Finite Group Databasebeta

← #13284708

Subgroups — #13284708

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×C7×#399C211yesmaximal, characteristic
2#11729319C212yesmaximal, complemented
3#11729319C212yesmaximal, complemented
4#11729318C212yesmaximal, complemented
5#11729302C212yesmaximal
6#11729302C211yesmaximal, characteristic
7#11729300C212yesmaximal
8#11729299C211yesmaximal, characteristic
9#11727170C212yesmaximal, complemented
10C22×C7×C4⋊C4C2212yes—
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2212yes—
18C22×C4×Dic7C2211yescharacteristic
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⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32#14646—71nomaximal, Sylow
33C25×C7C2311yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C23×C4×C7C2312yes—
36C23×C4×C7C2312yes—
37C23×C4×C7C2311yescharacteristic
38C23×Dic7C2312yes—
39C23×Dic7C2312yes—
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C2×C7×C4⋊C4—42no—
43C2×C7×C4⋊C4—41no—
44C2×C7×C4⋊C4—42no—
45C2×C7×C4⋊C4—41no—
46C2×C7×C4⋊C4C2×C418yes—
47C2×C7×C4⋊C4C2×C414yes—
48C2×C7×C22⋊C4—41no—
49C2×C7×C22⋊C4—42no—
50C2×C7×C22⋊C4—41no—
51C2×C7×C22⋊C4—42no—
52C2×C7×C22⋊C4C2×C414yes—
53C2×C7×C22⋊C4C2×C418yes—
54C2×C22⋊Dic7—24no—
55C2×C22⋊Dic7—42no—
56C2×C22⋊Dic7—42no—
57C2×C22⋊Dic7—42no—
58C2×C22⋊Dic7—41no—
59C2×C22⋊Dic7—42no—
60C2×C22⋊Dic7—41no—
61C2×C4⋊Dic7—42no—
62C2×C4⋊Dic7—42no—
63C2×C4⋊Dic7—42no—
64C2×C4×Dic7—41no—
65C2×C4×Dic7—42no—
66C7×#95C2×C414yes—
67C7⋊#95—24no—
68C7⋊#95—24no—
69C7⋊#95—42no—
70C7⋊#95—42no—
71C7⋊#95—42no—
72C7⋊#95—42no—
73C7⋊#95—42no—
74C7⋊#95—42no—
75C7⋊#95—41no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C2×#399D711yescharacteristic
79#1443—72no—
80#1441—72no—
81#1441—72no—
82#1440—72no—
83#1440—71no—
84#1439—72no—
85#1437—71no—
86#1436—72no—
87C24×C7—22no—
88C24×C7—22no—
89C24×C7—22no—
90C24×C7—21no—
91C24×C7—22no—
92C24×C7—21no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7C22×C412yes—
96C24×C7C2411yescharacteristic
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
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—21no—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7C22×C414yes—
118C22×C4×C7C22×C412yes—
119C22×C4×C7C22×C412yes—
120C22×C4×C7C22×C414yes—
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×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7#4114yes—
149C7×C4⋊C4—48no—
150C7×C4⋊C4—44no—
151C7×C22⋊C4—48no—
152C7×C22⋊C4—44no—
153C22⋊Dic7—48no—
154C22⋊Dic7—44no—
155C22⋊Dic7—44no—
156C4⋊Dic7—48no—
157C4×Dic7—44no—
158C22×C4⋊C4—72no—
159C22×C4⋊C4C2×D712yes—
160C22×C4⋊C4C2×D711yescharacteristic
161C22×C22⋊C4—72no—
162C22×C22⋊C4—71no—
163C22×C22⋊C4—71no—
164C22×C22⋊C4C2×D711yescharacteristic
165C22×C22⋊C4C2×D712yes—
166C22×C42—71no—
167#399Dic718yescomplemented
168C2×#95—72no—
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×#95—72no—
179C2×#95C2×D711yescharacteristic
180C23×C7—22no—
181C23×C7—22no—
182C23×C7—24no—
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—41no—
194C23×C7—42no—
195C23×C7—42no—
196C23×C7—42no—
197C23×C7—42no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—41no—
201C23×C7C23×C411yescharacteristic
202C23×C7C2×#4112yes—
203C23×C7H(5,𝔽2)12yes—
204C23×C7H(5,𝔽2)11yescharacteristic
205C23×C7H(5,𝔽2)12yes—
206C23×C7H(5,𝔽2)11yescharacteristic
207C23×C7H(5,𝔽2)12yes—
208C23×C72−1+412yes—
209C23×C72−1+412yes—
210C2×C4×C7—24no—
211C2×C4×C7—24no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—41no—
230C2×C4×C7—42no—
231C2×C4×C7—41no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×Dic7—24no—
235C2×Dic7—24no—
236C2×Dic7—24no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239C2×Dic7—42no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—42no—
