Finite Group Databasebeta

← #13283382

Subgroups — #13283382

Order 1792.

1554 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#393C211yesmaximal, characteristic
2C2×#90002C212yesmaximal, complemented
3C2×#89723C212yesmaximal
4#11729302C211yesmaximal, characteristic
5#11729300C212yesmaximal
6#11729299C211yesmaximal, characteristic
7#11727166C212yesmaximal, complemented
8#11727161C212yesmaximal, complemented
9#11727161C212yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C22×C7×C22⋊C4C2212yes—
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C22⋊Dic7C2211yescharacteristic
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2212yes—
17C22×C4⋊Dic7C2212yes—
18C22×C4×Dic7C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C7×#393C418yescomplemented
21C2×C7×#95C2212yes—
22C2×C7×#95C2211yescharacteristic
23#90002—28no—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33#89723—28no—
34#14537—71nomaximal, Sylow
35C25×C7C2311yescharacteristic
36C22×C7×Q8—24no—
37C22×C7×Q8—24no—
38C22×C7×Q8—41no—
39C22×C7×Q8—41no—
40C22×C7×Q8C2×C414yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C2×C7×C22⋊C4—24no—
50C2×C7×C22⋊C4—24no—
51C2×C7×C22⋊C4—41no—
52C2×C7×C22⋊C4—41no—
53C2×C7×C22⋊C4C2×C418yes—
54C2×C7×C22⋊C4C2×C414yes—
55C2×C22⋊Dic7—24no—
56C2×C22⋊Dic7—24no—
57C2×C22⋊Dic7—24no—
58C2×C22⋊Dic7—41no—
59C2×C22⋊Dic7—42no—
60C2×C22⋊Dic7—42no—
61C2×C22⋊Dic7—41no—
62C2×C4⋊Dic7—24no—
63C2×C4⋊Dic7—24no—
64C2×C4⋊Dic7—24no—
65C2×C4⋊Dic7—24no—
66C2×C4⋊Dic7—42no—
67C2×C4⋊Dic7—42no—
68C2×C4×Dic7—24no—
69C2×C4×Dic7—24no—
70C2×C4×Dic7—42no—
71C2×C4×Dic7—42no—
72C2×C4×Dic7—41no—
73C7×#95C2×C414yes—
74C7×#95C2×C418yes—
75C7⋊#95—24no—
76C7⋊#95—24no—
77C7⋊#95—24no—
78C7⋊#95—24no—
79C7⋊#95—24no—
80C7⋊#95—41no—
81C7⋊#95—42no—
82C7⋊#95—42no—
83C7⋊#95D418yes—
84C2×#393D711yescharacteristic
85C2×#391—72no—
86C2×#388—72no—
87#1442—72no—
88#1440—71no—
89#1439—72no—
90#1437—71no—
91#1436—72no—
92#1436—72no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—21no—
98C24×C7—21no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7C22×C412yes—
102C24×C7C2411yescharacteristic
103C2×C7×Q8—216no—
104C2×C7×Q8—44no—
105C2×C7×Q8—44no—
106C2×C7×Q8—48no—
107C2×C7×Q8—44no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7C22×C414yes—
127C22×C4×C7C22×C412yes—
128C22×C4×C7C22×C412yes—
129C22×C4×C7C22×C414yes—
130C22×C4×C7C2×D414yes—
131C22×C4×C7#4114yes—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
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—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7C2×D414yes—
158C22×Dic7C2×D414yes—
159C22×Dic7#4114yes—
160C7×C22⋊C4—216no—
161C7×C22⋊C4—44no—
162C22⋊Dic7—44no—
163C22⋊Dic7—48no—
164C22⋊Dic7—44no—
165C4⋊Dic7—48no—
166C4⋊Dic7—48no—
167C4×Dic7—48no—
168C4×Dic7—44no—
169C23×Q8C2×D711yescharacteristic
170C22×C4⋊C4—72no—
171C22×C4⋊C4—72no—
172C22×C22⋊C4—72no—
173C22×C22⋊C4—71no—
174C22×C22⋊C4—71no—
175C22×C22⋊C4C2×D712yes—
176C22×C22⋊C4C2×D711yescharacteristic
177C22×C42—72no—
178C22×C42—71no—
179#393Dic718yescomplemented
180#391—148no—
181#388—148no—
182C2×#95—72no—
183C2×#95—72no—
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—72no—
187C2×#95—72no—
188C2×#95—71no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95C2×D712yes—
192C2×#95C2×D711yescharacteristic
193C23×C7—24no—
194C23×C7—22no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—24no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—41no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—41no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—41no—
212C23×C7—42no—
213C23×C7—41no—
214C23×C7C23×C411yescharacteristic
215C23×C7C22×D412yes—
216C23×C7C2×#4112yes—
217C23×C7C2×#4112yes—
218C23×C7H(5,𝔽2)11yescharacteristic
219C23×C7H(5,𝔽2)11yescharacteristic
220C23×C7H(5,𝔽2)12yes—
221C23×C72−1+412yes—
222C23×C72−1+412yes—
223C7×Q8—416no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—41no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—41no—
247C2×C4×C7C4×D418yes—
248C2×Dic7—24no—
249C2×Dic7—24no—
250C2×Dic7—24no—
251C2×Dic7—24no—
252C2×Dic7—24no—
253C2×Dic7—24no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C25C22×D711yescharacteristic
277C22×Q8—24no—
278C22×Q8—24no—
279C22×Q8—41no—
280C22×Q8—41no—
281C22×Q8C2×Dic714yes—
282C23×C4—72no—
283C23×C4—72no—
284C23×C4—72no—
285C23×C4—72no—
286C23×C4C22×D712yes—
