Finite Group Databasebeta

← #13280980

Subgroups — #13280980

Order 1792.

1926 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6435C211yesmaximal, characteristic
2C2×#89759C211yesmaximal, characteristic
3C2×#89759C211yesmaximal, characteristic
4C2×#89748C211yesmaximal, characteristic, complemented
5C2×#89749C211yesmaximal, characteristic, complemented
6C7×C4⋊#95C211yesmaximal, characteristic, complemented
7#11729289C211yesmaximal, characteristic, complemented
8#11727182C214yesmaximal
9#11727163C214yesmaximal, complemented
10C23×C4×D7C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×(C2×D7)⋊C4C2212yes—
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2212yes—
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2212yes—
20C22×Dic7⋊C4C2214yes—
21C22×C4×Dic7C2211yescharacteristic
22C2×#6435—24no—
23C2×#6435—24no—
24C2×#6435C418yescomplemented
25C2×C7×#95C2214yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2214yes—
29C2×C7⋊#95C2214yes—
30#89759—24no—
31#89759—24no—
32#89749—24no—
33#89748—24no—
34#14340—71nomaximal, Sylow
35C24×D7C2311yescharacteristic
36C23×C4×C7C2312yes—
37C23×C4×C7C2312yes—
38C23×C4×C7C2312yes—
39C23×C4×C7C2311yescharacteristic
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C23×Dic7C2312yes—
43C23×Dic7C2311yescharacteristic
44C22×C4×D7—22no—
45C22×C4×D7—22no—
46C22×C4×D7—22no—
47C22×C4×D7—22no—
48C22×C4×D7C2×C414yes—
49C22×C4×D7D414yes—
50C22×C4×D7Q814yes—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C42×C7—22no—
58C2×C42×C7—22no—
59C2×C42×C7C2×C414yes—
60C2×(C2×D7)⋊C4—24no—
61C2×(C2×D7)⋊C4—24no—
62C2×(C2×D7)⋊C4—24no—
63C2×(C2×D7)⋊C4—24no—
64C2×(C2×D7)⋊C4—44no—
65C2×(C2×D7)⋊C4—44no—
66C2×(C2×D7)⋊C4C2×C418yes—
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—22no—
73C2×Dic7⋊C4—24no—
74C2×Dic7⋊C4—24no—
75C2×Dic7⋊C4—28no—
76C2×Dic7⋊C4—44no—
77C2×Dic7⋊C4—44no—
78C2×Dic7⋊C4C2×C418yes—
79C2×C4×Dic7C2×C414yes—
80C2×C4×Dic7D414yes—
81C2×C4×Dic7D414yes—
82#6435—232no—
83C7×#95—44no—
84C7⋊#95—28no—
85C7⋊#95—24no—
86C7⋊#95—24no—
87C7⋊#95—44no—
88C22×#117—71no—
89C2×#384—71no—
90C2×#382—71no—
91C2×#380—71no—
92C2×#380—71no—
93#1441—74no—
94#1439—74no—
95C4⋊#95—71no—
96C4⋊#95D711yescharacteristic, complemented
97C24×C7C2411yescharacteristic
98C23×D7—21no—
99C23×D7—21no—
100C23×D7—21no—
101C23×D7—21no—
102C23×D7—22no—
103C23×D7—22no—
104C23×D7—22no—
105C23×D7—22no—
106C23×D7C22×C412yes—
107C23×D7C2×D412yes—
108C23×D7C2×Q812yes—
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—22no—
121C22×C4×C7—24no—
122C22×C4×C7—24no—
123C22×C4×C7—24no—
124C22×C4×C7—24no—
125C22×C4×C7C22×C414yes—
126C22×C4×C7C22×C412yes—
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×Q812yes—
131C22×C4×C7#4112yes—
132C22×C4×C7#4112yes—
133C22×Dic7—21no—
134C22×Dic7—21no—
135C22×Dic7—21no—
136C22×Dic7—21no—
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—24no—
