Finite Group Databasebeta

← #13284480

Subgroups — #13284480

Order 1792.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#393C211yesmaximal, characteristic
2C2×#90002C211yesmaximal, characteristic, complemented
3C2×#89723C211yesmaximal, characteristic
4#11729888C211yesmaximal, characteristic, complemented
5#11729302C212yesmaximal
6#11729300C211yesmaximal, characteristic
7#11729300C212yesmaximal
8#11727173C211yesmaximal, characteristic, complemented
9#11727166C212yesmaximal, complemented
10#11727166C212yesmaximal, complemented
11#11727155C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×C4⋊Dic7C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C7×#393C418yescomplemented
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2212yes—
25#90002—24no—
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89723—24no—
38#14625—71nomaximal, Sylow
39C25×C7C2311yescharacteristic
40C22×C7×Q8—22no—
41C22×C7×Q8—22no—
42C22×C7×Q8—42no—
43C22×C7×Q8—42no—
44C22×C7×Q8C2×C414yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×Dic7C2312yes—
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2311yescharacteristic
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—22no—
57C2×C7×C22⋊C4—42no—
58C2×C7×C22⋊C4—42no—
59C2×C7×C22⋊C4C2×C414yes—
60C2×C7×C22⋊C4C2×C418yes—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—42no—
64C2×C22⋊Dic7—42no—
65C2×C22⋊Dic7—42no—
66C2×C22⋊Dic7—42no—
67C2×C22⋊Dic7—42no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—42no—
71C2×C4⋊Dic7—42no—
72C2×C4⋊Dic7—42no—
73C2×C4⋊Dic7—42no—
74C2×C4⋊Dic7—42no—
75C2×C4×Dic7—42no—
76C2×C4×Dic7D414yes—
77C7×#95C2×C418yes—
78C7×#95C2×C414yes—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C7⋊#95—42no—
89C7⋊#95—42no—
90C2×#393D711yescharacteristic
91C2×#391—71no—
92C2×#388—71no—
93#1446—71no—
94#1444—71no—
95#1442—72no—
96#1442—72no—
97#1440—72no—
98#1439—72no—
99#1439—71no—
100#1434—71no—
101C24×C7—21no—
102C24×C7—21no—
103C24×C7—22no—
104C24×C7—21no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—21no—
110C24×C7—22no—
111C24×C7C22×C412yes—
112C24×C7C2411yescharacteristic
113C2×C7×Q8—28no—
114C2×C7×Q8—48no—
115C2×C7×Q8—44no—
116C2×C7×Q8—44no—
117C2×C7×Q8—48no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C22×C414yes—
138C22×C4×C7C22×C414yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7#4112yes—
142C22×Dic7—21no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—22no—
149C22×Dic7—21no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—22no—
171C22×Dic7—21no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7C2×D412yes—
175C22×Dic7C2×D412yes—
176C7×C22⋊C4—28no—
177C7×C22⋊C4—48no—
178C22⋊Dic7—44no—
179C22⋊Dic7—44no—
180C22⋊Dic7—48no—
181C4⋊Dic7—44no—
182C4⋊Dic7—44no—
183C4⋊Dic7—48no—
184C4×Dic7—44no—
185C23×Q8C2×D711yescharacteristic
186C22×C4⋊C4—71no—
187C22×C4⋊C4—71no—
188C22×C4⋊C4—72no—
189C22×C22⋊C4—71no—
190C22×C22⋊C4—71no—
191C22×C22⋊C4—72no—
192C22×C22⋊C4C2×D711yescharacteristic
193C22×C22⋊C4C2×D712yes—
194C22×C42—71no—
195#393Dic718yescomplemented
196#391—144no—
197#388—144no—
198C2×#95—71no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95—71no—
202C2×#95—71no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95C2×D711yescharacteristic
210C2×#95C2×D712yes—
211C23×C7—22no—
212C23×C7—24no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—24no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—42no—
223C23×C7—41no—
224C23×C7—42no—
225C23×C7—41no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7C23×C411yescharacteristic
235C23×C7C22×D411yescharacteristic
236C23×C7C2×#4111yescharacteristic
237C23×C7H(5,𝔽2)12yes—
238C23×C7H(5,𝔽2)12yes—
239C23×C7H(5,𝔽2)12yes—
240C23×C7H(5,𝔽2)11yescharacteristic
241C23×C72−1+411yescharacteristic
242C23×C72−1+411yescharacteristic
243C23×C72−1+411yescharacteristic
244C23×C72−1+412yes—
245C7×Q8—416no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—42no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—41no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7C4×D414yes—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—41no—
281C2×Dic7—42no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—42no—
286C2×Dic7—41no—
287C2×Dic7—42no—
288C2×Dic7—41no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C25C22×D711yescharacteristic
305C22×Q8—22no—
306C22×Q8—22no—
307C22×Q8—42no—
308C22×Q8—42no—
309C22×Q8C2×Dic714yes—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—72no—
314C23×C4—71no—
315C23×C4—72no—
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D712yes—
318C23×C4C22×D712yes—
319C23×C4C22×D711yescharacteristic
320C2×C4⋊C4—142no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—282no—
323C2×C4⋊C4—282no—
324C2×C4⋊C4—282no—
325C2×C4⋊C4—282no—
326C2×C4⋊C4—282no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—142no—
332C2×C22⋊C4—142no—
333C2×C22⋊C4—282no—
334C2×C22⋊C4—282no—
335C2×C22⋊C4—282no—
