Finite Group Databasebeta

← #13421853

Subgroups — #13421853

Order 1792.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2492C211yesmaximal, characteristic, complemented
2#11730912C211yesmaximal, characteristic
3#11729546C211yesmaximal, characteristic, complemented
4#11729533C211yesmaximal, characteristic, complemented
5#11729453C211yesmaximal, characteristic
6#11729446C211yesmaximal, characteristic, complemented
7#11727441C211yesmaximal, characteristic, complemented
8#11727434C211yesmaximal, characteristic, complemented
9#11727428C211yesmaximal, characteristic, complemented
10#11727420C211yesmaximal, characteristic, complemented
11#11727278C211yesmaximal, characteristic
12#11727255C211yesmaximal, characteristic, complemented
13C4×#6383C211yesmaximal, characteristic, complemented
14#11727213C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8C2211yescharacteristic
18C22×#1143C2211yescharacteristic
19C2×#6493C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C2×(C22⋊D7)⋊C4—21no—
22C2×C4×D28C2211yescharacteristic
23C7×#400—21no—
24C7×#393—21no—
25C7×#391—21no—
26C7×#391C2211yescharacteristic
27C7×#391C2211yescharacteristic
28C7×#388—21no—
29C7×#380C2211yescharacteristic
30C4×C7×C4⋊C4C2211yescharacteristic
31C4×C7×C22⋊C4C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×C22⋊(C7⋊C8)C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41#89730—21no—
42#89711C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89442—21no—
48#89440—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26722—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×C7×Q8C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C22×C22⋊D7—21no—
64C23×Dic7—21no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—41no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×#1143—22no—
84C2×#1143—41no—
85C2×#1143—41no—
86#6493—22no—
87#6493—22no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—41no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104(C22⋊D7)⋊C4—44no—
105C4×D28—22no—
106C4×D28—41no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95D411yescharacteristic
114C7×#95D411yescharacteristic
115#6409—22no—
116#6409—22no—
117#6409—41no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C22⋊(C7⋊C8)—41no—
121C22⋊(C7⋊C8)—41no—
122#6383—22no—
123#6383—22no—
124#6383—41no—
125#6382—22no—
126C4⋊(C7⋊C8)—41no—
127C4×C7⋊C8—41no—
128#2492D711yescharacteristic, complemented
129#2361—71no—
130#2055—71no—
131#2034—71no—
132#2023—71no—
133#2000—71no—
134#1987—71no—
135#1933—71no—
136#1927—71no—
137#1923—71no—
138#1919—71no—
139#1877—71no—
140#1872—71no—
141#1823—71no—
142C4×#102—71no—
143C24×C7C2×D411yescharacteristic
144C23×D7—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—42no—
152C2×C7×Q8—42no—
153C2×C7×Q8—81no—
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×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C2×C22⋊D7—24no—
171C2×C22⋊D7—42no—
172C2×C22⋊D7—42no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—42no—
177C22×Dic7—42no—
178C22×Dic7—81no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28—21no—
184C2×D28#4112yes—
185C2×C4×D7—42no—
186C2×C4×D7—42no—
187C2×C4×D7—81no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—42no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4#4112yes—
196C7×C22⋊C4—41no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—81no—
199C7×C22⋊C4SD1614yescomplemented
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7—81no—
205#1143—44no—
206(C2×D7)⋊C4—24no—
207(C2×D7)⋊C4—42no—
208(C2×D7)⋊C4—42no—
209(C2×D7)⋊C4—81no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—42no—
215C4⋊Dic7#4112yes—
216C4⋊Dic7#4112yes—
217Dic7⋊C4—42no—
