Finite Group Databasebeta

← C107×#18173 #41044137

Subgroups — C107×#18173 #41044137

Order 27392.

1242 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C107×#3496C211yesmaximal, characteristic
2C107×#3094C212yesmaximal, complemented
3C107×#2395C211yesmaximal, characteristic, complemented
4C107×#1616C212yesmaximal, complemented
5C107×#1613C212yesmaximal, complemented
6C107×#1610C212yesmaximal, complemented
7C107×#1607C212yesmaximal, complemented
8C107×#1507C212yesmaximal, complemented
9C107×#1465C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C107—21no—
11#547×C107—22no—
12#546×C107—24no—
13D42×C107—22no—
14#537×C107C2212yes—
15#534×C107—21no—
16#534×C107C2212yes—
17#532×C107—21no—
18C2×#121×C107—21no—
19C2×#121×C107C2212yes—
20C2×C4×D4×C107C2211yescharacteristic
21C2×#117×C107C2211yescharacteristic, complemented
22#482×C107—24no—
23#481×C107C2214yescomplemented
24#480×C107C2214yescomplemented
25#428×C107—22no—
26#418×C107—22no—
27#407×C107C2212yescomplemented
28#400×C107C2211yescharacteristic, complemented
29#399×C107C2211yescharacteristic, complemented
30#398×C107C2211yescharacteristic, complemented
31#397×C107—21no—
32#397×C107C2211yescharacteristic, complemented
33#394×C107—21no—
34C4×C4⋊C4×C107—21no—
35C4×C22⋊C4×C107—21no—
36C4×C22⋊C4×C107C2211yescharacteristic, complemented
37#327×C107C2214yescomplemented
38#327×C107C2214yescomplemented
39#324×C107C2212yescomplemented
40#323×C107C2212yescomplemented
41H(5,𝔽2)×C107—44no—
42H(5,𝔽2)×C107—44no—
43C2×#41×C107—24no—
44C2×#41×C107—22no—
45C22×D4×C107—24no—
46C22×D4×C107—22no—
47C22×D4×C107—22no—
48C22×D4×C107—21no—
49C22×D4×C107D411yescharacteristic
50C22×D4×C107D412yes—
51C23×C4×C107D411yescharacteristic
52C23×C4×C107C2311yescharacteristic
53C2×M16×C107—22no—
54C4⋊Q8×C107—21no—
55#127×C107—21no—
56#124×C107—22no—
57#123×C107—24no—
58#123×C107—24no—
59C22⋊Q8×C107—24no—
60#121×C107—22no—
61#121×C107—22no—
62#121×C107—24no—
63#121×C107—22no—
64#121×C107—22no—
65#121×C107—22no—
66#121×C107—22no—
67#121×C107D414yes—
68#121×C107C2314yes—
69#120×C107—24no—
70#120×C107—24no—
71C4×D4×C107—22no—
72C4×D4×C107—22no—
73C4×D4×C107—22no—
74C4×D4×C107—22no—
75#117×C107—22no—
76#117×C107—22no—
77C2×C4⋊C4×C107—21no—
78C2×C4⋊C4×C107—21no—
79C2×C4⋊C4×C107—21no—
80C2×C4⋊C4×C107D412yes—
81C2×C4⋊C4×C107C2311yescharacteristic
82C2×C4⋊C4×C107C2311yescharacteristic
83C2×C4⋊C4×C107C2311yescharacteristic
84C2×C22⋊C4×C107—22no—
85C2×C22⋊C4×C107—22no—
86C2×C22⋊C4×C107—21no—
87C2×C22⋊C4×C107—21no—
88C2×C22⋊C4×C107—21no—
89C2×C22⋊C4×C107—21no—
90C2×C22⋊C4×C107D411yescharacteristic
91C2×C22⋊C4×C107C2311yescharacteristic
92C2×C42×C107—21no—
93C2×C42×C107—21no—
94C2×C42×C107D411yescharacteristic
95#107×C107—24no—
96#106×C107—24no—
97#102×C107—24no—
98#102×C107—24no—
99C22⋊C8×C107—22no—
100C22⋊C8×C107C2314yes—
101#95×C107—21no—
102#95×C107—21no—
103#95×C107—21no—
104#95×C107—21no—
105#95×C107C2311yescharacteristic
106#95×C107C2311yescharacteristic
107C24×C107—22no—
108C24×C107—22no—
109C24×C107C2×D411yescharacteristic
110C24×C107C2×D411yescharacteristic
111#41×C107—44no—
