Finite Group Databasebeta

← #13284513

Subgroups — #13284513

Order 1792.

1326 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
1C2×#89745C211yesmaximal, characteristic, complemented
2C2×#89721C211yesmaximal, characteristic
3C2×#89723C211yesmaximal, characteristic
4C7×#1439C211yesmaximal, characteristic
5#11729321C211yesmaximal, characteristic, complemented
6#11729303C212yesmaximal
7#11729301C212yesmaximal
8#11727176C211yesmaximal, characteristic, complemented
9#11727164C212yesmaximal, complemented
10#11727163C212yesmaximal, complemented
11#11727155C211yesmaximal, characteristic, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×Dic7⋊C4C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35#89745—24no—
36#89723—24no—
37#89721C418yescomplemented
38#14628—71nomaximal, Sylow
39C25×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×Dic7C2311yescharacteristic
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—42no—
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4—42no—
55C2×C7×C22⋊C4—42no—
56C2×C7×C22⋊C4—42no—
57C2×C7×C22⋊C4C2×C414yes—
58C2×C22⋊Dic7—22no—
59C2×C22⋊Dic7—22no—
60C2×C22⋊Dic7—42no—
61C2×C22⋊Dic7—42no—
62C2×C22⋊Dic7—42no—
63C2×C22⋊Dic7—42no—
64C2×C22⋊Dic7C2×C418yes—
65C2×C4⋊Dic7—22no—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7—42no—
69C2×C4⋊Dic7C2×C414yes—
70C2×Dic7⋊C4—24no—
71C2×Dic7⋊C4—42no—
72C2×Dic7⋊C4—42no—
73C2×Dic7⋊C4—42no—
74C2×Dic7⋊C4—42no—
75C2×Dic7⋊C4C2×C418yes—
76C2×C4×Dic7—42no—
77C2×C4×Dic7D414yes—
78C7×#95—22no—
79C7×#95—42no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—24no—
83C7⋊#95—42no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C7⋊#95—42no—
89C7⋊#95C2×C414yes—
90C2×#399—71no—
91C2×#389—71no—
92C2×#388—71no—
93#1445—71no—
94#1442—71no—
95#1441—72no—
96#1441—72no—
97#1440—72no—
98#1439—72no—
99#1439D711yescharacteristic
100#1434—71no—
101C24×C7—21no—
102C24×C7—21no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—21no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—22no—
110C24×C7—21no—
111C24×C7C22×C412yes—
112C24×C7C2411yescharacteristic
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
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×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7#4112yes—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—22no—
166C22×Dic7—21no—
167C22×Dic7C22×C414yes—
168C22×Dic7C22×C414yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D412yes—
171C7×C4⋊C4—44no—
172C7×C22⋊C4—28no—
173C7×C22⋊C4—48no—
174C22⋊Dic7—48no—
175C22⋊Dic7—44no—
176C22⋊Dic7—44no—
177C4⋊Dic7—28no—
178C4⋊Dic7—44no—
179Dic7⋊C4—48no—
180Dic7⋊C4—48no—
181C4×Dic7—44no—
182C22×C4⋊C4—71no—
183C22×C4⋊C4—72no—
184C22×C4⋊C4—72no—
185C22×C4⋊C4—71no—
186C22×C4⋊C4C2×D711yescharacteristic
187C22×C22⋊C4—72no—
188C22×C22⋊C4—71no—
189C22×C22⋊C4—71no—
190C22×C22⋊C4C2×D711yescharacteristic
191C22×C22⋊C4C2×D712yes—
192C22×C42—71no—
193#399—78no—
194#389—144no—
195#388—144no—
196C2×#95—71no—
197C2×#95—72no—
198C2×#95—71no—
199C2×#95—71no—
200C2×#95—72no—
201C2×#95—72no—
202C2×#95—72no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95C2×D711yescharacteristic
207C2×#95C2×D712yes—
208C23×C7—22no—
209C23×C7—24no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—24no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7C23×C411yescharacteristic
232C23×C7C22×D411yescharacteristic
233C23×C7C2×#4111yescharacteristic
234C23×C7H(5,𝔽2)12yes—
235C23×C7H(5,𝔽2)12yes—
236C23×C7H(5,𝔽2)11yescharacteristic
237C23×C7H(5,𝔽2)12yes—
238C23×C72−1+411yescharacteristic
239C23×C72−1+411yescharacteristic
240C23×C72−1+411yescharacteristic
241C23×C72−1+412yes—
242C2×C4×C7—24no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—24no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7C4×D414yes—
266C2×Dic7—24no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C25C22×D711yescharacteristic
301C23×C4—72no—
302C23×C4—71no—
303C23×C4—72no—
304C23×C4—71no—
305C23×C4—71no—
306C23×C4—71no—
307C23×C4C22×D712yes—
308C23×C4C22×D712yes—
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D711yescharacteristic
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—74no—
314C2×C4⋊C4—78no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—144no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—282no—
320C2×C4⋊C4—282no—
321C2×C4⋊C4—282no—
322C2×C4⋊C4—282no—
323C2×C4⋊C4—282no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—78no—
330C2×C22⋊C4—142no—
331C2×C22⋊C4—142no—
332C2×C22⋊C4—282no—
333C2×C22⋊C4—282no—
334C2×C22⋊C4—282no—
335C2×C22⋊C4—282no—
336C2×C22⋊C4C4×D714yes—
337C2×C42—74no—
338C2×C42—282no—
339#95—22no—
