Finite Group Databasebeta

← #13284709

Subgroups — #13284709

Order 1792.

1226 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89721C211yesmaximal, characteristic
2C7×#1440C211yesmaximal, characteristic
3#11729303C212yesmaximal
4#11729301C212yesmaximal
5#11729299C211yesmaximal, characteristic
6#11727171C212yesmaximal, complemented
7#11727165C212yesmaximal, complemented
8#11727163C212yesmaximal, complemented
9#11727159C212yesmaximal, complemented
10C22×C7×C22⋊C4C2211yescharacteristic
11C22×C7×C22⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C22⋊Dic7C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4C2212yes—
17C22×Dic7⋊C4C2212yes—
18C22×C4×Dic7C2211yescharacteristic
19C2×C7×#95C2212yes—
20C2×C7×#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2211yescharacteristic
31#89721C418yescomplemented
32#14646—71nomaximal, Sylow
33C25×C7C2311yescharacteristic
34C23×C4×C7C2312yes—
35C23×C4×C7C2312yes—
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C23×Dic7C2312yes—
39C23×Dic7C2312yes—
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C2×C7×C22⋊C4—41no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4—42no—
46C2×C7×C22⋊C4—42no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4C2×C414yes—
49C2×C22⋊Dic7—24no—
50C2×C22⋊Dic7—42no—
51C2×C22⋊Dic7—42no—
52C2×C22⋊Dic7—42no—
53C2×C22⋊Dic7—42no—
54C2×C22⋊Dic7C2×C418yes—
55C2×C4⋊Dic7—41no—
56C2×C4⋊Dic7—41no—
57C2×C4⋊Dic7C2×C414yes—
58C2×Dic7⋊C4—42no—
59C2×Dic7⋊C4—42no—
60C2×Dic7⋊C4—42no—
61C2×Dic7⋊C4—42no—
62C2×Dic7⋊C4—42no—
63C2×Dic7⋊C4C2×C418yes—
64C2×C4×Dic7—41no—
65C2×C4×Dic7—42no—
66C7×#95—42no—
67C7×#95—42no—
68C7⋊#95—24no—
69C7⋊#95—24no—
70C7⋊#95—42no—
71C7⋊#95—42no—
72C7⋊#95—42no—
73C7⋊#95—42no—
74C7⋊#95—42no—
75C7⋊#95—42no—
76C7⋊#95—41no—
77C7⋊#95C2×C414yes—
78C2×#399—71no—
79#1443—72no—
80#1441—72no—
81#1441—72no—
82#1440—72no—
83#1440D711yescharacteristic
84#1439—72no—
85#1437—71no—
86#1436—72no—
87C24×C7—22no—
88C24×C7—22no—
89C24×C7—22no—
90C24×C7—22no—
91C24×C7—21no—
92C24×C7—21no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7C22×C412yes—
96C24×C7C2411yescharacteristic
97C22×C4×C7—22no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
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×C7C22×C412yes—
120C22×C4×C7C22×C412yes—
121C22×Dic7—22no—
122C22×Dic7—22no—
123C22×Dic7—22no—
124C22×Dic7—22no—
125C22×Dic7—22no—
126C22×Dic7—22no—
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×Dic7C22×C414yes—
147C22×Dic7C22×C414yes—
148C22×Dic7#4114yes—
149C7×C22⋊C4—44no—
150C7×C22⋊C4—44no—
151C7×C22⋊C4—44no—
152C22⋊Dic7—48no—
153C22⋊Dic7—48no—
154C4⋊Dic7—44no—
155Dic7⋊C4—48no—
156Dic7⋊C4—48no—
157C4×Dic7—44no—
158C22×C4⋊C4—72no—
159C22×C4⋊C4—71no—
160C22×C4⋊C4—72no—
161C22×C22⋊C4—72no—
162C22×C22⋊C4—72no—
163C22×C22⋊C4C2×D711yescharacteristic
164C22×C22⋊C4C2×D711yescharacteristic
165C22×C22⋊C4C2×D711yescharacteristic
166C22×C42—71no—
167#399—78no—
168C2×#95—71no—
169C2×#95—72no—
170C2×#95—72no—
171C2×#95—72no—
172C2×#95—72no—
173C2×#95—71no—
174C2×#95—72no—
175C2×#95—72no—
176C2×#95—72no—
177C2×#95—72no—
178C2×#95C2×D712yes—
179C2×#95C2×D712yes—
180C23×C7—22no—
181C23×C7—24no—
182C23×C7—22no—
183C23×C7—22no—
184C23×C7—24no—
185C23×C7—42no—
186C23×C7—42no—
187C23×C7—42no—
188C23×C7—42no—
189C23×C7—42no—
190C23×C7—42no—
191C23×C7—41no—
192C23×C7—41no—
193C23×C7—42no—
194C23×C7—41no—
195C23×C7—42no—
196C23×C7—42no—
197C23×C7—42no—
198C23×C7—42no—
199C23×C7—41no—
200C23×C7—42no—
201C23×C7C23×C411yescharacteristic
202C23×C7C2×#4112yes—
203C23×C7H(5,𝔽2)11yescharacteristic
204C23×C7H(5,𝔽2)12yes—
205C23×C7H(5,𝔽2)11yescharacteristic
206C23×C7H(5,𝔽2)12yes—
207C23×C7H(5,𝔽2)12yes—
208C23×C72−1+412yes—
209C23×C72−1+412yes—
210C2×C4×C7—24no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—24no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—41no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×Dic7—24no—
235C2×Dic7—24no—
236C2×Dic7—24no—
237C2×Dic7—24no—
238C2×Dic7—24no—
239C2×Dic7—24no—
240C2×Dic7—24no—
241C2×Dic7—24no—
242C2×Dic7—24no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C25C22×D711yescharacteristic
263C23×C4—72no—
264C23×C4—72no—
265C23×C4—72no—
266C23×C4—72no—
267C23×C4C22×D712yes—
268C23×C4C22×D712yes—
269C23×C4C22×D711yescharacteristic
