Finite Group Databasebeta

← #14224248

Subgroups — #14224248

Order 1792.

1886 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3511C211yesmaximal, characteristic
2#11742898C211yesmaximal, characteristic, complemented
3#11740903C218yesmaximal, complemented
4#11740868C214yesmaximal, complemented
5#11740793C212yesmaximal, complemented
6#11740789C212yesmaximal, complemented
7#11740739C212yesmaximal, complemented
8#11740735C212yesmaximal, complemented
9#11740733C211yesmaximal, characteristic, complemented
10#11740726C211yesmaximal, characteristic, complemented
11#11740652C211yesmaximal, characteristic, complemented
12C2×#89865C211yesmaximal, characteristic, complemented
13#11740016C214yesmaximal, complemented
14C4×#6511C211yesmaximal, characteristic, complemented
15C7×#556—21no—
16C7×#555—21no—
17C7×#554—21no—
18C7×#553—22no—
19C7×#548—22no—
20C7×#547—21no—
21C7×#542C2214yes—
22C7×#537C2212yes—
23C7×#533C2212yes—
24C7×#532C2211yescharacteristic
25C7×#532C2212yes—
26C2×C7×C4⋊Q8C2211yescharacteristic
27C7×#520C2212yes—
28C4×C7×#41C2211yescharacteristic
29#90516C2212yes—
30C2×#6514C2214yes—
31C2×#6511C2212yes—
32#90258—21no—
33#90258—21no—
34#90233—21no—
35#90233—21no—
36#90232—22no—
37#90232—22no—
38#90209C2214yes—
39#90207C2212yes—
40#90161C2211yescharacteristic
41#90161C2212yes—
42#90161C2212yes—
43C2×#6432C2211yescharacteristic
44#90005C2214yes—
45#89865C2212yes—
46#89865C2212yes—
47#89865C2214yes—
48#89864C2212yes—
49#89853C2211yescharacteristic
50#89853C2211yescharacteristic
51#89852C2214yes—
52#89851C2212yes—
53#89850C2212yes—
54#89848C2212yes—
55#89842C2211yescharacteristic
56#89842C2211yescharacteristic
57#89832C2214yes—
58#89828C2214yes—
59#89824C2218yes—
60#89823C2218yes—
61#89814C2218yes—
62#89813C2218yes—
63#89802C2218yes—
64C4×#1144C2218yes—
65#89790C2218yes—
66C4×#1142C2218yes—
67#89769C2212yes—
68#89769C2214yes—
69C2×#6384C2214yes—
70#89736C2214yes—
71C4×#1136C2211yescharacteristic
72C4×#1136C2212yes—
73C2×C4×C7⋊C8C2211yescharacteristic
74C2×C4×C7⋊C8C2212yes—
75#60723—71nomaximal, Sylow
76C2×C7×#41—22no—
77C2×C7×#41C2312yes—
78C2×C7×#41C2311yescharacteristic
79C22×C7×Q8—21no—
80C22×Dic14C2312yes—
81C7×C4⋊Q8—21no—
82C7×C4⋊Q8—21no—
83C7×C4⋊Q8C2311yescharacteristic
84C7×C4⋊Q8C2312yes—
85C7×C4⋊Q8C2312yes—
86C7×#127C2311yescharacteristic
87C7×#126—24no—
88C7×#125—22no—
89C7×#124—24no—
90C7×#124—22no—
91C7×#124C2312yes—
92C7×#123—24no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—24no—
96C7×C22⋊Q8C2314yes—
97C7×#121—22no—
98C7×#121—24no—
99C7×#121—21no—
100C7×#121—21no—
101C7×#121—22no—
102C7×#121C2314yes—
103C4×C7×Q8—22no—
104C4×C7×Q8D411yescharacteristic
105C4×C7×Q8D411yescharacteristic
106C4×C7×Q8C2314yes—
107C4×C7×D4—24no—
108C4×C7×D4—21no—
109C4×C7×D4—22no—
110C4×C7×D4—21no—
111C4×C7×D4D412yes—
112C4×C7×D4D412yes—
113C4×C7×D4D411yescharacteristic
114C4×C7×D4D411yescharacteristic
115C4×C7×D4C2314yes—
116C7×#117—22no—