340#95—42no—
341#95—74no—
342#95—142no—
343#95—144no—
344#95—142no—
345#95—282no—
346#95—282no—
347#95—282no—
348#95—282no—
349#95—282no—
350#95—282no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—44no—
355C22×C7—44no—
356C22×C7—81no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7—82no—
360C22×C7—82no—
361C22×C7—81no—
362C22×C7C2×C4×D411yescharacteristic
363C22×C7#51812yes—
364C22×C7#51911yescharacteristic
365C22×C7#52011yescharacteristic
366C22×C7#52012yes—
367C22×C7#53411yescharacteristic
368C22×C7#53511yescharacteristic
369C22×C7#53511yescharacteristic
370C22×C7#53611yescharacteristic
371C22×C7#53611yescharacteristic
372C22×C7#53711yescharacteristic
373C22×C7#53912yes—
374C22×C7#54012yes—
375C22×C7#54111yescharacteristic
376C22×C7#54211yescharacteristic
377C22×C7#56012yes—
378C22×C7#56212yes—
379C22×C7#56212yes—
380C22×C7#56212yes—
381C22×C7#56212yes—
382C22×C7#56312yes—
383C22×C7#56312yes—
384C22×C7#56312yes—
385C4×C7—24no—
386C4×C7—42no—
387C4×C7—82no—
388C4×C7—82no—
389Dic7—42no—
390Dic7—44no—
391Dic7—44no—
392Dic7—42no—
393Dic7—81no—
394Dic7—81no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24—21no—
401C24—22no—
402C24—21no—
403C24—21no—
404C24—22no—
405C24C2×C4×D712yes—
406C24C23×D711yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—74no—
428C22×C4—74no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—141no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—142no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4C2×C4×D712yes—
462C22×C4C2×C4×D712yes—
463C22×C4D4×D712yes—
464C22×C4#115912yes—
465C4⋊C4—44no—
466C4⋊C4—148no—
467C4⋊C4—288no—
468C4⋊C4—288no—
469C4⋊C4—284no—
470C22⋊C4—28no—
471C22⋊C4—48no—
472C22⋊C4—288no—
473C22⋊C4—284no—
474C22⋊C4—284no—
475C42—284no—
476C2×C7—82no—
477C2×C7#232211yescharacteristic
478C2×C7#232311yescharacteristic
479C2×C7#232811yescharacteristic
480C2×C7#237612yes—
481C2×C7#237812yes—
482C2×C7#281311yescharacteristic
483C2×C7#281611yescharacteristic
484C2×C7#282011yescharacteristic
485C2×C7#282111yescharacteristic
486C2×C7#283612yes—
487C2×C7#283712yes—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23C22×C4×D711yescharacteristic
512C23C2×D4×D711yescharacteristic
513C23C2×#115911yescharacteristic
514C23#654812yes—
515C23#654812yes—
516C23#654811yescharacteristic
517C23#655111yescharacteristic
518C23#655312yes—
519C23#655411yescharacteristic
520C23#655411yescharacteristic
521C23#655412yes—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—142no—
546C2×C4—144no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—144no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—282no—
558C2×C4—282no—
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—282no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—282no—
575C2×C4—281no—
576C2×C4—282no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4(C22⋊D7)⋊C414yes—
580C7#1462811yescharacteristic, complemented, Sylow
581C22—44no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—82no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22C2×(C22⋊D7)⋊C411yescharacteristic
593C22#9019511yescharacteristic
594C22#9020711yescharacteristic
595C22#9023612yes—
596C22#9023712yes—
597C22#9028011yescharacteristic
598C22#9028612yes—
599C22#9029312yes—
600C22#9029411yescharacteristic
601C22#9029911yescharacteristic
602C22#9031012yes—
603C22#9031212yes—
604C22#9031611yescharacteristic
605C22#9031911yescharacteristic
606C22#9033712yes—
607C22#9033812yes—
608C22#9033911yescharacteristic
609C22#9034212yes—
610C22#9035812yes—
611C22#9036511yescharacteristic
612C22#9037512yes—
613C22#9040312yes—
614C22#9040611yescharacteristic
615C4—24no—
616C4—42no—
617C4—82no—
618C4—82no—
619C4—282no—
620C4—284no—
621C4—282no—
622C4—284no—
623C4—561no—
624C4—561no—
625C2—82no—
626C2#1173131611yescharacteristic
627C2#1173204312yes—
628C2#1173210911yescharacteristic
629C2#1173293712yes—
630C2#1173374311yescharacteristic
631C2#1173422011yescharacteristic
632C2#1173429811yescharacteristic
633C2#1173454712yes—
634C2#1173485212yes—
635C2#1173509211yescharacteristic
636C2#1173553711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7#14628
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#89745C2semidirect
5. #11729321C2semidirect
8. #11727176C2semidirect
9. #11727164C2semidirect
10. #11727163C2semidirect
11. #11727155C2semidirect
37. #89721C4semidirect
580. C7#14628semidirect

Maximal subgroups

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

subgroupindex
C2×#897452
C2×#897212
C2×#897232
C7×#14392
#117293212
#117293032
#117293012
#117271762
#117271642
#117271632
#117271552
#146287