Finite Group Databasebeta

← #28339828

Subgroups — #28339828

Order 13696.

1246 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D107C211yesmaximal, characteristic, complemented
2C2×#16812082C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic, complemented
4C2×#16812077C211yesmaximal, characteristic, complemented
5C2×#16812077C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
7#20292543C211yesmaximal, characteristic, complemented
8#20292541C211yesmaximal, characteristic, complemented
9#20292535C211yesmaximal, characteristic, complemented
10#20292505C211yesmaximal, characteristic, complemented
11#20292503C211yesmaximal, characteristic, complemented
12#20292503C211yesmaximal, characteristic, complemented
13#20292501C211yesmaximal, characteristic, complemented
14#20292482C211yesmaximal, characteristic, complemented
15#392×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C24×D107C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C22×C22⋊D107C2211yescharacteristic
22C22×C22⋊D107C2211yescharacteristic, complemented
23C22×C22⋊D107C2211yescharacteristic, complemented
24C2×D4×D107—24no—
25C2×D4×D107—24no—
26C2×D4×D107—24no—
27C22×C4×D107C2211yescharacteristic
28C22×C4×D107C2211yescharacteristic, complemented
29C22×C4×D107C2211yescharacteristic, complemented
30#16812082—24no—
31#16812080—24no—
32#16812077—24no—
33#16812077—24no—
34C2×C4⋊Dic107C2211yescharacteristic, complemented
35C2×Dic107⋊C4C2211yescharacteristic, complemented
36C2×C22⋊Dic107C2211yescharacteristic, complemented
37C2×C22⋊Dic107C2211yescharacteristic, complemented
38C2×C22⋊Dic107C2211yescharacteristic, complemented
39C2×C22⋊Dic107C2211yescharacteristic, complemented
40C2×C22⋊Dic107C2211yescharacteristic, complemented
41C2×(C2×D107)⋊C4C2211yescharacteristic
42C2×(C2×D107)⋊C4C2211yescharacteristic
43C2×(C2×D107)⋊C4C2211yescharacteristic
44C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
48C22⋊C4×D107—24no—
49C107⋊#95C2211yescharacteristic, complemented
50C107⋊#95C2211yescharacteristic, complemented
51C107⋊#95C2211yescharacteristic, complemented
52C22×D4×C107C2211yescharacteristic
53C22×D4×C107C2211yescharacteristic, complemented
54C22×D4×C107C2211yescharacteristic, complemented
55C2×C22⋊C4×C107C2211yescharacteristic, complemented
56C2×C22⋊C4×C107C2211yescharacteristic, complemented
57C2×C22⋊C4×C107C2211yescharacteristic, complemented
58#95×C107C2211yescharacteristic, complemented
59C23×D107—22no—
60C23×D107—22no—
61C23×D107—22no—
62C23×D107—22no—
63C23×D107—22no—
64C23×D107—22no—
65C23×D107—22no—
66C23×D107—22no—
67C23×D107—22no—
68C23×D107—41no—
69C23×D107—41no—
70C23×D107—41no—
71C23×D107—41no—
72C23×D107D414yescomplemented
73C23×D107C2311yescharacteristic
74C23×D107C2311yescharacteristic
75C23×D107C2311yescharacteristic
76C2×D428—22no—
77C2×D428—22no—
78C2×D428D414yescomplemented
79C2×C22⋊D107—22no—
80C2×C22⋊D107—22no—
81C2×C22⋊D107—22no—
82C2×C22⋊D107—22no—