218C4×Dic7—42no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C22×SD16—71no—
232C2×C4⋊Q8C2×D711yescharacteristic
233C2×C22⋊Q8C2×D711yescharacteristic
234C2×#121—71no—
235C2×C4×D4—71no—
236C2×C4×D4—141no—
237C22×C4⋊C4—71no—
238C2×#106—71no—
239C2×C4⋊C8—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#102—71no—
244C2×#102—71no—
245C2×#102—71no—
246C2×C22⋊C8—71no—
247C2×C22⋊C8—71no—
248C2×C4×C8—71no—
249#400—21no—
250#393—21no—
251#391—21no—
252#391C2×D711yescharacteristic
253#391C2×D711yescharacteristic
254#390—71no—
255#389—71no—
256#388—21no—
257#388—141no—
258#386—141no—
259#385—141no—
260#380C2×D711yescharacteristic
261C4×C4⋊C4—71no—
262C4×C4⋊C4C2×D711yescharacteristic
263C4×C22⋊C4C2×D711yescharacteristic
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#336—71no—
273#336—71no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7C2×SD1612yes—
281C23×C7C22×D411yescharacteristic
282C22×D7—22no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C7×Q8—41no—
290C7×Q8—41no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×Q8—84no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
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×C7C2×SD1612yes—
324C2×C4×C7C2×SD1612yes—
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7H(5,𝔽2)11yescharacteristic
330C2×C4×C7H(5,𝔽2)11yescharacteristic
331C22⋊D7—48no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—84no—
342D28—22no—
343D28—22no—
344D28—41no—
345D28—41no—
346C4×D7—84no—
347C7⋊C8—41no—
348C7⋊C8—41no—
349C7⋊C8—41no—
350C7⋊C8—41no—
351C22×Q8—21no—
352C22×Q8C22×D711yescharacteristic
353C22×D4—71no—
354C22×D4—141no—
355C23×C4—141no—
356C23×C4—141no—
357C23×C4C22×D711yescharacteristic
358C2×SD16—142no—
359C2×SD16—281no—
360C2×SD16—281no—
361C22×C8—71no—
362C22×C8—71no—
363C22×C8—71no—
364C22×C8—71no—
365C4⋊Q8—22no—
366C4⋊Q8—41no—
367C22⋊Q8—22no—
368C22⋊Q8—41no—
369#121—142no—
370#121—142no—
371C4×D4—142no—
372C4×D4—281no—
373C4×D4—284no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—142no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—281no—
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—141no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4C22⋊D711yescharacteristic
393C2×C22⋊C4C22⋊D711yescharacteristic
394C2×C42—21no—
395C2×C42—141no—
396C2×C42C22×D711yescharacteristic
397C2×C42C22×D711yescharacteristic
398#106—142no—
399C4⋊C8—281no—
400#103—142no—
401#103—142no—
402#103—281no—
403#102—142no—
404#102—142no—
405#102—281no—
406C22⋊C8—281no—
407C22⋊C8—281no—
408C4×C8—281no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—141no—
416#95—141no—
417#95C22⋊D711yescharacteristic
418#95C22⋊D711yescharacteristic
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—41no—
423C22×C7—41no—
424C22×C7—41no—
425C22×C7—41no—
426C22×C7#52511yescharacteristic
427C22×C7#54611yescharacteristic
428C22×C7#54611yescharacteristic
429C22×C7C22×SD1611yescharacteristic
430C22×C7#57511yescharacteristic
431C22×C7#57611yescharacteristic
432C22×C7#57811yescharacteristic
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—42no—
436C2×D7—81no—
437C2×D7—81no—
438C2×D7—81no—
439C2×D7—81no—
440C4×C7—22no—
441C4×C7—22no—
442C4×C7—41no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—81no—
447C4×C7—81no—
448C4×C7—81no—
449C4×C7#52911yescharacteristic
450C4×C7#53811yescharacteristic
451C4×C7#55511yescharacteristic
452C4×C7#55911yescharacteristic
453Dic7—42no—
454Dic7—42no—
455Dic7—81no—
456C24—141no—
457C24C2×C22⋊D711yescharacteristic
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—81no—
467C2×D4—72no—
468C2×D4—144no—
469C2×D4—141no—
470C2×D4—141no—
471C2×D4—141no—
472C2×D4—141no—
473C2×D4—141no—
474C2×D4—282no—
