Finite Group Databasebeta

← #13284111

Subgroups — #13284111

Order 1792.

1442 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#399C211yesmaximal, characteristic
2#11729319C212yesmaximal, complemented
3#11729318C212yesmaximal, complemented
4#11729318C212yesmaximal, complemented
5#11729300C211yesmaximal, characteristic
6#11729299C212yesmaximal
7#11729298C211yesmaximal, characteristic
8#11729298C212yesmaximal
9#11727170C212yesmaximal, complemented
10C24×Dic7C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4×Dic7C2212yes—
19C7×#399C418yescomplemented
20C2×C7×#95C2211yescharacteristic
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32#14592—71nomaximal, Sylow
33C25×C7C2311yescharacteristic
34C23×C4×C7C2312yes—
35C23×C4×C7C2311yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2312yes—
38C23×Dic7—22no—
39C23×Dic7—24no—
40C23×Dic7—41no—
41C23×Dic7—42no—
42C23×Dic7—41no—
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C2×C7×C4⋊C4—41no—
48C2×C7×C4⋊C4—42no—
49C2×C7×C4⋊C4—41no—
50C2×C7×C4⋊C4—42no—
51C2×C7×C4⋊C4C2×C414yescomplemented
52C2×C7×C4⋊C4C2×C418yescomplemented
53C2×C7×C22⋊C4—42no—
54C2×C7×C22⋊C4—41no—
55C2×C7×C22⋊C4—42no—
56C2×C7×C22⋊C4—41no—
57C2×C7×C22⋊C4C2×C418yes—
58C2×C7×C22⋊C4C2×C414yes—
59C2×C22⋊Dic7—24no—
60C2×C22⋊Dic7—24no—
61C2×C22⋊Dic7—41no—
62C2×C22⋊Dic7—41no—
63C2×C22⋊Dic7—42no—
64C2×C22⋊Dic7Q814yes—
65C2×C4⋊Dic7—41no—
66C2×C4⋊Dic7—42no—
67C2×C4×Dic7—42no—
68C2×C4×Dic7—42no—
69C2×C4×Dic7—42no—
70C7×#95C2×C414yescomplemented
71C7⋊#95—24no—
72C7⋊#95—24no—
73C7⋊#95—24no—
74C7⋊#95—24no—
75C7⋊#95—41no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95—42no—
81C7⋊#95—42no—
82C2×#399D711yescharacteristic
83#1443—72no—
84#1441—72no—
85#1439—71no—
86#1437—72no—
87#1436—72no—
88#1436—72no—
89C22⋊#95—71no—
90C22⋊#95—72no—
91C24×C7—22no—
92C24×C7—22no—
93C24×C7—21no—
94C24×C7—22no—
95C24×C7—21no—
96C24×C7—21no—
97C24×C7—22no—
98C24×C7—21no—
99C24×C7C22×C412yes—
100C24×C7C2×Q812yes—
101C24×C7#4112yes—
102C24×C7C2411yescharacteristic
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—21no—
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—21no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7C22×C412yes—
124C22×C4×C7C22×C414yes—
125C22×C4×C7C22×C414yes—
126C22×C4×C7C22×C412yes—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
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—44no—
153C22×Dic7—48no—
154C22×Dic7—48no—
155C22×Dic7—44no—
156C22×Dic7—44no—
157C22×Dic7C2×Q814yes—
158C22×Dic7#4114yes—
159C22×Dic7#4114yes—
160C7×C4⋊C4—48no—
161C7×C4⋊C4—44no—
162C7×C22⋊C4—48no—
163C7×C22⋊C4—44no—
164C22⋊Dic7—44no—
165C22⋊Dic7—48no—
166C4⋊Dic7—44no—
167C4×Dic7—48no—
168C24×C4—71no—
169C22×C4⋊C4—71no—
170C22×C4⋊C4C2×D711yescharacteristic
171C22×C4⋊C4C2×D712yes—
172C22×C22⋊C4—71no—
173C22×C22⋊C4—72no—
174C22×C22⋊C4C2×D712yes—
175C22×C22⋊C4C2×D711yescharacteristic
176C22×C42—72no—
177#399Dic718yescomplemented
178C2×#95—72no—
179C2×#95—72no—
180C2×#95—72no—
181C2×#95—72no—
182C2×#95—72no—
183C2×#95—72no—
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—71no—
187C2×#95—72no—
188C2×#95—72no—
189C2×#95C2×D711yescharacteristic
