Finite Group Databasebeta

← #13373274

Subgroups — #13373274

Order 1792.

740 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11736457C211yesmaximal, characteristic, complemented
2#11736445C211yesmaximal, characteristic, complemented
3#11735939C211yesmaximal, characteristic
4C7×#1771C211yesmaximal, characteristic, complemented
5#11727597C211yesmaximal, characteristic, complemented
6#11727595C211yesmaximal, characteristic, complemented
7#11727593C211yesmaximal, characteristic, complemented
8#11727547C211yesmaximal, characteristic, complemented
9#11727543C211yesmaximal, characteristic, complemented
10#11727248C211yesmaximal, characteristic, complemented
11#11727248C211yesmaximal, characteristic, complemented
12#11726449C211yesmaximal, characteristic
13#11726449C211yesmaximal, characteristic
14#11726399C211yesmaximal, characteristic, complemented
15#11726399C211yesmaximal, characteristic, complemented
16C2×#6467C2211yescharacteristic
17C2×#6464C2211yescharacteristic
18C2×#6464C2211yescharacteristic
19C4×#1157C2211yescharacteristic
20C4×#1157C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26#89885C2211yescharacteristic
27#89885C2211yescharacteristic
28#89882C2211yescharacteristic
29#89882C2211yescharacteristic
30#89878C2211yescharacteristic
31#89874C2211yescharacteristic
32#89874C2211yescharacteristic
33#89873C2211yescharacteristic
34#89873C2211yescharacteristic
35C2×#6379C2211yescharacteristic
36C2×#6379C2211yescharacteristic
37C2×#6379C2211yescharacteristic
38C2×#6379C2211yescharacteristic
39#89696C2211yescharacteristic
40#89696C2211yescharacteristic
41#89612—21no—
42#89612—21no—
43#89611—21no—
44#89611—21no—
45#89609—21no—
46#89607—21no—
47#89605—21no—
48D7×C4⋊C8—21no—
49#89488C411yescharacteristic, complemented
50#89488C411yescharacteristic, complemented
51#89483C411yescharacteristic, complemented
52#89483C411yescharacteristic, complemented
53#89482C411yescharacteristic, complemented
54#89482C411yescharacteristic, complemented
55#89482C411yescharacteristic, complemented
56#89482C411yescharacteristic, complemented
57C7×#336C2211yescharacteristic
58C7×#336C2211yescharacteristic
59#89347C2211yescharacteristic
60#89347C2211yescharacteristic
61#89347C2211yescharacteristic
62#89347C2211yescharacteristic
63#89318C2211yescharacteristic
64#89318C2211yescharacteristic
65#89318C2211yescharacteristic
66#89318C2211yescharacteristic
67#22784—71nomaximal, Sylow
68C2×#1157C2311yescharacteristic
69C2×#1157C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C2×C7×C4⋊C4—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C4×C22⋊D7—21no—
79C4×C22⋊D7—21no—
80C4×C22⋊D7—21no—
81C4×C22⋊D7—21no—
82#6489C2311yescharacteristic
83#6489C2311yescharacteristic
84C2×#1136C2311yescharacteristic
85C2×#1136C2311yescharacteristic
86C2×#1136C2311yescharacteristic
87C2×#1136C2311yescharacteristic
88C2×Dic28C2×C411yescharacteristic
89#6467—22no—
90#6467—22no—
91C2×D56C2×C411yescharacteristic
92C2×#1132C2×C411yescharacteristic
93C2×#1132C2×C411yescharacteristic
94#6464—22no—
95#6464—22no—
96#6464—41no—
97#6464—41no—
98C2×#1131—21no—
