Finite Group Databasebeta

← #14218494

Subgroups — #14218494

Order 1792.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742910C211yesmaximal, characteristic
2C7×#2971C211yesmaximal, characteristic, complemented
3#41×C7⋊C8C211yesmaximal, characteristic, complemented
4Q8×#1131C211yesmaximal, characteristic, complemented
5C8×Q8×D7C211yesmaximal, characteristic, complemented
6#11737350C211yesmaximal, characteristic, complemented
7C8×#1161C211yesmaximal, characteristic, complemented
8#11736986C213yesmaximal, complemented
9#11736983C213yesmaximal, complemented
10#11736966C213yesmaximal, complemented
11#11736963C213yesmaximal, complemented
12#11736779C213yesmaximal, complemented
13#11736770C213yesmaximal, complemented
14C4×#6470C213yesmaximal, complemented
15#11735956C213yesmaximal, complemented
16C4×C7×#41C2211yescharacteristic
17#90525C411yescharacteristic
18#90522C411yescharacteristic
19C2×#6470C2213yes—
20C2×#6470C2213yes—
21#90267C411yescharacteristic
22C4×#1161C2211yescharacteristic
23C4×Q8×D7C2211yescharacteristic
24#90260C411yescharacteristic
25#90250C413yes—
26#90249C413yes—
27#90248C2213yes—
28#90239C2213yes—
29#90224C413yes—
30#90221C413yes—
31#90170C2213yes—
32C4×#1157C2213yes—
33C7×C8×Q8C2211yescharacteristic
34C7×C8×Q8C2211yescharacteristic
35C7×#436C2213yes—
36C7×#436C2213yes—
37C7×#432C2213yes—
38C4×C7×M16C2213yes—
39#89900C2216yes—
40#89899C2216yes—
41#89894C2213yes—
42#89894C2213yes—
43Q8×C7⋊C8C2211yescharacteristic
44Q8×C7⋊C8C2211yescharacteristic
45D4×C7⋊C8C2213yes—
46D4×C7⋊C8C2213yes—
47#89697C2213yes—
48C2×C4×C7⋊C8C2213yes—
49#89610C2213yes—
50#89609C2213yes—
51#89607C2213yes—
52#89606C2213yes—
53#89605C2213yes—
54D7×C4⋊C8C2213yes—
55#89603C2213yes—
56#89602C2213yes—
57#89502C2216yes—
58#89499C2216yes—
59(C22⋊D7)⋊C8C2216yes—
60#89492C2216yes—
61#89481C2216yes—
62#89480C2213yes—
63#89480C2211yescharacteristic
64#89480C2213yes—
65#89480C2213yes—
66#89460C2213yes—
67C4×#1131C2213yes—
68C4×#1131C2213yes—
69C8×D28C2213yes—
70C4×C8×D7C2213yes—
71C4×C8×D7C2213yes—
72#89451C2213yes—
73C8×Dic14C2213yes—
74#60218—71nomaximal, Sylow
75C2×C7×#41C2×C411yescharacteristic
76C2×#1161C2×C411yescharacteristic
77C2×Q8×D7C2×C411yescharacteristic
78C2×#1157C2313yes—
79C2×C7×M16C2313yes—
80C2×C7×M16C2313yes—
81C4×C7×Q8C2×C411yescharacteristic
82C4×C7×Q8C2311yescharacteristic
83C4×C7×D4C2×C413yes—
84C4×C7×D4C2313yes—
85C7×#117C2×C413yes—
86C2×C42×C7C2313yes—
87Q8×Dic7C2×C411yescharacteristic
88Q8×Dic7C2×C411yescharacteristic
89#6503C2×C413yes—
90#6501C2×C413yes—
91#6501C2×C413yes—
92#6499C2×C413yes—
93#6493C2×C413yes—
94#6492C2×C416yes—
95C4×C22⋊D7C2×C413yes—
96C4×C22⋊D7C2×C413yes—
97C4×C22⋊D7C2×C413yes—
98C4×C22⋊D7C2×C413yes—
99C4×C22⋊D7C2316yes—
100#6489C2313yes—
