Finite Group Databasebeta

← #13283574

Subgroups — #13283574

Order 1792.

1642 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89982C212yesmaximal
2C2×#89721C211yesmaximal, characteristic
3C2×#89419C212yesmaximal, complemented
4C7×C22⋊#95C211yesmaximal, characteristic
5#11729300C211yesmaximal, characteristic
6#11729299C212yesmaximal
7#11727175C212yesmaximal, complemented
8#11727165C212yesmaximal, complemented
9#11727160C212yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C22×C7×C22⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C22⋊Dic7C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4C2212yes—
18C22×Dic7⋊C4C2212yes—
19C22×Dic7⋊C4C2212yes—
20C22×C4×Dic7C2212yes—
21C2×C7×#95C2212yes—
22C2×C7×#95C2212yes—
23#89982—28no—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32#89721C418yescomplemented
33#89419—28no—
34#14551—71nomaximal, Sylow
35C25×C7C2311yescharacteristic
36C23×C4×C7—22no—
37C23×C4×C7—24no—
38C23×C4×C7—42no—
39C23×C4×C7—41no—
40C23×C4×C7—41no—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C2×C7×C22⋊C4—41no—
50C2×C7×C22⋊C4—41no—
51C2×C7×C22⋊C4—41no—
52C2×C7×C22⋊C4—41no—
53C2×C7×C22⋊C4C2×C414yes—
54C2×C7×C22⋊C4Q814yes—
55C2×C22⋊Dic7—24no—
56C2×C22⋊Dic7—24no—
57C2×C22⋊Dic7—24no—
58C2×C22⋊Dic7—24no—
59C2×C22⋊Dic7—42no—
60C2×C22⋊Dic7C2×C418yes—
61C2×C4⋊Dic7—24no—
62C2×C4⋊Dic7—41no—
63C2×C4⋊Dic7—41no—
64C2×C4⋊Dic7—41no—
65C2×C4⋊Dic7C2×C414yescomplemented
66C2×Dic7⋊C4—24no—
67C2×Dic7⋊C4—24no—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4—24no—
70C2×Dic7⋊C4—42no—
71C2×Dic7⋊C4—42no—
72C2×Dic7⋊C4—42no—
73C2×Dic7⋊C4C2×C418yescomplemented
74C2×Dic7⋊C4D418yes—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—42no—
77C2×C4×Dic7—42no—
78C7×#95—24no—
79C7×#95—42no—
80C7⋊#95—24no—
81C7⋊#95—24no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—41no—
87C7⋊#95C2×C414yescomplemented
88C2×#399—71no—
89C2×#389—72no—
90C2×#385—72no—
91#1445—72no—
92#1441—72no—
93#1439—71no—
94#1437—72no—
95#1436—72no—
96C22⋊#95D711yescharacteristic
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—21no—
101C24×C7—21no—
102C24×C7—21no—
103C24×C7—21no—
104C24×C7—22no—
105C24×C7C22×C412yes—
106C24×C7C2×Q812yes—
107C24×C7#4112yes—
108C24×C7C2411yescharacteristic
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—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—44no—
130C22×C4×C7—48no—
131C22×C4×C7—48no—
132C22×C4×C7—44no—
133C22×C4×C7—44no—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C2×D414yes—
137C22×C4×C7C2×Q814yes—
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×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7C22×C414yes—
162C22×Dic7C22×C414yes—
163C22×Dic7C2×D414yes—
164C22×Dic7C2×D414yes—
165C22×Dic7#4114yes—
166C7×C22⋊C4—44no—
167C7×C22⋊C4—44no—
168C22⋊Dic7—216no—
169C22⋊Dic7—48no—
170C4⋊Dic7—44no—
171C4⋊Dic7—44no—
172Dic7⋊C4—216no—
173Dic7⋊C4—48no—
174Dic7⋊C4—48no—
175C4×Dic7—48no—
176C24×C4C2×D711yescharacteristic
177C22×C4⋊C4—72no—
178C22×C4⋊C4—71no—
179C22×C4⋊C4—72no—
180C22×C4⋊C4—72no—
181C22×C4⋊C4—71no—
182C22×C22⋊C4—72no—
183C22×C22⋊C4—72no—
184C22×C22⋊C4C2×D711yescharacteristic
185C22×C22⋊C4C2×D711yescharacteristic
186C22×C42—72no—
187#399—78no—
188#389—148no—
189#385—148no—
190C2×#95—71no—
