Finite Group Databasebeta

← #13280844

Subgroups — #13280844

Order 1792.

1650 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89994C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic
3C2×#89424C211yesmaximal, characteristic, complemented
4C7×#1438C211yesmaximal, characteristic
5#11729303C212yesmaximal, complemented
6#11729301C212yesmaximal, complemented
7#11729299C212yesmaximal, complemented
8#11729299C212yesmaximal, complemented
9#11727169C211yesmaximal, characteristic, complemented
10#11727164C211yesmaximal, characteristic, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2212yes—
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2211yescharacteristic
21C22×Dic7⋊C4C2211yescharacteristic
22C22×C4×Dic7C2212yes—
23C22×C4×Dic7C2211yescharacteristic
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2211yescharacteristic
26#89994C418yescomplemented
27#89993—24no—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2211yescharacteristic
37#89424—24no—
38#14325—71nomaximal, Sylow
39C25×C7C2312yes—
40C22×C7×D4—22no—
41C22×C7×D4—22no—
42C22×C7×D4—42no—
43C22×C7×D4—42no—
44C22×C7×D4C2×C414yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×Dic7C2312yes—
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2311yescharacteristic
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2311yescharacteristic
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4—42no—
58C2×C7×C22⋊C4—42no—
59C2×C7×C22⋊C4—42no—
60C2×C7×C22⋊C4—42no—
61C2×C22⋊Dic7—24no—
62C2×C22⋊Dic7—24no—
63C2×C22⋊Dic7—24no—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—42no—
66C2×C22⋊Dic7—42no—
67C2×C22⋊Dic7—42no—
68C2×C22⋊Dic7—42no—
69C2×C22⋊Dic7—42no—
70C2×C22⋊Dic7—42no—
71C2×C22⋊Dic7C2×C418yes—
72C2×C22⋊Dic7C2×C418yes—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—42no—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—22no—
77C2×Dic7⋊C4—22no—
78C2×Dic7⋊C4—42no—
79C2×Dic7⋊C4C2×C414yes—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—42no—
82C2×C4×Dic7—42no—
83C2×C4×Dic7—42no—
84C2×C4×Dic7D414yes—
85C7×#95—22no—
86C7×#95—22no—
87C7⋊#95—24no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91C7⋊#95—42no—
92C7⋊#95—42no—
93C7⋊#95—42no—
94C7⋊#95—42no—
95C7⋊#95C2×C414yes—
96C2×#394—71no—
97C2×#390—71no—
98C2×#387—71no—
99#1442—71no—
100#1441—71no—
101#1440—72no—
102#1439—72no—
103#1438D711yescharacteristic
104#1437—72no—
105#1437—72no—
106C4×#95—71no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7C22×C414yes—
122C24×C7C2411yescharacteristic
123C2×C7×D4—28no—
124C2×C7×D4—48no—
125C2×C7×D4—44no—
126C2×C7×D4—44no—
127C2×C7×D4—48no—
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—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C22×C412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7#4114yes—
149C22×C4×C7#4112yes—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—22no—
180C22×Dic7C22×C414yes—
181C22×Dic7C22×C412yes—
182C22×Dic7C22×C412yes—
183C22×Dic7C2×D412yes—
184C22×Dic7C2×D412yes—
185C22×Dic7#4114yes—
186C7×C22⋊C4—48no—
187C7×C22⋊C4—48no—
188C22⋊Dic7—48no—
189C22⋊Dic7—48no—
190C22⋊Dic7—48no—
191C22⋊Dic7—48no—
192C4⋊Dic7—44no—
193Dic7⋊C4—28no—
194Dic7⋊C4—44no—
195C4×Dic7—44no—
196C4×Dic7—48no—
197C23×D4C2×D711yescharacteristic
198C22×C4⋊C4—71no—
199C22×C4⋊C4—71no—
200C22×C4⋊C4—71no—
201C22×C22⋊C4—72no—
202C22×C22⋊C4—72no—
203C22×C22⋊C4—72no—
204C22×C22⋊C4—72no—
205C22×C22⋊C4C2×D712yes—
206C22×C22⋊C4C2×D712yes—
207C22×C42—71no—
208C22×C42—72no—
209#394—78no—
210#390—144no—
211#387—144no—