101#6487C2×C413yes—
102C22×C7⋊C8C2313yes—
103C22×C7⋊C8C2313yes—
104#6470—212no—
105#6470—212no—
106C2×#1131C2313yes—
107C2×#1131C2313yes—
108C2×#1131C2313yes—
109C2×#1131C2311yescharacteristic
110C2×C8×D7C2313yes—
111C2×C8×D7C2313yes—
112C2×C8×D7C2313yes—
113C2×C8×D7C2311yescharacteristic
114#6460C2×C413yes—
115#6458C2×C413yes—
116#6456C2×C413yes—
117#6455C2×C413yes—
118D7×C4⋊C4C2×C413yes—
119#6453C2×C413yes—
120#6451C2×C413yes—
121#6450C2×C413yes—
122#6448C2×C416yes—
123#6447C2×C416yes—
124#6446C2×C416yes—
125#6442C2×C416yes—
126#6439C2×C416yes—
127#6438C2×C413yes—
128C4×D28C2×C413yes—
129C4×D28C2×C413yes—
130C4×D28C2313yes—
131#6435C2311yescharacteristic
132#6435C2313yes—
133C42×D7C2313yes—
134C42×D7C2313yes—
135#6433C2×C413yes—
136C4×Dic14C2×C413yes—
137C4×Dic14C2×C413yes—
138C4×Dic14C2313yes—
139C7×C4⋊C8C2313yes—
140C7×C4⋊C8C2313yes—
141C7×C22⋊C8C2316yes—
142C7×#97C2311yescharacteristic
143C7×#97C2313yes—
144C4×C7×C8C2313yes—
145C4×C7×C8C2313yes—
146C22⋊(C7⋊C8)C2316yes—
147(C2×D7)⋊C8C2316yes—
148(C2×D7)⋊C8C2316yes—
149#6389C2313yes—
150#6389C2313yes—
151#6389C2311yescharacteristic
152#6389C2313yes—
153Dic7⋊C8C2316yes—
154Dic7⋊C8C2316yes—
155C8×Dic7C2313yes—
156C8×Dic7C2313yes—
157C8×Dic7C2311yescharacteristic
158C8×Dic7C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4⋊(C7⋊C8)C2313yes—
161C4×C7⋊C8C2313yes—
162C4×C7⋊C8C2313yes—
163C4×C7⋊C8C2311yescharacteristic
164C4×C7⋊C8C2313yes—
165#3480—71no—
166#2971—71no—
167#2971D711yescharacteristic, complemented
168#2969—73no—
169#2969—73no—
170#2966—73no—
171#2966—73no—
172C8×#41—71no—
173C8×#41—71no—
174Q8×M16—71no—
175C2×C8×Q8—71no—
176#2935—73no—
177#2931—73no—
178#2925—73no—
179C4×#131—73no—
180C7×#41—24no—
181C2×C7×Q8C22×C411yescharacteristic
182C2×C7×D4C22×C413yes—
183C22×C4×C7C22×C413yes—
184C22×C4×C7C22×C413yes—
185C22×C4×C7C2413yes—
186C2×C22⋊D7C22×C413yes—
187C2×C22⋊D7C22×C413yes—
188C2×C22⋊D7C22×C416yes—
189#1161—24no—
190Q8×D7—24no—
191#1157—26no—
192#1157—26no—
193C2×D28C22×C413yes—
194C2×D28C22×C413yes—
195C2×C4×D7C22×C413yes—
196C2×C4×D7C22×C413yes—
197C2×C4×D7C22×C413yes—
198C2×C4×D7C22×C413yes—
199C2×C4×D7C2413yes—
200C2×C4×D7C2411yescharacteristic
201C2×Dic14C22×C413yes—
202C2×Dic14C22×C413yes—
203C7×M16—26no—
204C7×M16—26no—
205C2×C7×C8C2413yes—
206C2×C7×C8C2413yes—
207C2×C7×C8C2411yescharacteristic
208C2×C7×C8C2411yescharacteristic
209C7×C4⋊C4C22×C413yes—
210C7×C4⋊C4C22×C413yes—
211C7×C22⋊C4C22×C416yes—
212C42×C7C22×C413yes—
213C42×C7C22×C413yes—
214C42×C7C2413yes—
215C42×C7C2411yescharacteristic
216C22⋊Dic7C22×C413yes—
217C22⋊Dic7C22×C416yes—
218C22⋊Dic7C22×C413yes—
219(C2×D7)⋊C4C22×C411yescharacteristic
220(C2×D7)⋊C4C22×C411yescharacteristic
221(C2×D7)⋊C4C22×C413yes—
222(C2×D7)⋊C4C22×C413yes—
223(C2×D7)⋊C4C22×C416yes—
