Finite Group Databasebeta

← C107×#12687 #41038651

Subgroups — C107×#12687 #41038651

Order 27392.

2508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 2508 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#120×C107C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)×C107C214yesmaximal, complemented
3C107×#1783C211yesmaximal, characteristic, complemented
4C107×#1781C211yesmaximal, characteristic, complemented
5C26×C107C411yescharacteristic, complemented
6C23×D4×C107—21no—
7C23×D4×C107C2211yescharacteristic
8C2×#120×C107—21no—
9C2×#120×C107—21no—
10C2×#120×C107—28no—
11C2×#120×C107C412yescomplemented
12C2×#120×C107C2212yes—
13C22×C22⋊C4×C107—21no—
14C22×C22⋊C4×C107C411yescharacteristic, complemented
15C2×#100×C107—21no—
16C2×#100×C107C2212yescomplemented
17C2×#99×C107—21no—
18C2×#99×C107C2212yescomplemented
19#407×C107—21no—
20#379×C107—21no—
21AΣL(1,𝔽16)×C107—28no—
22C25×C107—21no—
23C25×C107—21no—
24C25×C107—24no—
25C25×C107—48no—
26C25×C107—42no—
27C25×C107—42no—
28C25×C107C2×C411yescharacteristic
29C25×C107C2×C412yes—
30C25×C107D411yescharacteristic
31C22×D4×C107—28no—
32C22×D4×C107—22no—
33C22×D4×C107—22no—
34C22×D4×C107—24no—
35C22×D4×C107—42no—
36C22×D4×C107—42no—
37C22×D4×C107—48no—
38C22×D4×C107C2×C411yescharacteristic
39C22×D4×C107D411yescharacteristic
40C22×D4×C107D411yescharacteristic
41C22×D4×C107C2311yescharacteristic
42C23×C4×C107—21no—
43C23×C4×C107D411yescharacteristic
44C2×M16×C107—22no—
45#120×C107—28no—
46#120×C107—28no—
47#120×C107—24no—
48#120×C107—24no—
49#120×C107—24no—
50#120×C107D418yes—
51C2×C22⋊C4×C107—28no—
52C2×C22⋊C4×C107—21no—
53C2×C22⋊C4×C107—22no—
54C2×C22⋊C4×C107—22no—
55C2×C22⋊C4×C107—24no—
56C2×C22⋊C4×C107—22no—
57C2×C22⋊C4×C107—42no—
58C2×C22⋊C4×C107C2×C412yes—
59#100×C107—24no—
60#100×C107—42no—
61#99×C107—24no—
62#99×C107—42no—
63C22⋊C8×C107—42no—
64C24×C107—24no—
65C24×C107—28no—
66C24×C107—22no—
67C24×C107—22no—
68C24×C107—22no—
69C24×C107—22no—
70C24×C107—24no—
71C24×C107—22no—
72C24×C107—22no—
73C24×C107—22no—
74C24×C107—41no—
75C24×C107—44no—
76C24×C107—48no—
77C24×C107—42no—
78C24×C107—42no—
79C24×C107—44no—
80C24×C107—42no—
81C24×C107—48no—
82C24×C107—432no—
83C24×C107—416no—
84C24×C107—42no—
85C24×C107—44no—
86C24×C107—48no—
87C24×C107—416no—
88C24×C107—432no—
89C24×C107—48no—
90C24×C107—42no—
91C24×C107—42no—
92C24×C107C22⋊C411yescharacteristic
93C24×C107C22⋊C411yescharacteristic
94C24×C107C22⋊C411yescharacteristic
95C24×C107C22⋊C414yescomplemented
96C24×C107C22×C411yescharacteristic
97C24×C107C2×D411yescharacteristic
98C2×D4×C107—28no—
99C2×D4×C107—22no—
100C2×D4×C107—22no—
101C2×D4×C107—22no—
102C2×D4×C107—22no—
103C2×D4×C107—28no—
104C2×D4×C107—22no—
105C2×D4×C107—41no—
106C2×D4×C107—41no—
107C2×D4×C107—41no—
108C2×D4×C107—41no—
109C2×D4×C107—41no—
110C2×D4×C107—41no—
111C2×D4×C107—416no—
112C2×D4×C107—48no—
113C2×D4×C107—48no—
114C2×D4×C107—432no—
115C2×D4×C107C2×D414yes—
116C22×C4×C107—24no—
117C22×C4×C107—22no—
118C22×C4×C107—22no—
119C22×C4×C107—22no—
120C22×C4×C107—41no—
121C22×C4×C107—44no—
122C22×C4×C107—42no—
123C22×C4×C107C22⋊C411yescharacteristic
124C22×C4×C107C2×D411yescharacteristic
125M16×C107—42no—
126M16×C107—42no—
127C2×C8×C107—42no—
128C22⋊C4×C107—28no—
129C22⋊C4×C107—24no—
130C22⋊C4×C107—22no—
131C22⋊C4×C107—28no—
132C22⋊C4×C107—42no—
133C22⋊C4×C107—42no—
134C22⋊C4×C107—84no—
135C22⋊C4×C107C22⋊C414yescomplemented
136C23×C107—24no—
137C23×C107—21no—
138C23×C107—21no—
139C23×C107—216no—
140C23×C107—28no—
141C23×C107—28no—
142C23×C107—21no—
143C23×C107—24no—
144C23×C107—21no—
145C23×C107—21no—
146C23×C107—21no—
147C23×C107—21no—
148C23×C107—21no—
149C23×C107—21no—
150C23×C107—21no—