212C2×#95—71no—
213C2×#95—72no—
214C2×#95—71no—
215C2×#95—71no—
216C2×#95—71no—
217C2×#95—72no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95C2×D711yescharacteristic
222C2×#95C2×D711yescharacteristic
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—24no—
229C23×C7—24no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
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×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7C23×C411yescharacteristic
259C23×C7C22×D411yescharacteristic
260C23×C7C2×#4111yescharacteristic
261C23×C7C2×#4112yes—
262C23×C7C2×#4112yes—
263C23×C7H(5,𝔽2)12yes—
264C23×C7H(5,𝔽2)12yes—
265C23×C7H(5,𝔽2)11yescharacteristic
266C23×C7H(5,𝔽2)11yescharacteristic
267C23×C7H(5,𝔽2)11yescharacteristic
268C23×C72−1+411yescharacteristic
269C7×D4—416no—
270C2×C4×C7—24no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7C4×D414yes—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—24no—
299C2×Dic7—22no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—24no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314C2×Dic7—42no—
315C2×Dic7—41no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—41no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7#12414yes—
328C25C22×D712yes—
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4C4×D714yes—
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—71no—
337C23×C4—71no—
338C23×C4—72no—
339C23×C4—72no—
340C23×C4C22×D711yescharacteristic
341C23×C4C22×D711yescharacteristic
342C23×C4C22×D712yes—
343C23×C4C22×D711yescharacteristic
344C2×C4⋊C4—74no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4—282no—
350C2×C4⋊C4—282no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—78no—
358C2×C22⋊C4—78no—
359C2×C22⋊C4—144no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—144no—
362C2×C22⋊C4—144no—
363C2×C22⋊C4—282no—
364C2×C22⋊C4—282no—
365C2×C22⋊C4—282no—
366C2×C22⋊C4—282no—
367C2×C22⋊C4—282no—
368C2×C22⋊C4—282no—
369C2×C42—74no—
370C2×C42—144no—
371C2×C42—282no—
372C2×C42—282no—
373C2×C42—282no—
374#95—22no—
375#95—22no—
376#95—74no—
377#95—142no—
378#95—142no—
379#95—142no—
380#95—144no—
381#95—282no—
382#95—282no—
383#95—282no—
384#95—282no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—44no—
389C22×C7—44no—
390C22×C7—44no—
391C22×C7—44no—
392C22×C7—82no—
393C22×C7—82no—
394C22×C7—82no—
395C22×C7—82no—
396C22×C7—82no—
397C22×C7—82no—
398C22×C7—82no—
399C22×C7—82no—
400C22×C7C2×C4×D411yescharacteristic
401C22×C7C4×#4112yes—
402C22×C7#51812yes—
403C22×C7#51811yescharacteristic
404C22×C7#52011yescharacteristic
405C22×C7C2×#12411yescharacteristic
406C22×C7#52912yes—
407C22×C7#53211yescharacteristic
408C22×C7#53411yescharacteristic
409C22×C7#53511yescharacteristic
410C22×C7#53511yescharacteristic
411C22×C7#53711yescharacteristic
412C22×C7#53812yes—
413C22×C7#53812yes—
414C22×C7#53812yes—
415C22×C7#53812yes—
416C22×C7#53912yes—
417C22×C7#54011yescharacteristic
418C22×C7#54012yes—
419C22×C7#54012yes—
420C22×C7#54012yes—
421C22×C7#54211yescharacteristic
422C22×C7#54212yes—
423C4×C7—24no—
424C4×C7—44no—
425C4×C7—81no—
426C4×C7—81no—
427Dic7—24no—
428Dic7—42no—
429Dic7—42no—
430Dic7—44no—
431Dic7—42no—
432Dic7—82no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24C2×C4×D714yes—
448C24C23×D711yescharacteristic
449C2×D4—28no—
450C2×D4—48no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—48no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—72no—
473C22×C4—74no—
474C22×C4—74no—
475C22×C4—72no—
476C22×C4—72no—