336C2×C22⋊C4—282no—
337C2×C22⋊C4—282no—
338C2×C22⋊C4C2×Dic714yes—
339C2×C22⋊C4C2×Dic718yes—
340C2×C42—74no—
341C2×C42—282no—
342#95—142no—
343#95—144no—
344#95—144no—
345#95—142no—
346#95—142no—
347#95—282no—
348#95—282no—
349#95—282no—
350#95—282no—
351#95—282no—
352#95—282no—
353#95C2×Dic718yes—
354#95C2×Dic714yes—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—44no—
359C22×C7—44no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—81no—
366C22×C7C2×C4×D411yescharacteristic
367C22×C7#51812yes—
368C22×C7#51911yescharacteristic
369C22×C7#52011yescharacteristic
370C22×C7#52012yes—
371C22×C7#53411yescharacteristic
372C22×C7#53511yescharacteristic
373C22×C7#53511yescharacteristic
374C22×C7#53611yescharacteristic
375C22×C7#53611yescharacteristic
376C22×C7#53711yescharacteristic
377C22×C7#53812yes—
378C22×C7#54111yescharacteristic
379C22×C7#54212yes—
380C22×C7#54211yescharacteristic
381C22×C7H(3,𝔽4)12yes—
382C22×C7#56212yes—
383C22×C7#56212yes—
384C22×C7#56212yes—
385C22×C7#56212yes—
386C22×C7#56312yes—
387C22×C7#56312yes—
388C22×C7#56312yes—
389C4×C7—24no—
390C4×C7—44no—
391C4×C7—42no—
392C4×C7—44no—
393Dic7—42no—
394Dic7—42no—
395Dic7—81no—
396Dic7—82no—
397Dic7—81no—
398Dic7—82no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—22no—
408C24—21no—
409C24C22×Dic712yes—
410C24C23×D711yescharacteristic
411C2×Q8—28no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—72no—
435C22×C4—72no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—142no—
442C22×C4—141no—
443C22×C4—142no—
444C22×C4—141no—
445C22×C4—142no—
446C22×C4—141no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—141no—
466C22×C4—142no—
467C22×C4—141no—
468C22×C4D4×D712yes—
469C22×C4#115912yes—
470C22×C4C22×Dic712yes—
471C22×C4C22×Dic714yes—
472C22×C4C22×Dic714yes—
473C22×C4C22×Dic712yes—
474C4⋊C4—284no—
475C4⋊C4—284no—
476C4⋊C4—288no—
477C22⋊C4—28no—
478C22⋊C4—48no—
479C22⋊C4—284no—
480C22⋊C4—284no—
481C22⋊C4—288no—
482C42—284no—
483C2×C7—82no—
484C2×C7#232111yescharacteristic
485C2×C7#232311yescharacteristic
486C2×C7#232611yescharacteristic
487C2×C7#237812yes—
488C2×C7#237912yes—
489C2×C7#281411yescharacteristic
490C2×C7#281611yescharacteristic
491C2×C7#281911yescharacteristic
492C2×C7#282111yescharacteristic
493C2×C7#283112yes—
494C2×C7#283212yes—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23C2×D4×D711yescharacteristic
519C23C2×#115911yescharacteristic
520C23#654812yes—
521C23#654812yes—
522C23#655111yescharacteristic
523C23#655111yescharacteristic
524C23#655312yes—
525C23#655311yescharacteristic
526C23#655412yes—
527C23#655411yescharacteristic
528C23C23×Dic711yescharacteristic
529Q8—416no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—282no—
560C2×C4—282no—
561C2×C4—282no—
562C2×C4—282no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—281no—
571C2×C4—282no—
572C2×C4—281no—
573C2×C4—282no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—282no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4—282no—
580C2×C4—282no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—282no—
587C2×C4D4×Dic714yes—
588C7#1462511yescharacteristic, complemented, Sylow
589C22—42no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—81no—
595C22—81no—
596C22—82no—
597C22—82no—
598C22—81no—
599C22—81no—
600C22#9028011yescharacteristic
601C22#9028311yescharacteristic
602C22#9028412yes—
603C22#9031611yescharacteristic
604C22#9031911yescharacteristic
605C22#9032011yescharacteristic
606C22#9033411yescharacteristic
607C22#9033712yes—
608C22#9033712yes—
609C22#9033812yes—
610C22#9033812yes—
611C22#9036312yes—
612C22#9036511yescharacteristic
613C22#9036611yescharacteristic
614C22#9037312yes—
615C22#9037412yes—
616C22#9037512yes—
617C22#9037512yes—
618C22C2×D4×Dic711yescharacteristic
619C22#9048912yes—
620C22#9050311yescharacteristic
621C22#9051811yescharacteristic
622C22#9051812yes—
623C4—24no—
624C4—44no—
625C4—42no—
626C4—44no—
627C4—282no—
628C4—282no—
629C4—561no—
630C4—562no—
631C4—562no—
632C4—561no—
633C2—82no—
634C2#1173429811yescharacteristic
635C2#1173429911yescharacteristic
636C2#1173430112yes—
637C2#1173430212yes—
638C2#1173432911yescharacteristic
639C2#1173433011yescharacteristic
640C2#1174002311yescharacteristic
641C2#1174018911yescharacteristic
642C2#1174019512yes—
643C2#1174039811yescharacteristic
644C2#1174040212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#393C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#393D7
hypocenterC7#14625
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#90002C2semidirect
4. #11729888C2semidirect
8. #11727173C2semidirect
9. #11727166C2semidirect
10. #11727166C2semidirect
11. #11727155C2semidirect
22. C7×#393C4semidirect
195. #393Dic7semidirect
588. C7#14625semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C2×#900022
C2×#897232
#117298882
#117293022
#117293002
#117271732
#117271662
#117271552
#146257