Finite Group Databasebeta

← #14219827

Subgroups — #14219827

Order 1792.

1920 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#11742913C211yesmaximal, characteristic
2C7×#2971C211yesmaximal, characteristic, complemented
3#11739965C211yesmaximal, characteristic, complemented
4#11737352C211yesmaximal, characteristic, complemented
5#11737351C211yesmaximal, characteristic, complemented
6#11737350C211yesmaximal, characteristic, complemented
7C8×#1161C211yesmaximal, characteristic, complemented
8#11736992C213yesmaximal, complemented
9#11736991C213yesmaximal, complemented
10#11736986C213yesmaximal, complemented
11#11736983C213yesmaximal, complemented
12#11736785C213yesmaximal, complemented
13#11736783C213yesmaximal, complemented
14#11735968C213yesmaximal, complemented
15#11735967C213yesmaximal, complemented
16C4×C7×#41C2211yescharacteristic
17#90525C411yescharacteristic
18#90525C411yescharacteristic
19C2×#6470C2216yes—
20#90268C411yescharacteristic
21C4×#1161C2211yescharacteristic
22C4×#1161C2211yescharacteristic
23Q8×Dic14C411yescharacteristic
24#90254C413yes—
25#90244C413yes—
26#90239C2213yes—
27#90239C2213yes—
28#90223C413yes—
29#90223C413yes—
30#90168C2213yes—
31C4×#1157C2213yes—
32C7×C8×Q8C2211yescharacteristic
33C7×C8×Q8C2211yescharacteristic
34C7×#436C2213yes—
35C7×#436C2213yes—
36C7×#432C2213yes—
37C4×C7×M16C2213yes—
38#89901C2216yes—
39#89900C2213yes—
40#89900C2213yes—
41#89899C2213yes—
42#89899C2213yes—
43Q8×C7⋊C8C2211yescharacteristic
44Q8×C7⋊C8C2211yescharacteristic
45D4×C7⋊C8C2213yes—
46D4×C7⋊C8C2213yes—
47#89697C2213yes—
48C2×C4⋊(C7⋊C8)C2213yes—
49#89610C2213yes—
50#89610C2213yes—
51#89609C2213yes—
52#89607C2213yes—
53#89606C2213yes—
54#89606C2213yes—
55#89605C2213yes—
56D7×C4⋊C8C2213yes—
57#89502C2216yes—
58#89499C2216yes—
59#89499C2216yes—
60(C22⋊D7)⋊C8C2216yes—
61#89478C2213yes—
62#89478C2216yes—
63#89478C2213yes—
64#89473C2211yescharacteristic
65#89473C2213yes—
66#89460C2213yes—
67#89460C2213yes—
68C4×#1131C2213yes—
69C8×D28C2213yes—
70C8×D28C2213yes—
71#89457C2213yes—
72#89457C2213yes—
73C4×C8×D7C2213yes—
74#60328—71nomaximal, Sylow
75C2×C7×#41C2×C411yescharacteristic
76C2×#1161C2×C411yescharacteristic
77C2×#1161C2×C411yescharacteristic
78C2×#1157C2×C413yes—
79C2×#1157C2316yes—
80C2×C7×M16C2313yes—
81C2×C7×M16C2313yes—
82C4×C7×Q8C2×C411yescharacteristic
83C4×C7×Q8C2311yescharacteristic
84C4×C7×D4C2×C413yes—
85C4×C7×D4C2313yes—
86C7×#117C2×C413yes—
87C2×C42×C7C2313yes—
88Q8×Dic7C2×C411yescharacteristic
89Q8×Dic7C2×C411yescharacteristic
90#6503C2×C413yes—
91#6503C2×C413yes—
92#6501C2×C413yes—
93#6501C2×C413yes—
94#6493C2×C413yes—
95#6493C2×C413yes—
96#6493C2×C416yes—
97C4×C22⋊D7C2×C413yes—