117C7×#117—22no—
118C7×#117C2311yescharacteristic
119C7×#117C2312yes—
120C2×C7×C4⋊C4—21no—
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4C2312yes—
123C2×C42×C7C2311yescharacteristic
124C2×C42×C7C2312yes—
125#6514—216no—
126#6511—28no—
127#6507—22no—
128C2×#1144C2318yes—
129#6499—22no—
130#6498—24no—
131C2×#1142C2318yes—
132#6489—22no—
133#6489—22no—
134#6489C2312yes—
135C2×C4⋊Dic7—21no—
136C2×C4⋊Dic7—21no—
137#6487—21no—
138#6487—21no—
139#6487—22no—
140#6487—22no—
141#6487C2314yes—
142#6487C2314yes—
143C2×#1136C2314yes—
144C2×#1136C2312yes—
145C22×C7⋊C8C2312yes—
146C22×C7⋊C8C2314yes—
147#6452—24no—
148#6452—24no—
149#6450—22no—
150#6450—22no—
151#6442—24no—
152#6442—24no—
153#6440—24no—
154#6433—22no—
155#6433—22no—
156#6433C2312yes—
157#6432—21no—
158#6432—21no—
159#6432C2312yes—
160#6432C2312yes—
161#6432C2311yescharacteristic
162#6432C2311yescharacteristic
163C4×Dic14—22no—
164C4×Dic14—22no—
165C4×Dic14—21no—
166C4×Dic14—21no—
167C4×Dic14C2314yes—
168C4×Dic14C2314yes—
169#6409C2314yes—
170#6409C2314yes—
171#6406C2314yes—
172#6406C2314yes—
173#6384C2318yes—
174#6384C2318yes—
175#6382C2318yes—
176#6381C2318yes—
177#6377C2314yes—
178#6377C2312yes—
179C4×C7⋊C8C2312yes—
180C4×C7⋊C8C2311yescharacteristic
181C4×C7⋊C8C2311yescharacteristic
182C4×C7⋊C8C2312yes—
183C4×C7⋊C8C2314yes—
184#3511D711yescharacteristic
185#3509—71no—
186#3185—72no—
187#3182—72no—
188#3178—78no—
189#3170—72no—
190#3167—72no—
191#3165—71no—
192#3152—71no—
193#3131—71no—
194C2×#487—71no—
195#3063—74no—
196#2947—74no—
197C4×#131—71no—
198C7×#41—24no—
199C7×#41—28no—
200C7×#41—84no—
201C2×C7×Q8—22no—
202C2×C7×Q8—22no—
203C2×C7×Q8—82no—
204C2×C7×Q8C2×D411yescharacteristic
205C2×C7×Q8C2412yes—
206C2×C7×D4—22no—
207C2×C7×D4—24no—
208C2×C7×D4—21no—
209C2×C7×D4—21no—
210C2×C7×D4C2×D412yes—
211C2×C7×D4C2×D411yescharacteristic
212C2×C7×D4C2412yes—
213C22×C4×C7—22no—
214C22×C4×C7—22no—
215C22×C4×C7—24no—
216C22×C4×C7C2×D411yescharacteristic
217C22×C4×C7C2×D411yescharacteristic
218C22×C4×C7C2×D412yes—
219C22×C4×C7C2×D412yes—
220C22×C4×C7C2412yes—
221C22×C4×C7C2411yescharacteristic
222C22×Dic7—22no—
223C2×Dic14—28no—
224C2×Dic14—21no—
225C2×Dic14—21no—
226C2×Dic14—22no—
227C2×Dic14C2414yes—
228C7×C4⋊C4—22no—
229C7×C4⋊C4—22no—
230C7×C4⋊C4—24no—
231C7×C4⋊C4—22no—
232C7×C4⋊C4—21no—
233C7×C4⋊C4—21no—
234C7×C4⋊C4—24no—
235C7×C4⋊C4C2×D411yescharacteristic
236C7×C4⋊C4C2×D411yescharacteristic
237C7×C4⋊C4C2×D412yes—
238C7×C4⋊C4C2×D412yes—
239C7×C4⋊C4C2414yes—
240C7×C22⋊C4—24no—
241C7×C22⋊C4—24no—
242C7×C22⋊C4—24no—
243C7×C22⋊C4—22no—
244C7×C22⋊C4—22no—
245C7×C22⋊C4C2×D412yes—
246C7×C22⋊C4C2×D412yes—
247C7×C22⋊C4C2×D412yes—
248C42×C7—24no—
249C42×C7C2×D411yescharacteristic
250C42×C7C2×D411yescharacteristic
251C42×C7C2×D412yes—
252C42×C7C2×D412yes—
253C42×C7C2412yes—
254C42×C7C2411yescharacteristic
255C42×C7C2411yescharacteristic
256C22⋊Dic7—22no—
257C22⋊Dic7—24no—
258C22⋊Dic7—24no—
