Finite Group Databasebeta

← #13414384

Subgroups — #13414384

Order 1792.

844 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2414C211yesmaximal, characteristic
2#11739780C211yesmaximal, characteristic, complemented
3#11729435C211yesmaximal, characteristic, complemented
4#11729433C211yesmaximal, characteristic, complemented
5#11729432C211yesmaximal, characteristic, complemented
6#11729431C211yesmaximal, characteristic, complemented
7#11729364C211yesmaximal, characteristic, complemented
8#11729334C211yesmaximal, characteristic, complemented
9#11729330C211yesmaximal, characteristic, complemented
10#11727395C211yesmaximal, characteristic, complemented
11#11727391C211yesmaximal, characteristic, complemented
12#11727367C211yesmaximal, characteristic, complemented
13#11727354C211yesmaximal, characteristic, complemented
14#11727261C211yesmaximal, characteristic, complemented
15#11727256C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C7×#121C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×C7×#117C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C23×C7⋊C8C2211yescharacteristic
23C7×#394—21no—
24C7×#392—21no—
25C7×#390—21no—
26C7×#390C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#388—21no—
29C7×#380C2211yescharacteristic
30#89992—21no—
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89984C2211yescharacteristic
36#89984C2211yescharacteristic
37C2×C22⋊(C7⋊C8)C2211yescharacteristic
38C2×C22⋊(C7⋊C8)C2211yescharacteristic
39C2×C22⋊(C7⋊C8)C2211yescharacteristic
40#89753—21no—
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89723—21no—
44C22⋊C4×Dic7—21no—
45#89708C2211yescharacteristic
46C4×C4⋊Dic7C2211yescharacteristic
47#89324C2211yescharacteristic
48#89324C2211yescharacteristic
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26199—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4—21no—
62C22×C7×D4D411yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C23×C4×C7C2311yescharacteristic
66C23×Dic7—21no—
67C7×#121—22no—
68C7×#121—22no—
69C7×#121—22no—
70C7×#121—41no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C7×#117—22no—
74C7×#117—41no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C7×C22⋊C4D411yescharacteristic
88C2×C42×C7—21no—
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—21no—
91C2×C22⋊Dic7—21no—
92C2×C22⋊Dic7—21no—
93#6489—22no—
94#6489—41no—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—41no—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C22×C7⋊C8—22no—
105C22×C7⋊C8—41no—
106C22×C7⋊C8—41no—
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C7×#95—21no—
112C7×#95—21no—
113#6406—22no—
114#6406—41no—
115#6406—41no—
116#6406—41no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C22⋊(C7⋊C8)—22no—
123C22⋊(C7⋊C8)—41no—
124C22⋊(C7⋊C8)—41no—
125#6382—41no—
126#6381—22no—
127#2414D711yescharacteristic
128#2352—71no—
129#2101—71no—
130#2077—71no—
131#2073—71no—
132#2033—71no—
133#1926—71no—
134#1918—71no—
135#1893—71no—
136#1875—71no—
137#1874—71no—
138#1869—71no—
139#1854—71no—
140#1824—71no—
141#1735—71no—
142C24×C7—21no—
143C24×C7C2×D411yescharacteristic
144C24×C7C2×D411yescharacteristic
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—41no—
154C2×C7×D4—42no—
155C2×C7×D4—42no—
156C2×C7×D4—42no—
157C2×C7×D4—81no—
158C2×C7×D4—81no—
159C2×C7×D4—81no—
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×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7#4112yes—
178C22×C4×C7#4112yes—
179C22×C4×C7C2411yescharacteristic
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—21no—
184C22×Dic7—42no—
185C22×Dic7—42no—
186C22×Dic7—81no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—81no—
194C7×C4⋊C4#4112yes—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—42no—
199C7×C22⋊C4—42no—
200C7×C22⋊C4—42no—
201C7×C22⋊C4—81no—
202C7×C22⋊C4—81no—
203C42×C7—21no—
204C42×C7—21no—
205C42×C7—42no—
206C22⋊Dic7—42no—
207C22⋊Dic7—42no—
208C22⋊Dic7—81no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—42no—
217C4⋊Dic7#4112yes—