98C4×C22⋊D7C2×C413yes—
99C4×C22⋊D7C2313yes—
100C4×C22⋊D7C2313yes—
101#6489C2×C413yes—
102C22×C7⋊C8C2316yes—
103#6470—412no—
104C2×#1131C2313yes—
105C2×#1131C2316yes—
106C2×#1131C2313yes—
107C2×C8×D7C2313yes—
108C2×C8×D7C2316yes—
109C2×C8×D7C2313yes—
110#6458C2×C413yes—
111#6458C2×C413yes—
112#6456C2×C413yes—
113#6456C2×C413yes—
114#6455C2×C413yes—
115#6455C2×C413yes—
116#6451C2×C413yes—
117#6451C2×C413yes—
118#6448C2×C416yes—
119#6448C2×C416yes—
120#6447C2×C416yes—
121#6447C2×C416yes—
122#6438C2×C416yes—
123#6437C2×C413yes—
124C4×D28C2×C413yes—
125C4×D28C2313yes—
126C4×D28C2313yes—
127C4×D28C2313yes—
128#6435C2×C413yes—
129#6435C2×C413yes—
130C42×D7C2313yes—
131C42×D7C2313yes—
132#6432C2×C413yes—
133C4×Dic14C2×C413yes—
134C4×Dic14C2311yescharacteristic
135C7×C4⋊C8C2313yes—
136C7×C4⋊C8C2313yes—
137C7×C22⋊C8C2316yes—
138C7×#97C2311yescharacteristic
139C7×#97C2313yes—
140C4×C7×C8C2313yes—
141C4×C7×C8C2313yes—
142C22⋊(C7⋊C8)C2313yes—
143C22⋊(C7⋊C8)C2313yes—
144(C2×D7)⋊C8C2316yes—
145(C2×D7)⋊C8C2313yes—
146(C2×D7)⋊C8C2316yes—
147(C2×D7)⋊C8C2313yes—
148(C2×D7)⋊C8C2313yes—
149(C2×D7)⋊C8C2313yes—
150#6389C2313yes—
151#6389C2311yescharacteristic
152Dic7⋊C8C2313yes—
153Dic7⋊C8C2313yes—
154Dic7⋊C8C2311yescharacteristic
155Dic7⋊C8C2311yescharacteristic
156C8×Dic7C2313yes—
157C8×Dic7C2311yescharacteristic
158C4⋊(C7⋊C8)C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4⋊(C7⋊C8)C2313yes—
161C4⋊(C7⋊C8)C2311yescharacteristic
162C4×C7⋊C8C2313yes—
163C4×C7⋊C8C2313yes—
164#3483—71no—
165#2991—71no—
166#2990—73no—
167#2988—71no—
168#2988—71no—
169#2981—73no—
170#2981—73no—
171#2979—73no—
172#2971—71no—
173#2971D711yescharacteristic, complemented
174#2965—73no—
175C8×#41—71no—
176#2935—73no—
177#2935—73no—
178#2931—73no—
179C7×#41—24no—
180C2×C7×Q8C22×C411yescharacteristic
181C2×C7×D4C22×C413yes—
182C22×C4×C7C22×C413yes—
183C22×C4×C7C22×C413yes—
184C22×C4×C7C2413yes—
185C2×C22⋊D7C22×C413yes—
186C2×C22⋊D7C22×C413yes—
187C2×C22⋊D7C22×C416yes—
188C2×C22⋊D7C22×C416yes—
189#1161—24no—
190#1161—24no—
191#1157—46no—
192#1157—46no—
193#1157—46no—
194C2×D28C22×C413yes—
195C2×D28C22×C413yes—
196C2×D28C22×C413yes—
197C2×D28C22×C416yes—
198C2×C4×D7C22×C413yes—
199C2×C4×D7C22×C413yes—
200C2×C4×D7C22×C413yes—
201C2×C4×D7C22×C413yes—
202C2×C4×D7C2413yes—
203C2×C4×D7C2413yes—
204C2×Dic14C22×C411yescharacteristic
205C2×Dic14C22×C416yes—
206C7×M16—26no—
207C7×M16—43no—
208C2×C7×C8C2411yescharacteristic
209C2×C7×C8C2413yes—
210C2×C7×C8C2413yes—
211C2×C7×C8C2411yescharacteristic
212C7×C4⋊C4C22×C413yes—
213C7×C4⋊C4C22×C413yes—
214C7×C22⋊C4C22×C416yes—
215C42×C7C22×C413yes—
216C42×C7C22×C413yes—
217C42×C7C2413yes—
218C42×C7C2411yescharacteristic
