Finite Group Databasebeta

← #13284849

Subgroups — #13284849

Order 1792.

1530 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6442C211yesmaximal, characteristic
2C2×#89748C211yesmaximal, characteristic, complemented
3C2×#89423C211yesmaximal, characteristic, complemented
4C7×#1439C211yesmaximal, characteristic
5#11729320C211yesmaximal, characteristic, complemented
6#11729301C212yesmaximal
7#11729301C212yesmaximal
8#11729300C211yesmaximal, characteristic
9#11727165C212yesmaximal, complemented
10#11727164C211yesmaximal, characteristic, complemented
11#11727163C212yesmaximal, 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⋊C4C2211yescharacteristic
21C22×Dic7⋊C4C2211yescharacteristic
22C22×Dic7⋊C4C2212yes—
23C22×Dic7⋊C4C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C2×#6442—24no—
26C2×#6442—24no—
27C2×#6442C418yescomplemented
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89748—24no—
39#89423—24no—
40#14665—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—42no—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—42no—
57C2×C7×C22⋊C4—42no—
58C2×C7×C22⋊C4—42no—
59C2×C7×C22⋊C4C2×C414yes—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—42no—
65C2×C22⋊Dic7—42no—
66C2×C22⋊Dic7—42no—
67C2×C22⋊Dic7C2×C414yes—
68C2×C22⋊Dic7C2×C414yes—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—42no—
73C2×C4⋊Dic7C2×C414yes—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—22no—
77C2×Dic7⋊C4—22no—
78C2×Dic7⋊C4—24no—
79C2×Dic7⋊C4—24no—
80C2×Dic7⋊C4—42no—
81C2×Dic7⋊C4—42no—
82C2×Dic7⋊C4—42no—
83C2×Dic7⋊C4—42no—
84C2×Dic7⋊C4C2×C414yes—
85C2×Dic7⋊C4C2×C414yes—
86C2×C4×Dic7C2×C414yes—
87C2×C4×Dic7D414yes—
88C2×C4×Dic7D414yes—
89#6442—232no—
90C7×#95—22no—
91C7×#95—42no—
92C7⋊#95—22no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—42no—
96C7⋊#95—42no—
97C7⋊#95—42no—
98C7⋊#95—42no—
99C7⋊#95—42no—
100C22×#126—71no—
101C2×#384—71no—
102C2×#383—71no—
103#1441—72no—
104#1441—71no—
105#1441—72no—
106#1441—71no—
107#1439—71no—
108#1439—72no—
109#1439—72no—
110#1439D711yescharacteristic
111C24×C7—21no—
112C24×C7—21no—
113C24×C7—21no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—21no—
117C24×C7—21no—
118C24×C7—21no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7C22×C412yes—
122C24×C7C2411yescharacteristic
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
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×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D412yes—
147C22×Dic7—22no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7C22×C412yes—
178C22×Dic7C22×C412yes—
179C22×Dic7C22×C412yes—
180C22×Dic7C22×C412yes—
181C22×Dic7C2×D412yes—
182C22×Dic7C2×D412yes—
183C22×Dic7C2×D412yes—
184C22×Dic7C2×D412yes—
185C7×C4⋊C4—44no—
186C7×C22⋊C4—28no—
187C7×C22⋊C4—48no—
188C22⋊Dic7—28no—
189C22⋊Dic7—28no—
190C22⋊Dic7—48no—
191C4⋊Dic7—28no—
192C4⋊Dic7—44no—
193Dic7⋊C4—28no—
194Dic7⋊C4—28no—
195Dic7⋊C4—48no—
196Dic7⋊C4—48no—
197C4×Dic7C22⋊C4116yescomplemented
198C2×#126—78no—
199C2×#126—144no—
200C2×#126—144no—
201C22×C4⋊C4—71no—
202C22×C4⋊C4—71no—
203C22×C4⋊C4—71no—
204C22×C4⋊C4—72no—
205C22×C4⋊C4—72no—
206C22×C4⋊C4—71no—
207C22×C4⋊C4C2×D711yescharacteristic
208C22×C22⋊C4—71no—
209C22×C22⋊C4—71no—
210C22×C22⋊C4—72no—
211C22×C22⋊C4C2×D711yescharacteristic
212C22×C22⋊C4C2×D712yes—
213C22×C42—71no—
214#384—144no—
215#383—144no—
216C2×#95—72no—
217C2×#95—72no—