151C23×C107—21no—
152C23×C107—44no—
153C23×C107—42no—
154C23×C107—48no—
155C23×C107—44no—
156C23×C107—42no—
157C23×C107—48no—
158C23×C107—48no—
159C23×C107—48no—
160C23×C107—416no—
161C23×C107—44no—
162C23×C107—48no—
163C23×C107—44no—
164C23×C107—48no—
165C23×C107—48no—
166C23×C107—416no—
167C23×C107—48no—
168C23×C107—44no—
169C23×C107—42no—
170C23×C107—42no—
171C23×C107—48no—
172C23×C107—48no—
173C23×C107—416no—
174C23×C107—416no—
175C23×C107—464no—
176C23×C107—464no—
177C23×C107—48no—
178C23×C107—416no—
179C23×C107—48no—
180C23×C107—44no—
181C23×C107—42no—
182C23×C107—42no—
183C23×C107—44no—
184C23×C107—82no—
185C23×C107—84no—
186C23×C107#9912yes—
187C23×C107#9912yes—
188C23×C107C2×C22⋊C411yescharacteristic
189C23×C107C2×C22⋊C412yes—
190C23×C107#12012yes—
191D4×C107—44no—
192D4×C107—44no—
193D4×C107—44no—
194D4×C107—416no—
195D4×C107—416no—
196D4×C107—44no—
197C2×C4×C107—21no—
198C2×C4×C107—21no—
199C2×C4×C107—21no—
200C2×C4×C107—24no—
201C2×C4×C107—21no—
202C2×C4×C107—21no—
203C2×C4×C107—44no—
204C2×C4×C107—48no—
205C2×C4×C107—44no—
206C2×C4×C107—82no—
207C2×C4×C107—84no—
208C2×C4×C107#12012yes—
209C8×C107—82no—
210C22×C107—24no—
211C22×C107—24no—
212C22×C107—22no—
213C22×C107—22no—
214C22×C107—22no—
215C22×C107—28no—
216C22×C107—22no—
217C22×C107—22no—
218C22×C107—22no—
219C22×C107—42no—
220C22×C107—42no—
221C22×C107—42no—
222C22×C107—42no—
223C22×C107—44no—
224C22×C107—432no—
225C22×C107—416no—
226C22×C107—48no—
227C22×C107—42no—
228C22×C107—42no—
229C22×C107—48no—
230C22×C107—48no—
231C22×C107—416no—
232C22×C107—432no—
233C22×C107—42no—
234C22×C107—48no—
235C22×C107—42no—
236C22×C107—42no—
237C22×C107—44no—
238C22×C107—42no—
239C22×C107—41no—
240C22×C107—81no—
241C22×C107—88no—
242C22×C107—81no—
243C22×C107AΣL(1,𝔽16)14yescomplemented
244C22×C107#37911yescharacteristic
245C22×C107C2×#9911yescharacteristic
246C22×C107C2×#9911yescharacteristic
247C4×C107—22no—
248C4×C107—44no—
249C4×C107—41no—
250C4×C107—162no—
251#12687C10711yesmaximal, characteristic, direct product factor, Sylow
252C2×C107—21no—
253C2×C107—24no—
254C2×C107—21no—
255C2×C107—42no—
256C2×C107—48no—
257C2×C107—42no—
258C2×C107—42no—
259C2×C107—81no—
260C2×C107#178111yescharacteristic
261C2×C107C2×AΣL(1,𝔽16)12yes—
262C22×#120C2×C10711yescharacteristic
263C2×AΣL(1,𝔽16)C2×C10714yescomplemented
264#1783C2×C10711yescharacteristic, complemented
265#1781C2×C10711yescharacteristic, complemented
266C107#1268711yescharacteristic, direct product factor, Sylow
267C26C4×C10711yescharacteristic, complemented
268C23×D4—21no—
269C23×D4C22×C10711yescharacteristic
270C2×#120—21no—
271C2×#120—21no—
272C2×#120—28no—
273C2×#120C4×C10712yescomplemented
274C2×#120C22×C10712yes—
275C22×C22⋊C4—21no—
276C22×C22⋊C4C4×C10711yescharacteristic, complemented
277C2×#100—21no—
278C2×#100C22×C10712yescomplemented
279C2×#99—21no—
280C2×#99C22×C10712yescomplemented
281#407—21no—
282#379—21no—
283AΣL(1,𝔽16)—28no—
284C25—21no—
285C25—21no—
286C25—24no—
287C25—48no—
288C25—42no—
289C25—42no—
290C25C2×C4×C10711yescharacteristic
291C25C2×C4×C10712yes—
292C25D4×C10711yescharacteristic
293C22×D4—28no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4—24no—
297C22×D4—42no—
298C22×D4—42no—
299C22×D4—48no—
300C22×D4C2×C4×C10711yescharacteristic
301C22×D4D4×C10711yescharacteristic
302C22×D4D4×C10711yescharacteristic
303C22×D4C23×C10711yescharacteristic
304C23×C4—21no—
305C23×C4D4×C10711yescharacteristic
306C2×M16—22no—
307#120—28no—
308#120—28no—
309#120—24no—
310#120—24no—
311#120—24no—
312#120D4×C10718yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4C2×C4×C10712yes—
321#100—24no—
322#100—42no—
323#99—24no—
324#99—42no—
325C22⋊C8—42no—