219C22⋊Dic7C22×C413yes—
220C22⋊Dic7C22×C413yes—
221(C2×D7)⋊C4C22×C413yes—
222(C2×D7)⋊C4C22×C413yes—
223(C2×D7)⋊C4C22×C413yes—
224(C2×D7)⋊C4C22×C413yes—
225(C2×D7)⋊C4C22×C416yes—
226(C2×D7)⋊C4C22×C416yes—
227C4⋊Dic7C22×C413yes—
228C4⋊Dic7C22×C411yescharacteristic
229C4⋊Dic7C22×C413yes—
230C4⋊Dic7C22×C413yes—
231Dic7⋊C4C22×C413yes—
232Dic7⋊C4C22×C413yes—
233Dic7⋊C4C22×C411yescharacteristic
234Dic7⋊C4C22×C411yescharacteristic
235C4×Dic7C22×C413yes—
236C4×Dic7C22×C413yes—
237C4×Dic7C22×C413yes—
238C4×Dic7C22×C413yes—
239C4×Dic7C2411yescharacteristic
240C4×Dic7C2411yescharacteristic
241C2×C7⋊C8—46no—
242C2×C7⋊C8C2411yescharacteristic
243C2×C7⋊C8C2411yescharacteristic
244C2×C7⋊C8C2413yes—
245C2×C7⋊C8C2413yes—
246#1131—26no—
247#1131—43no—
248#1131—46no—
249C8×D7—26no—
250C8×D7—43no—
251C8×D7—46no—
252C2×#131—76no—
253Q82—71no—
254#555—73no—
255#555—73no—
256#555—73no—
257#550—71no—
258#550—71no—
259#550—71no—
260#550—73no—
261#532—73no—
262#532—73no—
263#532—73no—
264C4×#41—71no—
265C4×#41—71no—
266C4×#41—73no—
267C4×#41C2×D711yescharacteristic
268#446—71no—
269#446—73no—
270C8×Q8—71no—
271C8×Q8—71no—
272C8×Q8C2×D711yescharacteristic
273C8×Q8C2×D711yescharacteristic
274#436—73no—
275#436—73no—
276#436—73no—
277#436—73no—
278#436—73no—
279#436—73no—
280#436C2×D713yes—
281#436C2×D713yes—
282#435—73no—
283#435—73no—
284#435—76no—
285#435—73no—
286#435—76no—
287#435—73no—
288C8×D4—73no—
289C8×D4—73no—
290C8×D4—73no—
291C8×D4—73no—
292C8×D4—76no—
293C8×D4—73no—
294C8×D4—73no—
295#432—73no—
296#432C2×D713yes—
297#431—73no—
298#431—73no—
299#431—73no—
300#431—73no—
301#423—73no—
302C2×C4⋊C8—73no—
303C2×C4⋊C8—73no—
304#408—76no—
305#408—76no—
306#408—76no—
307C4×M16—73no—
308C4×M16C2×D713yes—
309C2×C4×C8—73no—
310C23×C7C23×C413yes—
311C22×D7C23×C413yes—
312C22×D7C23×C413yes—
313C7×Q8#13114yescomplemented
314C7×D4—26no—
315C2×C4×C7—26no—
316C2×C4×C7—23no—
317C2×C4×C7—23no—
318C2×C4×C7—43no—
319C2×C4×C7C23×C413yes—
320C2×C4×C7C23×C411yescharacteristic
321C2×C4×C7C23×C413yes—
322C2×C4×C7C23×C411yescharacteristic
323C2×C4×C7C23×C413yes—
324C2×C4×C7C2511yescharacteristic
325C22⋊D7—26no—
326C22⋊D7—46no—
327C22⋊D7—46no—
328C22⋊D7—43no—
329C2×Dic7C23×C411yescharacteristic
330C2×Dic7C23×C411yescharacteristic
331C2×Dic7C23×C411yescharacteristic
332C2×Dic7C23×C413yes—
333C2×Dic7C23×C413yes—
334C2×Dic7C23×C411yescharacteristic
335D28—26no—
336D28—26no—
337D28—46no—
338D28—43no—
339C4×D7—26no—
340C4×D7—26no—
341C4×D7—23no—
342C4×D7—23no—
343C4×D7—23no—
344C4×D7—23no—
345C4×D7—26no—
346C4×D7—43no—
347Dic14—22no—
348Dic14—46no—
349C7×C8—23no—
350C7×C8—23no—
351C7×C8—21no—
352C7×C8H(5,𝔽2)12yes—
353C7⋊C8—23no—
354C7⋊C8—23no—
355C7⋊C8—21no—
356C7⋊C8—21no—