224(C2×D7)⋊C4C22×C416yes—
225C4⋊Dic7C22×C413yes—
226C4⋊Dic7C22×C413yes—
227C4⋊Dic7C22×C413yes—
228C4⋊Dic7C22×C413yes—
229Dic7⋊C4C22×C411yescharacteristic
230Dic7⋊C4C22×C411yescharacteristic
231Dic7⋊C4C22×C413yes—
232Dic7⋊C4C22×C413yes—
233Dic7⋊C4C22×C416yes—
234Dic7⋊C4C22×C416yes—
235C4×Dic7C22×C413yes—
236C4×Dic7C22×C413yes—
237C4×Dic7C22×C413yes—
238C4×Dic7C22×C413yes—
239C4×Dic7C2413yes—
240C4×Dic7C2411yescharacteristic
241C2×C7⋊C8—26no—
242C2×C7⋊C8—26no—
243C2×C7⋊C8C2413yes—
244C2×C7⋊C8C2413yes—
245C2×C7⋊C8C2411yescharacteristic
246C2×C7⋊C8C2411yescharacteristic
247#1131—26no—
248#1131—26no—
249#1131—26no—
250#1131#4114yescomplemented
251C8×D7—26no—
252C8×D7—26no—
253C8×D7—26no—
254C8×D7#4114yescomplemented
255C2×#131—73no—
256C2×#131—73no—
257#559—73no—
258#557—71no—
259#556—73no—
260#553—73no—
261#553—73no—
262#550—71no—
263#548—71no—
264#548—71no—
265#532—73no—
266#529—73no—
267#529—73no—
268C4×#41—71no—
269C4×#41—73no—
270C4×#41C2×D711yescharacteristic
271C2×C4×Q8—71no—
272#446—73no—
273#446—73no—
274C8×Q8—71no—
275C8×Q8—71no—
276C8×Q8—73no—
277C8×Q8—73no—
278C8×Q8C2×D711yescharacteristic
279C8×Q8C2×D711yescharacteristic
280#436—76no—
281#436—73no—
282#436—73no—
283#436C2×D713yes—
284#436C2×D713yes—
285#435—76no—
286#435—76no—
287C8×D4—73no—
288C8×D4—73no—
289C8×D4—73no—
290C8×D4—76no—
291C8×D4—73no—
292#433—76no—
293#433—76no—
294#432—73no—
295#432C2×D713yes—
296#431—73no—
297#431—73no—
298#423—73no—
299C2×C4⋊C8—73no—
300#408—76no—
301#405—73no—
302#405—73no—
303#405—71no—
304#405—73no—
305#405—73no—
306#405—73no—
307C4×M16—73no—
308C4×M16—73no—
309C4×M16C2×D713yes—
310C2×C4×C8—73no—
311C2×C4×C8—73no—
312C2×C4×C8—73no—
313C23×C7C23×C413yes—
314C22×D7C23×C413yes—
315C22×D7C23×C411yescharacteristic
316C7×Q8#13114yescomplemented
317C7×D4—26no—
318C2×C4×C7—26no—
319C2×C4×C7—23no—
320C2×C4×C7—23no—
321C2×C4×C7—43no—
322C2×C4×C7C23×C413yes—
323C2×C4×C7C23×C411yescharacteristic
324C2×C4×C7C23×C411yescharacteristic
325C2×C4×C7C23×C413yes—
326C2×C4×C7C23×C413yes—
327C2×C4×C7C2511yescharacteristic
328C22⋊D7—26no—
329C22⋊D7—26no—
330C22⋊D7—46no—
331C2×Dic7C23×C411yescharacteristic
332C2×Dic7C23×C411yescharacteristic
333C2×Dic7C23×C413yes—
334C2×Dic7C23×C413yes—
335C2×Dic7C23×C413yes—
336C2×Dic7C23×C411yescharacteristic
337D28—26no—
338D28—26no—
339C4×D7—26no—
340C4×D7—26no—
341C4×D7—23no—
342C4×D7—23no—
343C4×D7—43no—
344C4×D7—43no—
345C4×D7C2×#4112yes—
346C4×D7C2×#4112yes—
347Dic14—26no—
348Dic14—26no—
349C7×C8—23no—
350C7×C8—23no—
351C7×C8C2×#4112yes—
352C7×C8C2×#4112yes—
353C7⋊C8—23no—
354C7⋊C8—23no—
355C7⋊C8C2×#4112yes—
356C7⋊C8C2×#4112yes—
357C2×#41—71no—
358C2×#41—73no—
359C2×#41C4×D711yescharacteristic
360C22×Q8—71no—
361#131—1412no—
362#131—1412no—
363C2×M16—71no—