218C2×#95—71no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95C2×D711yescharacteristic
225C2×#95C2×D712yes—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—24no—
230C23×C7—24no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7C23×C411yescharacteristic
250C23×C7C22×D411yescharacteristic
251C23×C7C22×D411yescharacteristic
252C23×C7H(5,𝔽2)12yes—
253C23×C7H(5,𝔽2)11yescharacteristic
254C23×C7H(5,𝔽2)12yes—
255C23×C7H(5,𝔽2)11yescharacteristic
256C23×C72−1+411yescharacteristic
257C23×C72−1+412yes—
258C23×C72−1+412yes—
259C23×C72−1+411yescharacteristic
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—24no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7C2×C22⋊C414yes—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—24no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—24no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—42no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7C2×C22⋊C414yes—
321C2×Dic7C2×C22⋊C414yes—
322C25C22×D711yescharacteristic
323C23×C4—71no—
324C23×C4—71no—
325C23×C4—71no—
326C23×C4—71no—
327C23×C4—72no—
328C23×C4—72no—
329C23×C4C22×D712yes—
330C23×C4C22×D712yes—
331C23×C4C22×D711yescharacteristic
332C23×C4C22×D711yescharacteristic
333#126—1432no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4—74no—
338C2×C4⋊C4—74no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—142no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4—142no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—144no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—282no—
349C2×C4⋊C4—282no—
350C2×C4⋊C4—282no—
351C2×C4⋊C4—282no—
352C2×C4⋊C4—282no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—74no—
359C2×C22⋊C4—74no—
360C2×C22⋊C4—142no—
361C2×C22⋊C4—142no—
362C2×C22⋊C4—142no—
363C2×C22⋊C4—142no—
364C2×C22⋊C4—282no—
365C2×C22⋊C4—282no—
366C2×C22⋊C4—282no—
367C2×C22⋊C4C4×D714yes—
368C2×C42—74no—
369C2×C42—74no—
370C2×C42—74no—
371#95—22no—
372#95—42no—
373#95—144no—
374#95—144no—
375#95—142no—
376#95—282no—
377#95—282no—
378#95—282no—
379#95—282no—
380#95—282no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—44no—
385C22×C7—44no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7—81no—
390C22×C7—82no—
391C22×C7—82no—
392C22×C7C22×C22⋊C411yescharacteristic
393C22×C7#52011yescharacteristic
394C22×C7#52011yescharacteristic
395C22×C7#52012yes—
396C22×C7#52012yes—
397C22×C7#53511yescharacteristic
398C22×C7#53511yescharacteristic
399C22×C7#53511yescharacteristic
400C22×C7#53611yescharacteristic
401C22×C7#53611yescharacteristic
402C22×C7#53611yescharacteristic
403C22×C7#53712yes—
404C22×C7#53711yescharacteristic
405C22×C7#53712yes—
406C22×C7#53711yescharacteristic
407C22×C7#56012yes—
408C22×C7#56012yes—
409C22×C7#56312yes—
410C22×C7#56312yes—
411C22×C7#56312yes—
412C22×C7#56312yes—
413C22×C7#56312yes—
414C22×C7#56312yes—
415C4×C7—24no—
416C4×C7—42no—
417C4×C7—82no—
418C4×C7—82no—
419Dic7—24no—
420Dic7—24no—
421Dic7—42no—
422Dic7—42no—
423Dic7—82no—
424Dic7—82no—
425C24—21no—
426C24—21no—
427C24—21no—
428C24—22no—
429C24—22no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24C2×C4×D712yes—
436C24C23×D711yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—72no—
458C22×C4—72no—
459C22×C4—72no—
460C22×C4—72no—
461C22×C4—72no—
462C22×C4—72no—
463C22×C4—72no—
464C22×C4—72no—
465C22×C4—142no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—142no—
471C22×C4—141no—