83C2×C22⋊D107—22no—
84C2×C22⋊D107—22no—
85C2×C22⋊D107—22no—
86C2×C22⋊D107—22no—
87C2×C22⋊D107—22no—
88C2×C22⋊D107—22no—
89C2×C22⋊D107—22no—
90C2×C22⋊D107—22no—
91C2×C22⋊D107—22no—
92C2×C22⋊D107—41no—
93C2×C22⋊D107D414yescomplemented
94D4×D107—232no—
95C22×Dic107C2311yescharacteristic
96C22×Dic107C2311yescharacteristic
97C22×Dic107C2311yescharacteristic
98C22×Dic107C2311yescharacteristic
99C22×Dic107C2311yescharacteristic
100C2×C4×D107—22no—
101C2×C4×D107—22no—
102C2×C4×D107—22no—
103C2×C4×D107—22no—
104C2×C4×D107—22no—
105C2×C4×D107—22no—
106C2×C4×D107—22no—
107C2×C4×D107—41no—
108C2×C4×D107—41no—
109C4⋊Dic107—22no—
110Dic107⋊C4—22no—
111C22⋊Dic107—22no—
112C22⋊Dic107—22no—
113C22⋊Dic107—22no—
114C22⋊Dic107—22no—
115C22⋊Dic107—22no—
116(C2×D107)⋊C4—22no—
117(C2×D107)⋊C4—22no—
118(C2×D107)⋊C4—22no—
119(C2×D107)⋊C4—22no—
120(C2×D107)⋊C4—22no—
121(C2×D107)⋊C4—41no—
122(C2×D107)⋊C4D414yescomplemented
123C24×C107C2311yescharacteristic
124C24×C107C2311yescharacteristic
125C24×C107C2311yescharacteristic
126C2×D4×C107—22no—
127C2×D4×C107—22no—
128C2×D4×C107—22no—
129C2×D4×C107—22no—
130C2×D4×C107—22no—
131C2×D4×C107—22no—
132C2×D4×C107—41no—
133C2×D4×C107—41no—
134C2×D4×C107—41no—
135C22×C4×C107C2311yescharacteristic
136C22×C4×C107C2311yescharacteristic
137C22×C4×C107C2311yescharacteristic
138C22×C4×C107C2311yescharacteristic
139C22⋊C4×C107—22no—
140C22⋊C4×C107—22no—
141C22⋊C4×C107—41no—
142C23×C107—21no—
143C23×C107—21no—
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—21no—
153C23×C107—21no—
154C23×C107—21no—
155C23×C107—21no—
156C23×C107—21no—
157C23×C107—21no—
158C23×C107—21no—
159C23×C107—21no—
160C23×C107—21no—
161C23×C107C2×D412yes—
162C23×C107C2×D412yes—
163C23×C107C2411yescharacteristic
164C22×D107—21no—
165C22×D107—21no—
166C22×D107—21no—
167C22×D107—21no—
168C22×D107—21no—
169C22×D107—21no—
170C22×D107—21no—
171C22×D107—21no—
172C22×D107—28no—
173C22×D107—21no—
174C22×D107—21no—
175C22×D107—21no—
176C22×D107—21no—
177C22×D107—21no—
178C22×D107—21no—
179C22×D107—28no—
180C22×D107—28no—
181C22×D107—44no—
182C22×D107—44no—
183C22×D107—44no—
184C22×D107—44no—
185C22×D107—44no—
186C22×D107—44no—
187C22×D107—44no—
188C22×D107—44no—
189C22×D107—44no—
190C22×D107—44no—
191C22×D107—44no—
192C22×D107C2×D412yes—
193C22×D107C2×D412yes—
194C22×D107C2×D412yes—
195C22×D107C2×D412yes—
196C22×D107C2×D412yes—
197C22×D107C2×D412yes—
198C22×D107#4112yes—
199D4×C107—28no—
200D4×C107—44no—
201D4×C107—44no—
202C2×C4×C107—21no—
203C2×C4×C107—21no—
204C2×C4×C107—21no—
205C2×C4×C107—21no—
206C2×C4×C107—21no—
207C2×C4×C107—21no—
208C2×C4×C107—21no—
209C2×C4×C107—21no—
210C2×C4×C107—21no—
211C2×C4×C107—21no—
212C2×C4×C107C2×D412yes—
213C2×C4×C107C2×D412yescomplemented
214C22⋊D107—28no—
215C22⋊D107—28no—
216C22⋊D107—44no—
217C22⋊D107—44no—
218C22⋊D107—44no—
219C2×Dic107—21no—
220C2×Dic107—21no—
221C2×Dic107—21no—
222C2×Dic107—21no—
223C2×Dic107—21no—
224C2×Dic107—21no—