357C2×#41—71no—
358C2×#41—71no—
359C2×#41—73no—
360C2×#41—76no—
361C2×#41C4×D711yescharacteristic
362#131—2812no—
363C2×M16—73no—
364C2×M16—76no—
365C2×M16—73no—
366C2×M16C22×D713yes—
367C2×M16C22×D713yes—
368C22×C8—76no—
369C22×C8—73no—
370C22×C8—76no—
371C22×C8—73no—
372C4⋊Q8—73no—
373C4⋊Q8—73no—
374C4⋊Q8—73no—
375#127—73no—
376#127—73no—
377#127—73no—
378#124—76no—
379#124—76no—
380#124—76no—
381#121—73no—
382#121—73no—
383#121—73no—
384#121—73no—
385#121—76no—
386#121—73no—
387#121—73no—
388#121—76no—
389#121—76no—
390C4×Q8—71no—
391C4×Q8—71no—
392C4×Q8—73no—
393C4×Q8—71no—
394C4×Q8C4×D711yescharacteristic
395C4×Q8C22×D711yescharacteristic
396C4×D4—73no—
397C4×D4—73no—
398C4×D4—73no—
399C4×D4—73no—
400C4×D4—73no—
401C4×D4—73no—
402C4×D4—73no—
403C4×D4—73no—
404C4×D4—73no—
405C4×D4—73no—
406C4×D4C4×D713yes—
407C4×D4C22×D713yes—
408#117—73no—
409#117—73no—
410#117—73no—
411#117—73no—
412#117—73no—
413#117C4×D713yes—
414C2×C42—73no—
415C2×C42—73no—
416C2×C42C22×D713yes—
417C4⋊C8—73no—
418C4⋊C8—73no—
419C4⋊C8—71no—
420C4⋊C8—71no—
421C4⋊C8—73no—
422C4⋊C8—73no—
423C4⋊C8—73no—
424C4⋊C8—71no—
425C4⋊C8C22×D713yes—
426C4⋊C8C22×D713yes—
427C22⋊C8—73no—
428C22⋊C8—73no—
429C22⋊C8—76no—
430C22⋊C8—73no—
431C22⋊C8—73no—
432C22⋊C8—73no—
433C22⋊C8—73no—
434C22⋊C8—76no—
435C22⋊C8C22×D716yes—
436#97—71no—
437#97—73no—
438#97C22×D711yescharacteristic
439#97C22×D713yes—
440C4×C8—71no—
441C4×C8—73no—
442C4×C8—73no—
443C4×C8—73no—
444C4×C8C22×D713yes—
445C4×C8C22×D713yes—
446C22×C7—23no—
447C22×C7—23no—
448C22×C7—23no—
449C22×C7C24×C411yescharacteristic
450C2×D7—23no—
451C2×D7—23no—
452C2×D7—23no—
453C2×D7—23no—
454C2×D7—23no—
455C2×D7—23no—
456C4×C7—23no—
457C4×C7—21no—
458C4×C7—21no—
459C4×C7—23no—
460C4×C7C2×#13116yes—
461C4×C7C2×H(5,𝔽2)11yescharacteristic
462C4×C7#58511yescharacteristic
463Dic7—23no—
464Dic7—23no—
465Dic7—21no—
466Dic7—21no—
467Dic7—21no—
468Dic7#56812yes—
469#41—24no—
470#41—144no—
471#41—144no—
472#41—286no—
473#41—286no—
474#41—286no—
475C2×Q8—71no—
476C2×Q8—76no—
477C2×Q8C2×C4×D711yescharacteristic
478C2×D4—73no—
479C2×D4—73no—
480C2×D4—76no—
481C2×D4—76no—
482C2×D4—73no—
483C2×D4—73no—
484C2×D4—73no—
485C2×D4—76no—
486C2×D4C2×C4×D713yes—
487C22×C4—73no—
488C22×C4—73no—
489C22×C4—73no—
490C22×C4—73no—
491C22×C4—73no—
492C22×C4—73no—
493C22×C4C2×C4×D713yes—
494C22×C4C2×C4×D713yes—
495C22×C4C23×D713yes—
496M16—26no—
497M16—43no—
498M16—146no—
499M16—283no—
500M16—286no—
501C2×C8—71no—
502C2×C8—71no—
503C2×C8—73no—
504C2×C8—73no—
505C2×C8—146no—
506C2×C8—283no—
507C2×C8—286no—
508C2×C8—286no—
509C2×C8C23×D713yes—
510C2×C8C23×D711yescharacteristic
511C2×C8C23×D713yes—
512C2×C8C23×D711yescharacteristic
