Finite Group Databasebeta

← #13372001

Subgroups — #13372001

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739686C211yesmaximal, characteristic
2C2×#89770C211yesmaximal, characteristic, complemented
3C2×#89653C211yesmaximal, characteristic, complemented
4C2×#89502C211yesmaximal, characteristic
5#11731457C211yesmaximal, characteristic
6C7×#1854C211yesmaximal, characteristic, complemented
7#11728370C211yesmaximal, characteristic, complemented
8#11728333C211yesmaximal, characteristic, complemented
9#11728310C211yesmaximal, characteristic, complemented
10#11727683C211yesmaximal, characteristic, complemented
11#11727265C211yesmaximal, characteristic, complemented
12#11726631C211yesmaximal, characteristic, complemented
13#11726624C211yesmaximal, characteristic, complemented
14#11726550C211yesmaximal, characteristic, complemented
15#11726540C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C2×C4×C22⋊D7C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C23×C7⋊C8C2211yescharacteristic
20C22×#1131C2211yescharacteristic
21C2×#6455C2211yescharacteristic
22C2×D7×C4⋊C4C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#106C2211yescharacteristic
25C2×C7×C22⋊C8C2211yescharacteristic
26C7×#380C2211yescharacteristic
27#89984C2211yescharacteristic
28C2×(C2×D7)⋊C8C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33#89770—24no—
34#89759C2211yescharacteristic
35#89759C2211yescharacteristic
36#89749C2211yescharacteristic
37#89748C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89729—21no—
47#89728—21no—
48#89716—21no—
49#89716—21no—
50#89712—21no—
51C4×Dic7⋊C4—21no—
52#89653—24no—
53#89502C418yescomplemented
54#89438—21no—
55#89424—21no—
56C7×#340C2211yescharacteristic
57C7×#340C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22662—71nomaximal, Sylow
63C23×C4×C7C2311yescharacteristic
64C22×C22⋊D7D411yescharacteristic
65C22×C4×D7—21no—
66C22×C4×D7C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×#117—22no—
70C7×#117—41no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7D411yescharacteristic
81C4×C22⋊D7C2×C414yes—
82C4×C22⋊D7Q814yes—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4D411yescharacteristic
87#6489—22no—
88#6489—41no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4D411yescharacteristic
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C22×C7⋊C8—22no—
103C22×C7⋊C8—22no—
104C22×C7⋊C8C2×C414yescomplemented
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C2×#1131—22no—
108C2×#1131—22no—
109C2×#1131C2×C414yes—
110#6455—22no—
111#6455—41no—
112D7×C4⋊C4—22no—
113D7×C4⋊C4—41no—
114C7×#107—22no—
115C7×#106—41no—
116C7×C22⋊C8C2×C414yescomplemented
117C7×#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125(C2×D7)⋊C8C2×C414yes—
126#6392—41no—
127#6391—22no—
128#6389C2×C414yescomplemented
129Dic7⋊C8C2×C414yescomplemented
130#6382—22no—
131#6382—22no—
132#6382—22no—
133#6382—22no—
134#6381—22no—
135#6381D414yescomplemented
136C2×#435—71no—
137C2×#428—71no—
138C2×#427—71no—
139#2354—71no—
140#2352—71no—
141#1949—71no—
142#1948—71no—
143#1854—71no—
144#1854—71no—
145#1854D711yescharacteristic, complemented
146#1853—71no—
147#1847—71no—
148#1809—71no—
149#1807—71no—
150#1776—71no—
151C24×C7C2×D411yescharacteristic
152C23×D7C2×D411yescharacteristic
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×C7C22×C412yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×Q812yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—41no—
166C2×C22⋊D7—41no—
167C2×C22⋊D7C4⋊C414yes—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7C2×D411yescharacteristic
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C2×C4×D7—21no—
176C2×C4×D7—21no—
177C2×C4×D7—21no—
178C2×C4×D7—21no—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7—81no—
182C2×C4×D7C22×C412yes—
183C2×C4×D7C2×Q812yes—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8C22×C412yes—
189C2×C7×C8C22×C412yes—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4C2×D412yes—
198C7×C22⋊C4—42no—
199C7×C22⋊C4—81no—
200C42×C7—42no—
201C22⋊Dic7—42no—
202C22⋊Dic7—81no—
203C22⋊Dic7C4⋊C414yes—
204(C2×D7)⋊C4—42no—
205(C2×D7)⋊C4—42no—
206(C2×D7)⋊C4—81no—
