Finite Group Databasebeta

← #13411018

Subgroups — #13411018

Order 1792.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2341C211yesmaximal, characteristic
2C2×#90005C211yesmaximal, characteristic
3C2×#89794C211yesmaximal, characteristic, complemented
4#11730939C211yesmaximal, characteristic, complemented
5#11729889C211yesmaximal, characteristic
6#11729382C211yesmaximal, characteristic, complemented
7#11729382C211yesmaximal, characteristic, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11727409C211yesmaximal, characteristic, complemented
10#11727407C211yesmaximal, characteristic, complemented
11#11727406C211yesmaximal, characteristic, complemented
12#11727393C211yesmaximal, characteristic, complemented
13#11727266C211yesmaximal, characteristic, complemented
14#11727260C211yesmaximal, characteristic, complemented
15#11727203C211yesmaximal, characteristic, complemented
16C22×C7×#41C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C23×C7⋊C8C2211yescharacteristic
21C2×#6433C2211yescharacteristic
22C7×#390—21no—
23C7×#390—21no—
24C7×#390C2211yescharacteristic
25C7×#390C2211yescharacteristic
26C7×#380—21no—
27C7×#380C2211yescharacteristic
28C4×C7×C22⋊C4—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30#90005—24no—
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C2×#6406C2211yescharacteristic
37C2×#6406C2211yescharacteristic
38C2×#6406C2211yescharacteristic
39#89984C2211yescharacteristic
40C2×C22⋊(C7⋊C8)C2211yescharacteristic
41#89794—24no—
42C2×#6382C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89721—21no—
47#89720—21no—
48#89707C2211yescharacteristic
49#89700C2211yescharacteristic
50#89700C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89426—21no—
54#89419—21no—
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#25961—71nomaximal, Sylow
62C2×C7×#41—22no—
63C2×C7×#41—22no—
64C2×C7×#41—22no—
65C2×C7×#41—41no—
66C2×C7×#41—41no—
67C2×C7×#41Q814yes—
68C22×C7×Q8C2311yescharacteristic
69C22×C7×D4—21no—
70C22×C7×D4C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7C2311yescharacteristic
73C4×C7×Q8—22no—
74C4×C7×Q8—22no—
75C4×C7×Q8—41no—
76C4×C7×Q8—41no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
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×C7—21no—
90C2×C42×C7C2311yescharacteristic
91C2×C42×C7C2311yescharacteristic
92C2×C22⋊Dic7—21no—
93C2×C22⋊Dic7—21no—
94#6489—22no—
95#6489—41no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4—21no—
104C2×Dic7⋊C4—21no—
105C2×Dic7⋊C4—21no—
106C2×C4×Dic7—21no—
107C22×C7⋊C8—22no—
108C22×C7⋊C8—22no—
109C22×C7⋊C8—41no—
110C22×C7⋊C8C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C22×C7⋊C8C2311yescharacteristic
113C22×C7⋊C8C2311yescharacteristic
114#6433—22no—
115#6433—41no—
116C7×#95—21no—
117C7×#95D411yescharacteristic
118C7×#95D411yescharacteristic
119#6409—22no—
120#6409—22no—
121#6409—22no—
122#6409D414yes—
123#6406—22no—
124#6406—22no—
125#6406—41no—
126#6406—41no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95—21no—
130C22⋊(C7⋊C8)—22no—
131#6382—22no—
132#6381—22no—
133#6381—41no—
134#6381—41no—
135C4⋊(C7⋊C8)—22no—
136C4⋊(C7⋊C8)—41no—
137C2×#479—71no—
138C2×#416—71no—
139#2441—71no—
140#2341D711yescharacteristic
141#2093—71no—
142#2067—71no—
143#2065—71no—
144#2025—71no—
145#1863—71no—
146#1863—71no—
