Finite Group Databasebeta

← #13276692

Subgroups — #13276692

Order 1792.

2358 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic
3C2×#89993C211yesmaximal, characteristic
4C2×#89749C211yesmaximal, characteristic, complemented
5C2×#89700C211yesmaximal, characteristic, complemented
6#11729881C214yesmaximal, complemented
7#11729317C211yesmaximal, characteristic, complemented
8#11729301C214yesmaximal, complemented
9C4×C7⋊#95C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2212yes—
12C2×C4×C7×D4—24no—
13C2×C4×C7×D4—24no—
14C2×C4×C7×D4—24no—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2212yes—
17C22×C42×C7C2211yescharacteristic
18C22×C22⋊Dic7C2214yes—
19C22×C22⋊Dic7C2214yes—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C22×Dic7⋊C4C2212yes—
23C22×C4×Dic7C2211yescharacteristic
24C22×C4×Dic7C2211yescharacteristic
25#89993—24no—
26#89993C418yescomplemented
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2214yes—
31C2×C7⋊#95C2214yes—
32#89749—24no—
33#89700—24no—
34#13979—71nomaximal, Sylow
35C25×C7C2312yes—
36C22×C7×D4—22no—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4C2×C414yes—
41C22×C7×D4D414yes—
42C22×C7×D4Q814yes—
43C23×C4×C7—24no—
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—28no—
47C23×C4×C7—44no—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×Dic7C2312yes—
53C23×Dic7C2312yes—
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C4×C7×D4—232no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4D414yes—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C42×C7D414yes—
66C2×C22⋊Dic7—28no—
67C2×C22⋊Dic7—28no—
68C2×C22⋊Dic7—44no—
69C2×C22⋊Dic7—44no—
70C2×C22⋊Dic7—44no—
71C2×C22⋊Dic7C2×C4116yescomplemented
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7C2×C414yescomplemented
78C2×Dic7⋊C4—24no—
79C2×Dic7⋊C4—44no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—22no—
85C2×C4×Dic7C2×C414yescomplemented
86C7⋊#95—24no—
87C7⋊#95—28no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—44no—
91C22×C4×D4D711yescharacteristic
92C2×#390—71no—
93C2×#390—71no—
94C2×#389—71no—
95C2×#382—71no—
96#1439—74no—
97C4⋊#95—71no—
98C22⋊#95—74no—
99C4×#95—71no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—24no—
103C24×C7—24no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—24no—
107C24×C7—24no—
108C24×C7C22×C414yes—
109C24×C7C2×D414yes—
110C24×C7C2×Q814yes—
111C24×C7C2411yescharacteristic
112C2×C7×D4—28no—
113C2×C7×D4—28no—
114C2×C7×D4—28no—
115C2×C7×D4—28no—
116C2×C7×D4—28no—
117C2×C7×D4—28no—
118C2×C7×D4C4⋊C4116yescomplemented
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—216no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—24no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—416no—
132C22×C4×C7—416no—
133C22×C4×C7—48no—
134C22×C4×C7—48no—
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×C7C2×D412yes—
142C22×C4×C7C2×Q812yes—
143C22×C4×C7#4114yes—
144C22×C4×C7#4112yes—
145C22×C4×C7#4112yes—
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—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—24no—
158C22×Dic7—24no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7C22×C414yes—
168C22×Dic7C22×C414yes—
169C22×Dic7#4114yes—
170C7×C4⋊C4—28no—
171C7×C22⋊C4—216no—
172C42×C7—28no—
173C22⋊Dic7—416no—
174C22⋊Dic7—416no—
175C4⋊Dic7—28no—
176C4⋊Dic7—44no—
177Dic7⋊C4—48no—
178C4×Dic7—28no—
179C4×Dic7—44no—
180C23×D4C2×D711yescharacteristic
181C24×C4C2×D712yes—
182C2×C4×D4—24no—
183C2×C4×D4—24no—
184C2×C4×D4—24no—
185C22×C4⋊C4—71no—
186C22×C4⋊C4—71no—
187C22×C4⋊C4—72no—
188C22×C4⋊C4C2×D711yescharacteristic
189C22×C22⋊C4—74no—
190C22×C22⋊C4—74no—
191C22×C22⋊C4C2×D712yes—
192C22×C42—71no—
193C22×C42—71no—
194C22×C42C2×D711yescharacteristic
195#390—78no—
196#390—144no—
197#389—144no—
198#382—144no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95—72no—