99C2×#1131—21no—
100C2×C8×D7—21no—
101C2×C8×D7—21no—
102C4×D28C2×C411yescharacteristic
103C4×D28C2×C411yescharacteristic
104C4×D28C2×C411yescharacteristic
105C4×D28C2×C411yescharacteristic
106#6435—21no—
107#6435—21no—
108C42×D7—21no—
109C42×D7—21no—
110C4×Dic14C2×C411yescharacteristic
111C4×Dic14C2×C411yescharacteristic
112C4×Dic14C2×C411yescharacteristic
113C4×Dic14C2×C411yescharacteristic
114C7×C4⋊C8D411yescharacteristic
115C7×C4⋊C8D411yescharacteristic
116C7×C4⋊C8D411yescharacteristic
117C7×C4⋊C8D411yescharacteristic
118C7×#97—41no—
119C7×#97C2×C411yescharacteristic
120C7×#97C2×C411yescharacteristic
121C7×#97C2×C411yescharacteristic
122C7×#97C2×C411yescharacteristic
123C4×C7×C8—41no—
124C7×#95—21no—
125#6395C2×C411yescharacteristic
126#6395C2×C411yescharacteristic
127#6395C2×C411yescharacteristic
128#6395C2×C411yescharacteristic
129(C2×D7)⋊C8—21no—
130(C2×D7)⋊C8—21no—
131(C2×D7)⋊C8—21no—
132(C2×D7)⋊C8—21no—
133#6392C2×C411yescharacteristic
134#6392C2×C411yescharacteristic
135#6391C2×C411yescharacteristic
136#6391C2×C411yescharacteristic
137#6390C2×C411yescharacteristic
138#6390C2×C411yescharacteristic
139#6390C2×C411yescharacteristic
140#6390C2×C411yescharacteristic
141#6389—21no—
142#6389—21no—
143Dic7⋊C8—21no—
144Dic7⋊C8—21no—
145Dic7⋊C8—21no—
146Dic7⋊C8—21no—
147C8×Dic7—21no—
148C8×Dic7—21no—
149#6379—22no—
150#6379—22no—
151#6379—41no—
152#6379—41no—
153C4⋊(C7⋊C8)—21no—
154C4⋊(C7⋊C8)—21no—
155C4⋊(C7⋊C8)—21no—
156C4⋊(C7⋊C8)—21no—
157#2943—71no—
158#2925—71no—
159#2906—71no—
160#1938—71no—
161#1937—71no—
162#1922—71no—
163#1922—71no—
164#1876—71no—
165#1876—71no—
166#1838—71no—
167#1806—71no—
168#1806—71no—
169#1771D711yescharacteristic, complemented
170#1765—71no—
171#1764—71no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7#4111yescharacteristic
175C22×C4×C7#4111yescharacteristic
176C22×C4×C7C2411yescharacteristic
177C2×C22⋊D7—21no—
178C2×C22⋊D7—21no—
179#1157—21no—
180#1157—21no—
181#1157—21no—
182#1157—21no—
183#1157—21no—
184#1157—21no—
185#1157#4112yes—
186#1157#4112yes—
187C2×D28C22×C411yescharacteristic
188C2×D28C22×C411yescharacteristic
189C2×C4×D7—21no—
190C2×C4×D7—21no—
191C2×C4×D7—21no—
192C2×C4×D7—21no—
193C2×C4×D7—21no—
194C2×C4×D7—21no—
195C2×Dic14C22×C411yescharacteristic
196C2×Dic14C22×C411yescharacteristic
197C2×C7×C8—21no—
198C2×C7×C8—21no—
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8—21no—
202C2×C7×C8—21no—
203C2×C7×C8—21no—
204C2×C7×C8—21no—
205C2×C7×C8C22×C411yescharacteristic
206C2×C7×C8C22×C411yescharacteristic
207C2×C7×C8C22×C411yescharacteristic
208C2×C7×C8C22×C411yescharacteristic
209C2×C7×C8C2×D411yescharacteristic
210C2×C7×C8C2×D411yescharacteristic
211C2×C7×C8C2×D411yescharacteristic
212C2×C7×C8C2×D411yescharacteristic
213C7×C4⋊C4—42no—
214C42×C7—21no—
215C42×C7—21no—
216C42×C7—21no—
217C42×C7—21no—
218C42×C7C22×C411yescharacteristic
219C42×C7C22×C411yescharacteristic
220C42×C7C2×D411yescharacteristic
221C42×C7C2×D411yescharacteristic
222C22⋊Dic7—21no—
223C22⋊Dic7—21no—
224(C2×D7)⋊C4—21no—
225(C2×D7)⋊C4—21no—
226(C2×D7)⋊C4—21no—
227(C2×D7)⋊C4—21no—
