Finite Group Databasebeta

← #13277278

Subgroups — #13277278

Order 1792.

2358 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6499C211yesmaximal, characteristic
2C22×#6450C211yesmaximal, characteristic, complemented
3C2×C7×#390C211yesmaximal, characteristic
4C2×#89993C211yesmaximal, characteristic
5C2×#89701C211yesmaximal, characteristic, complemented
6#11729317C211yesmaximal, characteristic, complemented
7#11729301C214yesmaximal, complemented
8#11729299C214yesmaximal, complemented
9C4×C7⋊#95C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2214yes—
14C22×C42×C7C2211yescharacteristic
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2214yes—
17C22×C22⋊Dic7C2212yes—
18C2×#6499—24no—
19C2×#6499—24no—
20C2×#6499C418yescomplemented
21C22×C4⋊Dic7C2211yescharacteristic
22C22×Dic7⋊C4C2212yes—
23C22×C4×Dic7C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C22×C4×Dic7C2211yescharacteristic
26C2×#6450—24no—
27C2×#6450—24no—
28C2×#6450—24no—
29C7×#390—24no—
30#89993—24no—
31C2×C7⋊#95C2214yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2214yes—
35#89701—24no—
36#14026—71nomaximal, Sylow
37C25×C7C2312yes—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4—22no—
42C22×C7×D4—22no—
43C22×C7×D4—22no—
44C22×C7×D4C2×C414yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2312yes—
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2312yes—
52C23×Dic7C2311yescharacteristic
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—22no—
56C22×Dic14—22no—
57C22×Dic14C2×C414yes—
58C22×Dic14D414yes—
59C22×Dic14D414yes—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4D414yes—
63C2×C7×C22⋊C4—28no—
64C2×C7×C22⋊C4—28no—
65C2×C7×C22⋊C4—44no—
66C2×C42×C7—22no—
67C2×C42×C7—22no—
68C2×C42×C7—22no—
69C2×C22⋊Dic7—28no—
70C2×C22⋊Dic7—28no—
71C2×C22⋊Dic7—24no—
72C2×C22⋊Dic7—24no—
73C2×C22⋊Dic7—28no—
74C2×C22⋊Dic7—44no—
75C2×C22⋊Dic7C2×C4116yes—
76#6499—232no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—22no—
80C2×Dic7⋊C4—24no—
81C2×Dic7⋊C4—24no—
82C2×Dic7⋊C4—24no—
83C2×C4×Dic7—24no—
84C2×C4×Dic7—22no—
85C2×C4×Dic7—22no—
86C2×C4×Dic7—22no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7—24no—
89C2×C4×Dic7—24no—
90C2×C4×Dic7C2×C414yes—
91C2×C4×Dic7D414yes—
92#6450—232no—
93C7⋊#95—24no—
94C7⋊#95—28no—
95C7⋊#95—24no—
96C7⋊#95—44no—
97C22×#124—71no—
98C22×C4×Q8—71no—
99C2×#391—71no—
100C2×#390—71no—
101C2×#390D711yescharacteristic
102#1439—74no—
103#1437—74no—
104C4⋊#95—71no—
105C4×#95—71no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—24no—
111C24×C7—24no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—24no—
115C24×C7—24no—
116C24×C7C22×C414yes—
117C24×C7C2411yescharacteristic
118C2×C7×D4—28no—
119C2×C7×D4—28no—
120C2×C7×D4—28no—
121C2×C7×D4—44no—
122C2×C7×D4—44no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
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—24no—
132C22×C4×C7—24no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C2×D414yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7#4114yes—
142C22×C4×C7#4112yes—
143C22×C4×C7#4112yes—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
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—24no—
165C22×Dic7—24no—
166C22×Dic7C22×C414yes—
167C22×Dic7C22×C414yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×D414yes—
170C22×Dic7C2×D414yes—
171C22×Dic7C2×D412yes—
172C22×Dic7#4114yes—
173C2×Dic14—28no—
174C2×Dic14—28no—
175C2×Dic14—28no—
176C2×Dic14—28no—
177C2×Dic14—28no—
178C2×Dic14—28no—
179C2×Dic14C22⋊C4116yescomplemented
180C7×C4⋊C4—28no—
181C7×C22⋊C4—416no—
182C42×C7—44no—
183C22⋊Dic7—232no—
184C22⋊Dic7—48no—
185C22⋊Dic7—48no—
186C4⋊Dic7—44no—
187Dic7⋊C4—216no—
188C4×Dic7—216no—
189C4×Dic7—28no—
190C4×Dic7—28no—
191C23×Q8—71no—
192C23×D4C2×D711yescharacteristic
193C2×#124—78no—
194C2×#124—144no—
195C2×#124—144no—
196C2×C4×Q8—144no—
197C2×C4×Q8—144no—
198C2×C4×Q8—144no—