202C2×#95—74no—
203C2×#95—74no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—28no—
207C23×C7—28no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—28no—
213C23×C7—28no—
214C23×C7—28no—
215C23×C7—28no—
216C23×C7—42no—
217C23×C7—44no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—44no—
222C23×C7—44no—
223C23×C7—44no—
224C23×C7—44no—
225C23×C7—44no—
226C23×C7C2×C4⋊C418yes—
227C23×C7C23×C411yescharacteristic
228C23×C7C22×D411yescharacteristic
229C23×C7C22×D411yescharacteristic
230C23×C7C22×D411yescharacteristic
231C23×C7C22×Q811yescharacteristic
232C23×C7C2×#4111yescharacteristic
233C23×C7C2×#4111yescharacteristic
234C23×C7C2×#4112yes—
235C23×C7C2×#4112yes—
236C23×C7H(5,𝔽2)12yes—
237C23×C7H(5,𝔽2)12yes—
238C7×D4—232no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—22no—
244C2×C4×C7—24no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—28no—
248C2×C4×C7—432no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—44no—
253C2×C4×C7—42no—
254C2×C4×C7C2×C4⋊C414yes—
255C2×C4×C7C4×D414yes—
256C2×C4×C7C4×D414yes—
257C2×C4×C7#12014yes—
258C2×C4×C7C22⋊Q814yes—
259C2×C4×C7C22⋊Q814yes—
260C2×C4×C7#12314yes—
261C2×C4×C7#12414yes—
262C2×Dic7—24no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—28no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—44no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—44no—
285C2×Dic7—44no—
286C25C22×D712yes—
287C22×D4—22no—
288C22×D4—22no—
289C22×D4—22no—
290C22×D4—22no—
291C22×D4Dic1414yes—
292C22×D4C4×D714yes—
293C22×D4C22⋊D714yes—
294C23×C4—24no—
295C23×C4—24no—
296C23×C4—24no—
297C23×C4—28no—
298C23×C4—44no—
299C23×C4—72no—
300C23×C4—72no—
301C23×C4—72no—
302C23×C4—72no—
303C23×C4C22×D711yescharacteristic
304C23×C4C22×D712yes—
305C23×C4C22×D711yescharacteristic
306C23×C4C22×D711yescharacteristic
307C4×D4—232no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—74no—
311C2×C4⋊C4—142no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—144no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—284no—
318C2×C4⋊C4C22⋊D714yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—716no—
323C2×C22⋊C4—148no—
324C2×C22⋊C4—148no—
325C2×C22⋊C4—284no—
326C2×C22⋊C4—284no—
327C2×C22⋊C4—284no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—74no—
331C2×C42—142no—
332C2×C42—142no—
333C2×C42—142no—
334C2×C42—142no—
335C2×C42—142no—
336C2×C42C22⋊D714yes—
337#95—144no—
338#95—144no—
339#95—148no—
340#95—144no—
341#95—284no—
342C22×C7—216no—
343C22×C7—28no—
344C22×C7—44no—
345C22×C7—44no—
346C22×C7—44no—
347C22×C7—48no—
348C22×C7—48no—
349C22×C7—44no—
350C22×C7—44no—
351C22×C7—48no—
352C22×C7—44no—
353C22×C7C22×C4⋊C411yescharacteristic
354C22×C7C2×C4×D411yescharacteristic
355C22×C7C2×C4×D411yescharacteristic
356C22×C7C4×#4112yes—
357C22×C7#51812yes—
358C22×C7C2×#12011yescharacteristic
359C22×C7C2×C22⋊Q811yescharacteristic
360C22×C7C2×C22⋊Q811yescharacteristic
361C22×C7C2×#12311yescharacteristic
362C22×C7C2×#12411yescharacteristic
363C22×C7#52912yes—
364C22×C7#53211yescharacteristic
365C22×C7#53511yescharacteristic
366C22×C7#53511yescharacteristic
367C22×C7#54012yes—
368C22×C7#54612yes—
369C22×C7#54612yes—
370C22×C7#54612yes—
371C22×C7#55012yes—
372C22×C7#55312yes—
373C22×C7#55412yes—
374C22×C7#55412yes—
375C22×C7#55512yes—
376C4×C7—24no—
377C4×C7—42no—
378C4×C7—44no—
379C4×C7#38518yes—
380Dic7—44no—
381Dic7—44no—
382Dic7—44no—
383Dic7—82no—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24C2×Dic1414yes—
393C24C2×C4×D714yes—
394C24C2×C22⋊D714yes—
395C24C23×D711yescharacteristic
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4Dic7⋊C4116yescomplemented
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—216no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—74no—
420C22×C4—74no—