477C22×C4—72no—
478C22×C4—142no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—141no—
499C22×C4—141no—
500C22×C4—141no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—142no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4C2×C4×D712yes—
509C22×C4D4×D712yes—
510C22×C4#115914yes—
511C22×C4#115912yes—
512C4⋊C4—148no—
513C4⋊C4—284no—
514C4⋊C4—284no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—288no—
518C22⋊C4—288no—
519C22⋊C4—288no—
520C22⋊C4—288no—
521C42—284no—
522C42—288no—
523C2×C7—84no—
524C2×C7C4×#12411yescharacteristic
525C2×C7#232111yescharacteristic
526C2×C7#232411yescharacteristic
527C2×C7#233312yes—
528C2×C7#233512yes—
529C2×C7#263911yescharacteristic
530C2×C7#264111yescharacteristic
531C2×C7#264812yes—
532C2×C7#265112yes—
533C2×C7#266111yescharacteristic
534C2×C7#266211yescharacteristic
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C22×C4×D711yescharacteristic
571C23C2×D4×D711yescharacteristic
572C23C2×#115911yescharacteristic
573C23C2×#115912yes—
574C23#654811yescharacteristic
575C23#654811yescharacteristic
576C23#654812yes—
577C23#654812yes—
578C23D7×#4112yes—
579C23#655311yescharacteristic
580C23#655411yescharacteristic
581D4—416no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—74no—
604C2×C4—142no—
605C2×C4—142no—
606C2×C4—144no—
607C2×C4—144no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—144no—
612C2×C4—144no—
613C2×C4—144no—
614C2×C4—142no—
615C2×C4—144no—
616C2×C4—142no—
617C2×C4—142no—
618C2×C4—142no—
619C2×C4—281no—
620C2×C4—281no—
621C2×C4—281no—
622C2×C4—281no—
623C2×C4—282no—
624C2×C4—282no—
625C2×C4—282no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4—282no—
639C2×C4(C22⋊D7)⋊C414yes—
640C7#1432511yescharacteristic, complemented, Sylow
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C2×(C22⋊D7)⋊C411yescharacteristic
657C22#9017712yes—
658C22C4×#115912yes—
659C22#9023611yescharacteristic
660C22#9023711yescharacteristic
661C22#9028011yescharacteristic
662C22#9028311yescharacteristic
663C22#9028412yes—
664C22#9028412yes—
665C22#9028412yes—
666C22#9029112yes—
667C22#9029412yes—
668C22#9029811yescharacteristic
669C22#9029911yescharacteristic
670C22#9030412yes—
671C22#9030612yes—
672C22#9031012yes—
673C22#9035112yes—
674C22#9035412yes—
675C22D7×#12411yescharacteristic
676C22#9037311yescharacteristic
677C22#9040711yescharacteristic
678C22#9041011yescharacteristic
679C4—24no—
680C4—44no—
681C4—81no—
682C4—81no—
683C4—144no—
684C4—282no—
685C4—282no—
686C4—282no—
687C4—284no—
688C4—562no—
689C2—84no—
690C2#1173126611yescharacteristic
691C2#1173211312yes—
692C2#1173211512yes—
693C2#1173373711yescharacteristic
694C2#1173387511yescharacteristic
695C2#1173415012yes—
696C2#1173415311yescharacteristic
697C2#1173415411yescharacteristic
698C2#1173442911yescharacteristic
699C2#1173443512yes—
700C2#1173457111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1438C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenter#1438D7
hypocenterC7#14325
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
3. C2×#89424C2semidirect
5. #11729303C2semidirect
6. #11729301C2semidirect
7. #11729299C2semidirect
8. #11729299C2semidirect
9. #11727169C2semidirect
10. #11727164C2semidirect
11. Dic7×#95C2semidirect
26. #89994C4semidirect
640. C7#14325semidirect

Maximal subgroups

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

subgroupindex
C2×#899942
C2×#899932
C2×#894242
C7×#14382
#117293032
#117293012
#117292992
#117271692
#117271642
Dic7×#952
#143257