259#1144—216no—
260#1142—216no—
261C4⋊Dic7—22no—
262C4⋊Dic7—22no—
263C4⋊Dic7—21no—
264C4⋊Dic7—21no—
265C4⋊Dic7—22no—
266C4⋊Dic7—22no—
267C4⋊Dic7C2412yes—
268C4⋊Dic7C2412yes—
269Dic7⋊C4—24no—
270Dic7⋊C4—24no—
271Dic7⋊C4—22no—
272Dic7⋊C4—22no—
273Dic7⋊C4—24no—
274Dic7⋊C4—24no—
275C4×Dic7—22no—
276C4×Dic7—24no—
277C4×Dic7—24no—
278#1136—28no—
279#1136#4118yescomplemented
280C2×C7⋊C8—24no—
281C2×C7⋊C8—28no—
282C2×C7⋊C8C2412yes—
283C2×C7⋊C8C2412yes—
284C2×C7⋊C8C2414yes—
285C2×#137—74no—
286C2×#131—72no—
287#557—141no—
288#557—141no—
289#556—21no—
290#556—141no—
291#556—141no—
292#555—21no—
293#554—21no—
294#553—22no—
295#552—142no—
296#552—142no—
297#548—22no—
298#547—21no—
299#542C2×D714yes—
300#541—74no—
301#537C2×D712yes—
302#536—72no—
303#533—71no—
304#533—72no—
305#533—72no—
306#533C2×D712yes—
307#532C2×D711yescharacteristic
308#532C2×D712yes—
309C2×C4⋊Q8—71no—
310C2×C4⋊Q8C2×D711yescharacteristic
311#520C2×D712yes—
312#519—72no—
313C4×#41C2×D711yescharacteristic
314#501—74no—
315#500—72no—
316#498—72no—
317#497—74no—
318#494—72no—
319#492—72no—
320#488—74no—
321#487—72no—
322#487—74no—
323#487—72no—
324#487—71no—
325#487—71no—
326#486—71no—
327#486—71no—
328#470—78no—
329#468—78no—
330#467—78no—
331#465—78no—
332#441—78no—
333#440—78no—
334C4×Q16—78no—
335C4×SD16—78no—
336#428—74no—
337#416—74no—
338C2×#103—74no—
339#405—74no—
340#405—72no—
341C4×M16—71no—
342C4×M16—72no—
343C2×C4×C8—71no—
344C2×C4×C8—72no—
345C23×C7—22no—
346C23×C7C22×D412yes—
347C23×C7C22×D411yescharacteristic
348C7×Q8—22no—
349C7×Q8—24no—
350C7×Q8—44no—
351C7×Q8—82no—
352C7×D4—24no—
353C7×D4—22no—
354C7×D4—24no—
355C7×D4—44no—
356C7×D4—42no—
357C7×D4—42no—
358C7×D4—84no—
359C2×C4×C7—22no—
360C2×C4×C7—24no—
361C2×C4×C7—22no—
362C2×C4×C7—24no—
363C2×C4×C7—24no—
364C2×C4×C7—22no—
365C2×C4×C7—24no—
366C2×C4×C7—44no—
367C2×C4×C7—84no—
368C2×C4×C7—82no—
369C2×C4×C7C22×D412yes—
370C2×C4×C7C22×D412yes—
371C2×C4×C7C22×D412yes—
372C2×C4×C7C22×D411yescharacteristic
373C2×C4×C7C22×D411yescharacteristic
374C2×C4×C7C22×D411yescharacteristic
375C2×C4×C7C22×D411yescharacteristic
376C2×C4×C7C22×D411yescharacteristic
377C2×C4×C7C2×#4114yes—
378C2×C4×C7C2511yescharacteristic
379C2×Dic7—22no—
380C2×Dic7—22no—
381C2×Dic7—24no—
382C2×Dic7—44no—
383Dic14—28no—
384Dic14—44no—
385Dic14—42no—
386Dic14—42no—
387C7⋊C8—22no—
388C7⋊C8—22no—
389C7⋊C8C2×#4118yes—
390C2×#41—22no—
391C2×#41C22×D712yes—
392C2×#41C22×D711yescharacteristic
393C22×Q8—21no—
394C22×Q8—72no—
395#137—1416no—
396C2×Q16—78no—
397C2×SD16—78no—
398#131—148no—
399C2×M16—72no—
400C2×M16—74no—
401C22×C8—72no—
402C22×C8—74no—
403C4⋊Q8—21no—
404C4⋊Q8—21no—
405C4⋊Q8—71no—
406C4⋊Q8—71no—
407C4⋊Q8—72no—
408C4⋊Q8—72no—
409C4⋊Q8—142no—
410C4⋊Q8—141no—
411C4⋊Q8—141no—
412C4⋊Q8C22×D711yescharacteristic
413C4⋊Q8C22×D712yes—
414C4⋊Q8C22×D712yes—
415#127C22×D711yescharacteristic
416#126—24no—
417#126—144no—
418#126—144no—
419#125—22no—
420#125—72no—