191C2×#95—72no—
192C2×#95—72no—
193C2×#95—72no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—71no—
198C2×#95C2×D712yes—
199C2×#95C2×D712yes—
200C23×C7—24no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—24no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—41no—
217C23×C7—42no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—42no—
221C23×C7—41no—
222C23×C7C4×Q814yes—
223C23×C7C23×C411yescharacteristic
224C23×C7C22×D412yes—
225C23×C7C22×Q811yescharacteristic
226C23×C7C2×#4111yescharacteristic
227C23×C7C2×#4112yes—
228C23×C7H(5,𝔽2)11yescharacteristic
229C23×C7H(5,𝔽2)11yescharacteristic
230C23×C7H(5,𝔽2)11yescharacteristic
231C23×C7H(5,𝔽2)12yes—
232C23×C72−1+411yescharacteristic
233C23×C72−1+412yes—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—416no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
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—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
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—42no—
285C2×Dic7—42no—
286C2×Dic7C4×D418yes—
287C25C22×D711yescharacteristic
288C23×C4—22no—
289C23×C4—24no—
290C23×C4—42no—
291C23×C4—41no—
292C23×C4—41no—
293C23×C4—72no—
294C23×C4—72no—
295C23×C4—72no—
296C23×C4—72no—
297C23×C4C22×D712yes—
298C23×C4C22×D711yescharacteristic
299C23×C4C22×D711yescharacteristic
300C23×C4C22×D712yes—
301C2×C4⋊C4—78no—
302C2×C4⋊C4—74no—
303C2×C4⋊C4—78no—
304C2×C4⋊C4—144no—
305C2×C4⋊C4—144no—
306C2×C4⋊C4—144no—
307C2×C4⋊C4—144no—
308C2×C4⋊C4—144no—
309C2×C4⋊C4—282no—
310C2×C4⋊C4—282no—
311C2×C4⋊C4—282no—
312C2×C4⋊C4—281no—
313C2×C4⋊C4—281no—
314C2×C4⋊C4—281no—
315C2×C22⋊C4—41no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4—41no—
318C2×C22⋊C4—41no—
319C2×C22⋊C4—78no—
320C2×C22⋊C4—144no—
321C2×C22⋊C4—144no—
322C2×C22⋊C4—144no—
323C2×C22⋊C4—144no—
324C2×C22⋊C4—282no—
325C2×C22⋊C4Dic1414yes—
326C2×C22⋊C4C4×D714yes—
327C2×C42—144no—
328C2×C42—282no—
329C2×C42—282no—
330#95—24no—
331#95—42no—
332#95—74no—
333#95—144no—
334#95—144no—
335#95—144no—
336#95—144no—
337#95—282no—
338#95—282no—
339#95—281no—
340C22×C7—24no—
341C22×C7—28no—
342C22×C7—44no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—44no—
348C22×C7—44no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7C2×C4×D412yes—
352C22×C7C2×C4×Q811yescharacteristic
353C22×C7#51811yescharacteristic
354C22×C7#52011yescharacteristic
355C22×C7#52012yes—
356C22×C7#53412yes—
357C22×C7#53612yes—
358C22×C7#53811yescharacteristic
359C22×C7#53911yescharacteristic
360C22×C7#53912yes—
361C22×C7#53912yes—
362C22×C7#54311yescharacteristic
363C22×C7#54411yescharacteristic
364C22×C7D4211yescharacteristic
365C22×C7#54612yes—
366C22×C7D4×Q812yes—
367C22×C7#55012yes—
368C22×C7#55212yes—
369C22×C7#55412yes—
370C22×C7#55911yescharacteristic
371C22×C7#56011yescharacteristic
372C22×C7#56212yes—
373C22×C7#56311yescharacteristic
374C4×C7—42no—
375C4×C7—42no—
376C4×C7—44no—
377C4×C7—44no—
378Dic7—28no—
379Dic7—44no—
380Dic7—44no—
381Dic7—82no—
382C24—22no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—22no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24C2×Dic1412yes—
391C24C2×C4×D712yes—
392C24#115712yes—
393C24C23×D711yescharacteristic
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—74no—
