Finite Group Databasebeta

← #13421883

Subgroups — #13421883

Order 1792.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2491C211yesmaximal, characteristic
2#11730910C211yesmaximal, characteristic, complemented
3#11729548C211yesmaximal, characteristic, complemented
4#11729534C211yesmaximal, characteristic, complemented
5#11729439C211yesmaximal, characteristic
6#11729431C211yesmaximal, characteristic
7#11727414C211yesmaximal, characteristic, complemented
8#11727405C211yesmaximal, characteristic, complemented
9#11727398C211yesmaximal, characteristic, complemented
10#11727391C211yesmaximal, characteristic, complemented
11#11727280C211yesmaximal, characteristic, complemented
12#11727256C211yesmaximal, characteristic, complemented
13C4×#6384C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#124—21no—
18C2×C7×#121C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C2×#6487C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#394—21no—
24C7×#393—21no—
25C7×#390C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C7×#388—21no—
28C7×#380C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40#89718—21no—
41C22⋊C4×Dic7—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89427—21no—
48#89419—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26724—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×C7×D4—21no—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—21no—
65C22×Dic14C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—41no—
68C7×#124—44no—
69C7×#121—22no—
70C7×#121—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×#1142—22no—
88C2×#1142—41no—
89C2×#1142—41no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—41no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96#6487—22no—
97#6487—22no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C4×Dic14—22no—
108C4×Dic14—41no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95D411yescharacteristic
112C7×#95D411yescharacteristic
113#6406—22no—
114#6406—22no—
115#6406—41no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C22⋊(C7⋊C8)—41no—
121C22⋊(C7⋊C8)—41no—
122#6384—22no—
123#6384—22no—
124#6384—41no—
125#6382—22no—
126C4⋊(C7⋊C8)—41no—
127C4×C7⋊C8—41no—
128#2491D711yescharacteristic
129#2360—71no—
130#2055—71no—
131#2035—71no—
132#2024—71no—
133#2000—71no—
134#1988—71no—
135#1934—71no—
136#1926—71no—
137#1923—71no—
138#1920—71no—
139#1880—71no—
140#1874—71no—
141#1824—71no—
142C4×#103—71no—
143C24×C7—21no—
144C24×C7C2×D411yescharacteristic
145C2×C7×Q8—42no—
146C2×C7×Q8—42no—
147C2×C7×Q8—81no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—42no—
155C2×C7×D4—42no—
156C2×C7×D4—81no—
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×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—42no—
177C22×Dic7—42no—
178C22×Dic7—81no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14#4112yes—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4#4112yes—
192C7×C22⋊C4—41no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—42no—
196C7×C22⋊C4—42no—
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—42no—
205C22⋊Dic7—24no—
206#1142—44no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—42no—
212C4⋊Dic7#4112yes—
213C4⋊Dic7#4112yes—
214Dic7⋊C4—24no—
215Dic7⋊C4—42no—
216Dic7⋊C4—81no—
217C4×Dic7—42no—
218C4×Dic7—81no—
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×#124—21no—
234C2×C22⋊Q8—71no—
235C2×#121C2×D711yescharacteristic
236C2×C4×Q8—71no—
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#394—21no—
250#393—21no—
251#391—71no—
252#390C2×D711yescharacteristic
253#390C2×D711yescharacteristic
254#389—71no—
255#389—141no—
256#388—21no—
257#387—141no—
258#385—141no—
259#380C2×D711yescharacteristic
260C4×C4⋊C4—71no—
261C4×C4⋊C4C2×D711yescharacteristic
262C4×C22⋊C4—141no—
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×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7C2×SD1612yes—
288C23×C7C22×D411yescharacteristic
289C7×Q8—84no—
290C7×D4—41no—
291C7×D4—41no—
292C7×D4—41no—
293C7×D4—41no—
294C7×D4—84no—
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—21no—
303C2×C4×C7—21no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
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×C7C2×SD1612yes—
322C2×C4×C7C2×SD1612yes—
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7H(5,𝔽2)11yescharacteristic
328C2×C4×C7H(5,𝔽2)11yescharacteristic
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—84no—
342Dic14—22no—
343Dic14—22no—
344Dic14—41no—
345Dic14—41no—
346C7⋊C8—41no—
347C7⋊C8—41no—
348C7⋊C8—41no—
349C7⋊C8—41no—
350C22×Q8—21no—
351C22×Q8—71no—
352C22×D4—21no—
353C22×D4C22×D711yescharacteristic
354C23×C4—141no—
355C23×C4C22×D711yescharacteristic
356C2×SD16—142no—
357C2×SD16—281no—
358C2×SD16—281no—
359C22×C8—71no—
360C22×C8—71no—
361C22×C8—71no—
362C22×C8—71no—
363C4⋊Q8—22no—
364C4⋊Q8—41no—
365#124—44no—
366C22⋊Q8—142no—
367C22⋊Q8—142no—
368#121—22no—
369#121—41no—
370C4×Q8—142no—
371C4×Q8—281no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—142no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—281no—
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4C22⋊D711yescharacteristic