207(C2×D7)⋊C4C4⋊C414yes—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7C2×D412yes—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218Dic7⋊C4—81no—
219Dic7⋊C4C4⋊C414yes—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C4×Dic7—81no—
224C4×Dic7C22×C412yes—
225C4×Dic7C2×Q812yes—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—28no—
229C2×C7⋊C8C22×C412yes—
230C2×C7⋊C8C22×C412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8#4112yes—
233#1131—28no—
234C22×M16—71no—
235C23×C8—71no—
236C2×C4×D4—71no—
237C2×#117—71no—
238C2×#117—71no—
239C2×#117C2×D711yescharacteristic
240C22×C4⋊C4—71no—
241#435—78no—
242#428—144no—
243#427—144no—
244C2×#107—71no—
245C2×#107—71no—
246C2×#107—71no—
247C2×#107C2×D711yescharacteristic
248C2×#106—71no—
249C2×#106—71no—
250C2×#106—71no—
251C2×#106—71no—
252C2×#106—71no—
253C2×#106C2×D711yescharacteristic
254C2×C4⋊C8—71no—
255C2×C22⋊C8—71no—
256C2×C22⋊C8C2×D711yescharacteristic
257C2×#97—71no—
258#390—141no—
259#388—141no—
260#388—141no—
261#388—141no—
262#387—141no—
263#386—141no—
264#384—71no—
265#382—71no—
266#380—71no—
267#380—71no—
268#380—71no—
269#380C2×D711yescharacteristic
270C4×C4⋊C4—71no—
271C4×C4⋊C4—71no—
272C4×C4⋊C4—141no—
273C4×C22⋊C4—141no—
274#340—71no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340C2×D711yescharacteristic
279#340C2×D711yescharacteristic
280C23×C7—21no—
281C23×C7—21no—
282C23×C7—21no—
283C23×C7—21no—
284C23×C7—21no—
285C23×C7—21no—
286C23×C7C2×C4⋊C412yes—
287C23×C7C22×D411yescharacteristic
288C22×D7—21no—
289C22×D7—21no—
290C22×D7—21no—
291C22×D7—21no—
292C22×D7—21no—
293C22×D7—21no—
294C22×D7C2×C4⋊C412yes—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
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×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7C2×C4⋊C412yes—
314C2×C4×C7C23×C411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×Q811yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7H(5,𝔽2)11yescharacteristic
319C2×C4×C72−1+411yescharacteristic
320C22⋊D7—44no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—22no—
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—
338C2×Dic7—41no—
339C2×Dic7C2×C4⋊C412yes—
340C2×Dic7C2×C4⋊C412yes—
341C2×Dic7C2×C4⋊C412yes—
342C4×D7—22no—
343C4×D7—22no—
344C4×D7—22no—
345C4×D7—22no—
346C4×D7—84no—
347C7×C8—22no—
348C7×C8—22no—
349C7⋊C8—22no—
350C7⋊C8C4×D414yes—
351C22×D4—71no—
352C23×C4—71no—
353C23×C4—141no—
354C23×C4C22×D711yescharacteristic
355C2×M16—74no—
356C2×M16—142no—
357C2×M16—142no—
358C22×C8—71no—
359C22×C8—74no—
360C22×C8—71no—
361C22×C8—142no—
362C22×C8—142no—
363C22×C8C22×D711yescharacteristic
364C22×C8C22×D711yescharacteristic
365C4×D4—74no—
366C4×D4—74no—
367#117—22no—
368#117—41no—
369#117—142no—
370#117—142no—
371#117—281no—
372#117—281no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—142no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—281no—
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—71no—
390C2×C22⋊C4—71no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C22⋊C4—141no—
396C2×C42—21no—
397C2×C42—71no—
398C2×C42—141no—
399C2×C42—141no—
400C2×C42—141no—
401C2×C42—141no—
402#107—22no—
403#107—74no—
404#107—142no—
405#107—281no—
406#106—41no—
407#106—142no—
408#106—142no—
409#106—142no—
410#106—142no—
411#106—142no—
412C4⋊C8—74no—
413C22⋊C8—74no—
414C22⋊C8C4×D714yescomplemented
415#97—74no—
416#95—21no—
417#95—141no—
418#95—141no—
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—41no—
428C22×C7—41no—
429C22×C7—41no—
430C22×C7—41no—
431C22×C7C22×C4⋊C411yescharacteristic
432C22×C7D4211yescharacteristic
433C22×C7#54711yescharacteristic
434C22×C7#57511yescharacteristic
435C22×C7#57511yescharacteristic
436C22×C7#57711yescharacteristic
437C22×C7#57711yescharacteristic
438C2×D7—22no—
439C2×D7—22no—
440C2×D7—22no—
441C2×D7—41no—
442C2×D7—41no—
443C2×D7—41no—
444C2×D7—41no—
445C4×C7—22no—
446C4×C7—42no—
447C4×C7—81no—
448C4×C7—81no—
449C4×C7—81no—
450C4×C7C2×C4×D411yescharacteristic
451C4×C7#52011yescharacteristic
452C4×C7D4×Q811yescharacteristic
453C4×C7#55411yescharacteristic
454Dic7—22no—
455Dic7—22no—
456Dic7—22no—
457Dic7—42no—
