Finite Group Databasebeta

← #13421913

Subgroups — #13421913

Order 1792.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2492C211yesmaximal, characteristic
2#11730910C211yesmaximal, characteristic
3#11729548C211yesmaximal, characteristic, complemented
4#11729533C211yesmaximal, characteristic, complemented
5#11729454C211yesmaximal, characteristic
6#11729446C211yesmaximal, characteristic
7#11727444C211yesmaximal, characteristic, complemented
8#11727435C211yesmaximal, characteristic, complemented
9#11727428C211yesmaximal, characteristic, complemented
10#11727421C211yesmaximal, characteristic, complemented
11#11727281C211yesmaximal, characteristic
12#11727256C211yesmaximal, characteristic
13C4×#6384C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8C2211yescharacteristic
18C22×#1144C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
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×#6381C2211yescharacteristic
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#26726—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×C7×Q8C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—41no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4D411yescharacteristic
77C2×C7×C22⋊C4D411yescharacteristic
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—21no—
82C2×#1144—22no—
83C2×#1144—41no—
84C2×#1144—41no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—41no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91#6487—22no—
92#6487—22no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—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#6381—22no—
125C4⋊(C7⋊C8)—41no—
126C4×C7⋊C8—41no—
127#2492D711yescharacteristic
128#2360—71no—
129#2056—71no—
130#2036—71no—
131#2024—71no—
132#2001—71no—
133#1987—71no—
134#1933—71no—
135#1928—71no—
136#1923—71no—
137#1921—71no—
138#1878—71no—
139#1872—71no—
140#1824—71no—
141C4×#103—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—
172C22×Dic7—42no—
173C22×Dic7—42no—
174C22×Dic7—81no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14#4112yes—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4—42no—
188C7×C4⋊C4#4112yes—
189C7×C22⋊C4—41no—
190C7×C22⋊C4—42no—
191C7×C22⋊C4—81no—
192C7×C22⋊C4Q1614yescomplemented
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—81no—
198C22⋊Dic7—24no—
199#1144—44no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—42no—
205C4⋊Dic7#4112yes—
206C4⋊Dic7#4112yes—
207Dic7⋊C4—24no—
208Dic7⋊C4—42no—
209Dic7⋊C4—81no—
210C4×Dic7—42no—
211C4×Dic7—81no—
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—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C22×Q16—71no—
225C2×C4⋊Q8C2×D711yescharacteristic
226C2×C22⋊Q8—71no—
227C2×C22⋊Q8C2×D711yescharacteristic
228C2×C4×Q8—71no—
229C22×C4⋊C4—71no—
230C2×#107—71no—
231C2×C4⋊C8—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#103—71no—
235C2×#103—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×C22⋊C8—71no—
239C2×C22⋊C8—71no—
240C2×C4×C8—71no—
241#400—21no—
242#393—21no—
243#391—21no—
244#391—71no—
245#391C2×D711yescharacteristic
246#391C2×D711yescharacteristic
247#389—71no—
248#389—141no—
249#388—21no—
250#387—141no—
251#385—141no—
252#380C2×D711yescharacteristic
253C4×C4⋊C4—71no—
254C4×C4⋊C4C2×D711yescharacteristic
255C4×C22⋊C4—141no—
256C4×C22⋊C4C2×D711yescharacteristic
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#336—71no—
266#336—71no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7C2×Q1612yes—
274C23×C7C22×D411yescharacteristic
275C7×Q8—41no—
276C7×Q8—41no—
277C7×Q8—41no—
278C7×Q8—41no—
279C7×Q8—84no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
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—41no—
309C2×C4×C7C2×Q1612yes—
310C2×C4×C7C2×Q1612yes—
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7H(5,𝔽2)11yescharacteristic
316C2×C4×C7H(5,𝔽2)11yescharacteristic
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—84no—
330Dic14—22no—
331Dic14—22no—
332Dic14—41no—
333Dic14—41no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C7⋊C8—41no—
337C7⋊C8—41no—
338C22×Q8—21no—
339C22×Q8—71no—
340C22×Q8C22×D711yescharacteristic
341C23×C4—141no—
342C23×C4C22×D711yescharacteristic
343C2×Q16—142no—
344C2×Q16—281no—
345C2×Q16—281no—
346C22×C8—71no—
347C22×C8—71no—
348C22×C8—71no—
349C22×C8—71no—
350C4⋊Q8—22no—
351C4⋊Q8—41no—
352C22⋊Q8—22no—
353C22⋊Q8—41no—
354C22⋊Q8—142no—
355C22⋊Q8—142no—
356C4×Q8—142no—
357C4×Q8—281no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—142no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—281no—
