Finite Group Databasebeta

← #13284491

Subgroups — #13284491

Order 1792.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#395C211yesmaximal, characteristic
2C2×#90002C211yesmaximal, characteristic
3C2×#89753C211yesmaximal, characteristic
4#11729888C211yesmaximal, characteristic
5#11729323C212yesmaximal
6#11729319C211yesmaximal, characteristic
7#11729319C212yesmaximal
8#11727170C212yesmaximal
9#11727170C212yesmaximal
10#11727166C211yesmaximal, characteristic
11#11727155C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C7×#395C418yescomplemented
19C2×C7×#95C2211yescharacteristic
20C2×C7×#95C2212yes—
21#90002—24no—
22C2×C7⋊#95C2211yescharacteristic
23C2×C7⋊#95C2211yescharacteristic
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2211yescharacteristic
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89753—24no—
38#14626—71nomaximal, Sylow
39C22×C7×Q8—22no—
40C22×C7×Q8—22no—
41C22×C7×Q8—42no—
42C22×C7×Q8—42no—
43C22×C7×Q8C2×C414yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×Dic7C2312yes—
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2311yescharacteristic
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—42no—
58C2×C7×C4⋊C4—42no—
59C2×C7×C4⋊C4C2×C414yes—
60C2×C7×C4⋊C4C2×C418yes—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7—22no—
63C2×C4⋊Dic7—42no—
64C2×C4⋊Dic7—42no—
65C2×C4×Dic7—42no—
66C2×C4×Dic7D414yes—
67C7×#95C2×C418yes—
68C7×#95C2×C414yes—
69C7⋊#95—22no—
70C7⋊#95—22no—
71C7⋊#95—24no—
72C7⋊#95—24no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75C7⋊#95—22no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95—42no—
81C7⋊#95—42no—
82C7⋊#95—42no—
83C7⋊#95—42no—
84C2×#395D711yescharacteristic
85C2×#391—71no—
86C2×#387—71no—
87#1446—71no—
88#1445—72no—
89#1443—72no—
90#1443—72no—
91#1442—71no—
92#1441—72no—
93#1441—71no—
94#1434—71no—
95C24×C7C2411yescharacteristic
96C2×C7×Q8—28no—
97C2×C7×Q8—48no—
98C2×C7×Q8—44no—
99C2×C7×Q8—44no—
100C2×C7×Q8—48no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
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—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—22no—
125C22×C4×C7C22×C412yes—
126C22×C4×C7C22×C414yes—
127C22×C4×C7C22×C414yes—
128C22×C4×C7C22×C412yes—
129C22×C4×C7C22×C412yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7#4112yes—
132C22×Dic7—21no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—21no—
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—21no—
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—21no—
158C22×Dic7—21no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7C2×D412yes—
165C22×Dic7C2×D412yes—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—48no—
168C4⋊Dic7—44no—
169C4⋊Dic7—44no—
170C4×Dic7—44no—
171C23×Q8C2×D711yescharacteristic
172C22×C4⋊C4—71no—
173C22×C4⋊C4—71no—
174C22×C4⋊C4C2×D711yescharacteristic
175C22×C4⋊C4C2×D712yes—
176C22×C42—71no—
177#395Dic718yescomplemented
178#391—144no—
179#387—144no—
180C2×#95—71no—
181C2×#95—72no—
182C2×#95—71no—
183C2×#95—71no—
184C2×#95—71no—
185C2×#95—72no—
186C2×#95—71no—
187C2×#95—72no—
188C2×#95—72no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95—72no—
192C2×#95—72no—
193C2×#95—72no—
194C2×#95—72no—
195C2×#95C2×D711yescharacteristic
196C2×#95C2×D712yes—
197C23×C7C23×C411yescharacteristic
198C23×C7C22×D411yescharacteristic
199C23×C7C2×#4111yescharacteristic
200C23×C7H(5,𝔽2)12yes—
201C23×C7H(5,𝔽2)12yes—
202C23×C7H(5,𝔽2)11yescharacteristic
203C23×C72−1+412yes—
204C23×C72−1+412yes—
205C23×C72−1+411yescharacteristic
206C23×C72−1+411yescharacteristic
207C23×C72−1+411yescharacteristic
208C7×Q8—416no—
209C2×C4×C7—24no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—24no—
213C2×C4×C7—22no—
214C2×C4×C7—22no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—22no—
221C2×C4×C7—24no—
222C2×C4×C7—22no—
223C2×C4×C7—24no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7C4×D414yes—
240C2×Dic7—22no—
241C2×Dic7—22no—
242C2×Dic7—22no—
243C2×Dic7—22no—
244C2×Dic7—22no—
245C2×Dic7—22no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—41no—
251C2×Dic7—42no—
252C2×Dic7—41no—
253C2×Dic7—42no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8—42no—
277C22×Q8—42no—
278C22×Q8C2×Dic714yes—
279C23×C4—71no—
280C23×C4—71no—
281C23×C4—71no—
282C23×C4—72no—
283C23×C4—71no—
284C23×C4—72no—
285C23×C4C22×D711yescharacteristic
286C23×C4C22×D712yes—
287C23×C4C22×D711yescharacteristic
