Finite Group Databasebeta

← #13275391

Subgroups — #13275391

Order 1792.

2022 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89716C212yesmaximal
2C2×#89718C211yesmaximal, characteristic
3C2×C22⋊C4×Dic7C212yesmaximal
4C2×#89421C212yesmaximal, complemented
5C7×#1437C211yesmaximal, characteristic
6#11729299C211yesmaximal, characteristic
7C4×C7⋊#95C212yesmaximal, complemented
8#11727160C212yesmaximal, complemented
9#11727160C212yesmaximal, complemented
10C24×Dic7C2211yescharacteristic
11C22×C7×C22⋊C4C2212yes—
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4C2212yes—
17C22×Dic7⋊C4C2212yes—
18C22×C4×Dic7C2212yes—
19C22×C4×Dic7C2212yes—
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C2×C7×#95C2212yes—
23C2×C7×#95C2211yescharacteristic
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31#89718C418yescomplemented
32#89716C4116yescomplemented
33C22⋊C4×Dic7—28no—
34#89421—28no—
35#13823—71nomaximal, Sylow
36C25×C7C2311yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×Dic7—24no—
42C23×Dic7—24no—
43C23×Dic7—41no—
44C23×Dic7—41no—
45C23×Dic7C2×C414yes—
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C23×Dic7C2312yes—
50C2×C7×C22⋊C4—24no—
51C2×C7×C22⋊C4—24no—
52C2×C7×C22⋊C4—41no—
53C2×C7×C22⋊C4—41no—
54C2×C7×C22⋊C4C2×C418yes—
55C2×C7×C22⋊C4C2×C414yes—
56C2×C42×C7—24no—
57C2×C42×C7—41no—
58C2×C22⋊Dic7—24no—
59C2×C22⋊Dic7—24no—
60C2×C22⋊Dic7C2×C418yes—
61C2×C22⋊Dic7C2×C418yes—
62C2×C22⋊Dic7C2×C414yes—
63C2×Dic7⋊C4—24no—
64C2×Dic7⋊C4—24no—
65C2×Dic7⋊C4—24no—
66C2×Dic7⋊C4—24no—
67C2×Dic7⋊C4C2×C418yes—
68C2×Dic7⋊C4C2×C418yescomplemented
69C2×C4×Dic7—24no—
70C2×C4×Dic7—24no—
71C2×C4×Dic7—24no—
72C2×C4×Dic7—24no—
73C2×C4×Dic7—24no—
74C2×C4×Dic7—24no—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—41no—
77C2×C4×Dic7—42no—
78C2×C4×Dic7—42no—
79C2×C4×Dic7C2×C418yescomplemented
80C7×#95—24no—
81C7×#95—41no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—41no—
85C7⋊#95C2×C418yes—
86C7⋊#95C2×C418yescomplemented
87C7⋊#95C2×C418yescomplemented
88C7⋊#95D418yes—
89C2×#388—72no—
90C2×#385—71no—
91C2×C4×C4⋊C4—72no—
92C2×C4×C22⋊C4—72no—
93#1437—71no—
94#1437D711yescharacteristic
95#1436—72no—
96#1436—72no—
97C4×#95—72no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—21no—
103C24×C7—21no—
104C24×C7—22no—
105C24×C7C22×C414yes—
106C24×C7C22×C412yes—
107C24×C7C2411yescharacteristic
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—22no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7C22×C412yes—
127C22×C4×C7C22×C414yes—
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—216no—
151C22×Dic7—44no—
152C22×Dic7—44no—
153C22×Dic7—48no—
154C22×Dic7—44no—
155C22×Dic7C22×C414yes—
156C22×Dic7C22×C414yes—
157C22×Dic7C22×C414yes—
158C22×Dic7C22×C414yes—
159C22×Dic7C22×C414yes—
160C22×Dic7C22×C414yes—
161C22×Dic7C2×D414yes—
162C22×Dic7C2×D414yes—
163C22×Dic7#4114yes—
164C22×Dic7#4114yes—
165C7×C22⋊C4—216no—
166C7×C22⋊C4—44no—
167C42×C7—44no—
168C22⋊Dic7—216no—
169C22⋊Dic7C42116yescomplemented
170Dic7⋊C4—216no—
171Dic7⋊C4—216no—
172C4×Dic7—216no—
173C4×Dic7—44no—
174C4×Dic7—48no—
175C4×Dic7—48no—
176C24×C4—71no—
177C22×C4⋊C4—72no—
178C22×C4⋊C4—72no—
179C22×C22⋊C4—71no—