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—141no—
375C2×C22⋊C4C22⋊D711yescharacteristic
376C2×C22⋊C4C22⋊D711yescharacteristic
377C2×C42—21no—
378C2×C42—141no—
379C2×C42—141no—
380C2×C42C22×D711yescharacteristic
381C2×C42C22×D711yescharacteristic
382#107—142no—
383C4⋊C8—281no—
384#103—142no—
385#103—142no—
386#103—142no—
387#103—142no—
388#103—281no—
389#103—281no—
390C22⋊C8—281no—
391C22⋊C8—281no—
392C4×C8—281no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95C22⋊D711yescharacteristic
404#95C22⋊D711yescharacteristic
405C22×C7—22no—
406C22×C7—22no—
407C22×C7—22no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7—41no—
411C22×C7—41no—
412C22×C7#52511yescharacteristic
413C22×C7#54611yescharacteristic
414C22×C7#54611yescharacteristic
415C22×C7C22×Q1611yescharacteristic
416C22×C7#57511yescharacteristic
417C22×C7#57711yescharacteristic
418C22×C7#57711yescharacteristic
419C4×C7—22no—
420C4×C7—22no—
421C4×C7—41no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7—42no—
425C4×C7—81no—
426C4×C7—81no—
427C4×C7—81no—
428C4×C7#52911yescharacteristic
429C4×C7#53811yescharacteristic
430C4×C7#55511yescharacteristic
431C4×C7#55911yescharacteristic
432Dic7—42no—
433Dic7—42no—
434Dic7—42no—
435Dic7—81no—
436C24C2×C22⋊D711yescharacteristic
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—72no—
446C2×Q8—81no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—141no—
450C2×Q8—141no—
451C2×Q8—141no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—282no—
467C22×C4—282no—
468C22×C4—561no—
469C22×C4C2×C22⋊D711yescharacteristic
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C2×C22⋊D711yescharacteristic
474C22×C4C23×D711yescharacteristic
475Q16—284no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—72no—
496C4⋊C4—72no—
497C4⋊C4—144no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—141no—
501C4⋊C4—141no—
502C4⋊C4—282no—
503C4⋊C4—282no—
504C4⋊C4—561no—
505C4⋊C4#115912yes—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—144no—
510C22⋊C4#114414yescomplemented
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—81no—
516C42—282no—
517C42—561no—
518C2×C7—42no—
519C2×C7#245911yescharacteristic
520C2×C7#310411yescharacteristic
521C2×C7#312511yescharacteristic
522C2×C7#329911yescharacteristic
523C2×C7#329911yescharacteristic
524C2×C7#331111yescharacteristic
525C2×C7#332311yescharacteristic
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23C2×#114412yes—
533C23C22×C22⋊D711yescharacteristic
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—84no—
539Q8—142no—
540Q8—142no—
541Q8—281no—
542Q8—281no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
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—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—564no—
585C2×C4C2×#114412yes—
586C2×C4C2×#114412yes—
587C2×C4C2×#115911yescharacteristic
588C2×C4#654811yescharacteristic
589C2×C4D7×#4111yescharacteristic
590C2×C4D7×#4111yescharacteristic
591C2×C4D7×#4111yescharacteristic
592C2×C4#655311yescharacteristic
593C8—281no—
594C8—281no—
595C8—281no—
596C8—281no—
597C7#2672611yescharacteristic, complemented, Sylow
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#9049711yescharacteristic
606C22C22×#114411yescharacteristic
607C22#9051311yescharacteristic
608C22#9051911yescharacteristic
609C22#9052011yescharacteristic
610C22#9052711yescharacteristic
611C22#9052811yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—282no—
622C4—282no—
623C4—282no—
624C4—561no—
625C4#9036111yescharacteristic
626C4#9036911yescharacteristic
627C4#9037111yescharacteristic
628C4#9037211yescharacteristic
629C2—42no—
630C2#1174044211yescharacteristic
631C2#1174081311yescharacteristic
632C2#1174090711yescharacteristic
633C2#1174094611yescharacteristic
634C2#1174099611yescharacteristic
635C2#1174108511yescharacteristic
636C2#1174111011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2492C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2492D7
hypocenterC7#26726
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. #11727428C2semidirect
10. #11727421C2semidirect
13. C4×#6384C2semidirect
14. #11727215C2semidirect
192. C7×C22⋊C4Q16semidirect
510. C22⋊C4#1144semidirect
597. C7#26726semidirect

Maximal subgroups

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

subgroupindex
C7×#24922
#117309102
#117295482
#117295332
#117294542
#117294462
#117274442
#117274352
#117274282
#117274212
#117272812
#117272562
C4×#63842
#117272152
#117272052
#267267