287C23×C4C22×D711yescharacteristic
288C23×C4C22×D711yescharacteristic
289C23×C4C22×D712yes—
290C2×C4⋊C4—144no—
291C2×C4⋊C4—144no—
292C2×C4⋊C4—144no—
293C2×C4⋊C4—144no—
294C2×C4⋊C4—282no—
295C2×C4⋊C4—282no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—144no—
301C2×C22⋊C4—144no—
302C2×C22⋊C4—144no—
303C2×C22⋊C4—281no—
304C2×C22⋊C4—282no—
305C2×C22⋊C4—281no—
306C2×C22⋊C4—282no—
307C2×C22⋊C4C2×Dic718yes—
308C2×C22⋊C4C2×Dic714yes—
309C2×C42—144no—
310C2×C42—144no—
311C2×C42—282no—
312C2×C42—281no—
313C2×C42—282no—
314#95—78no—
315#95—144no—
316#95—144no—
317#95—144no—
318#95—144no—
319#95—144no—
320#95—281no—
321#95—282no—
322#95—282no—
323#95C2×Dic714yes—
324#95C2×Dic718yes—
325C22×C7—44no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7—44no—
329C22×C7—42no—
330C22×C7—82no—
331C22×C7—82no—
332C22×C7—82no—
333C22×C7—82no—
334C22×C7C2×C4×D412yes—
335C22×C7C4×#4111yescharacteristic
336C22×C7#51811yescharacteristic
337C22×C7#51911yescharacteristic
338C22×C7#52012yes—
339C22×C7#53612yes—
340C22×C7#53712yes—
341C22×C7#53912yes—
342C22×C7#54112yes—
343C22×C7#54212yes—
344C22×C7#54212yes—
345C22×C7D4211yescharacteristic
346C22×C7#54612yes—
347C22×C7#54612yes—
348C22×C7#54812yes—
349C22×C7#55111yescharacteristic
350C22×C7#55612yes—
351C22×C7#55912yes—
352C22×C7H(3,𝔽4)11yescharacteristic
353C22×C7#56212yes—
354C22×C7#56311yescharacteristic
355C4×C7—28no—
356C4×C7—42no—
357C4×C7—44no—
358C4×C7—42no—
359Dic7—44no—
360Dic7—44no—
361Dic7—44no—
362Dic7—82no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24C22×Dic712yes—
372C24C23×D711yescharacteristic
373C2×Q8—216no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—48no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—74no—
397C22×C4—74no—
398C22×C4—74no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4—142no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4D4×D714yes—
425C22×C4#115914yes—
426C22×C4C22×Dic714yes—
427C22×C4C22×Dic712yes—
428C22×C4C22×Dic712yes—
429C22×C4C22×Dic714yes—
430C4⋊C4—288no—
431C4⋊C4—288no—
432C22⋊C4—216no—
433C22⋊C4—44no—
434C22⋊C4—284no—
435C22⋊C4—288no—
436C22⋊C4—284no—
437C42—288no—
438C42—284no—
439C2×C7—82no—
440C2×C7#232312yes—
441C2×C7#233211yescharacteristic
442C2×C7#235811yescharacteristic
443C2×C7#236712yes—
444C2×C7#237911yescharacteristic
445C2×C7#268112yes—
446C2×C7#271212yes—
447C2×C7#271612yes—
448C2×C7#279812yes—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23C2×D4×D712yes—
471C23C2×#115912yes—
472C23#654811yescharacteristic
473C23#654811yescharacteristic
474C23#655112yes—
475C23D7×#4112yes—
476C23#655312yes—
477C23#655412yes—
478C23C23×Dic711yescharacteristic
479Q8—416no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—144no—
504C2×C4—144no—
505C2×C4—144no—
506C2×C4—144no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—144no—
510C2×C4—144no—
511C2×C4—144no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4D4×Dic718yes—
532C7#1453711yescharacteristic, complemented, Sylow
533C22—44no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22#9027811yescharacteristic
543C22#9028111yescharacteristic
544C22#9028212yes—
545C22#9031612yes—
546C22#9031912yes—
547C22#9032012yes—
548C22#9032912yes—
549C22#9033112yes—
550C22#9033412yes—
551C22#9036212yes—
552C22#9036311yescharacteristic
553C22#9036712yes—
554C22#9036812yes—
555C22#9037212yes—
556C22#9037411yescharacteristic
557C22#9037512yes—
558C22C2×D4×Dic712yes—
559C22#9048911yescharacteristic
560C22#9050311yescharacteristic
561C22#41×Dic711yescharacteristic
562C22#9051812yes—
563C4—28no—
564C4—42no—
565C4—44no—
566C4—42no—
567C4—284no—
568C4—284no—
569C4—284no—
570C4—562no—
571C2—82no—
572C2#1173427412yes—
573C2#1173427512yes—
574C2#1173429512yes—
575C2#1173432012yes—
576C2#1174002411yescharacteristic
577C2#1174018912yes—
578C2#1174019111yescharacteristic
579C2#1174040012yes—
580C2#1174040211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#393C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#393D7
hypocenterC7#14537
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#90002C2semidirect
7. #11727166C2semidirect
8. #11727161C2semidirect
9. #11727161C2semidirect
20. C7×#393C4semidirect
179. #393Dic7semidirect
532. C7#14537semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C2×#900022
C2×#897232
#117293022
#117293002
#117292992
#117271662
#117271612
#145377