112#41×C107—416no—
113#41×C107—44no—
114C2×Q8×C107—22no—
115C2×D4×C107—22no—
116C2×D4×C107—22no—
117C2×D4×C107—24no—
118C2×D4×C107—22no—
119C2×D4×C107—24no—
120C2×D4×C107—24no—
121C2×D4×C107—22no—
122C2×D4×C107—22no—
123C2×D4×C107—22no—
124C2×D4×C107—22no—
125C2×D4×C107—44no—
126C2×D4×C107—416no—
127C2×D4×C107—44no—
128C2×D4×C107—44no—
129C2×D4×C107—44no—
130C2×D4×C107—44no—
131C2×D4×C107—82no—
132C2×D4×C107C2×D414yes—
133C22×C4×C107—22no—
134C22×C4×C107—24no—
135C22×C4×C107—22no—
136C22×C4×C107—21no—
137C22×C4×C107—21no—
138C22×C4×C107—21no—
139C22×C4×C107—21no—
140C22×C4×C107—21no—
141C22×C4×C107—21no—
142C22×C4×C107—42no—
143C22×C4×C107—42no—
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—22no—
154C4⋊C4×C107—22no—
155C4⋊C4×C107—22no—
156C4⋊C4×C107—22no—
157C4⋊C4×C107—22no—
158C4⋊C4×C107—22no—
159C4⋊C4×C107—42no—
160C4⋊C4×C107—42no—
161C4⋊C4×C107C2×D414yes—
162C22⋊C4×C107—24no—
163C22⋊C4×C107—24no—
164C22⋊C4×C107—22no—
165C22⋊C4×C107—24no—
166C22⋊C4×C107—24no—
167C22⋊C4×C107—22no—
168C22⋊C4×C107—42no—
169C22⋊C4×C107—42no—
170C22⋊C4×C107—42no—
171C22⋊C4×C107—42no—
172C42×C107—21no—
173C42×C107—21no—
174C42×C107—42no—
175C42×C107—41no—
176C42×C107—41no—
177C23×C107—21no—
178C23×C107—21no—
179C23×C107—21no—
180C23×C107—21no—
181C23×C107—24no—
182C23×C107—41no—
183C23×C107—41no—
184C23×C107—41no—
185C23×C107—41no—
186C23×C107—44no—
187C23×C107—44no—
188C23×C107—82no—
189C23×C107—82no—
190C23×C107—82no—
191C23×C107—82no—
192C23×C107#12012yes—
193C23×C107AGL(1,ℤ/8)12yes—
194C23×C107C22×D411yescharacteristic
195Q8×C107—44no—
196D4×C107—28no—
197D4×C107—44no—
198D4×C107—44no—
199D4×C107—48no—
200D4×C107—48no—
201D4×C107—44no—
202D4×C107—44no—
203D4×C107—44no—
204D4×C107—82no—
205D4×C107—82no—
206C2×C4×C107—21no—
207C2×C4×C107—21no—
208C2×C4×C107—22no—
209C2×C4×C107—24no—
210C2×C4×C107—22no—
211C2×C4×C107—21no—
212C2×C4×C107—21no—
213C2×C4×C107—24no—
214C2×C4×C107—42no—
215C2×C4×C107—41no—
216C2×C4×C107—48no—
217C2×C4×C107—41no—
218C2×C4×C107—44no—
219C2×C4×C107—44no—
220C2×C4×C107—41no—
221C2×C4×C107—42no—
222C2×C4×C107—41no—
223C2×C4×C107—41no—
224C2×C4×C107—41no—
225C2×C4×C107—41no—
226C2×C4×C107—41no—
227C2×C4×C107—41no—
228C2×C4×C107—41no—
229C2×C4×C107—41no—
230C2×C4×C107—41no—
231C2×C4×C107—41no—
232C2×C4×C107—41no—
233C2×C4×C107—81no—
234C2×C4×C107#12011yescharacteristic
235C2×C4×C107#12011yescharacteristic
236C2×C4×C107AGL(1,ℤ/8)12yes—
237C2×C4×C107C22×D412yes—
238C8×C107—44no—
239C22×C107—21no—
240C22×C107—21no—
241C22×C107—22no—
242C22×C107—22no—
243C22×C107—22no—
244C22×C107—22no—
245C22×C107—42no—
246C22×C107—41no—
247C22×C107—41no—
248C22×C107—44no—
249C22×C107—44no—
250C22×C107—44no—
251C22×C107—81no—
252C22×C107—81no—
253C22×C107—81no—
254C22×C107—164no—
255C22×C107—164no—
256C22×C107C2×#12011yescharacteristic
257C22×C107C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C107C2×AGL(1,ℤ/8)11yescharacteristic
259C4×C107—22no—
260C4×C107—22no—
261C4×C107—21no—
262C4×C107—42no—
263C4×C107—44no—
264C4×C107—41no—
265C4×C107—41no—
266C4×C107—81no—
267C4×C107—81no—
268C4×C107—81no—