393C2×C22⋊C4C22⋊D711yescharacteristic
394C2×C42—21no—
395C2×C42—141no—
396C2×C42—141no—
397C2×C42C22×D711yescharacteristic
398C2×C42C22×D711yescharacteristic
399#106—142no—
400C4⋊C8—281no—
401#103—142no—
402#103—142no—
403#103—281no—
404#102—142no—
405#102—142no—
406#102—281no—
407C22⋊C8—281no—
408C22⋊C8—281no—
409C4×C8—281no—
410#95—21no—
411#95—21no—
412#95—141no—
413#95—141no—
414#95—141no—
415#95—141no—
416#95C22⋊D711yescharacteristic
417#95C22⋊D711yescharacteristic
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7—41no—
424C22×C7—41no—
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7—81no—
429C22×C7—81no—
430C22×C7—81no—
431C22×C7—81no—
432C22×C7#52511yescharacteristic
433C22×C7#54611yescharacteristic
434C22×C7#54611yescharacteristic
435C22×C7C22×SD1611yescharacteristic
436C22×C7#57511yescharacteristic
437C22×C7#57611yescharacteristic
438C22×C7#57811yescharacteristic
439C4×C7—22no—
440C4×C7—22no—
441C4×C7—41no—
442C4×C7—42no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7—81no—
446C4×C7—81no—
447C4×C7#52911yescharacteristic
448C4×C7#53811yescharacteristic
449C4×C7#55511yescharacteristic
450C4×C7#55911yescharacteristic
451Dic7—42no—
452Dic7—42no—
453Dic7—42no—
454Dic7—81no—
455C24—21no—
456C24C2×C22⋊D711yescharacteristic
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—72no—
460C2×Q8—81no—
461C2×Q8—141no—
462C2×Q8—141no—
463C2×Q8—141no—
464C2×Q8—141no—
465C2×Q8—141no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—81no—
475C22×C4—21no—
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—141no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—282no—
489C22×C4—282no—
490C22×C4—561no—
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C2×C22⋊D711yescharacteristic
493C22×C4C2×C22⋊D711yescharacteristic
494C22×C4C2×C22⋊D711yescharacteristic
495C22×C4C2×C22⋊D711yescharacteristic
496C22×C4C23×D711yescharacteristic
497SD16—284no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8—141no—
509C2×C8—141no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—72no—
517C4⋊C4—72no—
518C4⋊C4—144no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—141no—
522C4⋊C4—141no—
523C4⋊C4—282no—
524C4⋊C4—282no—
525C4⋊C4—561no—
526C4⋊C4#115912yes—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—144no—
535C22⋊C4#114214yescomplemented
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—282no—
542C42—561no—
543C2×C7—42no—
544C2×C7—161no—
545C2×C7#245911yescharacteristic
546C2×C7#310311yescharacteristic
547C2×C7#312411yescharacteristic
548C2×C7#329511yescharacteristic
549C2×C7#329811yescharacteristic
550C2×C7#331911yescharacteristic
551C2×C7#332711yescharacteristic
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×#114212yes—
566C23C22×C22⋊D711yescharacteristic
567Q8—84no—
568Q8—142no—
569Q8—142no—
570Q8—281no—
571Q8—281no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—84no—
577C2×C4—21no—
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—41no—
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—142no—
604C2×C4—142no—
605C2×C4—142no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—564no—
616C2×C4C2×#114212yes—
617C2×C4C2×#114212yes—
618C2×C4C2×#115911yescharacteristic
619C2×C4#654811yescharacteristic
620C2×C4D7×#4111yescharacteristic
621C2×C4D7×#4111yescharacteristic
622C2×C4D7×#4111yescharacteristic
623C2×C4#655311yescharacteristic
624C8—281no—
625C8—281no—
626C8—281no—
627C8—281no—
628C7#2672411yescharacteristic, complemented, Sylow
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C22×#114211yescharacteristic
644C22#9049711yescharacteristic
645C22#9051311yescharacteristic
646C22#9051911yescharacteristic
647C22#9052011yescharacteristic
648C22#9052611yescharacteristic
649C22#9052911yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—282no—
659C4—282no—
660C4—282no—
661C4—561no—
662C4#9036111yescharacteristic
663C4#9036911yescharacteristic
664C4#9037111yescharacteristic
665C4#9037211yescharacteristic
666C2—42no—
667C2—161no—
668C2#1174044211yescharacteristic
669C2#1174076011yescharacteristic
670C2#1174090611yescharacteristic
671C2#1174092811yescharacteristic
672C2#1174097611yescharacteristic
673C2#1174098611yescharacteristic
674C2#1174109811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2491C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2491D7
hypocenterC7#26724
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2491

Complements

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

NH ≅ G/Ntype
2. #11730910C2semidirect
3. #11729548C2semidirect
4. #11729534C2semidirect
7. #11727414C2semidirect
8. #11727405C2semidirect
9. #11727398C2semidirect
10. #11727391C2semidirect
11. #11727280C2semidirect
12. #11727256C2semidirect
13. C4×#6384C2semidirect
14. #11727214C2semidirect
15. #11727205C2semidirect
199. C7×C22⋊C4SD16semidirect
535. C22⋊C4#1142semidirect
628. C7#26724semidirect

Maximal subgroups

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

subgroupindex
C7×#24912
#117309102
#117295482
#117295342
#117294392
#117294312
#117274142
#117274052
#117273982
#117273912
#117272802
#117272562
C4×#63842
#117272142
#117272052
#267247