
62C22⋊Q8×C107—22no—
63C22⋊Q8×C107—22no—
64C22⋊Q8×C107—24no—
65C22⋊Q8×C107D414yescomplemented
66C22⋊Q8×C107C2314yes—
67#121×C107—44no—
68#120×C107—44no—
69C4×Q8×C107—22no—
70C4×Q8×C107—22no—
71C4×D4×C107—24no—
72C4×D4×C107—24no—
73C4×D4×C107—22no—
74C4×D4×C107—22no—
75#117×C107—22no—
76C2×C4⋊C4×C107—21no—
77C2×C4⋊C4×C107—22no—
78C2×C4⋊C4×C107—22no—
79C2×C4⋊C4×C107—21no—
80C2×C4⋊C4×C107D412yescomplemented
81C2×C4⋊C4×C107C2311yescharacteristic
82C2×C22⋊C4×C107—22no—
83C2×C22⋊C4×C107—22no—
84C2×C22⋊C4×C107—21no—
85C2×C22⋊C4×C107—22no—
86C2×C22⋊C4×C107—22no—
87C2×C22⋊C4×C107—22no—
88C2×C22⋊C4×C107D411yescharacteristic
89C2×C22⋊C4×C107C2311yescharacteristic
90C2×C42×C107D411yescharacteristic
91#102×C107—28no—
92C22⋊C8×C107—22no—
93C22⋊C8×C107C2314yes—
94#95×C107C2312yes—
95C24×C107—22no—
96C24×C107—44no—
97C24×C107—42no—
98C24×C107—42no—
99C24×C107—42no—
100C24×C107—82no—
101C24×C107C2×D411yescharacteristic
102C24×C107C2×D411yescharacteristic
103C2×Q8×C107—22no—
104C2×Q8×C107—22no—
105C2×Q8×C107—22no—
106C2×Q8×C107—44no—
107C2×Q8×C107—82no—
108C2×Q8×C107C2×D414yes—
109C2×D4×C107—22no—
110C2×D4×C107—22no—
111C2×D4×C107—28no—
112C2×D4×C107—24no—
113C2×D4×C107—24no—
114C2×D4×C107—42no—
115C2×D4×C107—44no—
116C2×D4×C107—44no—
117C2×D4×C107—48no—
118C2×D4×C107—44no—
119C2×D4×C107—44no—
120C2×D4×C107—41no—
121C2×D4×C107—41no—
122C2×D4×C107—82no—
123C2×D4×C107SD1614yescomplemented
124C22×C4×C107—22no—
125C22×C4×C107—22no—
126C22×C4×C107—22no—
127C22×C4×C107—22no—
128C22×C4×C107—21no—
129C22×C4×C107—21no—
130C22×C4×C107—22no—
131C22×C4×C107—42no—
132C22×C4×C107—42no—
133C22×C4×C107C2×D411yescharacteristic
134C22×C4×C107C2×D412yes—
135C22×C4×C107C2×D411yescharacteristic
136C22×C4×C107C2×D411yescharacteristic
137C22×C4×C107C2×D411yescharacteristic
138C22×C4×C107C2×D412yes—
139C22×C4×C107C2411yescharacteristic
140SD16×C107—416no—
141C2×C8×C107—24no—
142C2×C8×C107—44no—
143C4⋊C4×C107—22no—
144C4⋊C4×C107—24no—
145C4⋊C4×C107—22no—
146C4⋊C4×C107—22no—
147C4⋊C4×C107—24no—
148C4⋊C4×C107—22no—
149C4⋊C4×C107—24no—
150C4⋊C4×C107—22no—
151C4⋊C4×C107—24no—
152C4⋊C4×C107—22no—
153C4⋊C4×C107—42no—
154C4⋊C4×C107C2×D414yes—
155C22⋊C4×C107—24no—
156C22⋊C4×C107—24no—
157C22⋊C4×C107—22no—
158C22⋊C4×C107—24no—
159C22⋊C4×C107—22no—
160C22⋊C4×C107—44no—
161C22⋊C4×C107—42no—
162C22⋊C4×C107—44no—
163C22⋊C4×C107—82no—
164C42×C107—24no—
165C42×C107—21no—
166C42×C107—21no—
167C23×C107—21no—
168C23×C107—21no—
169C23×C107—24no—
170C23×C107—41no—
171C23×C107—41no—
172C23×C107—41no—
173C23×C107—41no—
174C23×C107—42no—
175C23×C107—48no—
176C23×C107—42no—
177C23×C107—42no—
178C23×C107—42no—
179C23×C107—42no—
180C23×C107—42no—
181C23×C107—84no—
182C23×C107—88no—
183C23×C107—88no—
184C23×C107—84no—
185C23×C107#12012yes—
186C23×C107C2×SD1612yes—
187C23×C107C2×SD1612yescomplemented
188C23×C107#13712yes—
189C23×C107C22×D411yescharacteristic
190Q8×C107—44no—
191Q8×C107—44no—
192Q8×C107—82no—
193Q8×C107—82no—
194D4×C107—28no—
195D4×C107—28no—
196D4×C107—44no—
197D4×C107—44no—
198D4×C107—44no—
199D4×C107—88no—
200C2×C4×C107—21no—
201C2×C4×C107—21no—
202C2×C4×C107—22no—
203C2×C4×C107—24no—
204C2×C4×C107—21no—
205C2×C4×C107—24no—
206C2×C4×C107—24no—
207C2×C4×C107—21no—
208C2×C4×C107—42no—
209C2×C4×C107—41no—
210C2×C4×C107—42no—
211C2×C4×C107—41no—
212C2×C4×C107—41no—
213C2×C4×C107—42no—
214C2×C4×C107—42no—