475C2×D4—282no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4—282no—
490C22×C4—282no—
491C22×C4—282no—
492C22×C4—282no—
493C22×C4—561no—
494C22×C4—561no—
495C22×C4C2×C22⋊D711yescharacteristic
496C22×C4C2×C22⋊D711yescharacteristic
497C22×C4C2×C22⋊D711yescharacteristic
498C22×C4C2×C22⋊D711yescharacteristic
499C22×C4C2×C22⋊D711yescharacteristic
500C22×C4C23×D711yescharacteristic
501SD16—284no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8—141no—
509C2×C8—141no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8—141no—
513C2×C8—141no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—72no—
522C4⋊C4—72no—
523C4⋊C4—141no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—282no—
528C4⋊C4—282no—
529C4⋊C4#115912yes—
530C22⋊C4—41no—
531C22⋊C4—42no—
532C22⋊C4—81no—
533C22⋊C4—144no—
534C22⋊C4—282no—
535C22⋊C4—282no—
536C22⋊C4—561no—
537C22⋊C4#114314yescomplemented
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—81no—
543C42—282no—
544C2×C7—42no—
545C2×C7#245911yescharacteristic
546C2×C7#310111yescharacteristic
547C2×C7#312411yescharacteristic
548C2×C7#329511yescharacteristic
549C2×C7#329811yescharacteristic
550C2×C7#331311yescharacteristic
551C2×C7#332211yescharacteristic
552D7—82no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—142no—
560C23—281no—
561C23—281no—
562C23—281no—
563C23—281no—
564C23—281no—
565C23—281no—
566C23C2×#114312yes—
567C23C22×C22⋊D711yescharacteristic
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—84no—
573D4—142no—
574D4—142no—
575D4—288no—
576D4—281no—
577D4—281no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—564no—
617C2×C4—564no—
618C2×C4C2×#114312yes—
619C2×C4C2×#114312yes—
620C2×C4C2×#115911yescharacteristic
621C2×C4#654811yescharacteristic
622C2×C4D7×#4111yescharacteristic
623C2×C4D7×#4111yescharacteristic
624C2×C4D7×#4111yescharacteristic
625C2×C4#655311yescharacteristic
626C8—281no—
627C8—281no—
628C8—281no—
629C8—281no—
630C7#2672211yescharacteristic, complemented, Sylow
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—282no—
639C22—282no—
640C22—282no—
641C22—561no—
642C22—561no—
643C22—561no—
644C22—561no—
645C22#9049711yescharacteristic
646C22C22×#114311yescharacteristic
647C22#9051311yescharacteristic
648C22#9051911yescharacteristic
649C22#9052011yescharacteristic
650C22#9052611yescharacteristic
651C22#9052911yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—282no—
662C4—282no—
663C4—561no—
664C4#9036111yescharacteristic
665C4#9036911yescharacteristic
666C4#9037111yescharacteristic
667C4#9037211yescharacteristic
668C2—42no—
669C2—562no—
670C2#1174044211yescharacteristic
671C2#1174079611yescharacteristic
672C2#1174090611yescharacteristic
673C2#1174094511yescharacteristic
674C2#1174099511yescharacteristic
675C2#1174107911yescharacteristic
676C2#1174111911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2492C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2492D7
hypocenterC7#26722
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
1. C7×#2492C2semidirect
3. #11729546C2semidirect
4. #11729533C2semidirect
6. #11729446C2semidirect
7. #11727441C2semidirect
8. #11727434C2semidirect
9. #11727428C2semidirect
10. #11727420C2semidirect
12. #11727255C2semidirect
13. C4×#6383C2semidirect
14. #11727213C2semidirect
15. #11727205C2semidirect
128. #2492D7semidirect
199. C7×C22⋊C4SD16semidirect
537. C22⋊C4#1143semidirect
630. C7#26722semidirect

Maximal subgroups

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

subgroupindex
C7×#24922
#117309122
#117295462
#117295332
#117294532
#117294462
#117274412
#117274342
#117274282
#117274202
#117272782
#117272552
C4×#63832
#117272132
#117272052
#267227