148C22×Dic7—24no—
149C22×Dic7—24no—
150C22×Dic7—24no—
151C22×Dic7C22×C414yes—
152C22×Dic7C22×C412yes—
153C22×Dic7C2×D414yes—
154C22×Dic7C2×D414yes—
155C22×Dic7C2×D412yes—
156C22×Dic7C2×Q812yes—
157C2×C4×D7—28no—
158C2×C4×D7—28no—
159C2×C4×D7—28no—
160C2×C4×D7—28no—
161C2×C4×D7—28no—
162C2×C4×D7—28no—
163C2×C4×D7C4⋊C4116yescomplemented
164C7×C4⋊C4—44no—
165C7×C4⋊C4—44no—
166C42×C7—28no—
167(C2×D7)⋊C4—216no—
168(C2×D7)⋊C4—48no—
169(C2×D7)⋊C4—48no—
170C4⋊Dic7—44no—
171C4⋊Dic7—44no—
172Dic7⋊C4—216no—
173Dic7⋊C4—416no—
174C4×Dic7C22⋊C4116yescomplemented
175C24×C4—71no—
176C2×#117—78no—
177C2×#117—144no—
178C2×#117—144no—
179C22×C4⋊C4—72no—
180C22×C4⋊C4—71no—
181C22×C4⋊C4—71no—
182C22×C4⋊C4—74no—
183C22×C4⋊C4C2×D711yescharacteristic
184C22×C4⋊C4C2×D711yescharacteristic
185C22×C22⋊C4—72no—
186C22×C22⋊C4—72no—
187C22×C22⋊C4—72no—
188C22×C42—71no—
189C22×C42C2×D711yescharacteristic
190#384—144no—
191#382—144no—
192#380—144no—
193#380—144no—
194C2×#95—74no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—74no—
198C2×#95C2×D714yes—
199C23×C7C23×C411yescharacteristic
200C23×C7C22×D411yescharacteristic
201C23×C7C22×D411yescharacteristic
202C23×C7C22×D411yescharacteristic
203C23×C7C22×Q811yescharacteristic
204C23×C7C2×#4111yescharacteristic
205C23×C7C2×#4111yescharacteristic
206C23×C7H(5,𝔽2)12yes—
207C23×C7H(5,𝔽2)12yes—
208C23×C72−1+412yes—
209C23×C72−1+412yes—
210C22×D7—22no—
211C22×D7—22no—
212C22×D7—24no—
213C22×D7—24no—
214C22×D7—22no—
215C22×D7—22no—
216C22×D7—22no—
217C22×D7—22no—
218C22×D7—22no—
219C22×D7—22no—
220C22×D7—22no—
221C22×D7—22no—
222C22×D7—24no—
223C22×D7—24no—
224C22×D7—42no—
225C22×D7—42no—
226C22×D7—42no—
227C22×D7—42no—
228C22×D7—42no—
229C22×D7—42no—
230C22×D7—42no—
231C22×D7—42no—
232C22×D7C2×C4⋊C414yes—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—44no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—44no—
248C2×C4×C7—44no—
249C2×C4×C7—44no—
250C2×C4×C7C2×C22⋊C414yes—
251C2×C4×C7C2×C4⋊C414yes—
252C2×C4×C7#11714yes—
253C2×C4×C7#12114yes—
254C2×C4×C7#12114yes—
255C2×C4×C7C22⋊Q814yes—
256C2×C4×C7C22⋊Q814yes—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—44no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—44no—
277C2×Dic7—44no—
278C2×Dic7—44no—
279C2×Dic7C2×C22⋊C418yes—
280C2×Dic7C2×C4⋊C414yes—
281C4×D7—232no—
282C25—71no—
283C23×C4—72no—
284C23×C4—71no—
285C23×C4—74no—
286C23×C4—74no—
287C23×C4—74no—
288C23×C4—72no—
289C23×C4—72no—
290C23×C4—142no—
291C23×C4—142no—
292C23×C4—142no—
293C23×C4—142no—
294C23×C4C22×D712yes—
295C23×C4C22×D712yes—
296C23×C4C22×D712yes—
297C23×C4C22×D711yescharacteristic
298#117—1432no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—78no—
306C2×C4⋊C4—144no—
307C2×C4⋊C4—144no—
308C2×C4⋊C4—148no—
309C2×C4⋊C4—142no—
310C2×C4⋊C4—142no—