215C2×C4×C107—42no—
216C2×C4×C107—81no—
217C2×C4×C107—82no—
218C2×C4×C107—82no—
219C2×C4×C107—82no—
220C2×C4×C107—82no—
221C2×C4×C107#12011yescharacteristic
222C2×C4×C107#12011yescharacteristic
223C2×C4×C107C2×SD1612yescomplemented
224C2×C4×C107C22×D412yes—
225C8×C107—44no—
226C22×C107—21no—
227C22×C107—22no—
228C22×C107—21no—
229C22×C107—22no—
230C22×C107—22no—
231C22×C107—42no—
232C22×C107—42no—
233C22×C107—42no—
234C22×C107—44no—
235C22×C107—44no—
236C22×C107—41no—
237C22×C107—41no—
238C22×C107—44no—
239C22×C107—81no—
240C22×C107—82no—
241C22×C107—82no—
242C22×C107—816no—
243C22×C107—82no—
244C22×C107—82no—
245C22×C107#44814yes—
246C22×C107C2×#12011yescharacteristic
247C22×C107C22×SD1611yescharacteristic
248C22×C107C2×#13711yescharacteristic
249C4×C107—22no—
250C4×C107—22no—
251C4×C107—21no—
252C4×C107—41no—
253C4×C107—44no—
254C4×C107—41no—
255C4×C107—44no—
256C4×C107—82no—
257C4×C107B(4,𝔽2)12yes—
258#18196C10711yesmaximal, characteristic, direct product factor, Sylow
259C2×C107—21no—
260C2×C107—21no—
261C2×C107—24no—
262C2×C107—42no—
263C2×C107—84no—
264C2×C107C2×#44811yescharacteristic
265C2×C107#302111yescharacteristic
266C2×C107C2×B(4,𝔽2)11yescharacteristic
267#3501C2×C10711yescharacteristic, complemented
268C2×#450C2×C10712yescomplemented
269#2408C2×C10711yescharacteristic
270#1599C2×C10714yescomplemented
271#1596C2×C10714yescomplemented
272#1520C2×C10712yescomplemented
273#1459C2×C10711yescharacteristic, complemented
274C107#1819611yescharacteristic, direct product factor, Sylow
275C23×D4C22×C10711yescharacteristic
276C22×SD16—22no—
277#554—24no—
278#554—22no—
279D4×Q8—22no—
280#544C22×C10712yescomplemented
281#543—21no—
282#543C22×C10712yescomplemented
283#533—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8C22×C10712yes—
286C2×#121—21no—
287C2×#120—21no—
288#518—21no—
289C2×C4×D4C22×C10711yescharacteristic
290#450—24no—
291#450C22×C10718yes—
292C2×#102—22no—
293C2×C22⋊C8C22×C10712yes—
294#394C22×C10712yes—
295#392C22×C10712yes—
296#390—21no—
297#380C22×C10711yescharacteristic
298#379—21no—
299#328C22×C10718yescomplemented
300#324C22×C10712yescomplemented
301#323C22×C10712yescomplemented
302C25—21no—
303C22×Q8—21no—
304C22×Q8D4×C10712yescomplemented
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—21no—
308C22×D4—22no—
309C22×D4—44no—
310C22×D4D4×C10711yescharacteristic
311C22×D4C23×C10712yes—
312C23×C4D4×C10711yescharacteristic
313C23×C4C23×C10711yescharacteristic
314C2×SD16—28no—
315C2×SD16—48no—
316C22×C8—22no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—24no—
320#125—22no—
321#125—24no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332C22⋊Q8D4×C10714yescomplemented
333C22⋊Q8C23×C10714yes—
334#121—44no—
335#120—44no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×D4—24no—
339C4×D4—24no—
340C4×D4—22no—
341C4×D4—22no—
342#117—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4D4×C10712yescomplemented
348C2×C4⋊C4C23×C10711yescharacteristic
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4D4×C10711yescharacteristic
356C2×C22⋊C4C23×C10711yescharacteristic
357C2×C42D4×C10711yescharacteristic
358#102—28no—
359C22⋊C8—22no—
360C22⋊C8C23×C10714yes—
361#95C23×C10712yes—
362C24—22no—
363C24—44no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—82no—
368C24C2×D4×C10711yescharacteristic