180C22×C22⋊C4—72no—
181C22×C22⋊C4C2×D712yes—
182C22×C22⋊C4C2×D711yescharacteristic
183C22×C42—72no—
184C22×C42—72no—
185C22×C42—72no—
186C22×C42—71no—
187C22×C42C2×D711yescharacteristic
188#388—716no—
189#385—78no—
190C4×C4⋊C4—148no—
191C4×C22⋊C4—148no—
192C2×#95—72no—
193C2×#95—72no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—72no—
198C2×#95—71no—
199C2×#95C2×D712yes—
200C2×#95C2×D711yescharacteristic
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—41no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—41no—
221C23×C7C2×C4214yes—
222C23×C7C23×C411yescharacteristic
223C23×C7C23×C412yes—
224C23×C7C22×D412yes—
225C23×C7C2×#4112yes—
226C23×C7C2×#4112yes—
227C23×C7C2×#4112yes—
228C23×C7H(5,𝔽2)11yescharacteristic
229C23×C7H(5,𝔽2)11yescharacteristic
230C23×C72−1+412yes—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—41no—
254C2×C4×C7C4×D418yes—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
258C2×Dic7—24no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—24no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—24no—
273C2×Dic7—24no—
274C2×Dic7—24no—
275C2×Dic7—24no—
276C2×Dic7—416no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7C2×C4218yes—
282C2×Dic7C4×D418yes—
283C2×Dic7C4×D418yes—
284C25C22×D711yescharacteristic
285C23×C4—72no—
286C23×C4—72no—
287C23×C4—72no—
288C23×C4—72no—
289C23×C4—74no—
290C23×C4—144no—
291C23×C4—144no—
292C23×C4—281no—
293C23×C4—281no—
294C23×C4C22×D711yescharacteristic
295C23×C4C22×D712yes—
296C23×C4C22×D711yescharacteristic
297C23×C4C22×D712yes—
298C2×C4⋊C4—78no—
299C2×C4⋊C4—78no—
300C2×C4⋊C4—144no—
301C2×C4⋊C4—144no—
302C2×C4⋊C4—144no—
303C2×C4⋊C4—144no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—74no—
309C2×C22⋊C4—78no—
310C2×C22⋊C4—78no—
311C2×C22⋊C4—144no—
312C2×C22⋊C4—144no—
313C2×C22⋊C4C4×D718yes—
314C2×C22⋊C4C4×D714yes—
315C2×C42—24no—
316C2×C42—41no—
317C2×C42—78no—
318C2×C42—144no—
319C2×C42—144no—
320C2×C42—144no—
321C2×C42—144no—
322C2×C42—144no—
323C2×C42—144no—
324C2×C42—144no—
325C2×C42—282no—
326C2×C42—282no—
327C2×C42—281no—
328#95—24no—
329#95—41no—
330#95—78no—
331#95—78no—
332#95—78no—
333#95—78no—
334#95—144no—
335#95—144no—
336#95—281no—
337C22×C7—28no—
338C22×C7—24no—
339C22×C7—42no—
340C22×C7—44no—
341C22×C7—44no—
342C22×C7—44no—
343C22×C7—44no—
344C22×C7—44no—
345C22×C7—42no—
346C22×C7C22×C4211yescharacteristic
347C22×C7C2×C4×D412yes—
348C22×C7C2×C4×D412yes—
349C22×C7C2×C4×D412yes—
350C22×C7C4×#4112yes—
351C22×C7C4×#4112yes—
352C22×C7C4×#4112yes—
353C22×C7#51811yescharacteristic
354C22×C7#51911yescharacteristic
355C22×C7#52012yes—
356C22×C7#52012yes—
357C22×C7D4211yescharacteristic
358C22×C7#54612yes—
359C22×C7#54712yes—
360C22×C7#54812yes—
361C22×C7#55111yescharacteristic
362C22×C7#55111yescharacteristic
363C22×C7#55212yes—
364C22×C7#55312yes—
365C22×C7#55612yes—
366C22×C7#55911yescharacteristic
367C4×C7—28no—
368C4×C7—42no—
369C4×C7—44no—
370C4×C7—42no—
371Dic7—28no—
372Dic7—28no—
373Dic7—28no—
374Dic7—44no—
375C24—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C2×C4×D714yes—
383C24C2×C4×D712yes—
384C24C23×D711yescharacteristic
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—74no—
404C22×C4—74no—