190C23×C7—24no—
191C23×C7—22no—
192C23×C7—24no—
193C23×C7—22no—
194C23×C7—22no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—22no—
198C23×C7—24no—
199C23×C7—24no—
200C23×C7—24no—
201C23×C7—24no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—41no—
207C23×C7—42no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7—41no—
211C23×C7—42no—
212C23×C7C4×Q814yes—
213C23×C7C23×C411yescharacteristic
214C23×C7C22×Q811yescharacteristic
215C23×C7C2×#4111yescharacteristic
216C23×C7C2×#4112yes—
217C23×C7C2×#4112yes—
218C23×C7H(5,𝔽2)12yes—
219C23×C7H(5,𝔽2)11yescharacteristic
220C23×C7H(5,𝔽2)11yescharacteristic
221C23×C7H(5,𝔽2)12yes—
222C23×C7H(5,𝔽2)11yescharacteristic
223C23×C72−1+411yescharacteristic
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—42no—
242C2×C4×C7—41no—
243C2×C4×C7—42no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
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—416no—
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—
276C2×Dic7—42no—
277C25C22×D711yescharacteristic
278C23×C4—72no—
279C23×C4—72no—
280C23×C4—72no—
281C23×C4—72no—
282C23×C4—144no—
283C23×C4—142no—
284C23×C4—281no—
285C23×C4—282no—
286C23×C4—281no—
287C23×C4C22×D711yescharacteristic
288C23×C4C22×D712yes—
289C23×C4C22×D711yescharacteristic
290C23×C4C22×D712yes—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—281no—
296C2×C4⋊C4—282no—
297C2×C4⋊C4C2×Dic714yescomplemented
298C2×C4⋊C4C2×Dic718yescomplemented
299C2×C22⋊C4—42no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—74no—
304C2×C22⋊C4—144no—
305C2×C22⋊C4—144no—
306C2×C22⋊C4—281no—
307C2×C22⋊C4—281no—
308C2×C22⋊C4—282no—
309C2×C22⋊C4C2×Dic718yes—
310C2×C22⋊C4C2×Dic714yes—
311C2×C42—282no—
312C2×C42—282no—
313C2×C42—282no—
314#95—144no—
315#95—144no—
316#95—144no—
317#95—144no—
318#95—281no—
319#95—282no—
320#95—282no—
321#95—282no—
322#95—282no—
323#95—282no—
324#95—282no—
325#95C2×Dic714yescomplemented
326C22×C7—24no—
327C22×C7—28no—
328C22×C7—44no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—44no—
334C22×C7—42no—
335C22×C7—44no—
336C22×C7—42no—
337C22×C7C2×C4×Q811yescharacteristic
338C22×C7C4×#4112yes—
339C22×C7#51811yescharacteristic
340C22×C7#51812yes—
341C22×C7#52011yescharacteristic
342C22×C7#53811yescharacteristic
343C22×C7#53812yes—
344C22×C7#53911yescharacteristic
345C22×C7#53912yes—
346C22×C7#54012yes—
347C22×C7#54212yes—
348C22×C7#54311yescharacteristic
349C22×C7#54411yescharacteristic
350C22×C7#55112yes—
351C22×C7#55112yes—
352C22×C7#55111yescharacteristic
353C22×C7#55312yes—
354C22×C7#55412yes—
355C22×C7#55412yes—
356C22×C7#55911yescharacteristic
357C22×C7#56011yescharacteristic
358C22×C7#56011yescharacteristic
359C22×C7#56012yes—
360C4×C7—44no—
361C4×C7—42no—
362C4×C7—44no—
363C4×C7—42no—
364Dic7—44no—
365Dic7—44no—
366Dic7—44no—
367Dic7—82no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24Q8×D712yes—
377C24#116112yes—
378C24C22×Dic712yes—
379C24C23×D711yescharacteristic
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—74no—
401C22×C4—74no—
402C22×C4—74no—
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×C4—142no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—284no—