364C2×M16—73no—
365C2×M16—73no—
366C2×M16—73no—
367C2×M16C22×D713yes—
368C2×M16C22×D713yes—
369C22×C8—71no—
370C22×C8—73no—
371C22×C8—73no—
372C22×C8—73no—
373C22×C8—73no—
374C22×C8—73no—
375C4⋊Q8—73no—
376#127—73no—
377#126—76no—
378#126—76no—
379#125—73no—
380#125—73no—
381#124—73no—
382#124—76no—
383#124—73no—
384#123—76no—
385#123—76no—
386C22⋊Q8—73no—
387C22⋊Q8—73no—
388#121—73no—
389#121—73no—
390#121—73no—
391#121—73no—
392#121—76no—
393C4×Q8—71no—
394C4×Q8—71no—
395C4×Q8—73no—
396C4×Q8—73no—
397C4×Q8—73no—
398C4×Q8—73no—
399C4×Q8C4×D711yescharacteristic
400C4×Q8C22×D711yescharacteristic
401C4×D4—73no—
402C4×D4—73no—
403C4×D4—73no—
404C4×D4—73no—
405C4×D4—76no—
406C4×D4—73no—
407C4×D4—73no—
408C4×D4—73no—
409C4×D4—73no—
410C4×D4C4×D713yes—
411C4×D4C22×D713yes—
412#117—73no—
413#117—73no—
414#117—71no—
415#117—73no—
416#117C4×D713yes—
417C2×C4⋊C4—73no—
418C2×C42—73no—
419C2×C42—73no—
420C2×C42C22×D713yes—
421C4⋊C8—76no—
422C4⋊C8—76no—
423C4⋊C8—73no—
424C4⋊C8—73no—
425C4⋊C8C22×D713yes—
426C4⋊C8C22×D713yes—
427C22⋊C8—76no—
428C22⋊C8—76no—
429C22⋊C8—76no—
430C22⋊C8C22×D716yes—
431#97—71no—
432#97—73no—
433#97—73no—
434#97—73no—
435#97C22×D711yescharacteristic
436#97C22×D713yes—
437C4×C8—71no—
438C4×C8—73no—
439C4×C8—73no—
440C4×C8—73no—
441C4×C8—71no—
442C4×C8—73no—
443C4×C8—73no—
444C4×C8—73no—
445C4×C8C22×D713yes—
446C4×C8C22×D713yes—
447C22×C7—23no—
448C22×C7—23no—
449C22×C7—23no—
450C22×C7C24×C411yescharacteristic
451C2×D7—21no—
452C2×D7—21no—
453C2×D7—23no—
454C2×D7—23no—
455C2×D7—23no—
456C2×D7C4×#4112yes—
457C4×C7—23no—
458C4×C7—21no—
459C4×C7—23no—
460C4×C7—21no—
461C4×C7C2×#13116yes—
462C4×C7C22×#4111yescharacteristic
463C4×C7C22×#4111yescharacteristic
464Dic7—23no—
465Dic7—23no—
466Dic7—21no—
467Dic7—21no—
468Dic7—23no—
469Dic7C4×#4112yes—
470#41—24no—
471#41—146no—
472#41—146no—
473#41—144no—
474C2×Q8—73no—
475C2×Q8—73no—
476C2×Q8—144no—
477C2×Q8C2×C4×D711yescharacteristic
478C2×D4—73no—
479C2×D4—73no—
480C2×D4—76no—
481C2×D4—73no—
482C2×D4—73no—
483C2×D4C2×C4×D713yes—
484C22×C4—71no—
485C22×C4—73no—
486C22×C4—73no—
487C22×C4—73no—
488C22×C4—73no—
489C22×C4—73no—
490C22×C4C2×C4×D713yes—
491C22×C4C2×C4×D713yes—
492C22×C4C23×D713yes—
493M16—26no—
494M16—26no—
495M16—74no—
496M16—146no—
497M16—146no—
498M16—146no—
499C2×C8—71no—
500C2×C8—71no—
501C2×C8—74no—
502C2×C8—73no—
503C2×C8—73no—
504C2×C8—146no—
505C2×C8—146no—
506C2×C8—146no—
507C2×C8—146no—
508C2×C8—146no—
509C2×C8C23×D713yes—
510C2×C8C23×D713yes—
511C2×C8C23×D711yescharacteristic
512C2×C8C23×D711yescharacteristic
513C4⋊C4—76no—
514C4⋊C4—71no—
515C4⋊C4—71no—
516C4⋊C4—73no—
517C4⋊C4—73no—
518C4⋊C4—76no—
519C4⋊C4—73no—
520C4⋊C4—73no—
521C4⋊C4—73no—
522C4⋊C4—73no—
523C4⋊C4C2×C4×D713yes—
524C4⋊C4C2×C4×D713yes—
