Finite Group Databasebeta

← #13422273

Subgroups — #13422273

Order 1792.

736 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2492C211yesmaximal, characteristic
2#11730913C211yesmaximal, characteristic
3#11729548C211yesmaximal, characteristic, complemented
4#11729533C211yesmaximal, characteristic, complemented
5#11729454C211yesmaximal, characteristic
6#11729447C211yesmaximal, characteristic
7#11727444C211yesmaximal, characteristic, complemented
8#11727435C211yesmaximal, characteristic, complemented
9#11727429C211yesmaximal, characteristic, complemented
10#11727420C211yesmaximal, characteristic, complemented
11#11727280C211yesmaximal, characteristic
12#11727257C211yesmaximal, characteristic
13#11727229C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727206C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8C2211yescharacteristic
18C22×#1144C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#400—21no—
23C7×#393—21no—
24C7×#391—21no—
25C7×#391C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#388—21no—
28C7×#380C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89424—21no—
48#89421—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26750—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×C7×Q8C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C22×Dic14C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8—41no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—41no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4D411yescharacteristic
76C2×C7×C22⋊C4D411yescharacteristic
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×#1144—22no—
83C2×#1144—41no—
84C2×#1144—41no—
85#6489—22no—
86#6489—41no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90#6487—22no—
91#6487—22no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C4×Dic14—22no—
103C4×Dic14—41no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95D411yescharacteristic
111C7×#95D411yescharacteristic
112#6409—22no—
113#6409—22no—
114#6409—41no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C22⋊(C7⋊C8)—41no—
120C22⋊(C7⋊C8)—41no—
121#6384—22no—
122#6384—22no—
123#6384—41no—
124#6382—22no—
125C4⋊(C7⋊C8)—41no—
126#6377—41no—
127#2492D711yescharacteristic
128#2362—71no—
129#2056—71no—
130#2035—71no—
131#2024—71no—
132#2001—71no—
133#1987—71no—
134#1935—71no—
135#1927—71no—
136#1924—71no—
137#1920—71no—
138#1878—71no—
139#1873—71no—
140#1825—71no—
141#1763—71no—
142C24×C7C2×D411yescharacteristic
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—42no—
150C2×C7×Q8—42no—
151C2×C7×Q8—81no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14#4112yes—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4#4112yes—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—41no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—81no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—81no—
195C22⋊Dic7—24no—
196C22⋊Dic7—42no—
197#1144—44no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7#4112yes—
203C4⋊Dic7#4112yes—
204Dic7⋊C4—24no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C4×Dic7—42no—
208C4×Dic7—42no—
209C4×Dic7—42no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C22×Q16—71no—
223C2×C4⋊Q8C2×D711yescharacteristic
224C2×C22⋊Q8—71no—
225C2×C22⋊Q8C2×D711yescharacteristic
226C2×C4×Q8—71no—
227C2×#117—71no—
228C2×#106—71no—
229C2×C4⋊C8—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#103—71no—
235C2×#103—71no—
236C2×C22⋊C8—71no—
237C2×C22⋊C8—71no—
238C2×#97—71no—
239#400—21no—
240#393—21no—
241#391—21no—
242#391—71no—
243#391C2×D711yescharacteristic
244#391C2×D711yescharacteristic
245#389—71no—
246#388—21no—
247#388—141no—
248#388—141no—
249#387—141no—
250#380C2×D711yescharacteristic
251C4×C4⋊C4—71no—
252C4×C4⋊C4—141no—
253C4×C4⋊C4C2×D711yescharacteristic
254C4×C22⋊C4C2×D711yescharacteristic
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#336—71no—
264#336—71no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7C22×D411yescharacteristic
273C7×Q8—41no—
274C7×Q8—41no—
275C7×Q8—41no—
276C7×Q8—41no—
277C7×Q8—84no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7#13712yes—
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7H(5,𝔽2)11yescharacteristic
314C2×C4×C7H(5,𝔽2)11yescharacteristic
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327Dic14—22no—
328Dic14—22no—
329Dic14—41no—
330Dic14—41no—
331C7⋊C8—41no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C22×Q8—21no—
336C22×Q8—71no—
337C22×Q8C22×D711yescharacteristic
338C23×C4C22×D711yescharacteristic