269C4×C107#45812yes—
270#18173C10711yesmaximal, characteristic, direct product factor, Sylow
271C2×C107—21no—
272C2×C107—21no—
273C2×C107—42no—
274C2×C107—42no—
275C2×C107—84no—
276C2×C107#300311yescharacteristic
277C2×C107#301811yescharacteristic
278C2×C107C2×#45811yescharacteristic
279#3496C2×C10711yescharacteristic
280#3094C2×C10712yescomplemented
281#2395C2×C10711yescharacteristic, complemented
282#1616C2×C10712yescomplemented
283#1613C2×C10712yescomplemented
284#1610C2×C10712yescomplemented
285#1607C2×C10712yescomplemented
286#1507C2×C10712yescomplemented
287#1465C2×C10711yescharacteristic, complemented
288C107#1817311yescharacteristic, direct product factor, Sylow
289C2×H(5,𝔽2)—21no—
290#547—22no—
291#546—24no—
292D42—22no—
293#537C22×C10712yes—
294#534—21no—
295#534C22×C10712yes—
296#532—21no—
297C2×#121—21no—
298C2×#121C22×C10712yes—
299C2×C4×D4C22×C10711yescharacteristic
300C2×#117C22×C10711yescharacteristic, complemented
301#482—24no—
302#481C22×C10714yescomplemented
303#480C22×C10714yescomplemented
304#428—22no—
305#418—22no—
306#407C22×C10712yescomplemented
307#400C22×C10711yescharacteristic, complemented
308#399C22×C10711yescharacteristic, complemented
309#398C22×C10711yescharacteristic, complemented
310#397—21no—
311#397C22×C10711yescharacteristic, complemented
312#394—21no—
313C4×C4⋊C4—21no—
314C4×C22⋊C4—21no—
315C4×C22⋊C4C22×C10711yescharacteristic, complemented
316#327C22×C10714yescomplemented
317#327C22×C10714yescomplemented
318#324C22×C10712yescomplemented
319#323C22×C10712yescomplemented
320H(5,𝔽2)—44no—
321H(5,𝔽2)—44no—
322C2×#41—24no—
323C2×#41—22no—
324C22×D4—24no—
325C22×D4—22no—
326C22×D4—22no—
327C22×D4—21no—
328C22×D4D4×C10711yescharacteristic
329C22×D4D4×C10712yes—
330C23×C4D4×C10711yescharacteristic
331C23×C4C23×C10711yescharacteristic
332C2×M16—22no—
333C4⋊Q8—21no—
334#127—21no—
335#124—22no—
336#123—24no—
337#123—24no—
338C22⋊Q8—24no—
339#121—22no—
340#121—22no—
341#121—24no—
342#121—22no—
343#121—22no—
344#121—22no—
345#121—22no—
346#121D4×C10714yes—
347#121C23×C10714yes—
348#120—24no—
349#120—24no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354#117—22no—
355#117—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4D4×C10712yes—
360C2×C4⋊C4C23×C10711yescharacteristic
361C2×C4⋊C4C23×C10711yescharacteristic
362C2×C4⋊C4C23×C10711yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4D4×C10711yescharacteristic
370C2×C22⋊C4C23×C10711yescharacteristic
371C2×C42—21no—
372C2×C42—21no—
373C2×C42D4×C10711yescharacteristic
374#107—24no—
375#106—24no—
376#102—24no—
377#102—24no—
378C22⋊C8—22no—
379C22⋊C8C23×C10714yes—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95C23×C10711yescharacteristic
385#95C23×C10711yescharacteristic
386C24—22no—
387C24—22no—
388C24C2×D4×C10711yescharacteristic
389C24C2×D4×C10711yescharacteristic
390#41—44no—
391#41—416no—
392#41—44no—
393C2×Q8—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—416no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4C2×D4×C10714yes—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4C2×D4×C10711yescharacteristic