199C22×C4⋊C4—72no—
200C22×C4⋊C4—71no—
201C22×C4⋊C4C2×D711yescharacteristic
202C22×C22⋊C4—74no—
203C22×C22⋊C4—72no—
204C22×C22⋊C4—72no—
205C22×C22⋊C4C2×D714yes—
206C22×C42—71no—
207C22×C42—71no—
208C22×C42—72no—
209C22×C42C2×D711yescharacteristic
210#391—144no—
211#390—24no—
212#390—144no—
213C2×#95—72no—
214C2×#95—72no—
215C2×#95—74no—
216C2×#95—74no—
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—28no—
221C23×C7—28no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—44no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—44no—
232C23×C7—44no—
233C23×C7—44no—
234C23×C7—44no—
235C23×C7—44no—
236C23×C7—44no—
237C23×C7—44no—
238C23×C7—44no—
239C23×C7—44no—
240C23×C7C23×C411yescharacteristic
241C23×C7C22×D411yescharacteristic
242C23×C7C22×D411yescharacteristic
243C23×C7C22×D411yescharacteristic
244C23×C7C22×D411yescharacteristic
245C23×C7C2×#4111yescharacteristic
246C23×C7C2×#4111yescharacteristic
247C23×C7C2×#4112yes—
248C23×C7C2×#4112yes—
249C23×C7H(5,𝔽2)12yes—
250C23×C72−1+412yes—
251C7×D4—416no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—28no—
256C2×C4×C7—24no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—44no—
263C2×C4×C7—42no—
264C2×C4×C7C2×C22⋊C414yes—
265C2×C4×C7C4×D414yes—
266C2×C4×C7C4×D414yes—
267C2×C4×C7#12014yes—
268C2×C4×C7#12114yes—
269C2×C4×C7#12114yes—
270C2×C4×C7#12314yes—
271C2×Dic7—24no—
272C2×Dic7—24no—
273C2×Dic7—22no—
274C2×Dic7—24no—
275C2×Dic7—24no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—22no—
279C2×Dic7—24no—
280C2×Dic7—24no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—44no—
297C2×Dic7—44no—
298C2×Dic7C2×C22⋊C418yes—
299C2×Dic7#12414yescomplemented
300Dic14—232no—
301C25C22×D712yes—
302C22×Q8—74no—
303C22×Q8—74no—
304C22×Q8—74no—
305C22×Q8—142no—
306C22×Q8—142no—
307C22×Q8—142no—
308C22×Q8—142no—
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—22no—
314C22×D4—22no—
315C22×D4C4×D714yes—
316C23×C4—72no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—72no—
320C23×C4—72no—
321C23×C4C22×D712yes—
322C23×C4C22×D712yes—
323C23×C4C22×D711yescharacteristic
324#124—1432no—
325C4×Q8—1432no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—144no—
329C2×C4⋊C4—144no—
330C2×C4⋊C4—144no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4C22⋊D714yes—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—716no—
339C2×C22⋊C4—148no—
340C2×C22⋊C4—148no—
341C2×C22⋊C4—148no—
342C2×C22⋊C4—144no—
343C2×C22⋊C4—144no—
344C2×C22⋊C4—284no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—74no—
349C2×C42—74no—
350C2×C42—144no—
351C2×C42—144no—
352C2×C42—142no—
353C2×C42—142no—
354C2×C42—142no—
355C2×C42—144no—
356C2×C42—142no—
357#95—144no—
358#95—148no—
359#95—144no—
360#95—284no—
361C22×C7—44no—
362C22×C7—48no—
363C22×C7—48no—
364C22×C7—44no—
365C22×C7—44no—
366C22×C7—84no—
367C22×C7—82no—
368C22×C7—82no—
369C22×C7—84no—
370C22×C7—82no—
371C22×C7—82no—
372C22×C7C22×C22⋊C411yescharacteristic
373C22×C7C2×C4×D411yescharacteristic
374C22×C7C2×C4×D411yescharacteristic
375C22×C7C4×#4112yes—
376C22×C7#52012yes—
377C22×C7C2×#12011yescharacteristic
378C22×C7C2×#12111yescharacteristic
379C22×C7C2×#12111yescharacteristic
380C22×C7C2×#12311yescharacteristic
381C22×C7C2×#12411yescharacteristic
382C22×C7#52912yes—
383C22×C7#53211yescharacteristic
384C22×C7#53511yescharacteristic
385C22×C7#53611yescharacteristic
386C22×C7#54212yes—
387C22×C7#54612yes—
388C22×C7#54812yes—
389C22×C7#54812yes—
390C22×C7#54812yes—
391C22×C7#55012yes—
392C22×C7#55012yes—
393C22×C7#55612yes—
394C22×C7#55912yes—
395C4×C7—44no—
396C4×C7—44no—
397C4×C7#38618yes—
398Dic7—24no—
399Dic7—28no—
400Dic7—44no—
401Dic7—42no—
402Dic7—82no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—24no—
408C24—22no—
409C24—24no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24C2×C4×D714yes—
414C24C23×D711yescharacteristic
415C2×Q8—716no—
416C2×Q8—148no—
417C2×Q8—148no—