218C4×Dic7—42no—
219C4×Dic7—81no—
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—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—44no—
233C23×C8—71no—
234C2×#121C2×D711yescharacteristic
235C2×#121C2×D711yescharacteristic
236C2×C4×D4C2×D711yescharacteristic
237C2×#117—71no—
238C2×#117C2×D711yescharacteristic
239C22×C4⋊C4—71no—
240C2×#107—71no—
241C2×#106—71no—
242C2×#102—71no—
243C2×#102—71no—
244C2×#102—71no—
245C2×#102—71no—
246C2×C22⋊C8—71no—
247C2×C22⋊C8—71no—
248C2×C22⋊C8—71no—
249#394—21no—
250#392—21no—
251#390—21no—
252#390C2×D711yescharacteristic
253#389C2×D711yescharacteristic
254#388—21no—
255#388—141no—
256#387—141no—
257#386—141no—
258#382—71no—
259#380—71no—
260#380—71no—
261#380C2×D711yescharacteristic
262C4×C4⋊C4—71no—
263C4×C22⋊C4—141no—
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#340—71no—
273#340—71no—
274#336—71no—
275#336—71no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7—21no—
281C23×C7—21no—
282C23×C7—21no—
283C23×C7—21no—
284C23×C7—21no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7—41no—
292C23×C7—41no—
293C23×C7—41no—
294C23×C7—41no—
295C23×C7#12312yes—
296C23×C7#13512yes—
297C23×C7C22×D411yescharacteristic
298C7×D4—44no—
299C7×D4—41no—
300C7×D4—41no—
301C7×D4—41no—
302C7×D4—41no—
303C7×D4—84no—
304C7×D4—84no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—22no—
316C2×C4×C7—21no—
317C2×C4×C7—21no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—41no—
333C2×C4×C7#12312yes—
334C2×C4×C7C2×#4111yescharacteristic
335C2×C4×C7C2×#4111yescharacteristic
336C2×C4×C7C2×#4111yescharacteristic
337C2×C4×C7C2×#4111yescharacteristic
338C2×C4×C7H(5,𝔽2)11yescharacteristic
339C2×C4×C7H(5,𝔽2)11yescharacteristic
340C2×Dic7—22no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C2×Dic7—84no—
353C7⋊C8—41no—
354C7⋊C8—41no—
355C7⋊C8—41no—
356C7⋊C8—41no—
357C22×D4—21no—
358C22×D4—21no—
359C22×D4C22⋊D711yescharacteristic
360C22×D4C22×D711yescharacteristic
361C23×C4—141no—
362C23×C4C22×D711yescharacteristic
363C23×C4C22×D711yescharacteristic
364C22×C8—71no—
365C22×C8—71no—
366C22×C8—71no—
367C22×C8—71no—
368C22×C8—142no—
369C22×C8—281no—
370C22×C8—281no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—41no—
375C4×D4—22no—
376C4×D4—22no—
377#117—22no—
378#117—41no—
379#117—142no—
380#117—281no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—71no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—71no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—142no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—281no—
389C2×C4⋊C4C22⋊D711yescharacteristic
390C2×C4⋊C4C22×D711yescharacteristic
391C2×C4⋊C4C22×D711yescharacteristic
392C2×C4⋊C4C22×D711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—141no—
401C2×C22⋊C4—141no—
402C2×C22⋊C4C22⋊D711yescharacteristic
403C2×C22⋊C4C22⋊D711yescharacteristic
404C2×C42—21no—
405C2×C42—141no—
406C2×C42—141no—
407C2×C42C22×D711yescharacteristic
408#107—142no—
409#106—281no—
410#102—142no—
411#102—281no—
412#102—281no—
413#102—281no—
414C22⋊C8—142no—
415C22⋊C8—281no—
416C22⋊C8—281no—
417#95—21no—
418#95—21no—
419#95—141no—
420#95—141no—
421#95—141no—
422#95—141no—
423#95—141no—
424C22×C7—22no—
425C22×C7—22no—
426C22×C7—22no—
427C22×C7—22no—
428C22×C7—22no—
429C22×C7—22no—
430C22×C7—41no—
431C22×C7—41no—
432C22×C7—41no—
433C22×C7—41no—
434C22×C7—41no—
435C22×C7—41no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7—81no—
439C22×C7—81no—
440C22×C7—81no—
441C22×C7—81no—
442C22×C7—81no—
443C22×C7—81no—
444C22×C7—81no—
445C22×C7C2×#12311yescharacteristic
446C22×C7#54611yescharacteristic
447C22×C7#54611yescharacteristic
448C22×C7C2×#13511yescharacteristic
449C22×C7#57511yescharacteristic
450C22×C7#57611yescharacteristic
451C22×C7#57711yescharacteristic
452C4×C7—22no—
453C4×C7—41no—
454C4×C7—41no—
455C4×C7—41no—
456C4×C7—42no—
457C4×C7—81no—
458C4×C7—81no—
459C4×C7#52911yescharacteristic
460C4×C7#53811yescharacteristic
461C4×C7#55311yescharacteristic
462C4×C7#55311yescharacteristic
463Dic7—42no—