311C2×C4⋊C4—142no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—142no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—284no—
316C2×C4⋊C4—284no—
317C2×C22⋊C4—78no—
318C2×C22⋊C4—144no—
319C2×C22⋊C4—144no—
320C2×C22⋊C4—144no—
321C2×C22⋊C4—144no—
322C2×C22⋊C4—284no—
323C2×C22⋊C4—284no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—74no—
327C2×C42—74no—
328C2×C42—74no—
329C2×C42C4×D714yes—
330#95—44no—
331#95—144no—
332#95—144no—
333#95—148no—
334#95—284no—
335C22×C7C22×C22⋊C411yescharacteristic
336C22×C7C22×C4⋊C411yescharacteristic
337C22×C7C2×#11711yescharacteristic
338C22×C7#52012yes—
339C22×C7#52012yes—
340C22×C7C2×#12111yescharacteristic
341C22×C7C2×#12111yescharacteristic
342C22×C7C2×C22⋊Q811yescharacteristic
343C22×C7C2×C22⋊Q811yescharacteristic
344C22×C7#53511yescharacteristic
345C22×C7#53512yes—
346C22×C7#53511yescharacteristic
347C22×C7#53611yescharacteristic
348C22×C7#53612yes—
349C22×C7#53611yescharacteristic
350C22×C7#53712yes—
351C22×C7#53712yes—
352C22×C7#54012yes—
353C22×C7#54112yes—
354C22×C7#54212yes—
355C22×C7#54212yes—
356C22×C7#54412yes—
357C22×C7#54412yes—
358C2×D7—24no—
359C2×D7—24no—
360C2×D7—24no—
361C2×D7—44no—
362C2×D7—44no—
363C2×D7—42no—
364C2×D7—42no—
365C2×D7—42no—
366C2×D7—42no—
367C2×D7—44no—
368C2×D7—44no—
369C4×C7—44no—
370C4×C7—82no—
371C4×C7—82no—
372C4×C7#38018yes—
373Dic7—28no—
374Dic7—24no—
375Dic7—82no—
376Dic7—82no—
377C24—72no—
378C24—72no—
379C24—72no—
380C24—141no—
381C24—141no—
382C24—141no—
383C24—141no—
384C24—142no—
385C24—142no—
386C24—142no—
387C24—142no—
388C24C23×D711yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—74no—
406C22×C4—72no—
407C22×C4—72no—
408C22×C4—74no—
409C22×C4—74no—
410C22×C4—72no—
411C22×C4—716no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—148no—
421C22×C4—148no—
422C22×C4—148no—
423C22×C4—148no—
424C22×C4—148no—
425C22×C4—148no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—144no—
433C22×C4—144no—
434C22×C4—144no—
435C22×C4—144no—
436C22×C4C2×C4×D714yes—
437C22×C4C2×C4×D712yes—
438C22×C4#115712yes—
439C22×C4#115712yes—
440C22×C4D4×D712yes—
441C22×C4D4×D712yes—
442C22×C4D4×D712yes—
443C22×C4Q8×D712yes—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—1416no—
447C4⋊C4—2816no—
448C4⋊C4—284no—
449C4⋊C4—284no—
450C22⋊C4—1416no—
451C22⋊C4—288no—
452C22⋊C4—288no—
453C42—28no—
454C42—716no—
455C2×C7C2×#38011yescharacteristic
456C2×C7#231112yes—
457C2×C7#231412yes—
458C2×C7#231711yescharacteristic
459C2×C7#231711yescharacteristic
460C2×C7#261911yescharacteristic
461C2×C7#262011yescharacteristic
462C2×C7#262011yescharacteristic
463C2×C7#262111yescharacteristic
464C2×C7#262812yes—
465C2×C7#262912yes—
466D7—44no—
467C23—74no—
468C23—144no—
469C23—142no—
470C23—142no—
471C23—142no—
472C23—144no—
473C23—142no—
474C23—142no—
475C23—142no—
476C23—144no—
477C23—142no—
478C23—142no—
479C23—144no—
480C23—142no—