424C22×C4C2×D4×C10712yes—
425C22×C4C2×D4×C10711yescharacteristic
426C22×C4C2×D4×C10711yescharacteristic
427C22×C4C2×D4×C10711yescharacteristic
428C22×C4C2×D4×C10712yes—
429C22×C4C24×C10711yescharacteristic
430M16—44no—
431C2×C8—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4C2×D4×C10714yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—41no—
455C42—41no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—44no—
466C23—44no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23#120×C10712yes—
472C23AGL(1,ℤ/8)×C10712yes—
473C23C22×D4×C10711yescharacteristic
474Q8—44no—
475D4—28no—
476D4—44no—
477D4—44no—
478D4—48no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—82no—
484D4—82no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—48no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—81no—
513C2×C4#120×C10711yescharacteristic
514C2×C4#120×C10711yescharacteristic
515C2×C4AGL(1,ℤ/8)×C10712yes—
516C2×C4C22×D4×C10712yes—
517C8—44no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—164no—
534C22—164no—
535C22C2×#120×C10711yescharacteristic
536C22C2×AGL(1,ℤ/8)×C10711yescharacteristic
537C22C2×AGL(1,ℤ/8)×C10711yescharacteristic
538C4—22no—
539C4—22no—
540C4—21no—
541C4—42no—
542C4—44no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4#458×C10712yes—
549C2—21no—
550C2—21no—
551C2—42no—
552C2—42no—
553C2—84no—
554C2C107×#300311yescharacteristic
555C2C107×#301811yescharacteristic
556C2C2×#458×C10711yescharacteristic
subgroupG/H
center, socleC22×C107C2×#120
commutator, Frattini, cosocleC22×C4C24×C107
prime radicalC107×#3496C2
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×#3094C2semidirect
3. C107×#2395C2semidirect
4. C107×#1616C2semidirect
5. C107×#1613C2semidirect
6. C107×#1610C2semidirect
7. C107×#1607C2semidirect
8. C107×#1507C2semidirect
9. C107×#1465C2semidirect
21. C2×#117×C107C22semidirect
23. #481×C107C22semidirect
24. #480×C107C22semidirect
27. #407×C107C22semidirect
28. #400×C107C22semidirect
29. #399×C107C22semidirect
30. #398×C107C22semidirect
32. #397×C107C22semidirect
36. C4×C22⋊C4×C107C22semidirect
37. #327×C107C22semidirect
38. #327×C107C22semidirect
39. #324×C107C22semidirect
40. #323×C107C22semidirect
270. #18173C107direct (N × H)
280. #3094C2×C107semidirect
281. #2395C2×C107semidirect
282. #1616C2×C107semidirect
283. #1613C2×C107semidirect
284. #1610C2×C107semidirect
285. #1607C2×C107semidirect
286. #1507C2×C107semidirect
287. #1465C2×C107semidirect
288. C107#18173direct (N × H)
300. C2×#117C22×C107semidirect
302. #481C22×C107semidirect
303. #480C22×C107semidirect
306. #407C22×C107semidirect
307. #400C22×C107semidirect
308. #399C22×C107semidirect
309. #398C22×C107semidirect
311. #397C22×C107semidirect
315. C4×C22⋊C4C22×C107semidirect
316. #327C22×C107semidirect
317. #327C22×C107semidirect
318. #324C22×C107semidirect
319. #323C22×C107semidirect

Maximal subgroups

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

subgroupindex
C107×#34962
C107×#30942
C107×#23952
C107×#16162
C107×#16132
C107×#16102
C107×#16072
C107×#15072
C107×#14652
#18173107