Finite Group Databasebeta

← #13422243

Subgroups — #13422243

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×#2491C211yesmaximal, characteristic
2#11730913C211yesmaximal, characteristic, complemented
3#11729548C211yesmaximal, characteristic, complemented
4#11729534C211yesmaximal, characteristic, complemented
5#11729439C211yesmaximal, characteristic
6#11729432C211yesmaximal, characteristic
7#11727414C211yesmaximal, characteristic, complemented
8#11727405C211yesmaximal, characteristic, complemented
9#11727399C211yesmaximal, characteristic, complemented
10#11727390C211yesmaximal, characteristic, complemented
11#11727281C211yesmaximal, characteristic, complemented
12#11727257C211yesmaximal, characteristic, complemented
13#11727229C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727206C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#124—21no—
18C2×C7×#121C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×#6489C2211yescharacteristic
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×#6381C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89424—21no—
48#89421—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26748—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×C7×D4—21no—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—41no—
67C7×#124—44no—
68C7×#121—22no—
69C7×#121—41no—
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⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×#1142—22no—
88C2×#1142—41no—
89C2×#1142—41no—
90#6489—22no—
91#6489—41no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95#6487—22no—
96#6487—22no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—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#6381—22no—
126C4⋊(C7⋊C8)—41no—
127#6377—41no—
128#2491D711yescharacteristic
129#2362—71no—
130#2055—71no—
131#2036—71no—
132#2024—71no—
133#2000—71no—
134#1988—71no—
135#1936—71no—
136#1925—71no—
137#1924—71no—
138#1921—71no—
139#1880—71no—
140#1875—71no—
141#1825—71no—
142#1763—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—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14#4112yes—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—42no—
188C7×C4⋊C4#4112yes—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—41no—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—42no—
196C7×C22⋊C4—81no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—42no—
202C22⋊Dic7—24no—
203C22⋊Dic7—42no—
204#1142—44no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7#4112yes—
210C4⋊Dic7#4112yes—
211Dic7⋊C4—24no—
212Dic7⋊C4—42no—
213Dic7⋊C4—42no—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C4×Dic7—42no—
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—
228C2×C7⋊C8—21no—
229C22×SD16—71no—
230C2×C4⋊Q8C2×D711yescharacteristic
231C2×#124—21no—
232C2×C22⋊Q8—71no—
233C2×#121C2×D711yescharacteristic
234C2×C4×Q8—71no—
235C2×#117—71no—
236C2×#107—71no—
237C2×C4⋊C8—71no—
238C2×#103—71no—
239C2×#103—71no—
240C2×#103—71no—
241C2×#102—71no—
242C2×#102—71no—
243C2×#102—71no—
244C2×C22⋊C8—71no—
245C2×C22⋊C8—71no—
246C2×#97—71no—
247#394—21no—
248#393—21no—
249#391—71no—
250#390C2×D711yescharacteristic
251#390C2×D711yescharacteristic
252#389—71no—
253#388—21no—
254#388—141no—
255#388—141no—
256#387—141no—
257#380C2×D711yescharacteristic
258C4×C4⋊C4—71no—
259C4×C4⋊C4—141no—
260C4×C4⋊C4C2×D711yescharacteristic
261C4×C22⋊C4C2×D711yescharacteristic
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#336—71no—
271#336—71no—
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×C7—41no—
286C23×C7C22×D411yescharacteristic
287C7×Q8—84no—
288C7×D4—41no—
289C7×D4—41no—
290C7×D4—41no—
291C7×D4—41no—
292C7×D4—84no—
293C2×C4×C7—21no—
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—22no—
302C2×C4×C7—21no—
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×C7AGL(1,ℤ/8)12yes—
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7H(5,𝔽2)11yescharacteristic
326C2×C4×C7H(5,𝔽2)11yescharacteristic
327C2×Dic7—22no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339Dic14—22no—
340Dic14—22no—
341Dic14—41no—
342Dic14—41no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C7⋊C8—41no—
346C7⋊C8—41no—
347C22×Q8—21no—
348C22×Q8—71no—
349C22×D4—21no—
350C22×D4C22×D711yescharacteristic
351C23×C4C22×D711yescharacteristic
352C2×SD16—142no—
353C2×SD16—281no—
354C2×SD16—281no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8—71no—
358C22×C8—71no—
359C4⋊Q8—22no—
360C4⋊Q8—41no—
361#124—44no—
362C22⋊Q8—142no—
363C22⋊Q8—142no—
364#121—22no—
365#121—41no—
366C4×Q8—142no—
367C4×Q8—281no—
368#117—142no—
369#117—281no—
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⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4C22⋊D711yescharacteristic
389C2×C22⋊C4C22⋊D711yescharacteristic