147#1854—71no—
148#1850—71no—
149#1808—71no—
150#1803—71no—
151#1788—71no—
152C24×C7—21no—
153C24×C7C2×D411yescharacteristic
154C7×#41—28no—
155C7×#41—44no—
156C7×#41—44no—
157C7×#41—44no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8C2×D412yes—
162C2×C7×Q8C2×Q812yes—
163C2×C7×Q8#4112yes—
164C2×C7×D4—24no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—42no—
171C2×C7×D4—42no—
172C2×C7×D4C2×Q812yes—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—24no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—42no—
182C22×C4×C7—42no—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×Q812yes—
189C22×C4×C7#4112yes—
190C22×C4×C7C2411yescharacteristic
191C22×Dic7—21no—
192C22×Dic7—21no—
193C22×Dic7—21no—
194C22×Dic7—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—42no—
200C7×C4⋊C4—42no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—41no—
203C7×C22⋊C4—42no—
204C7×C22⋊C4—42no—
205C7×C22⋊C4—81no—
206C7×C22⋊C4—81no—
207C7×C22⋊C4—81no—
208C42×C7—21no—
209C42×C7—21no—
210C42×C7—21no—
211C42×C7—21no—
212C42×C7—42no—
213C42×C7—42no—
214C22⋊Dic7—42no—
215C22⋊Dic7—81no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—42no—
224C4⋊Dic7—81no—
225C4⋊Dic7C2×D412yes—
226Dic7⋊C4—42no—
227Dic7⋊C4—42no—
228Dic7⋊C4—42no—
229Dic7⋊C4—81no—
230C4×Dic7—42no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8—44no—
243C2×C7⋊C8C2×D412yes—
244C22×#41C2×D711yescharacteristic
245C23×C8—71no—
246C2×#125—71no—
247C2×C4×Q8C2×D711yescharacteristic
248C2×C4×Q8C2×D711yescharacteristic
249C2×#117—71no—
250#479—144no—
251C2×#107—71no—
252C2×#107—71no—
253C2×#107—71no—
254C2×#106—71no—
255C2×C4⋊C8—71no—
256C2×C4⋊C8—71no—
257#416—144no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#102—71no—
263C2×#102—71no—
264C2×#102—71no—
265C2×#102—71no—
266C2×C22⋊C8—71no—
267#399—141no—
268#396—141no—
269#390—21no—
270#390—21no—
271#390C2×D711yescharacteristic
272#390C2×D711yescharacteristic
273#389—71no—
274#389—71no—
275#389—141no—
276#387—141no—
277#384—71no—
278#380—21no—
279#380—71no—
280#380C2×D711yescharacteristic
281C4×C22⋊C4—21no—
282C4×C22⋊C4C2×D711yescharacteristic
283#340—71no—
284#340—71no—
285#340—71no—
286#340—71no—
287#336—71no—
288#336—71no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—21no—
293C23×C7—21no—
294C23×C7—21no—
295C23×C7—22no—
296C23×C7—41no—
297C23×C7—41no—
298C23×C7—41no—
299C23×C7—41no—
300C23×C7—41no—
301C23×C7—41no—
302C23×C7C22⋊Q812yes—
303C23×C7C22×D411yescharacteristic
304C7×Q8—22no—
305C7×Q8—22no—
306C7×Q8—22no—
307C7×Q8C22⋊Q814yes—
308C7×D4—22no—
309C7×D4—22no—
310C7×D4—48no—
311C7×D4—41no—
312C7×D4—41no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—22no—
316C2×C4×C7—21no—
317C2×C4×C7—22no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—22no—
321C2×C4×C7—22no—
322C2×C4×C7—22no—
323C2×C4×C7—22no—
324C2×C4×C7—21no—
325C2×C4×C7—21no—
326C2×C4×C7—21no—
327C2×C4×C7—21no—
328C2×C4×C7—21no—
329C2×C4×C7—21no—
330C2×C4×C7—21no—
331C2×C4×C7—21no—
332C2×C4×C7—21no—
333C2×C4×C7—48no—
334C2×C4×C7—41no—
335C2×C4×C7—41no—
336C2×C4×C7—41no—
337C2×C4×C7—41no—
338C2×C4×C7—41no—
339C2×C4×C7—41no—
340C2×C4×C7C22⋊Q812yes—
341C2×C4×C7#13512yes—
342C2×C4×C7C22×D411yescharacteristic
343C2×C4×C7C22×Q811yescharacteristic
344C2×C4×C7C2×#4111yescharacteristic
345C2×C4×C7C2×#4111yescharacteristic