458Dic7—81no—
459Dic7—81no—
460Dic7—81no—
461C24—71no—
462C24D4×D711yescharacteristic
463C2×D4—74no—
464C2×D4—281no—
465C2×D4—281no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—72no—
478C22×C4—72no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4—561no—
490C22×C4C2×C4×D712yes—
491C22×C4D4×D711yescharacteristic
492C22×C4Q8×D712yes—
493C22×C4C23×D711yescharacteristic
494M16—148no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—72no—
500C2×C8—72no—
501C2×C8—72no—
502C2×C8—72no—
503C2×C8—148no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8C2×C4×D712yes—
507C2×C8C2×C4×D712yes—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—74no—
516C4⋊C4—72no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—141no—
522C4⋊C4—141no—
523C4⋊C4—141no—
524C4⋊C4—282no—
525C4⋊C4—282no—
526C4⋊C4—561no—
527C4⋊C4D4×D712yes—
528C22⋊C4—42no—
529C22⋊C4—74no—
530C22⋊C4—74no—
531C22⋊C4—81no—
532C22⋊C4—282no—
533C22⋊C4—282no—
534C22⋊C4—282no—
535C22⋊C4—561no—
536C22⋊C4—561no—
537C42—42no—
538C42—72no—
539C42—72no—
540C42—282no—
541C42—282no—
542C42—282no—
543C42—561no—
544C2×C7—42no—
545C2×C7D4×C4⋊C411yescharacteristic
546C2×C7#291111yescharacteristic
547C2×C7#291211yescharacteristic
548C2×C7#326711yescharacteristic
549C2×C7#327111yescharacteristic
550C2×C7#334811yescharacteristic
551C2×C7#335011yescharacteristic
552D7—42no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—72no—
560C23—141no—
561C23—141no—
562C23—141no—
563C23—141no—
564C23—141no—
565C23—141no—
566C23D7×C4⋊C412yes—
567C23C2×D4×D711yescharacteristic
568D4—284no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—72no—
588C2×C4—72no—
589C2×C4—72no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—141no—
595C2×C4—141no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—142no—
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—281no—
611C2×C4—281no—
612C2×C4—564no—
613C2×C4D7×C4⋊C412yes—
614C2×C4C22×C4×D711yescharacteristic
615C2×C4C2×D4×D711yescharacteristic
616C2×C4#654811yescharacteristic
617C2×C4C2×Q8×D711yescharacteristic
618C2×C4D7×#4111yescharacteristic
619C2×C4#655411yescharacteristic
620C8—22no—
621C8—22no—
622C8—74no—
623C8—142no—
624C7#2266211yescharacteristic, complemented, Sylow
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—142no—
633C22—142no—
634C22—142no—
635C22—281no—
636C22—281no—
637C22—281no—
638C22—281no—
639C22C2×D7×C4⋊C411yescharacteristic
640C22#9028111yescharacteristic
641C22#9034611yescharacteristic
642C22#9045911yescharacteristic
643C22#9045911yescharacteristic
644C22#9046611yescharacteristic
645C22#9046911yescharacteristic
646C4—22no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—142no—
652C4—142no—
653C4—142no—
654C4—282no—
655C4—561no—
656C4—561no—
657C4—561no—
658C4C4×D4×D711yescharacteristic
659C4#9023711yescharacteristic
660C4#9032311yescharacteristic
661C4#9032711yescharacteristic
662C2—42no—
663C2—282no—
664C2#1173183811yescharacteristic
665C2#1173662211yescharacteristic
666C2#1173665411yescharacteristic
667C2#1173756711yescharacteristic
668C2#1173771811yescharacteristic
669C2#1173883911yescharacteristic
670C2#1173898111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1854C2
socleC23×C7C22×D4
prime radicalC22×C22⋊D7D4
cosocleC22×C4×C7C24
hypercenter#1854D7
hypocenterC7#22662
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#89770C2semidirect
3. C2×#89653C2semidirect
6. C7×#1854C2semidirect
7. #11728370C2semidirect
8. #11728333C2semidirect
9. #11728310C2semidirect
10. #11727683C2semidirect
11. #11727265C2semidirect
12. #11726631C2semidirect
13. #11726624C2semidirect
14. #11726550C2semidirect
15. #11726540C2semidirect
53. #89502C4semidirect
104. C22×C7⋊C8C2×C4semidirect
116. C7×C22⋊C8C2×C4semidirect
128. #6389C2×C4semidirect
129. Dic7⋊C8C2×C4semidirect
135. #6381D4semidirect
145. #1854D7semidirect
414. C22⋊C8C4×D7semidirect
624. C7#22662semidirect

Maximal subgroups

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

subgroupindex
#117396862
C2×#897702
C2×#896532
C2×#895022
#117314572
C7×#18542
#117283702
#117283332
#117283102
#117276832
#117272652
#117266312
#117266242
#117265502
#117265402
#226627