421#125—144no—
422#125—144no—
423#125—142no—
424#125—142no—
425#124—24no—
426#124—22no—
427#124—142no—
428#124C22×D712yes—
429#123—24no—
430#123—144no—
431C22⋊Q8—22no—
432C22⋊Q8—22no—
433C22⋊Q8—24no—
434C22⋊Q8—74no—
435C22⋊Q8—74no—
436C22⋊Q8—141no—
437C22⋊Q8—141no—
438C22⋊Q8—142no—
439C22⋊Q8—142no—
440C22⋊Q8C22×D714yes—
441#121—22no—
442#121—24no—
443#121—21no—
444#121—21no—
445#121—22no—
446#121C22×D714yes—
447C4×Q8—22no—
448C4×Q8—74no—
449C4×Q8—74no—
450C4×Q8—142no—
451C4×Q8—142no—
452C4×Q8—142no—
453C4×Q8—142no—
454C4×Q8—141no—
455C4×Q8—141no—
456C4×Q8C22⋊D711yescharacteristic
457C4×Q8C22⋊D711yescharacteristic
458C4×Q8C22×D714yes—
459C4×D4—24no—
460C4×D4—21no—
461C4×D4—22no—
462C4×D4—21no—
463C4×D4C22⋊D711yescharacteristic
464C4×D4C22⋊D712yes—
465C4×D4C22⋊D712yes—
466C4×D4C22⋊D711yescharacteristic
467C4×D4C22×D714yes—
468#117—22no—
469#117—22no—
470#117—72no—
471#117—142no—
472#117—142no—
473#117—144no—
474#117C22×D711yescharacteristic
475#117C22×D712yes—
476C2×C4⋊C4—21no—
477C2×C4⋊C4—22no—
478C2×C4⋊C4—141no—
479C2×C4⋊C4—141no—
480C2×C4⋊C4C22×D712yes—
481C2×C42C22×D711yescharacteristic
482C2×C42C22×D712yes—
483#107—78no—
484#106—78no—
485#103—74no—
486#103—74no—
487#103—78no—
488#103—78no—
489#102—74no—
490#102—74no—
491#97—74no—
492#97—72no—
493C4×C8—71no—
494C4×C8—71no—
495C4×C8—72no—
496C4×C8—74no—
497C4×C8—72no—
498C22×C7—21no—
499C22×C7—21no—
500C22×C7—22no—
501C22×C7—22no—
502C22×C7—22no—
503C22×C7—42no—
504C22×C7—42no—
505C22×C7—42no—
506C22×C7#53212yes—
507C22×C7C23×D411yescharacteristic
508C4×C7—22no—
509C4×C7—22no—
510C4×C7—21no—
511C4×C7—22no—
512C4×C7—21no—
513C4×C7—42no—
514C4×C7—44no—
515C4×C7#53212yes—
516C4×C7#57712yes—
517C4×C7#57812yes—
518C4×C7C22×#4112yes—
519Dic7—44no—
520Dic7—42no—
521Dic7—42no—
522#41—24no—
523#41—28no—
524#41—84no—
525C2×Q8—22no—
526C2×Q8—22no—
527C2×Q8—74no—
528C2×Q8—82no—
529C2×Q8—141no—
530C2×Q8—141no—
531C2×Q8—142no—
532C2×Q8—148no—
533C2×Q8C2×C22⋊D711yescharacteristic
534C2×Q8C23×D712yes—
535C2×D4—22no—
536C2×D4—24no—
537C2×D4—21no—
538C2×D4—21no—
539C2×D4C2×C22⋊D712yes—
540C2×D4C2×C22⋊D711yescharacteristic
541C2×D4C23×D712yes—
542C22×C4—22no—
543C22×C4—22no—
544C22×C4—24no—
545C22×C4—142no—
546C22×C4C2×C22⋊D711yescharacteristic
547C22×C4C2×C22⋊D711yescharacteristic
548C22×C4C2×C22⋊D712yes—
549C22×C4C2×C22⋊D712yes—
550C22×C4C23×D712yes—
551C22×C4C23×D711yescharacteristic
552Q16—1416no—
553SD16—1416no—
554M16—78no—
555M16—148no—
556C2×C8—74no—
557C2×C8—72no—
558C2×C8—72no—
559C2×C8—148no—
560C2×C8—144no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—24no—
564C4⋊C4—22no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—24no—
568C4⋊C4—72no—
569C4⋊C4—72no—
570C4⋊C4—144no—
571C4⋊C4—144no—
572C4⋊C4—144no—
573C4⋊C4—144no—
574C4⋊C4—142no—
575C4⋊C4—142no—
576C4⋊C4—142no—
577C4⋊C4—142no—
578C4⋊C4—142no—
579C4⋊C4—142no—
580C4⋊C4—141no—
581C4⋊C4—141no—
582C4⋊C4C2×C22⋊D711yescharacteristic