228C4⋊Dic7C22×C411yescharacteristic
229C4⋊Dic7C22×C411yescharacteristic
230C4⋊Dic7C22×C411yescharacteristic
231C4⋊Dic7C22×C411yescharacteristic
232Dic7⋊C4—21no—
233Dic7⋊C4—21no—
234Dic7⋊C4—21no—
235Dic7⋊C4—21no—
236C4×Dic7—21no—
237C4×Dic7—21no—
238C4×Dic7—21no—
239C4×Dic7—21no—
240C4×Dic7—21no—
241C4×Dic7—21no—
242#1136—21no—
243#1136—21no—
244#1136—21no—
245#1136—21no—
246#1136—21no—
247#1136—21no—
248#1136—21no—
249#1136—21no—
250C2×C7⋊C8—21no—
251C2×C7⋊C8—21no—
252C2×C7⋊C8—21no—
253C2×C7⋊C8—21no—
254Dic28—22no—
255D56—22no—
256#1132—22no—
257#1132—22no—
258#1131—42no—
259#1131—42no—
260C8×D7—42no—
261C8×D7—42no—
262C2×#135—71no—
263C2×#131—71no—
264C2×#131—71no—
265C4×#41—71no—
266C4×#41—71no—
267#442—71no—
268#442—71no—
269#441—71no—
270#441—71no—
271#440—71no—
272#440—71no—
273#440—71no—
274#440—71no—
275#433—141no—
276#433—141no—
277#432—141no—
278#431—141no—
279#427—71no—
280#424—71no—
281#424—71no—
282#424—141no—
283#424—141no—
284#423—71no—
285#423—71no—
286#423—141no—
287C2×C4⋊C8—141no—
288C2×C4⋊C8C2×D711yescharacteristic
289C2×#104—71no—
290C2×#104—71no—
291C2×#104—71no—
292C2×#104—71no—
293#416—71no—
294#416—71no—
295#408—71no—
296#408—71no—
297#405—71no—
298#405—71no—
299C2×#97C2×D711yescharacteristic
300C2×#97C2×D711yescharacteristic
301C2×C4×C8C2×D711yescharacteristic
302C4×C4⋊C4C2×D711yescharacteristic
303#341—71no—
304#341—71no—
305#341—71no—
306#341—71no—
307#339—71no—
308#339—71no—
309#339—71no—
310#339—71no—
311#336C2×D711yescharacteristic
312#336C2×D711yescharacteristic
313C23×C7C2×#4111yescharacteristic
314C22×D7—21no—
315C22×D7—21no—
316C2×C4×C7—22no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7C4×D411yescharacteristic
325C2×C4×C7C4×D411yescharacteristic
326C2×C4×C7C4×D411yescharacteristic
327C2×C4×C7C4×D411yescharacteristic
328C2×C4×C7C23×C411yescharacteristic
329C2×C4×C7C22×D411yescharacteristic
330C2×C4×C7C2×#4111yescharacteristic
331C2×C4×C7C2×#4111yescharacteristic
332C2×C4×C7H(5,𝔽2)11yescharacteristic
333C2×C4×C72−1+411yescharacteristic
334C22⋊D7—22no—
335C22⋊D7—22no—
336C22⋊D7—41no—
337C22⋊D7—41no—
338C2×Dic7—21no—
339C2×Dic7—21no—
340C2×Dic7—21no—
341C2×Dic7—21no—
342C2×Dic7—21no—
343C2×Dic7—21no—
344D28—21no—
345D28—21no—
346D28—21no—
347D28#11712yes—
348C4×D7—22no—
349C4×D7—22no—
350C4×D7—42no—
351C4×D7—42no—
352C4×D7—41no—
353C4×D7—41no—
354C4×D7—81no—
355C4×D7—81no—
356Dic14—21no—
357Dic14—21no—
358Dic14—21no—
359Dic14#11712yes—
360C7×C8—21no—
361C7×C8—21no—
362C7×C8—21no—
363C7×C8—21no—
364C7×C8—21no—
365C7×C8—21no—
366C7×C8—21no—
367C7×C8—21no—
368C7⋊C8—41no—
369C7⋊C8—41no—
370C7⋊C8—41no—
371C7⋊C8—41no—
372C2×#41—71no—
373C2×#41—71no—
374#135—142no—
375#135—142no—
376C2×Q16—71no—
377C2×SD16—71no—
378C2×SD16—71no—
379C2×D8—71no—
380#131—142no—
381#131—142no—
382#131—281no—
383#131—281no—
384C2×M16—71no—
385C2×M16—71no—
386C2×M16—71no—
387C2×M16—71no—
388C2×M16—141no—
389C2×M16—141no—
390C22×C8—141no—
391C22×C8—141no—
392C22×C8C22×D711yescharacteristic
393C22×C8C22×D711yescharacteristic
394C22×C8C22×D711yescharacteristic
395C22×C8C22×D711yescharacteristic
396C4×Q8—71no—
397C4×Q8—71no—