270C23×C4C22×D711yescharacteristic
271C2×C4⋊C4—78no—
272C2×C4⋊C4—74no—
273C2×C4⋊C4—282no—
274C2×C4⋊C4—282no—
275C2×C4⋊C4—282no—
276C2×C4⋊C4—282no—
277C2×C4⋊C4—282no—
278C2×C4⋊C4—281no—
279C2×C4⋊C4—281no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—78no—
287C2×C22⋊C4—144no—
288C2×C22⋊C4—282no—
289C2×C22⋊C4—282no—
290C2×C22⋊C4—282no—
291C2×C22⋊C4—282no—
292C2×C22⋊C4C4×D714yes—
293C2×C42—281no—
294C2×C42—282no—
295#95—42no—
296#95—42no—
297#95—74no—
298#95—144no—
299#95—144no—
300#95—282no—
301#95—281no—
302#95—282no—
303#95—282no—
304#95—282no—
305#95—282no—
306#95—282no—
307C22×C7—42no—
308C22×C7—44no—
309C22×C7—42no—
310C22×C7—44no—
311C22×C7—42no—
312C22×C7—82no—
313C22×C7—82no—
314C22×C7—82no—
315C22×C7—82no—
316C22×C7C4×#4111yescharacteristic
317C22×C7#51811yescharacteristic
318C22×C7#51812yes—
319C22×C7#51911yescharacteristic
320C22×C7#52012yes—
321C22×C7#53812yes—
322C22×C7#53912yes—
323C22×C7#53912yes—
324C22×C7#54012yes—
325C22×C7#54212yes—
326C22×C7#54212yes—
327C22×C7#56011yescharacteristic
328C22×C7#56011yescharacteristic
329C22×C7#56012yes—
330C22×C7#56011yescharacteristic
331C22×C7#56212yes—
332C22×C7#56212yes—
333C22×C7#56212yes—
334C22×C7#56312yes—
335C22×C7#56311yescharacteristic
336C22×C7#56312yes—
337C4×C7—42no—
338C4×C7—42no—
339C4×C7—82no—
340C4×C7—82no—
341Dic7—44no—
342Dic7—44no—
343Dic7—44no—
344Dic7—82no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C2×C4×D712yes—
354C24C23×D711yescharacteristic
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—74no—
378C22×C4—74no—
379C22×C4—74no—
380C22×C4—142no—
381C22×C4—142no—
382C22×C4—142no—
383C22×C4—142no—
384C22×C4—142no—
385C22×C4—142no—
386C22×C4—142no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—142no—
390C22×C4—142no—
391C22×C4—142no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4C2×C4×D712yes—
406C22×C4C2×C4×D712yes—
407C4⋊C4—288no—
408C4⋊C4—284no—
409C4⋊C4—288no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—288no—
414C22⋊C4—288no—
415C42—284no—
416C2×C7—82no—
417C2×C7#233211yescharacteristic
418C2×C7#233512yes—
419C2×C7#237611yescharacteristic
420C2×C7#237812yes—
421C2×C7#238011yescharacteristic
422C2×C7#282712yes—
423C2×C7#282912yes—
424C2×C7#283712yes—
425C2×C7#283812yes—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23C22×C4×D711yescharacteristic
448C23#654812yes—
449C23#654811yescharacteristic
450C23#654811yescharacteristic
451C23#654812yes—
452C23D7×#4112yes—
453C23#655312yes—
454C23#655412yes—
455C23#655412yes—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—144no—
481C2×C4—144no—
482C2×C4—144no—
483C2×C4—144no—
484C2×C4—144no—
485C2×C4—144no—
486C2×C4—144no—
487C2×C4—144no—
488C2×C4—144no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C7#1464611yescharacteristic, complemented, Sylow
509C22—44no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22#9017711yescharacteristic
519C22#9020711yescharacteristic
520C22#9021311yescharacteristic
521C22#9023612yes—
522C22#9023712yes—
523C22#9028611yescharacteristic
524C22#9028611yescharacteristic
525C22#9030612yes—
526C22#9031112yes—
527C22#9031212yes—
528C22#9031212yes—
529C22#9034211yescharacteristic
530C22#9035112yes—
531C22#9035312yes—
532C22#9035312yes—
533C22#9035711yescharacteristic
534C22#9035812yes—
535C22#9036312yes—
536C22#9036912yes—
537C22#9039812yes—
538C22#9040312yes—
539C4—42no—
540C4—42no—
541C4—82no—
542C4—82no—
543C4—284no—
544C4—284no—
545C4—284no—
546C4—562no—
547C2—82no—
548C2#1173150211yescharacteristic
549C2#1173212711yescharacteristic
550C2#1173213011yescharacteristic
551C2#1173293612yes—
552C2#1173327612yes—
553C2#1173454312yes—
554C2#1173454812yes—
555C2#1173511112yes—
556C2#1173517912yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1440C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1440D7
hypocenterC7#14646
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
6. #11727171C2semidirect
7. #11727165C2semidirect
8. #11727163C2semidirect
9. #11727159C2semidirect
31. #89721C4semidirect
508. C7#14646semidirect

Maximal subgroups

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

subgroupindex
C2×#897212
C7×#14402
#117293032
#117293012
#117292992
#117271712
#117271652
#117271632
#117271592
#146467