418C2×Q8—148no—
419C2×Q8—148no—
420C2×Q8—148no—
421C2×Q8—148no—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—74no—
440C22×C4—74no—
441C22×C4—72no—
442C22×C4—74no—
443C22×C4—74no—
444C22×C4—74no—
445C22×C4—72no—
446C22×C4—142no—
447C22×C4—141no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—141no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—141no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—144no—
467C22×C4—144no—
468C22×C4C2×C4×D712yes—
469C22×C4#115712yes—
470C22×C4D4×D712yes—
471C22×C4D4×D712yes—
472C22×C4D4×D712yes—
473C22×C4#115914yes—
474C22×C4#115912yes—
475C22×C4C2×C22⋊D714yes—
476C22×C4C2×C22⋊D712yes—
477C4⋊C4—28no—
478C4⋊C4—1416no—
479C4⋊C4—284no—
480C22⋊C4—416no—
481C22⋊C4—1432no—
482C22⋊C4—288no—
483C22⋊C4—288no—
484C42—44no—
485C42—1416no—
486C42—148no—
487C42—148no—
488C2×C7—84no—
489C2×C7C2×#38611yescharacteristic
490C2×C7C4×#12411yescharacteristic
491C2×C7#232611yescharacteristic
492C2×C7#234112yes—
493C2×C7#236712yes—
494C2×C7#242612yes—
495C2×C7#243111yescharacteristic
496C2×C7#243611yescharacteristic
497C2×C7#244812yes—
498C2×C7#254511yescharacteristic
499C2×C7#254811yescharacteristic
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23C22×C4×D711yescharacteristic
524C23C2×#115711yescharacteristic
525C23C2×D4×D711yescharacteristic
526C23C2×D4×D711yescharacteristic
527C23C2×D4×D711yescharacteristic
528C23C2×#115912yes—
529C23C2×#115911yescharacteristic
530C23#654812yes—
531C23D7×#4112yes—
532C23#655412yes—
533C23C22×C22⋊D711yescharacteristic
534Q8—1432no—
535D4—416no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—78no—
549C2×C4—74no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—144no—
554C2×C4—144no—
555C2×C4—142no—
556C2×C4—144no—
557C2×C4—144no—
558C2×C4—144no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—142no—
565C2×C4—144no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—284no—
575C2×C4—284no—
576C2×C4—282no—
577C2×C4D7×C22⋊C414yes—
578C2×C4(C22⋊D7)⋊C414yes—
579C2×C4#644614yes—
580C2×C4#644714yes—
581C2×C4C4×C22⋊D714yes—
582C2×C4#650014yes—
583C2×C4#650114yes—
584C7#1402611yescharacteristic, complemented, Sylow
585C22—48no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22C2×D7×C22⋊C411yescharacteristic
597C22C2×(C22⋊D7)⋊C411yescharacteristic
598C22C2×#644611yescharacteristic
599C22C2×#644711yescharacteristic
600C22C4×#115912yes—
601C22#9023312yes—
602C22#9023712yes—
603C22#9025012yes—
604C22#9028912yes—
605C22#9029012yes—
606C22#9029112yes—
607C22#9029412yes—
608C22D7×#12411yescharacteristic
609C22#9036611yescharacteristic
610C22#9036812yes—
611C22#9040611yescharacteristic
612C22#9040711yescharacteristic
613C22#9040912yes—
614C22C2×C4×C22⋊D711yescharacteristic
615C22C2×#650011yescharacteristic
616C22C2×#650111yescharacteristic
617C22#9049712yes—
618C22#9052212yes—
619C4—44no—
620C4—44no—
621C4—148no—
622C4—144no—
623C4—282no—
624C4—284no—
625C4—562no—
626C4#8972818yes—
627C2—84no—
628C2#1173373711yescharacteristic
629C2#1173374211yescharacteristic
630C2#1173374612yes—
631C2#1173376612yes—
632C2#1173379611yescharacteristic
633C2#1173379711yescharacteristic
634C2C2×#8972811yescharacteristic
635C2#1173978912yes—
636C2#1174009112yes—
637C2#1174040611yescharacteristic
638C2#1174063211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#390C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7#14026
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#6450C2semidirect
5. C2×#89701C2semidirect
6. #11729317C2semidirect
7. #11729301C2semidirect
8. #11729299C2semidirect
9. C4×C7⋊#95C2semidirect
20. C2×#6499C4semidirect
179. C2×Dic14C22⋊C4semidirect
299. C2×Dic7#124semidirect
584. C7#14026semidirect

Maximal subgroups

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

subgroupindex
C22×#64992
C22×#64502
C2×C7×#3902
C2×#899932
C2×#897012
#117293172
#117293012
#117292992
C4×C7⋊#952
#140267