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×D711yescharacteristic
269C23×C4C22×D712yes—
270C23×C4C22×D711yescharacteristic
271C2×C4⋊C4—42no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—282no—
276C2×C4⋊C4—282no—
277C2×C4⋊C4—282no—
278C2×C4⋊C4C2×Dic718yes—
279C2×C4⋊C4C2×Dic714yes—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—144no—
285C2×C22⋊C4—282no—
286C2×C22⋊C4—282no—
287C2×C22⋊C4—281no—
288C2×C22⋊C4—282no—
289C2×C22⋊C4—281no—
290C2×C22⋊C4—282no—
291C2×C22⋊C4C2×Dic714yes—
292C2×C22⋊C4C2×Dic718yes—
293C2×C42—281no—
294C2×C42—282no—
295#95—144no—
296#95—144no—
297#95—282no—
298#95—281no—
299#95—282no—
300#95—282no—
301#95—282no—
302#95—282no—
303#95—282no—
304#95—282no—
305#95—282no—
306#95C2×Dic714yes—
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#56012yes—
329C22×C7#56011yescharacteristic
330C22×C7#56011yescharacteristic
331C22×C7#56212yes—
332C22×C7#56212yes—
333C22×C7#56212yes—
334C22×C7#56312yes—
335C22×C7#56312yes—
336C22×C7#56311yescharacteristic
337C4×C7—44no—
338C4×C7—42no—
339C4×C7—42no—
340C4×C7—44no—
341Dic7—44no—
342Dic7—82no—
343Dic7—82no—
344Dic7—82no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C22×Dic712yes—
354C24C23×D711yescharacteristic
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
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—74no—
376C22×C4—142no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—142no—
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×C4C22×Dic714yes—
404C22×C4C22×Dic712yes—
405C22×C4C22×Dic714yes—
406C22×C4C22×Dic712yes—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4—288no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—288no—
413C22⋊C4—284no—
414C22⋊C4—284no—
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—24no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23#654811yescharacteristic
448C23#654811yescharacteristic
449C23#654812yes—
450C23#654812yes—
451C23#655112yes—
452C23D7×#4112yes—
453C23#655312yes—
454C23#655412yes—
455C23C23×Dic711yescharacteristic
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—144no—
481C2×C4—144no—
482C2×C4—144no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—282no—
488C2×C4—282no—
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—42no—
510C22—44no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22#9032912yes—
519C22#9033112yes—
520C22#9033712yes—
521C22#9033712yes—
522C22#9033812yes—
523C22#9033812yes—
524C22#9034211yescharacteristic
525C22#9034812yes—
526C22#9035112yes—
527C22#9035312yes—
528C22#9035412yes—
529C22#9035711yescharacteristic
530C22#9035812yes—
531C22#9041111yescharacteristic
532C22#9041112yes—
533C22#9041111yescharacteristic
534C22#9048911yescharacteristic
535C22#9048912yes—
536C22#9050311yescharacteristic
537C22#41×Dic711yescharacteristic
538C22#9051812yes—
539C4—44no—
540C4—42no—
541C4—42no—
542C4—44no—
543C4—284no—
544C4—562no—
545C4—562no—
546C4—562no—
547C2—82no—
548C2#1173553812yes—
549C2#1173554212yes—
550C2#1173554312yes—
551C2#1173557212yes—
552C2#1174019111yescharacteristic
553C2#1174019512yes—
554C2#1174028212yes—
555C2#1174028511yescharacteristic
556C2#1174061911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#399C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#399D7
hypocenterC7#14646
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. #11729319C2semidirect
4. #11729318C2semidirect
9. #11727170C2semidirect
19. C7×#399C4semidirect
167. #399Dic7semidirect
508. C7#14646semidirect

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
#117293022
#117293002
#117292992
#117271702
#146467