339C2×Q16—142no—
340C2×Q16—281no—
341C2×Q16—281no—
342C22×C8—71no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8—71no—
346C4⋊Q8—22no—
347C4⋊Q8—41no—
348C22⋊Q8—22no—
349C22⋊Q8—41no—
350C22⋊Q8—142no—
351C22⋊Q8—142no—
352C4×Q8—142no—
353C4×Q8—281no—
354#117—142no—
355#117—281no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4C22⋊D711yescharacteristic
372C2×C22⋊C4C22⋊D711yescharacteristic
373C2×C42—21no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42—141no—
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379#106—142no—
380C4⋊C8—281no—
381#103—142no—
382#103—142no—
383#103—142no—
384#103—142no—
385#103—281no—
386#103—281no—
387C22⋊C8—281no—
388C22⋊C8—281no—
389#97—281no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—141no—
397#95—141no—
398#95—141no—
399#95—141no—
400#95C22⋊D711yescharacteristic
401#95C22⋊D711yescharacteristic
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7—41no—
407C22×C7—41no—
408C22×C7—41no—
409C22×C7#52511yescharacteristic
410C22×C7#54611yescharacteristic
411C22×C7#54611yescharacteristic
412C22×C7C2×#13711yescharacteristic
413C22×C7#57511yescharacteristic
414C22×C7#57711yescharacteristic
415C22×C7#57811yescharacteristic
416C4×C7—22no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7—41no—
422C4×C7—81no—
423C4×C7—81no—
424C4×C7—81no—
425C4×C7#52911yescharacteristic
426C4×C7#53811yescharacteristic
427C4×C7#55511yescharacteristic
428C4×C7#55911yescharacteristic
429Dic7—42no—
430Dic7—42no—
431Dic7—42no—
432Dic7—81no—
433C24C2×C22⋊D711yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—72no—
443C2×Q8—81no—
444C2×Q8—141no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—141no—
460C22×C4—141no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C2×C22⋊D711yescharacteristic
466C22×C4C2×C22⋊D711yescharacteristic
467C22×C4C2×C22⋊D711yescharacteristic
468C22×C4C23×D711yescharacteristic
469Q16—284no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—72no—
490C4⋊C4—72no—
491C4⋊C4—144no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4#115912yes—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—81no—
503C22⋊C4—144no—
504C22⋊C4—282no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—81no—
510C42—282no—
511C42—282no—
512C42—282no—
513C2×C7—81no—
514C2×C7#245911yescharacteristic
515C2×C7#311611yescharacteristic
516C2×C7#312111yescharacteristic
517C2×C7#331111yescharacteristic
518C2×C7#331211yescharacteristic
519C2×C7#331311yescharacteristic
520C2×C7#333111yescharacteristic
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23C22×C22⋊D711yescharacteristic
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—84no—
534Q8—142no—
535Q8—142no—
536Q8—281no—
537Q8—281no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4#650512yes—
581C2×C4C2×#115911yescharacteristic
582C2×C4#654811yescharacteristic
583C2×C4D7×#4111yescharacteristic
584C2×C4D7×#4111yescharacteristic
585C2×C4D7×#4111yescharacteristic
586C2×C4#655311yescharacteristic
587C8—281no—
588C8—281no—
589C8—281no—
590C8—281no—
591C7#2675011yescharacteristic, complemented, Sylow
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#9049711yescharacteristic
600C22C2×#650511yescharacteristic
601C22#9051311yescharacteristic
602C22#9051911yescharacteristic
603C22#9052011yescharacteristic
604C22#9052711yescharacteristic
605C22#9052911yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—282no—
616C4—282no—
617C4—282no—
618C4—561no—
619C4#9036111yescharacteristic
620C4#9036911yescharacteristic
621C4#9037111yescharacteristic
622C4#9037211yescharacteristic
623C2—81no—
624C2#1174044211yescharacteristic
625C2#1174079911yescharacteristic
626C2#1174091211yescharacteristic
627C2#1174094211yescharacteristic
628C2#1174099711yescharacteristic
629C2#1174108111yescharacteristic
630C2#1174111911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2492C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2492D7
hypocenterC7#26750
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
3. #11729548C2semidirect
4. #11729533C2semidirect
7. #11727444C2semidirect
8. #11727435C2semidirect
9. #11727429C2semidirect
10. #11727420C2semidirect
13. #11727229C2semidirect
14. #11727214C2semidirect
591. C7#26750semidirect

Maximal subgroups

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

subgroupindex
C7×#24922
#117309132
#117295482
#117295332
#117294542
#117294472
#117274442
#117274352
#117274292
#117274202
#117272802
#117272572
#117272292
#117272142
#117272062
#267507