472C22×C4—141no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—142no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4C2×C4×D712yes—
496C22×C4C2×C4×D712yes—
497C22×C4D4×D712yes—
498C22×C4D4×D712yes—
499C4⋊C4—44no—
500C4⋊C4—148no—
501C4⋊C4—148no—
502C4⋊C4—148no—
503C4⋊C4—284no—
504C4⋊C4—288no—
505C4⋊C4—288no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—148no—
509C22⋊C4—148no—
510C22⋊C4—288no—
511C42—716no—
512C2×C7—82no—
513C2×C7#231111yescharacteristic
514C2×C7#231111yescharacteristic
515C2×C7#231111yescharacteristic
516C2×C7#238312yes—
517C2×C7#238312yes—
518C2×C7#281811yescharacteristic
519C2×C7#282111yescharacteristic
520C2×C7#282111yescharacteristic
521C2×C7#282111yescharacteristic
522C2×C7#282312yes—
523C2×C7#282312yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23C22×C4×D711yescharacteristic
548C23C2×D4×D711yescharacteristic
549C23C2×D4×D711yescharacteristic
550C23#654812yes—
551C23#654811yescharacteristic
552C23#654812yes—
553C23#655111yescharacteristic
554C23#655311yescharacteristic
555C23#655412yes—
556C23#655412yes—
557C23#655411yescharacteristic
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—74no—
582C2×C4—74no—
583C2×C4—142no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—144no—
587C2×C4—142no—
588C2×C4—144no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—144no—
598C2×C4—281no—
599C2×C4—282no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—282no—
606C2×C4—282no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4—282no—
610C2×C4—282no—
611C2×C4—282no—
612C2×C4—282no—
613C2×C4—282no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—282no—
617C2×C4—282no—
618C2×C4—282no—
619C2×C4D7×C22⋊C414yes—
620C7#1466511yescharacteristic, complemented, Sylow
621C22—44no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22—81no—
631C22—81no—
632C22C2×D7×C22⋊C411yescharacteristic
633C22#9019511yescharacteristic
634C22#9021411yescharacteristic
635C22#9023712yes—
636C22#9023712yes—
637C22#9028311yescharacteristic
638C22#9028612yes—
639C22#9028812yes—
640C22#9029811yescharacteristic
641C22#9029911yescharacteristic
642C22#9029912yes—
643C22#9031112yes—
644C22#9031911yescharacteristic
645C22#9032011yescharacteristic
646C22#9033712yes—
647C22#9033712yes—
648C22#9033911yescharacteristic
649C22#9034212yes—
650C22#9034212yes—
651C22#9036312yes—
652C22#9036611yescharacteristic
653C22#9040312yes—
654C22#9040611yescharacteristic
655C4—24no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—144no—
660C4—144no—
661C4—282no—
662C4—282no—
663C4—562no—
664C4—562no—
665C2—82no—
666C2#1173144811yescharacteristic
667C2#1173184812yes—
668C2#1173210711yescharacteristic
669C2#1173293912yes—
670C2#1173373911yescharacteristic
671C2#1173418911yescharacteristic
672C2#1173429911yescharacteristic
673C2#1173453412yes—
674C2#1173489512yes—
675C2#1173509311yescharacteristic
676C2#1173553711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7#14665
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#89748C2semidirect
3. C2×#89423C2semidirect
5. #11729320C2semidirect
9. #11727165C2semidirect
10. #11727164C2semidirect
11. #11727163C2semidirect
27. C2×#6442C4semidirect
197. C4×Dic7C22⋊C4semidirect
620. C7#14665semidirect

Maximal subgroups

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

subgroupindex
C22×#64422
C2×#897482
C2×#894232
C7×#14392
#117293202
#117293012
#117293002
#117271652
#117271642
#117271632
#146657