390C2×C42—21no—
391C2×C42—141no—
392C2×C42—141no—
393C2×C42—141no—
394C2×C42C22×D711yescharacteristic
395C2×C42C22×D711yescharacteristic
396#107—142no—
397C4⋊C8—281no—
398#103—142no—
399#103—142no—
400#103—281no—
401#102—142no—
402#102—142no—
403#102—281no—
404C22⋊C8—281no—
405C22⋊C8—281no—
406#97—281no—
407#95—21no—
408#95—21no—
409#95—141no—
410#95—141no—
411#95—141no—
412#95—141no—
413#95C22⋊D711yescharacteristic
414#95C22⋊D711yescharacteristic
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—41no—
418C22×C7—41no—
419C22×C7—41no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7—81no—
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7—81no—
429C22×C7#52511yescharacteristic
430C22×C7#54611yescharacteristic
431C22×C7#54611yescharacteristic
432C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
433C22×C7#57511yescharacteristic
434C22×C7#57711yescharacteristic
435C22×C7#57811yescharacteristic
436C4×C7—22no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—42no—
440C4×C7—41no—
441C4×C7—41no—
442C4×C7—81no—
443C4×C7—81no—
444C4×C7#52911yescharacteristic
445C4×C7#53811yescharacteristic
446C4×C7#55511yescharacteristic
447C4×C7#55911yescharacteristic
448Dic7—42no—
449Dic7—42no—
450Dic7—42no—
451Dic7—81no—
452C24—21no—
453C24C2×C22⋊D711yescharacteristic
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—72no—
457C2×Q8—81no—
458C2×Q8—141no—
459C2×Q8—141no—
460C2×Q8—141no—
461C2×Q8—141no—
462C2×Q8—141no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—81no—
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—21no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C23×D711yescharacteristic
491SD16—284no—
492C2×C8—141no—
493C2×C8—141no—
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—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—72no—
511C4⋊C4—72no—
512C4⋊C4—144no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—282no—
518C4⋊C4—282no—
519C4⋊C4#115912yes—
520C22⋊C4—22no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—144no—
529C22⋊C4—282no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—282no—
536C42—282no—
537C42—282no—
538C2×C7—81no—
539C2×C7—161no—
540C2×C7#245911yescharacteristic
541C2×C7#311511yescharacteristic
542C2×C7#312111yescharacteristic
543C2×C7#331011yescharacteristic
544C2×C7#331311yescharacteristic
545C2×C7#332111yescharacteristic
546C2×C7#333011yescharacteristic
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—
560C23—41no—
561C23C22×C22⋊D711yescharacteristic
562Q8—84no—
563Q8—142no—
564Q8—142no—
565Q8—281no—
566Q8—281no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—84no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
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—41no—
598C2×C4—41no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—142no—
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—281no—
611C2×C4#649512yes—
612C2×C4C2×#115911yescharacteristic
613C2×C4#654811yescharacteristic
614C2×C4D7×#4111yescharacteristic
615C2×C4D7×#4111yescharacteristic
616C2×C4D7×#4111yescharacteristic
617C2×C4#655311yescharacteristic
618C8—281no—
619C8—281no—
620C8—281no—
621C8—281no—
622C7#2674811yescharacteristic, complemented, Sylow
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#649511yescharacteristic
638C22#9049711yescharacteristic
639C22#9051311yescharacteristic
640C22#9051911yescharacteristic
641C22#9052011yescharacteristic
642C22#9052711yescharacteristic
643C22#9052911yescharacteristic
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—282no—
653C4—282no—
654C4—282no—
655C4—561no—
656C4#9036111yescharacteristic
657C4#9036911yescharacteristic
658C4#9037111yescharacteristic
659C4#9037211yescharacteristic
660C2—81no—
661C2—161no—
662C2#1174044211yescharacteristic
663C2#1174074611yescharacteristic
664C2#1174091211yescharacteristic
665C2#1174092911yescharacteristic
666C2#1174097211yescharacteristic
667C2#1174098211yescharacteristic
668C2#1174110111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2491C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2491D7
hypocenterC7#26748
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. #11730913C2semidirect
3. #11729548C2semidirect
4. #11729534C2semidirect
7. #11727414C2semidirect
8. #11727405C2semidirect
9. #11727399C2semidirect
10. #11727390C2semidirect
11. #11727281C2semidirect
12. #11727257C2semidirect
13. #11727229C2semidirect
14. #11727215C2semidirect
15. #11727206C2semidirect
622. C7#26748semidirect

Maximal subgroups

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

subgroupindex
C7×#24912
#117309132
#117295482
#117295342
#117294392
#117294322
#117274142
#117274052
#117273992
#117273902
#117272812
#117272572
#117272292
#117272152
#117272062
#267487