346C2×C4×C7H(5,𝔽2)11yescharacteristic
347C2×C4×C7H(5,𝔽2)11yescharacteristic
348C2×Dic7—22no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C2×Dic7—41no—
353C2×Dic7—41no—
354C2×Dic7—41no—
355C2×Dic7—41no—
356C2×Dic7—41no—
357C2×Dic7—41no—
358C2×Dic7—41no—
359C2×Dic7—41no—
360C7⋊C8—22no—
361C7⋊C8—41no—
362C7⋊C8—41no—
363C7⋊C8—41no—
364C2×#41—22no—
365C2×#41—22no—
366C2×#41—22no—
367C2×#41—41no—
368C2×#41—41no—
369C2×#41Dic1414yes—
370C22×Q8C22×D711yescharacteristic
371C22×D4—21no—
372C22×D4C22×D711yescharacteristic
373C23×C4—21no—
374C23×C4C22×D711yescharacteristic
375C22×C8—71no—
376C22×C8—71no—
377C22×C8—71no—
378C22×C8—71no—
379C22×C8—142no—
380C22×C8—142no—
381C22×C8—281no—
382#125—142no—
383#125—281no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—41no—
387C4×Q8—41no—
388#117—142no—
389#117—281no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—71no—
393C2×C4⋊C4—71no—
394C2×C4⋊C4—71no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—141no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4C22×D711yescharacteristic
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—141no—
410C2×C22⋊C4—141no—
411C2×C22⋊C4C22⋊D711yescharacteristic
412C2×C22⋊C4C22⋊D711yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—141no—
416C2×C42C22×D711yescharacteristic
417C2×C42C22×D711yescharacteristic
418#107—142no—
419#107—281no—
420#107—281no—
421#106—142no—
422C4⋊C8—142no—
423C4⋊C8—281no—
424#103—74no—
425#103—142no—
426#103—142no—
427#103—142no—
428#102—142no—
429#102—142no—
430#102—281no—
431#102—281no—
432C22⋊C8—142no—
433#95—21no—
434#95—141no—
435#95—141no—
436#95—141no—
437#95C22⋊D711yescharacteristic
438#95C22⋊D711yescharacteristic
439C22×C7—22no—
440C22×C7—22no—
441C22×C7—22no—
442C22×C7—41no—
443C22×C7—41no—
444C22×C7—41no—
445C22×C7—41no—
446C22×C7—42no—
447C22×C7—42no—
448C22×C7—42no—
449C22×C7—81no—
450C22×C7—81no—
451C22×C7—81no—
452C22×C7—81no—
453C22×C7C2×C22⋊Q811yescharacteristic
454C22×C7D4211yescharacteristic
455C22×C7#54611yescharacteristic
456C22×C7C2×#13511yescharacteristic
457C22×C7#57511yescharacteristic
458C22×C7#57611yescharacteristic
459C22×C7#57811yescharacteristic
460C4×C7—22no—
461C4×C7—24no—
462C4×C7—22no—
463C4×C7—41no—
464C4×C7—41no—
465C4×C7—41no—
466C4×C7—81no—
467C4×C7—81no—
468C4×C7C2×C22⋊Q811yescharacteristic
469C4×C7#54311yescharacteristic
470C4×C7#54611yescharacteristic
471C4×C7#55111yescharacteristic
472Dic7—42no—
473Dic7—81no—
474Dic7—81no—
475Dic7—81no—
476C24—21no—
477C24C2×C22⋊D711yescharacteristic
478#41—28no—
479#41—44no—
480#41—44no—
481#41—44no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8C2×Dic1412yes—
486C2×Q8D4×D712yes—
487C2×Q8#115912yes—
488C2×D4—24no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4C2×Dic1412yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—24no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—141no—
508C22×C4—141no—
509C22×C4—141no—
510C22×C4—141no—
511C22×C4C2×Dic1412yes—
512C22×C4#115712yes—
513C22×C4C2×C22⋊D711yescharacteristic
514C22×C4C2×C22⋊D711yescharacteristic
515C22×C4C2×C22⋊D711yescharacteristic
516C22×C4C2×C22⋊D711yescharacteristic
517C22×C4C2×C22⋊D711yescharacteristic
518C22×C4C23×D711yescharacteristic
519C2×C8—72no—
520C2×C8—141no—
521C2×C8—141no—
522C2×C8—141no—
523C2×C8—141no—