405C22×C4—74no—
406C22×C4—74no—
407C22×C4—74no—
408C22×C4—74no—
409C22×C4—74no—
410C22×C4—74no—
411C22×C4—74no—
412C22×C4—74no—
413C22×C4—1416no—
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—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—284no—
433C22×C4—284no—
434C22×C4—284no—
435C22×C4—288no—
436C22×C4C2×C4×D712yes—
437C22×C4C2×C4×D714yes—
438C22×C4C2×C4×D712yes—
439C22×C4C2×C4×D714yes—
440C22×C4D4×D714yes—
441C22×C4#115914yes—
442C4⋊C4—1416no—
443C4⋊C4—1416no—
444C22⋊C4—216no—
445C22⋊C4—44no—
446C22⋊C4—716no—
447C22⋊C4—1416no—
448C42—44no—
449C42—1416no—
450C42—288no—
451C42—284no—
452C42—288no—
453C2×C7—44no—
454C2×C7C42×D412yes—
455C2×C7#227411yescharacteristic
456C2×C7#235811yescharacteristic
457C2×C7#235912yes—
458C2×C7#236112yes—
459C2×C7#236212yes—
460C2×C7#236712yes—
461C2×C7#237312yes—
462C2×C7#237411yescharacteristic
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23C42×D714yes—
484C23C22×C4×D712yes—
485C23C22×C4×D711yescharacteristic
486C23C2×D4×D712yes—
487C23C2×#115912yes—
488C23#654811yescharacteristic
489C23#654811yescharacteristic
490C23D7×#4112yes—
491C23D7×#4112yes—
492C23#655412yes—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—78no—
517C2×C4—78no—
518C2×C4—78no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—144no—
524C2×C4—144no—
525C2×C4—144no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—2816no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4(C22⋊D7)⋊C418yes—
546C7#1382311yescharacteristic, complemented, Sylow
547C22—28no—
548C22—24no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22C2×C42×D711yescharacteristic
557C22C2×(C22⋊D7)⋊C412yes—
558C22#9017711yescharacteristic
559C22#9020711yescharacteristic
560C22#9021312yes—
561C22C4×#115912yes—
562C22C4×#115912yes—
563C22C4×D4×D712yes—
564C22C4×D4×D712yes—
565C22#9023712yes—
566C22#9023712yes—
567C22#9027811yescharacteristic
568C22#9030112yes—
569C22#9030212yes—
570C22#9030512yes—
571C22#9034112yes—
572C22#9035511yescharacteristic
573C22#9036812yes—
574C22#9037011yescharacteristic
575C22#9039312yes—
576C22#9040811yescharacteristic
577C4—28no—
578C4—42no—
579C4—44no—
580C4—42no—
581C4—148no—
582C4—148no—
583C4—148no—
584C4—284no—
585C2—44no—
586C2C4×(C22⋊D7)⋊C412yes—
587C2#1173088011yescharacteristic
588C2#1173212012yes—
589C2#1173212611yescharacteristic
590C2#1173292312yes—
591C2#1173292612yes—
592C2#1173293012yes—
593C2#1173387711yescharacteristic
594C2#1173387812yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC22×C7C22×C42
FittingC7×#1437C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1437D7
hypocenterC7#13823
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
4. C2×#89421C2semidirect
7. C4×C7⋊#95C2semidirect
8. #11727160C2semidirect
9. #11727160C2semidirect
31. #89718C4semidirect
32. #89716C4semidirect
68. C2×Dic7⋊C4C2×C4semidirect
79. C2×C4×Dic7C2×C4semidirect
86. C7⋊#95C2×C4semidirect
87. C7⋊#95C2×C4semidirect
169. C22⋊Dic7C42semidirect
546. C7#13823semidirect

Maximal subgroups

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

subgroupindex
C2×#897162
C2×#897182
C2×C22⋊C4×Dic72
C2×#894212
C7×#14372
#117292992
C4×C7⋊#952
#117271602
#138237