Finite Group Databasebeta

← #13421986

Subgroups — #13421986

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2363C211yesmaximal, characteristic
2#11730953C211yesmaximal, characteristic, complemented
3#11729439C211yesmaximal, characteristic
4#11729433C211yesmaximal, characteristic
5#11729366C211yesmaximal, characteristic, complemented
6C4×#6406C211yesmaximal, characteristic, complemented
7#11729339C211yesmaximal, characteristic, complemented
8#11729329C211yesmaximal, characteristic, complemented
9#11727405C211yesmaximal, characteristic, complemented
10#11727392C211yesmaximal, characteristic, complemented
11#11727368C211yesmaximal, characteristic, complemented
12#11727357C211yesmaximal, characteristic, complemented
13#11727244C211yesmaximal, characteristic, complemented
14#11727219C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C7×#390—21no—
23C7×#390C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#388—21no—
26C7×#386—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C22⋊C4—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89984C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89723—21no—
41#89719—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89701C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89432—21no—
48#89418—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26731—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C22×Dic14—21no—
65C22×Dic14C2311yescharacteristic
66C7×#121—22no—
67C7×#121—22no—
68C4×C7×D4—22no—
69C4×C7×D4—41no—
70C7×#117—22no—
71C7×#117—41no—
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⋊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×C7—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—21no—
89C2×#1142—22no—
90C2×#1142—41no—
91C2×#1142—41no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96#6487—22no—
97#6487—41no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104#6432—22no—
105#6432—41no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95D411yescharacteristic
109C7×#95D411yescharacteristic
110#6406—22no—
111#6406—22no—
112#6406—41no—
113C7⋊#95—21no—
114C7⋊#95—21no—
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#6381—22no—
125C4⋊(C7⋊C8)—41no—
126C4×C7⋊C8—41no—
127#2492—71no—
128#2363D711yescharacteristic
129#2055—71no—
130#2033—71no—
131#2023—71no—
132#2000—71no—
133#1988—71no—
134#1933—71no—
135#1928—71no—
136#1923—71no—
137#1921—71no—
138#1877—71no—
139#1873—71no—
140#1825—71no—
141C4×#102—71no—
142C24×C7—21no—
143C24×C7C2×D411yescharacteristic
144C2×C7×D4—24no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—81no—
151C2×C7×D4—81no—
152C2×C7×D4#4112yes—
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×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—81no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—42no—
182C2×Dic14—42no—
183C2×Dic14—81no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—81no—
189C7×C4⋊C4#4112yes—
190C7×C4⋊C4#4112yes—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—42no—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—81no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—42no—
203C42×C7—81no—
204C22⋊Dic7—42no—
205C22⋊Dic7—81no—
206#1142—44no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—42no—
213C4⋊Dic7#4112yes—
214Dic7⋊C4—42no—
215C4×Dic7—81no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
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—
228C22×SD16—71no—
229C2×C4⋊Q8—71no—
230C2×C22⋊Q8—71no—
231C2×#121C2×D711yescharacteristic
232C2×C4×D4C2×D711yescharacteristic
233C2×#117C2×D711yescharacteristic
234C2×#107—71no—
235C2×C4⋊C8—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#103—71no—
239C2×#102—71no—
240C2×#102—71no—
241C2×#102—71no—
242C2×C22⋊C8—71no—
243C2×C22⋊C8—71no—
244C2×C4×C8—71no—
245#400—141no—
246#393—141no—
247#391—71no—
248#391—71no—
249#391—141no—
250#390—21no—
251#390C2×D711yescharacteristic
252#389C2×D711yescharacteristic
253#388—21no—
254#388—141no—
255#386—21no—
256#380—71no—
257C4×C4⋊C4—71no—
258C4×C4⋊C4C2×D711yescharacteristic
259C4×C22⋊C4—21no—
260C4×C22⋊C4C2×D711yescharacteristic
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#336—71no—
270#336—71no—
271C23×C7—22no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7C22×D411yescharacteristic
286C7×D4—22no—
287C7×D4—22no—
288C7×D4—41no—
289C7×D4—41no—
290C7×D4—84no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
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—84no—
318C2×C4×C7#13512yes—
319C2×C4×C7#13712yes—
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7H(5,𝔽2)11yescharacteristic
325C2×C4×C7H(5,𝔽2)11yescharacteristic
326C2×Dic7—22no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338Dic14—41no—
339Dic14—41no—
340Dic14—41no—
341Dic14—41no—
342Dic14—84no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C7⋊C8—41no—
346C7⋊C8—41no—
347C22×Q8—71no—
348C22×Q8—141no—
349C22×D4—21no—
350C22×D4C22×D711yescharacteristic