421C22×C4—74no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—144no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—144no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—144no—
439C22×C4—144no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4C2×Dic1412yes—
444C22×C4C2×C4×D712yes—
445C22×C4#115712yes—
446C22×C4#115712yes—
447C22×C4#115914yes—
448C22×C4C2×C22⋊D714yes—
449C22×C4C2×C22⋊D712yes—
450C22×C4C2×C22⋊D712yes—
451C22×C4C2×C22⋊D712yes—
452C22×C4C2×C22⋊D712yes—
453C22×C4C2×C22⋊D712yes—
454C4⋊C4—28no—
455C4⋊C4—148no—
456C4⋊C4—288no—
457C4⋊C4—284no—
458C22⋊C4—216no—
459C22⋊C4—2816no—
460C22⋊C4—2816no—
461C42—28no—
462C42—148no—
463C42—284no—
464C2×C7—48no—
465C2×C7C2×#38511yescharacteristic
466C2×C7C4×#12411yescharacteristic
467C2×C7#232811yescharacteristic
468C2×C7#234912yes—
469C2×C7#236312yes—
470C2×C7#242012yes—
471C2×C7#243111yescharacteristic
472C2×C7#243511yescharacteristic
473C2×C7#245412yes—
474C2×C7#256311yescharacteristic
475C2×C7#256611yescharacteristic
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23C2×Dic7⋊C418yes—
499C23C22×Dic1411yescharacteristic
500C23C22×C4×D711yescharacteristic
501C23C2×#115711yescharacteristic
502C23C2×#115711yescharacteristic
503C23C2×#115912yes—
504C23#654812yes—
505C23D7×#4112yes—
506C23#655312yes—
507C23C22×C22⋊D711yescharacteristic
508C23C22×C22⋊D711yescharacteristic
509C23C22×C22⋊D711yescharacteristic
510D4—232no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—432no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—148no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—282no—
539C2×C4—284no—
540C2×C4—282no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—284no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—284no—
549C2×C4—282no—
550C2×C4C2×Dic7⋊C414yes—
551C2×C4#648714yes—
552C2×C4#648714yes—
553C2×C4C4×C22⋊D714yes—
554C2×C4C4×C22⋊D714yes—
555C2×C4#649214yes—
556C2×C4#649914yes—
557C2×C4#651614yes—
558C7#1397911yescharacteristic, complemented, Sylow
559C22—28no—
560C22—216no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—48no—
566C22—48no—
567C22—44no—
568C22—44no—
569C22—48no—
570C22C4×#115912yes—
571C22#9023012yes—
572C22#9023412yes—
573C22#9023612yes—
574C22#9024812yes—
575C22#9024912yes—
576C22#9025312yes—
577C22#9025412yes—
578C22C22×Dic7⋊C411yescharacteristic
579C22C2×#648711yescharacteristic
580C22C2×#648711yescharacteristic
581C22C2×C4×C22⋊D711yescharacteristic
582C22C2×C4×C22⋊D711yescharacteristic
583C22C2×#649211yescharacteristic
584C22C2×#649911yescharacteristic
585C22#9049611yescharacteristic
586C22#9049712yes—
587C22#9049712yes—
588C22#9052012yes—
589C22#9052111yescharacteristic
590C22#9052311yescharacteristic
591C22#9052512yes—
592C22C2×#651611yescharacteristic
593C4—24no—
594C4—44no—
595C4—42no—
596C4—284no—
597C4—284no—
598C4—284no—
599C4—562no—
600C4#8998218yes—
601C2—48no—
602C2C4×#649911yescharacteristic
603C2#1173977411yescharacteristic
604C2#1173977912yes—
605C2#1173978612yes—
606C2#1173980812yes—
607C2#1173982911yescharacteristic
608C2#1173983111yescharacteristic
609C2#1173985012yes—
610C2C2×#8998211yescharacteristic
611C2#1174149511yescharacteristic
612C2#1174153711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC22×C4×C7×D4C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC22×C4×D4D7
hypocenterC7#13979
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
4. C2×#89749C2semidirect
5. C2×#89700C2semidirect
6. #11729881C2semidirect
7. #11729317C2semidirect
8. #11729301C2semidirect
9. C4×C7⋊#95C2semidirect
26. #89993C4semidirect
71. C2×C22⋊Dic7C2×C4semidirect
77. C2×C4⋊Dic7C2×C4semidirect
85. C2×C4×Dic7C2×C4semidirect
118. C2×C7×D4C4⋊C4semidirect
402. C2×D4Dic7⋊C4semidirect
558. C7#13979semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C2×#899932
C2×#897492
C2×#897002
#117298812
#117293172
#117293012
C4×C7⋊#952
#139797