464Dic7—81no—
465Dic7—81no—
466Dic7—81no—
467C24—21no—
468C24C2×C22⋊D711yescharacteristic
469C24C2×C22⋊D711yescharacteristic
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—41no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—81no—
483C2×D4—81no—
484C2×D4—81no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—141no—
499C22×C4—141no—
500C22×C4—141no—
501C22×C4—141no—
502C22×C4—282no—
503C22×C4—282no—
504C22×C4—561no—
505C22×C4#115912yes—
506C22×C4#115912yes—
507C22×C4C2×C22⋊D711yescharacteristic
508C22×C4C2×C22⋊D711yescharacteristic
509C22×C4C2×C22⋊D711yescharacteristic
510C22×C4C2×C22⋊D711yescharacteristic
511C22×C4C23×D711yescharacteristic
512C2×C8—141no—
513C2×C8—141no—
514C2×C8—141no—
515C2×C8—141no—
516C2×C8—141no—
517C2×C8—141no—
518C2×C8—141no—
519C2×C8—141no—
520C2×C8—141no—
521C2×C8—141no—
522C2×C8—141no—
523C2×C8—141no—
524C2×C8—284no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—72no—
532C4⋊C4—81no—
533C4⋊C4—141no—
534C4⋊C4—141no—
535C4⋊C4—141no—
536C4⋊C4—141no—
537C4⋊C4—141no—
538C4⋊C4—141no—
539C4⋊C4—141no—
540C4⋊C4—282no—
541C4⋊C4#115912yes—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—282no—
551C22⋊C4—282no—
552C22⋊C4—561no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—282no—
557C42—561no—
558C2×C7—42no—
559C2×C7—42no—
560C2×C7—161no—
561C2×C7#244411yescharacteristic
562C2×C7#309111yescharacteristic
563C2×C7#309611yescharacteristic
564C2×C7#329711yescharacteristic
565C2×C7#330111yescharacteristic
566C2×C7#331711yescharacteristic
567C2×C7#331911yescharacteristic
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23#649812yes—
588C23#651312yes—
589C23C22×C22⋊D711yescharacteristic
590D4—44no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—84no—
596D4—84no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—142no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—281no—
633C2×C4—281no—
634C2×C4—281no—
635C2×C4—281no—
636C2×C4—281no—
637C2×C4—564no—
638C2×C4#649812yes—
639C2×C4C2×#115911yescharacteristic
640C2×C4C2×#115911yescharacteristic
641C2×C4C2×#115911yescharacteristic
642C2×C4#654811yescharacteristic
643C2×C4D7×#4111yescharacteristic
644C2×C4#655311yescharacteristic
645C8—281no—
646C8—281no—
647C8—281no—
648C8—281no—
649C7#2619911yescharacteristic, complemented, Sylow
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22C2×#649811yescharacteristic
672C22#9049711yescharacteristic
673C22C2×#651311yescharacteristic
674C22#9051311yescharacteristic
675C22#9052011yescharacteristic
676C22#9052611yescharacteristic
677C22#9052711yescharacteristic
678C4—22no—
679C4—42no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—81no—
684C4—81no—
685C4—282no—
686C4—561no—
687C4—561no—
688C4—561no—
689C4#9029111yescharacteristic
690C4#9029211yescharacteristic
691C4#9030811yescharacteristic
692C4#9030911yescharacteristic
693C2—42no—
694C2—42no—
695C2—161no—
696C2#1174010111yescharacteristic
697C2#1174035811yescharacteristic
698C2#1174082111yescharacteristic
699C2#1174094011yescharacteristic
700C2#1174101811yescharacteristic
701C2#1174106311yescharacteristic
702C2#1174109811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2414C2
socleC23×C7C22×D4
hypercenter#2414D7
hypocenterC7#26199
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2414

Complements

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

NH ≅ G/Ntype
2. #11739780C2semidirect
3. #11729435C2semidirect
4. #11729433C2semidirect
5. #11729432C2semidirect
6. #11729431C2semidirect
7. #11729364C2semidirect
8. #11729334C2semidirect
9. #11729330C2semidirect
10. #11727395C2semidirect
11. #11727391C2semidirect
12. #11727367C2semidirect
13. #11727354C2semidirect
14. #11727261C2semidirect
15. #11727256C2semidirect
649. C7#26199semidirect

Maximal subgroups

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

subgroupindex
C7×#24142
#117397802
#117294352
#117294332
#117294322
#117294312
#117293642
#117293342
#117293302
#117273952
#117273912
#117273672
#117273542
#117272612
#117272562
#261997