524C2×C8—141no—
525C2×C8—141no—
526C2×C8—141no—
527C2×C8—141no—
528C2×C8—141no—
529C2×C8—141no—
530C2×C8—141no—
531C2×C8—284no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—72no—
539C4⋊C4—141no—
540C4⋊C4—141no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—141no—
544C4⋊C4—141no—
545C4⋊C4—141no—
546C4⋊C4—282no—
547C4⋊C4—282no—
548C4⋊C4—282no—
549C4⋊C4—282no—
550C4⋊C4—561no—
551C4⋊C4—561no—
552C22⋊C4—22no—
553C22⋊C4—41no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—81no—
557C22⋊C4—81no—
558C22⋊C4—81no—
559C22⋊C4—282no—
560C22⋊C4—561no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—42no—
567C42—282no—
568C2×C7—42no—
569C2×C7—82no—
570C2×C7#243711yescharacteristic
571C2×C7#307611yescharacteristic
572C2×C7#308311yescharacteristic
573C2×C7#326911yescharacteristic
574C2×C7#328711yescharacteristic
575C2×C7#330011yescharacteristic
576C2×C7#330911yescharacteristic
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23#648712yes—
591C23C22×C22⋊D711yescharacteristic
592Q8—22no—
593Q8—22no—
594Q8—22no—
595Q8#644114yes—
596D4—22no—
597D4—22no—
598D4—48no—
599D4—41no—
600D4—41no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—48no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—142no—
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—281no—
638C2×C4—281no—
639C2×C4—281no—
640C2×C4#648712yes—
641C2×C4#651312yes—
642C2×C4C22×Dic1411yescharacteristic
643C2×C4C2×#115711yescharacteristic
644C2×C4C2×D4×D711yescharacteristic
645C2×C4C2×#115911yescharacteristic
646C2×C4#654811yescharacteristic
647C2×C4#654811yescharacteristic
648C8—142no—
649C8—281no—
650C8—281no—
651C8—281no—
652C7#2596111yescharacteristic, complemented, Sylow
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22C2×#648711yescharacteristic
668C22D4×C22⋊D711yescharacteristic
669C22#9049711yescharacteristic
670C22C2×#651311yescharacteristic
671C22#9051311yescharacteristic
672C22#9052611yescharacteristic
673C22#9052911yescharacteristic
674C4—22no—
675C4—22no—
676C4—24no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—81no—
681C4—81no—
682C4—282no—
683C4—561no—
684C4—561no—
685C4—561no—
686C4C2×#644111yescharacteristic
687C4#9017411yescharacteristic
688C4#9018111yescharacteristic
689C4#9018611yescharacteristic
690C2—42no—
691C2—82no—
692C2#1173955511yescharacteristic
693C2#1173995211yescharacteristic
694C2#1174000411yescharacteristic
695C2#1174097911yescharacteristic
696C2#1174098811yescharacteristic
697C2#1174101711yescharacteristic
698C2#1174103011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2341C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenter#2341D7
hypocenterC7#25961
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
3. C2×#89794C2semidirect
4. #11730939C2semidirect
6. #11729382C2semidirect
7. #11729382C2semidirect
8. #11729380C2semidirect
9. #11727409C2semidirect
10. #11727407C2semidirect
11. #11727406C2semidirect
12. #11727393C2semidirect
13. #11727266C2semidirect
14. #11727260C2semidirect
15. #11727203C2semidirect
652. C7#25961semidirect

Maximal subgroups

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

subgroupindex
C7×#23412
C2×#900052
C2×#897942
#117309392
#117298892
#117293822
#117293802
#117274092
#117274072
#117274062
#117273932
#117272662
#117272602
#117272032
#259617