225C2×Dic107—21no—
226C2×Dic107—21no—
227C2×Dic107—21no—
228C2×Dic107—21no—
229C2×Dic107—21no—
230C2×Dic107—21no—
231C2×Dic107—21no—
232C2×Dic107C2×D412yes—
233C2×Dic107C2×D412yescomplemented
234D428—28no—
235C4×D107—28no—
236C4×D107—44no—
237C4×D107—44no—
238C22×C107—22no—
239C22×C107—22no—
240C22×C107—22no—
241C22×C107—22no—
242C22×C107—22no—
243C22×C107—22no—
244C22×C107—41no—
245C22×C107—41no—
246C22×C107—41no—
247C22×C107—41no—
248C22×C107—41no—
249C22×C107—41no—
250C22×C107—41no—
251C22×C107—41no—
252C22×C107—41no—
253C22×C107—41no—
254C22×C107—41no—
255C22×C107—41no—
256C22×C107—41no—
257C22×C107—41no—
258C22×C107—41no—
259C22×C107C22×D411yescharacteristic
260C22×C107C22×D411yescharacteristic
261C22×C107C22×D411yescharacteristic
262C22×C107C22×D411yescharacteristic
263C22×C107C2×#4111yescharacteristic
264C22×C107H(5,𝔽2)11yescharacteristic
265C22×C107H(5,𝔽2)11yescharacteristic
266C2×D107—22no—
267C2×D107—22no—
268C2×D107—22no—
269C2×D107—22no—
270C2×D107—22no—
271C2×D107—22no—
272C2×D107—22no—
273C2×D107—22no—
274C2×D107—22no—
275C2×D107—22no—
276C2×D107—22no—
277C2×D107—22no—
278C2×D107—22no—
279C2×D107—22no—
280C2×D107—22no—
281C2×D107—416no—
282C2×D107—416no—
283C2×D107—41no—
284C2×D107—41no—
285C2×D107—41no—
286C2×D107—41no—
287C2×D107#12014yescomplemented
288C2×D107#12114yescomplemented
289C4×C107—22no—
290C4×C107—22no—
291C4×C107—41no—
292C4×C107—41no—
293Dic107—22no—
294Dic107—22no—
295Dic107—41no—
296Dic107—41no—
297Dic107—41no—
298C2×C107—42no—
299C2×C107—42no—
300C2×C107—81no—
301C2×C107C2×#12011yescharacteristic
302C2×C107C2×#12111yescharacteristic
303C2×C107#53411yescharacteristic
304C2×C107D4211yescharacteristic
305C2×C107D4211yescharacteristic
306C2×C107#54611yescharacteristic
307C2×C107#54611yescharacteristic
308D107—24no—
309D107—24no—
310D107—42no—
311#2410—1071nomaximal, Sylow
312C107#241011yescharacteristic, complemented, Sylow
313C23×D4—1071no—
314C2×#121—1071no—
315C2×#121—1071no—
316C2×#120—1071no—
317C2×#120—1071no—
318C22×C22⋊C4—1071no—
319#394—1071no—
320#394—1071no—
321#392—1071no—
322#392D10711yescharacteristic, complemented
323#386—1071no—
324#386—1071no—
325#386—1071no—
326#385—1071no—
327#379—1071no—
328C25—1071no—
329C25—1071no—
330C22×D4—1071no—
331C22×D4—1071no—
332C22×D4—1071no—
333C22×D4—1071no—
334C22×D4—1071no—
335C22×D4—1071no—
336C22×D4—2144no—
337C22×D4—2144no—
338C22×D4—2144no—
339C22×D4C2×D10711yescharacteristic
340C22×D4C2×D10711yescharacteristic, complemented
341C22×D4C2×D10711yescharacteristic, complemented
342C23×C4—1071no—
343C23×C4—1071no—
344C23×C4—1071no—
345#121—2144no—
346#121—2144no—
347#120—2144no—
348#120—2144no—
349C2×C4⋊C4—1071no—
350C2×C4⋊C4—1071no—
351C2×C22⋊C4—1071no—
352C2×C22⋊C4—1071no—
353C2×C22⋊C4—1071no—
354C2×C22⋊C4—1071no—
355C2×C22⋊C4—1071no—
356C2×C22⋊C4—1071no—
357C2×C22⋊C4—1071no—
358C2×C22⋊C4—1071no—
359C2×C22⋊C4—1071no—
360C2×C22⋊C4—1071no—
361C2×C22⋊C4—1071no—