369C24C2×D4×C10711yescharacteristic
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8C2×D4×C10714yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—82no—
390C2×D4SD16×C10714yescomplemented
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C2×D4×C10711yescharacteristic
401C22×C4C2×D4×C10712yes—
402C22×C4C2×D4×C10711yescharacteristic
403C22×C4C2×D4×C10711yescharacteristic
404C22×C4C2×D4×C10711yescharacteristic
405C22×C4C2×D4×C10712yes—
406C22×C4C24×C10711yescharacteristic
407SD16—416no—
408C2×C8—24no—
409C2×C8—44no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4C2×D4×C10714yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C42—24no—
432C42—21no—
433C42—21no—
434C23—21no—
435C23—21no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—48no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—88no—
450C23—88no—
451C23—84no—
452C23#120×C10712yes—
453C23C2×SD16×C10712yes—
454C23C2×SD16×C10712yescomplemented
455C23#137×C10712yes—
456C23C22×D4×C10711yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—82no—
460Q8—82no—
461D4—28no—
462D4—28no—
463D4—44no—
464D4—44no—
465D4—44no—
466D4—88no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4#120×C10711yescharacteristic
489C2×C4#120×C10711yescharacteristic
490C2×C4C2×SD16×C10712yescomplemented
491C2×C4C22×D4×C10712yes—
492C8—44no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—44no—
502C22—44no—
503C22—41no—
504C22—41no—
505C22—44no—
506C22—81no—
507C22—82no—
508C22—82no—
509C22—816no—
510C22—82no—
511C22—82no—
512C22#448×C10714yes—
513C22C2×#120×C10711yescharacteristic
514C22C22×SD16×C10711yescharacteristic
515C22C2×#137×C10711yescharacteristic
516C4—22no—
517C4—22no—
518C4—21no—
519C4—41no—
520C4—44no—
521C4—41no—
522C4—44no—
523C4—82no—
524C4B(4,𝔽2)×C10712yes—
525C2—21no—
526C2—21no—
527C2—24no—
528C2—42no—
529C2—84no—
530C2C2×#448×C10711yescharacteristic
531C2C107×#302111yescharacteristic
532C2C2×B(4,𝔽2)×C10711yescharacteristic
subgroupG/H
center, socleC22×C107C2×#120
commutator, Frattini, cosocleC22×C4C24×C107
prime radicalC107×#2408C2
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
1. C107×#3501C2semidirect
2. C2×#450×C107C2semidirect
4. C107×#1599C2semidirect
5. C107×#1596C2semidirect
6. C107×#1520C2semidirect
7. C107×#1459C2semidirect
13. #544×C107C22semidirect
15. #543×C107C22semidirect
32. #328×C107C22semidirect
33. #324×C107C22semidirect
34. #323×C107C22semidirect
37. C22×Q8×C107D4semidirect
65. C22⋊Q8×C107D4semidirect
80. C2×C4⋊C4×C107D4semidirect
123. C2×D4×C107SD16semidirect
187. C23×C107C2×SD16semidirect
223. C2×C4×C107C2×SD16semidirect
258. #18196C107direct (N × H)
267. #3501C2×C107semidirect
268. C2×#450C2×C107semidirect
270. #1599C2×C107semidirect
271. #1596C2×C107semidirect
272. #1520C2×C107semidirect
273. #1459C2×C107semidirect
274. C107#18196direct (N × H)
280. #544C22×C107semidirect
282. #543C22×C107semidirect
299. #328C22×C107semidirect
300. #324C22×C107semidirect
301. #323C22×C107semidirect
304. C22×Q8D4×C107semidirect
332. C22⋊Q8D4×C107semidirect
347. C2×C4⋊C4D4×C107semidirect
390. C2×D4SD16×C107semidirect
454. C23C2×SD16×C107semidirect
490. C2×C4C2×SD16×C107semidirect

Maximal subgroups

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

subgroupindex
C107×#35012
C2×#450×C1072
C107×#24082
C107×#15992
C107×#15962
C107×#15202
C107×#14592
#18196107