583C4⋊C4C2×C22⋊D711yescharacteristic
584C4⋊C4C2×C22⋊D712yes—
585C4⋊C4C2×C22⋊D712yes—
586C4⋊C4C23×D714yes—
587C22⋊C4—24no—
588C22⋊C4—24no—
589C22⋊C4—24no—
590C22⋊C4—22no—
591C22⋊C4—22no—
592C22⋊C4—142no—
593C22⋊C4—144no—
594C22⋊C4—144no—
595C22⋊C4C2×C22⋊D712yes—
596C22⋊C4C2×C22⋊D712yes—
597C22⋊C4C2×C22⋊D712yes—
598C42—24no—
599C42—142no—
600C42—144no—
601C42—144no—
602C42C2×C22⋊D712yes—
603C42C2×C22⋊D711yescharacteristic
604C42C2×C22⋊D711yescharacteristic
605C42C2×C22⋊D712yes—
606C42C23×D711yescharacteristic
607C42C23×D712yes—
608C42C23×D711yescharacteristic
609C2×C7—22no—
610C2×C7—42no—
611C2×C7—82no—
612C2×C7C2×#53211yescharacteristic
613C2×C7C2×#57711yescharacteristic
614C2×C7C2×#57811yescharacteristic
615C23—22no—
616C23C22×C22⋊D712yes—
617C23C22×C22⋊D711yescharacteristic
618Q8—22no—
619Q8—24no—
620Q8—44no—
621Q8—82no—
622Q8—148no—
623Q8—284no—
624Q8—282no—
625Q8—282no—
626D4—24no—
627D4—22no—
628D4—24no—
629D4—44no—
630D4—42no—
631D4—42no—
632D4—84no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—44no—
641C2×C4—84no—
642C2×C4—82no—
643C2×C4—142no—
644C2×C4—142no—
645C2×C4—144no—
646C2×C4—284no—
647C2×C4D7×#4114yes—
648C2×C4C22×C22⋊D712yes—
649C2×C4C22×C22⋊D712yes—
650C2×C4C22×C22⋊D712yes—
651C2×C4C22×C22⋊D711yescharacteristic
652C2×C4C22×C22⋊D711yescharacteristic
653C2×C4C22×C22⋊D711yescharacteristic
654C2×C4C22×C22⋊D711yescharacteristic
655C2×C4C22×C22⋊D711yescharacteristic
656C2×C4C24×D711yescharacteristic
657C8—78no—
658C8—142no—
659C8—142no—
660C7#6072311yescharacteristic, complemented, Sylow
661C22—21no—
662C22—21no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22#9052312yes—
670C22C23×C22⋊D711yescharacteristic
671C4—22no—
672C4—22no—
673C4—21no—
674C4—22no—
675C4—21no—
676C4—42no—
677C4—44no—
678C4—284no—
679C4—282no—
680C4—282no—
681C4#9052312yes—
682C4#9052712yes—
683C4#9052912yes—
684C4C2×D7×#4112yes—
685C2—22no—
686C2—42no—
687C2—82no—
688C2C2×#9052311yescharacteristic
689C2C2×#9052711yescharacteristic
690C2C2×#9052911yescharacteristic
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
Fitting, prime radicalC7×#3511C2
socleC22×C7C23×D4
hypercenter#3511D7
hypocenterC7#60723
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3511

Complements

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

NH ≅ G/Ntype
2. #11742898C2semidirect
3. #11740903C2semidirect
4. #11740868C2semidirect
5. #11740793C2semidirect
6. #11740789C2semidirect
7. #11740739C2semidirect
8. #11740735C2semidirect
9. #11740733C2semidirect
10. #11740726C2semidirect
11. #11740652C2semidirect
12. C2×#89865C2semidirect
13. #11740016C2semidirect
14. C4×#6511C2semidirect
279. #1136#41semidirect
660. C7#60723semidirect

Maximal subgroups

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

subgroupindex
C7×#35112
#117428982
#117409032
#117408682
#117407932
#117407892
#117407392
#117407352
#117407332
#117407262
#117406522
C2×#898652
#117400162
C4×#65112
#607237