525C22⋊C4—73no—
526C22⋊C4—73no—
527C22⋊C4—76no—
528C22⋊C4—71no—
529C22⋊C4—71no—
530C22⋊C4—73no—
531C22⋊C4—76no—
532C22⋊C4—73no—
533C22⋊C4—76no—
534C22⋊C4C2×C4×D716yes—
535C42—73no—
536C42—73no—
537C42—73no—
538C42—73no—
539C42—71no—
540C42—73no—
541C42C2×C4×D713yes—
542C42C2×C4×D713yes—
543C42C23×D711yescharacteristic
544C42C23×D713yes—
545C2×C7—43no—
546C2×C7C2×C4×#4111yescharacteristic
547C2×C7C22×#13111yescharacteristic
548C2×C7#357311yescharacteristic
549D7—22no—
550D7—43no—
551C23—71no—
552C23—73no—
553C23C22×C4×D713yes—
554Q8—146no—
555Q8—146no—
556Q8#647014yescomplemented
557D4—26no—
558D4—146no—
559D4—146no—
560D4—146no—
561D4—146no—
562D4—286no—
563C2×C4—26no—
564C2×C4—23no—
565C2×C4—23no—
566C2×C4—43no—
567C2×C4—71no—
568C2×C4—72no—
569C2×C4—72no—
570C2×C4—71no—
571C2×C4—71no—
572C2×C4—73no—
573C2×C4—73no—
574C2×C4—73no—
575C2×C4—146no—
576C2×C4—146no—
577C2×C4—143no—
578C2×C4—143no—
579C2×C4—283no—
580C2×C4—283no—
581C2×C4C22×C4×D713yes—
582C2×C4C22×C4×D711yescharacteristic
583C2×C4C22×C4×D711yescharacteristic
584C2×C4C22×C4×D713yes—
585C2×C4C22×C4×D713yes—
586C2×C4C24×D711yescharacteristic
587C8—23no—
588C8—23no—
589C8—72no—
590C8—72no—
591C8—143no—
592C8—143no—
593C8D7×#4112yes—
594C8D7×#4112yes—
595C7#6021811yescharacteristic, complemented, Sylow
596C22—23no—
597C22—23no—
598C22—23no—
599C22—72no—
600C22—141no—
601C22—141no—
602C22—143no—
603C22—143no—
604C22—143no—
605C22C23×C4×D711yescharacteristic
606C4—23no—
607C4—21no—
608C4—21no—
609C4—23no—
610C4—72no—
611C4—143no—
612C4—141no—
613C4—143no—
614C4—141no—
615C4—143no—
616C4C2×#647016yes—
617C4C2×D7×#4111yescharacteristic
618C4C2×D7×#4111yescharacteristic
619C2—43no—
620C2—142no—
621C2—283no—
622C2C4×D7×#4111yescharacteristic
623C2C22×#647011yescharacteristic
624C2#1174423411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2971C2
prime radical#11742910C2
cosocleC2×C4×C7C25
hypercenter#2971D7
hypocenterC7#60218
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. C7×#2971C2semidirect
3. #41×C7⋊C8C2semidirect
4. Q8×#1131C2semidirect
5. C8×Q8×D7C2semidirect
6. #11737350C2semidirect
7. C8×#1161C2semidirect
8. #11736986C2semidirect
9. #11736983C2semidirect
10. #11736966C2semidirect
11. #11736963C2semidirect
12. #11736779C2semidirect
13. #11736770C2semidirect
14. C4×#6470C2semidirect
15. #11735956C2semidirect
167. #2971D7semidirect
250. #1131#41semidirect
254. C8×D7#41semidirect
316. C7×Q8#131semidirect
556. Q8#6470semidirect
595. C7#60218semidirect

Maximal subgroups

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

subgroupindex
#117429102
C7×#29712
#41×C7⋊C82
Q8×#11312
C8×Q8×D72
#117373502
C8×#11612
#117369862
#117369832
#117369662
#117369632
#117367792
#117367702
C4×#64702
#117359562
#602187