398C4×Q8—71no—
399C4×Q8—71no—
400C4×D4—71no—
401C4×D4—71no—
402C4×D4—71no—
403C4×D4—71no—
404C4×D4—141no—
405C4×D4—141no—
406C4×D4—141no—
407C4×D4—141no—
408#117—71no—
409#117—71no—
410#117—141no—
411#117—141no—
412C2×C4⋊C4—21no—
413C2×C42—141no—
414C2×C42—141no—
415C2×C42C22×D711yescharacteristic
416C2×C42C22×D711yescharacteristic
417C2×C42C22×D711yescharacteristic
418#107—71no—
419#107—71no—
420#106—71no—
421#106—71no—
422C4⋊C8—141no—
423C4⋊C8—141no—
424C4⋊C8—141no—
425C4⋊C8—141no—
426C4⋊C8—141no—
427C4⋊C8—141no—
428C4⋊C8—141no—
429C4⋊C8—141no—
430C4⋊C8D2811yescharacteristic
431C4⋊C8D2811yescharacteristic
432C4⋊C8D2811yescharacteristic
433C4⋊C8D2811yescharacteristic
434#104—142no—
435#104—142no—
436#104—281no—
437#104—281no—
438#103—71no—
439#103—71no—
440#103—71no—
441#103—71no—
442#102—71no—
443#102—71no—
444#102—71no—
445#102—71no—
446C22⋊C8—141no—
447C22⋊C8—141no—
448C22⋊C8—141no—
449C22⋊C8—141no—
450#97—41no—
451#97—141no—
452#97—141no—
453#97C4×D711yescharacteristic
454#97C4×D711yescharacteristic
455#97C4×D711yescharacteristic
456#97C4×D711yescharacteristic
457C4×C8—41no—
458C4×C8—141no—
459C4×C8—141no—
460#95—21no—
461C22×C7—21no—
462C22×C7—21no—
463C22×C7C2×C4×D411yescharacteristic
464C22×C7#55211yescharacteristic
465C22×C7#55311yescharacteristic
466C2×D7—22no—
467C2×D7—41no—
468C2×D7—41no—
469C2×D7—41no—
470C2×D7—41no—
471C2×D7—41no—
472C4×C7—21no—
473C4×C7—21no—
474C4×C7—21no—
475C4×C7—21no—
476C4×C7—42no—
477C4×C7—81no—
478C4×C7C2×#11711yescharacteristic
479C4×C7#52011yescharacteristic
480C4×C7#54611yescharacteristic
481C4×C7#54811yescharacteristic
482Dic7—22no—
483Dic7—41no—
484Dic7—41no—
485Dic7—41no—
486Dic7—41no—
487Dic7—41no—
488#41—72no—
489#41—72no—
490#41—141no—
491#41—141no—
492#41—141no—
493#41—141no—
494#41—141no—
495#41—141no—
496C2×Q8—71no—
497C2×Q8—71no—
498C2×D4—71no—
499C2×D4—71no—
500C2×D4—141no—
501C2×D4—141no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—141no—
505C22×C4—141no—
506C22×C4—141no—
507C22×C4—141no—
508C22×C4—141no—
509C22×C4—141no—
510C22×C4#115711yescharacteristic
511C22×C4#115711yescharacteristic
512C22×C4C23×D711yescharacteristic
513Q16—142no—
514SD16—142no—
515SD16—142no—
516D8—142no—
517M16—141no—
518M16—141no—
519M16—141no—
520M16—141no—
521M16—141no—
522M16—141no—
523M16—141no—
524M16—141no—
525M16—282no—
526M16—282no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—141no—
536C2×C8—141no—
537C2×C8—141no—
538C2×C8—141no—
539C2×C8—282no—
540C2×C8—282no—
541C2×C8C2×C4×D711yescharacteristic
542C2×C8C2×C4×D711yescharacteristic
543C2×C8C2×C4×D711yescharacteristic
544C2×C8C2×C4×D711yescharacteristic
545C2×C8C2×D2811yescharacteristic
546C2×C8C2×D2811yescharacteristic
547C2×C8C2×D2811yescharacteristic
548C2×C8C2×D2811yescharacteristic
549C4⋊C4—42no—
550C4⋊C4—71no—
551C4⋊C4—71no—
552C4⋊C4—71no—
553C4⋊C4—71no—
554C4⋊C4—141no—
555C4⋊C4—141no—
556C4⋊C4—141no—
557C4⋊C4—141no—
558C22⋊C4—141no—
559C22⋊C4—141no—
560C22⋊C4—141no—
561C22⋊C4—141no—
562C22⋊C4—141no—
563C22⋊C4—141no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—141no—
569C42—141no—
570C42—141no—