481C23—142no—
482C23—282no—
483C23—282no—
484C23—282no—
485C23—282no—
486C23—282no—
487C23—282no—
488C23—282no—
489C23—282no—
490C23C22×C4×D711yescharacteristic
491C23C2×#115711yescharacteristic
492C23C2×#115711yescharacteristic
493C23C2×D4×D711yescharacteristic
494C23C2×D4×D711yescharacteristic
495C23C2×D4×D711yescharacteristic
496C23#654812yes—
497C23C2×Q8×D711yescharacteristic
498C23#655112yes—
499C23#655312yes—
500C23#655412yes—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—78no—
519C2×C4—74no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—142no—
524C2×C4—144no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—1432no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—284no—
538C2×C4—284no—
539C2×C4—284no—
540C2×C4—284no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4#643514yes—
544C2×C4D7×C22⋊C414yes—
545C2×C4#644714yes—
546C2×C4#644714yes—
547C2×C4D7×C4⋊C414yes—
548C2×C4#645914yes—
549C2×C4#645914yes—
550C7#1434011yescharacteristic, complemented, Sylow
551C22—144no—
552C22—144no—
553C22—144no—
554C22—284no—
555C22—282no—
556C22—282no—
557C22—282no—
558C22—282no—
559C22—284no—
560C22—284no—
561C22—284no—
562C22C2×#643511yescharacteristic
563C22C2×D7×C22⋊C411yescharacteristic
564C22C2×#644711yescharacteristic
565C22C2×#644711yescharacteristic
566C22C2×D7×C4⋊C411yescharacteristic
567C22#9019512yes—
568C22C2×#645911yescharacteristic
569C22C2×#645911yescharacteristic
570C22#9021412yes—
571C22#9025212yes—
572C22#9025312yes—
573C22#9027312yes—
574C22#9027412yes—
575C22#9029812yes—
576C22#9029912yes—
577C22#9031912yes—
578C22#9032012yes—
579C22#9036511yescharacteristic
580C22#9036611yescharacteristic
581C22#9038012yes—
582C22#9040611yescharacteristic
583C22#9041511yescharacteristic
584C22#9042012yes—
585C4—44no—
586C4—82no—
587C4—82no—
588C4—144no—
589C4—148no—
590C4—562no—
591C4—562no—
592C4#8943918yes—
593C2—284no—
594C2C2×#8943911yescharacteristic
595C2#1173120512yes—
596C2#1173144812yes—
597C2#1173167011yescharacteristic
598C2#1173180411yescharacteristic
599C2#1173369512yes—
600C2#1173369612yes—
601C2#1173390111yescharacteristic
602C2#1173390211yescharacteristic
603C2#1173408511yescharacteristic
604C2#1173408611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C4⋊#95C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenterC4⋊#95D7
hypocenterC7#14340
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. C2×#89748C2semidirect
5. C2×#89749C2semidirect
6. C7×C4⋊#95C2semidirect
7. #11729289C2semidirect
9. #11727163C2semidirect
24. C2×#6435C4semidirect
96. C4⋊#95D7semidirect
163. C2×C4×D7C4⋊C4semidirect
174. C4×Dic7C22⋊C4semidirect
550. C7#14340semidirect

Maximal subgroups

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

subgroupindex
C22×#64352
C2×#897592
C2×#897482
C2×#897492
C7×C4⋊#952
#117292892
#117271822
#117271632
#143407