351C23×C4—21no—
352C23×C4C22×D711yescharacteristic
353C2×SD16—142no—
354C2×SD16—281no—
355C2×SD16—281no—
356C22×C8—71no—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8—71no—
360C4⋊Q8—142no—
361C4⋊Q8—281no—
362C22⋊Q8—142no—
363C22⋊Q8—281no—
364#121—22no—
365#121—22no—
366C4×D4—22no—
367C4×D4—41no—
368#117—22no—
369#117—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4C22⋊D711yescharacteristic
387C2×C22⋊C4C22⋊D711yescharacteristic
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392C2×C42C22×D711yescharacteristic
393#107—142no—
394C4⋊C8—281no—
395#103—142no—
396#103—142no—
397#103—281no—
398#102—142no—
399#102—142no—
400#102—281no—
401C22⋊C8—281no—
402C22⋊C8—281no—
403C4×C8—281no—
404#95—21no—
405#95—21no—
406#95—141no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411#95—141no—
412#95C22⋊D711yescharacteristic
413#95C22⋊D711yescharacteristic
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—42no—
417C22×C7—41no—
418C22×C7—42no—
419C22×C7—41no—
420C22×C7—42no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7—41no—
424C22×C7—81no—
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7#52511yescharacteristic
429C22×C7#54611yescharacteristic
430C22×C7#54611yescharacteristic
431C22×C7C2×#13511yescharacteristic
432C22×C7C2×#13711yescharacteristic
433C22×C7#57611yescharacteristic
434C22×C7#57711yescharacteristic
435C4×C7—22no—
436C4×C7—22no—
437C4×C7—41no—
438C4×C7—42no—
439C4×C7—41no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—81no—
443C4×C7#52911yescharacteristic
444C4×C7#53811yescharacteristic
445C4×C7#55511yescharacteristic
446C4×C7#55911yescharacteristic
447Dic7—42no—
448Dic7—81no—
449Dic7—81no—
450Dic7—81no—
451C24—21no—
452C24C2×C22⋊D711yescharacteristic
453C2×Q8—141no—
454C2×Q8—141no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×Q8—141no—
459C2×Q8—282no—
460C2×Q8—282no—
461C2×Q8—561no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—81no—
469C2×D4—81no—
470C2×D4#115712yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C23×D711yescharacteristic
493SD16—284no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—72no—
511C4⋊C4—81no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—282no—
518C4⋊C4—282no—
519C4⋊C4#115712yes—
520C4⋊C4#115912yes—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—282no—
529C22⋊C4—561no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—81no—
536C42—561no—
537C2×C7—81no—
538C2×C7—82no—
539C2×C7#245911yescharacteristic
540C2×C7#310711yescharacteristic
541C2×C7#311811yescharacteristic
542C2×C7#329711yescharacteristic
543C2×C7#330411yescharacteristic
544C2×C7#331411yescharacteristic
545C2×C7#332911yescharacteristic
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C22×C22⋊D711yescharacteristic
561Q8—281no—
562Q8—281no—
563Q8—281no—
564Q8—281no—
565Q8—564no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—84no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—142no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4#651312yes—
611C2×C4#651412yes—
612C2×C4C2×#115711yescharacteristic
613C2×C4C2×#115711yescharacteristic
614C2×C4C2×#115911yescharacteristic
615C2×C4#654811yescharacteristic
616C2×C4D7×#4111yescharacteristic
617C2×C4#655311yescharacteristic
618C8—281no—
619C8—281no—
620C8—281no—
621C8—281no—
622C7#2673111yescharacteristic, complemented, Sylow
623C22—41no—
624C22—41no—
625C22—42no—
626C22—41no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#9048211yescharacteristic
638C22#9049711yescharacteristic
639C22C2×#651311yescharacteristic
640C22C2×#651411yescharacteristic
641C22#9052011yescharacteristic
642C22#9052611yescharacteristic
643C22#9052711yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—81no—
652C4—282no—
653C4—561no—
654C4—561no—
655C4—561no—
656C4#9022911yescharacteristic
657C4#9025011yescharacteristic
658C4#9025111yescharacteristic
659C4#9025411yescharacteristic
660C2—81no—
661C2—82no—
662C2#1173981011yescharacteristic
663C2#1173997911yescharacteristic
664C2#1173998611yescharacteristic
665C2#1174094311yescharacteristic
666C2#1174101511yescharacteristic
667C2#1174105911yescharacteristic
668C2#1174110211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2363C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2363D7
hypocenterC7#26731
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. #11730953C2semidirect
5. #11729366C2semidirect
6. C4×#6406C2semidirect
7. #11729339C2semidirect
8. #11729329C2semidirect
9. #11727405C2semidirect
10. #11727392C2semidirect
11. #11727368C2semidirect
12. #11727357C2semidirect
13. #11727244C2semidirect
14. #11727219C2semidirect
15. #11727205C2semidirect
622. C7#26731semidirect

Maximal subgroups

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

subgroupindex
C7×#23632
#117309532
#117294392
#117294332
#117293662
C4×#64062
#117293392
#117293292
#117274052
#117273922
#117273682
#117273572
#117272442
#117272192
#117272052
#267317