571C42—141no—
572C42—141no—
573C42—141no—
574C42C2×C4×D711yescharacteristic
575C42C2×C4×D711yescharacteristic
576C42C2×D2811yescharacteristic
577C42C2×D2811yescharacteristic
578C2×C7—21no—
579C2×C7—21no—
580C2×C7#235211yescharacteristic
581C2×C7#295511yescharacteristic
582C2×C7#295611yescharacteristic
583D7—42no—
584D7—81no—
585C23—141no—
586C23—141no—
587C23C2×#115711yescharacteristic
588Q8—72no—
589Q8—141no—
590Q8—141no—
591Q8—141no—
592D4—72no—
593D4—142no—
594D4—142no—
595D4—141no—
596D4—141no—
597D4—141no—
598D4—281no—
599D4—281no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—141no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—141no—
612C2×C4—141no—
613C2×C4—141no—
614C2×C4—141no—
615C2×C4—141no—
616C2×C4—282no—
617C2×C4—282no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4—561no—
621C2×C4—561no—
622C2×C4C4×D2811yescharacteristic
623C2×C4C4×D2811yescharacteristic
624C2×C4C4×D2811yescharacteristic
625C2×C4C4×D2811yescharacteristic
626C2×C4C22×C4×D711yescharacteristic
627C2×C4C22×D2811yescharacteristic
628C2×C4#654811yescharacteristic
629C2×C4#655111yescharacteristic
630C2×C4D7×#4111yescharacteristic
631C2×C4D7×#4111yescharacteristic
632C8—21no—
633C8—21no—
634C8—21no—
635C8—21no—
636C8—21no—
637C8—21no—
638C8—21no—
639C8—21no—
640C8—281no—
641C8—281no—
642C8—281no—
643C8—281no—
644C7#2278411yescharacteristic, complemented, Sylow
645C22—21no—
646C22—21no—
647C22—142no—
648C22—281no—
649C22—281no—
650C22—281no—
651C22—281no—
652C22—281no—
653C22C2×C4×D2811yescharacteristic
654C22#9024911yescharacteristic
655C22#9027211yescharacteristic
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—42no—
661C4—81no—
662C4—142no—
663C4—281no—
664C4—281no—
665C4—281no—
666C4—281no—
667C4—281no—
668C4#9019511yescharacteristic
669C4D7×#11711yescharacteristic
670C4#9024511yescharacteristic
671C4#9026711yescharacteristic
672C2—21no—
673C2—21no—
674C2—282no—
675C2—561no—
676C2#1173121611yescharacteristic
677C2#1173598611yescharacteristic
678C2#1173602411yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1771C2
socleC22×C7C2×C4×D4
prime radicalC2×#6467C22
cosocleC22×C4×C7C24
hypercenter#1771D7
hypocenterC7#22784
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #1771

Complements

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

NH ≅ G/Ntype
1. #11736457C2semidirect
2. #11736445C2semidirect
4. C7×#1771C2semidirect
5. #11727597C2semidirect
6. #11727595C2semidirect
7. #11727593C2semidirect
8. #11727547C2semidirect
9. #11727543C2semidirect
10. #11727248C2semidirect
11. #11727248C2semidirect
14. #11726399C2semidirect
15. #11726399C2semidirect
49. #89488C4semidirect
50. #89488C4semidirect
51. #89483C4semidirect
52. #89483C4semidirect
53. #89482C4semidirect
54. #89482C4semidirect
55. #89482C4semidirect
56. #89482C4semidirect
169. #1771D7semidirect
644. C7#22784semidirect

Maximal subgroups

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

subgroupindex
#117364572
#117364452
#117359392
C7×#17712
#117275972
#117275952
#117275932
#117275472
#117275432
#117272482
#117264492
#117263992
#227847