362C2×C22⋊C4—1071no—
363C2×C22⋊C4—2144no—
364C2×C22⋊C4C2×D10711yescharacteristic, complemented
365C2×C22⋊C4C2×D10711yescharacteristic, complemented
366C2×C22⋊C4C2×D10711yescharacteristic, complemented
367#95—1071no—
368#95—1071no—
369#95—1071no—
370#95C2×D10711yescharacteristic, complemented
371C24—1071no—
372C24—1071no—
373C24—1074no—
374C24—1071no—
375C24—4281no—
376C24—4281no—
377C24—4281no—
378C24—4281no—
379C24—2142no—
380C24—2142no—
381C24—2142no—
382C24—2142no—
383C24—2142no—
384C24—2142no—
385C24—2142no—
386C24—2142no—
387C24—2142no—
388C24C22×D10711yescharacteristic
389C24C22×D10711yescharacteristic
390C24C22×D10711yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—1074no—
401C2×D4—1074no—
402C2×D4—4281no—
403C2×D4—2142no—
404C2×D4—2142no—
405C2×D4—2142no—
406C2×D4—2142no—
407C2×D4—21432no—
408C2×D4—2142no—
409C2×D4—2142no—
410C2×D4—2142no—
411C2×D4—2142no—
412C2×D4—2142no—
413C2×D4—2142no—
414C2×D4—2142no—
415C2×D4—2142no—
416C2×D4—2142no—
417C2×D4—2142no—
418C2×D4—2142no—
419C22×C4—1071no—
420C22×C4—1071no—
421C22×C4—1071no—
422C22×C4—1071no—
423C22×C4—1071no—
424C22×C4—4281no—
425C22×C4—4281no—
426C22×C4—2142no—
427C22×C4—2142no—
428C22×C4—2142no—
429C22×C4—2142no—
430C22×C4—2142no—
431C22×C4—2142no—
432C22×C4—2142no—
433C22×C4C22×D10711yescharacteristic
434C22×C4C22×D10711yescharacteristic
435C22×C4C22×D10711yescharacteristic
436C22×C4C22×D10711yescharacteristic
437C4⋊C4—2142no—
438C4⋊C4—2142no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—1074no—
443C22⋊C4—4281no—
444C22⋊C4—2142no—
445C22⋊C4—2142no—
446C22⋊C4—2142no—
447C22⋊C4—2142no—
448C22⋊C4—2142no—
449C22⋊C4—2142no—
450C22⋊C4—2142no—
451C22⋊C4—2142no—
452C22⋊C4—2142no—
453C22⋊C4—2142no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—1072no—
474C23—1072no—
475C23—1072no—
476C23—1072no—
477C23—1072no—
478C23—1072no—
479C23—1072no—
480C23—4284no—
481C23—4284no—
482C23—4284no—
483C23—4284no—
484C23—4284no—
485C23—4284no—
486C23—4284no—
487C23—4284no—
488C23—4284no—
489C23—4284no—
490C23—4284no—
491C23—2148no—
492C23—2148no—
493C23—2148no—
494C23—2141no—
495C23—2141no—
496C23—2141no—
497C23—2141no—
498C23—2141no—
499C23—2141no—
500C23—2141no—
501C23—2141no—
502C23—2141no—
503C23—2141no—
504C23—2141no—
505C23—2141no—
506C23—2141no—
507C23—2141no—
508C23D4×D10712yes—
509C23D4×D10712yes—
510C23C23×D10711yescharacteristic
511D4—28no—
512D4—44no—
513D4—44no—
514D4—4284no—
515D4—4284no—
516D4—4284no—
517D4—2148no—
518D4—2148no—
519D4—2148no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—1072no—
531C2×C4—1072no—
532C2×C4—4284no—
533C2×C4—4284no—
534C2×C4—2148no—
535C2×C4—2141no—
536C2×C4—2141no—
537C2×C4—2141no—
538C2×C4—2141no—
539C2×C4—2141no—
540C2×C4—2141no—
541C2×C4—2141no—
542C2×C4—2141no—
543C2×C4—2141no—
544C2×C4—2141no—
545C2×C4—2141no—
546C2×C4—2141no—
547C2×C4—2141no—
548C2×C4D4×D10712yes—
549C2×C4D4×D10712yescomplemented
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—1074no—