420C22×C4—74no—
421C22×C4—74no—
422C22×C4—74no—
423C22×C4—74no—
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—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4C2×Dic1414yes—
448C22×C4C2×C4×D712yes—
449C22×C4C2×C4×D712yes—
450C22×C4D4×D714yes—
451C4⋊C4—1416no—
452C4⋊C4—288no—
453C4⋊C4—284no—
454C4⋊C4—288no—
455C4⋊C4—284no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—1416no—
459C22⋊C4—288no—
460C42—288no—
461C2×C7—44no—
462C2×C7#232212yes—
463C2×C7#232911yescharacteristic
464C2×C7#236012yes—
465C2×C7#236611yescharacteristic
466C2×C7#237611yescharacteristic
467C2×C7#266811yescharacteristic
468C2×C7#267711yescharacteristic
469C2×C7#269412yes—
470C2×C7#273212yes—
471C2×C7#277111yescharacteristic
472C2×C7#280711yescharacteristic
473C23—22no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23C4×Dic1414yes—
496C23C22×Dic1411yescharacteristic
497C23C22×C4×D711yescharacteristic
498C23C2×#115711yescharacteristic
499C23C2×D4×D712yes—
500C23#654811yescharacteristic
501C23#654811yescharacteristic
502C23#654811yescharacteristic
503C23#654812yes—
504C23#655111yescharacteristic
505C23D7×#4112yes—
506C23#655412yes—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—416no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—78no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—144no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—144no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—282no—
560C7#1455111yescharacteristic, complemented, Sylow
561C22—24no—
562C22—28no—
563C22—44no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22C2×C4×Dic1411yescharacteristic
573C22#9017411yescharacteristic
574C22#9017711yescharacteristic
575C22#9018112yes—
576C22#9018511yescharacteristic
577C22#9019111yescharacteristic
578C22#9019511yescharacteristic
579C22#9020611yescharacteristic
580C22#9022212yes—
581C22D4×Dic1412yes—
582C22#9023012yes—
583C22C4×D4×D712yes—
584C22#9023712yes—
585C22#9028012yes—
586C22#9028111yescharacteristic
587C22#9028611yescharacteristic
588C22#9029012yes—
589C22#9031212yes—
590C22#9031912yes—
591C22#9032912yes—
592C22#9034211yescharacteristic
593C22#9035112yes—
594C22#9035511yescharacteristic
595C4—44no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—148no—
600C4—284no—
601C4—284no—
602C4—562no—
603C2—44no—
604C2#1173077911yescharacteristic
605C2#1173091112yes—
606C2#1173130211yescharacteristic
607C2#1173140712yes—
608C2#1173155811yescharacteristic
609C2#1173195511yescharacteristic
610C2#1173202811yescharacteristic
611C2#1173203812yes—
612C2#1173212411yescharacteristic
613C2#1173212711yescharacteristic
614C2#1173227512yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C22⋊#95C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC22⋊#95D7
hypocenterC7#14551
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#89419C2semidirect
7. #11727175C2semidirect
8. #11727165C2semidirect
9. #11727160C2semidirect
32. #89721C4semidirect
65. C2×C4⋊Dic7C2×C4semidirect
73. C2×Dic7⋊C4C2×C4semidirect
87. C7⋊#95C2×C4semidirect
560. C7#14551semidirect

Maximal subgroups

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

subgroupindex
C2×#899822
C2×#897212
C2×#894192
C7×C22⋊#952
#117293002
#117292992
#117271752
#117271652
#117271602
#145517