429C22×C4—288no—
430C22×C4—288no—
431C22×C4—284no—
432C22×C4—284no—
433C22×C4C22×Dic712yes—
434C22×C4C22×Dic714yes—
435C22×C4C22×Dic712yes—
436C22×C4C22×Dic714yes—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—284no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—284no—
443C22⋊C4—288no—
444C42—288no—
445C2×C7—44no—
446C2×C7#232911yescharacteristic
447C2×C7#233512yes—
448C2×C7#236812yes—
449C2×C7#237411yescharacteristic
450C2×C7#238011yescharacteristic
451C2×C7#273511yescharacteristic
452C2×C7#274211yescharacteristic
453C2×C7#274612yes—
454C2×C7#275211yescharacteristic
455C2×C7#278012yes—
456C2×C7#280711yescharacteristic
457C23—22no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23Q8×Dic714yes—
480C23#654811yescharacteristic
481C23#654811yescharacteristic
482C23#654812yes—
483C23#654812yes—
484C23C2×Q8×D711yescharacteristic
485C23C2×#116111yescharacteristic
486C23D7×#4112yes—
487C23D7×#4112yes—
488C23#655311yescharacteristic
489C23#655411yescharacteristic
490C23C23×Dic711yescharacteristic
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—144no—
518C2×C4—144no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—144no—
524C2×C4—2816no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—282no—
539C2×C4—282no—
540C2×C4—282no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C7#1459211yescharacteristic, complemented, Sylow
545C22—24no—
546C22—28no—
547C22—42no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22#9031411yescharacteristic
557C22#9032511yescharacteristic
558C22#9032712yes—
559C22#9032712yes—
560C22#9032812yes—
561C22#9032812yes—
562C22#9033511yescharacteristic
563C22#9033611yescharacteristic
564C22#9034812yes—
565C22#9035011yescharacteristic
566C22#9035112yes—
567C22#9035212yes—
568C22#9035312yes—
569C22#9035412yes—
570C22#9035511yescharacteristic
571C22#9041111yescharacteristic
572C22#9041111yescharacteristic
573C22#9041112yes—
574C22#9048911yescharacteristic
575C22#9048912yes—
576C22C2×Q8×Dic711yescharacteristic
577C22#41×Dic712yes—
578C22#9051811yescharacteristic
579C4—44no—
580C4—42no—
581C4—44no—
582C4—42no—
583C4—284no—
584C4—284no—
585C4—284no—
586C4—562no—
587C2—44no—
588C2#1173552111yescharacteristic
589C2#1173552212yes—
590C2#1173552311yescharacteristic
591C2#1173553012yes—
592C2#1173553411yescharacteristic
593C2#1173556511yescharacteristic
594C2#1174019011yescharacteristic
595C2#1174019412yes—
596C2#1174028212yes—
597C2#1174028411yescharacteristic
598C2#1174061911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#399C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#399D7
hypocenterC7#14592
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. #11729319C2semidirect
3. #11729318C2semidirect
4. #11729318C2semidirect
9. #11727170C2semidirect
19. C7×#399C4semidirect
51. C2×C7×C4⋊C4C2×C4semidirect
52. C2×C7×C4⋊C4C2×C4semidirect
70. C7×#95C2×C4semidirect
177. #399Dic7semidirect
297. C2×C4⋊C4C2×Dic7semidirect
298. C2×C4⋊C4C2×Dic7semidirect
325. #95C2×Dic7semidirect
544. C7#14592semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3992
#117293192
#117293182
#117293002
#117292992
#117292982
#117271702
#145927