326C24—24no—
327C24—28no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—44no—
338C24—48no—
339C24—42no—
340C24—42no—
341C24—44no—
342C24—42no—
343C24—48no—
344C24—432no—
345C24—416no—
346C24—42no—
347C24—44no—
348C24—48no—
349C24—416no—
350C24—432no—
351C24—48no—
352C24—42no—
353C24—42no—
354C24C22⋊C4×C10711yescharacteristic
355C24C22⋊C4×C10711yescharacteristic
356C24C22⋊C4×C10711yescharacteristic
357C24C22⋊C4×C10714yescomplemented
358C24C22×C4×C10711yescharacteristic
359C24C2×D4×C10711yescharacteristic
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—432no—
377C2×D4C2×D4×C10714yes—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4C22⋊C4×C10711yescharacteristic
386C22×C4C2×D4×C10711yescharacteristic
387M16—42no—
388M16—42no—
389C2×C8—42no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—28no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4C22⋊C4×C10714yescomplemented
398C23—24no—
399C23—21no—
400C23—21no—
401C23—216no—
402C23—28no—
403C23—28no—
404C23—21no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—44no—
415C23—42no—
416C23—48no—
417C23—44no—
418C23—42no—
419C23—48no—
420C23—48no—
421C23—48no—
422C23—416no—
423C23—44no—
424C23—48no—
425C23—44no—
426C23—48no—
427C23—48no—
428C23—416no—
429C23—48no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—48no—
434C23—48no—
435C23—416no—
436C23—416no—
437C23—464no—
438C23—464no—
439C23—48no—
440C23—416no—
441C23—48no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—82no—
447C23—84no—
448C23#99×C10712yes—
449C23#99×C10712yes—
450C23C2×C22⋊C4×C10711yescharacteristic
451C23C2×C22⋊C4×C10712yes—
452C23#120×C10712yes—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—416no—
457D4—416no—
458D4—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4#120×C10712yes—
471C8—82no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—28no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—432no—
487C22—416no—
488C22—48no—
489C22—42no—
490C22—42no—
491C22—48no—
492C22—48no—
493C22—416no—
494C22—432no—
495C22—42no—
496C22—48no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—42no—
501C22—41no—
502C22—81no—
503C22—88no—
504C22—81no—
505C22AΣL(1,𝔽16)×C10714yescomplemented
506C22#379×C10711yescharacteristic
507C22C2×#99×C10711yescharacteristic
508C22C2×#99×C10711yescharacteristic
509C4—22no—
510C4—44no—
511C4—41no—
512C4—162no—
513C2—21no—
514C2—24no—
515C2—21no—
516C2—42no—
517C2—48no—
518C2—42no—
519C2—42no—
520C2—81no—
521C2C107×#178111yescharacteristic
522C2C2×AΣL(1,𝔽16)×C10712yes—
subgroupG/H
center, socleC22×C107C2×#99
commutatorC24C22×C4×C107
Frattini, cosocleC22×D4C23×C107
prime radicalC22×#120×C107C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C107 ◁ C24×C107 ◁ C23×D4×C107 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×AΣL(1,𝔽16)×C107C2semidirect
3. C107×#1783C2semidirect
4. C107×#1781C2semidirect
5. C26×C107C4semidirect
11. C2×#120×C107C4semidirect
14. C22×C22⋊C4×C107C4semidirect
16. C2×#100×C107C22semidirect
18. C2×#99×C107C22semidirect
95. C24×C107C22⋊C4semidirect
135. C22⋊C4×C107C22⋊C4semidirect
243. C22×C107AΣL(1,𝔽16)semidirect
251. #12687C107direct (N × H)
263. C2×AΣL(1,𝔽16)C2×C107semidirect
264. #1783C2×C107semidirect
265. #1781C2×C107semidirect
266. C107#12687direct (N × H)
267. C26C4×C107semidirect
273. C2×#120C4×C107semidirect
276. C22×C22⋊C4C4×C107semidirect
278. C2×#100C22×C107semidirect
280. C2×#99C22×C107semidirect
357. C24C22⋊C4×C107semidirect
397. C22⋊C4C22⋊C4×C107semidirect
505. C22AΣL(1,𝔽16)×C107semidirect

Maximal subgroups

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

subgroupindex
C22×#120×C1072
C2×AΣL(1,𝔽16)×C1072
C107×#17832
C107×#17812
#12687107