513C4⋊C4—71no—
514C4⋊C4—71no—
515C4⋊C4—73no—
516C4⋊C4—73no—
517C4⋊C4—71no—
518C4⋊C4—73no—
519C4⋊C4—73no—
520C4⋊C4—73no—
521C4⋊C4C2×C4×D713yes—
522C4⋊C4C2×C4×D713yes—
523C22⋊C4—73no—
524C22⋊C4—73no—
525C22⋊C4—73no—
526C22⋊C4—73no—
527C22⋊C4—76no—
528C22⋊C4—76no—
529C22⋊C4—73no—
530C22⋊C4—73no—
531C22⋊C4C2×C4×D716yes—
532C42—73no—
533C42—73no—
534C42—73no—
535C42—73no—
536C42—71no—
537C42—71no—
538C42C2×C4×D713yes—
539C42C2×C4×D713yes—
540C42C23×D713yes—
541C42C23×D711yescharacteristic
542C2×C7—43no—
543C2×C7C4×H(5,𝔽2)11yescharacteristic
544C2×C7C22×#13111yescharacteristic
545C2×C7C2×#56811yescharacteristic
546D7—43no—
547D7—43no—
548C23—73no—
549C23—73no—
550C23C22×C4×D713yes—
551Q8—142no—
552Q8—286no—
553Q8#647014yescomplemented
554D4—26no—
555D4—146no—
556D4—146no—
557D4—146no—
558D4—286no—
559D4—283no—
560D4—286no—
561D4—286no—
562D4—283no—
563C2×C4—26no—
564C2×C4—23no—
565C2×C4—23no—
566C2×C4—43no—
567C2×C4—71no—
568C2×C4—71no—
569C2×C4—71no—
570C2×C4—71no—
571C2×C4—73no—
572C2×C4—73no—
573C2×C4—146no—
574C2×C4—146no—
575C2×C4—146no—
576C2×C4—143no—
577C2×C4—143no—
578C2×C4—143no—
579C2×C4—143no—
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—21no—
590C8—143no—
591C8—143no—
592C8—141no—
593C8—141no—
594C8#655312yes—
595C7#6032811yescharacteristic, complemented, Sylow
596C22—23no—
597C22—23no—
598C22—23no—
599C22—143no—
600C22—143no—
601C22—143no—
602C22—143no—
603C22—143no—
604C22—143no—
605C22C23×C4×D711yescharacteristic
606C4—23no—
607C4—21no—
608C4—21no—
609C4—23no—
610C4—72no—
611C4—143no—
612C4—143no—
613C4—141no—
614C4—141no—
615C4—141no—
616C4C2×#647016yes—
617C4C2×#655311yescharacteristic
618C4#9061611yescharacteristic
619C2—43no—
620C2—283no—
621C2—283no—
622C2C4×#655311yescharacteristic
623C2C22×#647011yescharacteristic
624C2D7×#56811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2971C2
prime radical#11742913C2
cosocleC2×C4×C7C25
hypercenter#2971D7
hypocenterC7#60328
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. #11739965C2semidirect
4. #11737352C2semidirect
5. #11737351C2semidirect
6. #11737350C2semidirect
7. C8×#1161C2semidirect
8. #11736992C2semidirect
9. #11736991C2semidirect
10. #11736986C2semidirect
11. #11736983C2semidirect
12. #11736785C2semidirect
13. #11736783C2semidirect
14. #11735968C2semidirect
15. #11735967C2semidirect
173. #2971D7semidirect
313. C7×Q8#131semidirect
553. Q8#6470semidirect
595. C7#60328semidirect

Maximal subgroups

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

subgroupindex
#117429132
C7×#29712
#117399652
#117373522
#117373512
#117373502
C8×#11612
#117369922
#117369912
#117369862
#117369832
#117367852
#117367832
#117359682
#117359672
#603287