572C22—1074no—
573C22—42816no—
574C22—42816no—
575C22—4281no—
576C22—4281no—
577C22—4281no—
578C22—4281no—
579C22—2142no—
580C22—2142no—
581C22—2142no—
582C22—2142no—
583C22—2142no—
584C22—2142no—
585C22—2142no—
586C22—2142no—
587C22—2142no—
588C22—2142no—
589C22—2142no—
590C22—2142no—
591C22—2142no—
592C22—2142no—
593C22—2142no—
594C22C2×D4×D10711yescharacteristic
595C22C2×D4×D10711yescharacteristic
596C22C2×D4×D10711yescharacteristic
597C22C2×D4×D10711yescharacteristic
598C22#41×D10711yescharacteristic
599C22#1681215411yescharacteristic
600C22#1681215411yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—4281no—
606C4—4281no—
607C4—4281no—
608C4—2142no—
609C4—2142no—
610C2—42no—
611C2—42no—
612C2—81no—
613C2—4282no—
614C2—2144no—
615C2—2144no—
616C2#120×D10711yescharacteristic
617C2#121×D10711yescharacteristic
618C2#2029318711yescharacteristic
619C2#2029325911yescharacteristic
620C2#2029326211yescharacteristic
621C2#2029326611yescharacteristic
622C2#2029327411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2410
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D107C2semidirect
2. C2×#16812082C2semidirect
3. C2×#16812080C2semidirect
4. C2×#16812077C2semidirect
5. C2×#16812077C2semidirect
6. C2×C22⋊C4×D107C2semidirect
7. #20292543C2semidirect
8. #20292541C2semidirect
9. #20292535C2semidirect
10. #20292505C2semidirect
11. #20292503C2semidirect
12. #20292503C2semidirect
13. #20292501C2semidirect
14. #20292482C2semidirect
15. #392×C107C2semidirect
22. C22×C22⋊D107C22semidirect
23. C22×C22⋊D107C22semidirect
28. C22×C4×D107C22semidirect
29. C22×C4×D107C22semidirect
34. C2×C4⋊Dic107C22semidirect
35. C2×Dic107⋊C4C22semidirect
36. C2×C22⋊Dic107C22semidirect
37. C2×C22⋊Dic107C22semidirect
38. C2×C22⋊Dic107C22semidirect
39. C2×C22⋊Dic107C22semidirect
40. C2×C22⋊Dic107C22semidirect
44. C2×(C2×D107)⋊C4C22semidirect
45. C2×(C2×D107)⋊C4C22semidirect
46. C2×(C2×D107)⋊C4C22semidirect
47. C2×(C2×D107)⋊C4C22semidirect
49. C107⋊#95C22semidirect
50. C107⋊#95C22semidirect
51. C107⋊#95C22semidirect
53. C22×D4×C107C22semidirect
54. C22×D4×C107C22semidirect
55. C2×C22⋊C4×C107C22semidirect
56. C2×C22⋊C4×C107C22semidirect
57. C2×C22⋊C4×C107C22semidirect
58. #95×C107C22semidirect
72. C23×D107D4semidirect
78. C2×D428D4semidirect
93. C2×C22⋊D107D4semidirect
122. (C2×D107)⋊C4D4semidirect
213. C2×C4×C107C2×D4semidirect
233. C2×Dic107C2×D4semidirect
287. C2×D107#120semidirect
288. C2×D107#121semidirect
312. C107#2410semidirect
322. #392D107semidirect
340. C22×D4C2×D107semidirect
341. C22×D4C2×D107semidirect
364. C2×C22⋊C4C2×D107semidirect
365. C2×C22⋊C4C2×D107semidirect
366. C2×C22⋊C4C2×D107semidirect
370. #95C2×D107semidirect
549. C2×C4D4×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168120822
C2×#168120802
C2×#168120772
C2×C22⋊C4×D1072
#202925432
#202925412
#202925352
#202925052
#202925032
#202925012
#202924822
#392×C1072
#2410107