288C23×C4C22×D712yes—
289C23×C4C22×D711yescharacteristic
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—142no—
295C2×C4⋊C4—142no—
296C2×C4⋊C4—282no—
297C2×C4⋊C4—282no—
298C2×C4⋊C4C2×Dic714yes—
299C2×C4⋊C4C2×Dic718yes—
300C2×C42—74no—
301C2×C42—282no—
302#95—142no—
303#95—142no—
304#95—142no—
305#95—144no—
306#95—144no—
307#95—142no—
308#95—142no—
309#95—282no—
310#95—282no—
311#95—282no—
312#95—282no—
313#95—282no—
314#95—282no—
315#95—282no—
316#95—282no—
317#95C2×Dic718yes—
318#95C2×Dic714yes—
319C22×C7C2×C4×D411yescharacteristic
320C22×C7#51812yes—
321C22×C7#51911yescharacteristic
322C22×C7#51912yes—
323C22×C7#52011yescharacteristic
324C22×C7#53411yescharacteristic
325C22×C7#53511yescharacteristic
326C22×C7#53611yescharacteristic
327C22×C7#53611yescharacteristic
328C22×C7#53611yescharacteristic
329C22×C7#53711yescharacteristic
330C22×C7#53812yes—
331C22×C7#54112yes—
332C22×C7#54111yescharacteristic
333C22×C7#54211yescharacteristic
334C22×C7#56212yes—
335C22×C7#56212yes—
336C22×C7#56312yes—
337C22×C7#56312yes—
338C22×C7#56312yes—
339C22×C7#56312yes—
340C22×C7#56312yes—
341C22×C7#56312yes—
342C4×C7—24no—
343C4×C7—42no—
344C4×C7—44no—
345C4×C7—42no—
346C4×C7—44no—
347Dic7—42no—
348Dic7—42no—
349Dic7—82no—
350Dic7—81no—
351Dic7—82no—
352Dic7—81no—
353C24C23×D711yescharacteristic
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—48no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
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—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—72no—
384C22×C4—72no—
385C22×C4—142no—
386C22×C4—142no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—141no—
390C22×C4—141no—
391C22×C4—141no—
392C22×C4—141no—
393C22×C4—141no—
394C22×C4—141no—
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×C4—142no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—141no—
410C22×C4—141no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—141no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—141no—
417C22×C4D4×D712yes—
418C22×C4#115912yes—
419C22×C4C22×Dic712yes—
420C22×C4C22×Dic714yes—
421C22×C4C22×Dic714yes—
422C22×C4C22×Dic712yes—
423C22×C4C22×Dic712yes—
424C4⋊C4—28no—
425C4⋊C4—48no—
426C4⋊C4—284no—
427C4⋊C4—284no—
428C42—284no—
429C2×C7#232111yescharacteristic
430C2×C7#232311yescharacteristic
431C2×C7#232711yescharacteristic
432C2×C7#237812yes—
433C2×C7#238112yes—
434C2×C7#281411yescharacteristic
435C2×C7#281711yescharacteristic
436C2×C7#282011yescharacteristic
437C2×C7#282111yescharacteristic
438C2×C7#283112yes—
439C2×C7#283312yes—
440C23C2×D4×D711yescharacteristic
441C23C2×#115911yescharacteristic
442C23#654812yes—
443C23#654812yes—
444C23#655112yes—
445C23#655112yes—
446C23#655111yescharacteristic
447C23#655111yescharacteristic
448C23#655311yescharacteristic
449C23#655411yescharacteristic
450C23C23×Dic711yescharacteristic
451Q8—416no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—142no—
483C2×C4—142no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—142no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—281no—
493C2×C4—282no—
494C2×C4—281no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—281no—
503C2×C4—282no—
504C2×C4—281no—
505C2×C4—282no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4D4×Dic714yes—
517C7#1462611yescharacteristic, complemented, Sylow
518C22#9028011yescharacteristic
519C22#9028311yescharacteristic
520C22#9028412yes—
521C22#9031611yescharacteristic
522C22#9031911yescharacteristic
523C22#9032011yescharacteristic
524C22#9033411yescharacteristic
525C22#9033712yes—
526C22#9033712yes—
527C22#9033812yes—
528C22#9033812yes—
529C22#9041511yescharacteristic
530C22#9041511yescharacteristic
531C22#9042212yes—
532C22#9042612yes—
533C22#9042612yes—
534C22#9042612yes—
535C22#9042612yes—
536C22C2×D4×Dic711yescharacteristic
537C22#9048912yes—
538C22#9050312yes—
539C22#9050311yescharacteristic
540C22#9051811yescharacteristic
541C4—24no—
542C4—42no—
543C4—44no—
544C4—44no—
545C4—42no—
546C4—282no—
547C4—282no—
548C4—561no—
549C4—562no—
550C4—561no—
551C4—562no—
552C2#1173468611yescharacteristic
553C2#1173468812yes—
554C2#1173471911yescharacteristic
555C2#1173472011yescharacteristic
556C2#1173472111yescharacteristic
557C2#1173472312yes—
558C2#1174002311yescharacteristic
559C2#1174018911yescharacteristic
560C2#1174019512yes—
561C2#1174067211yescharacteristic
562C2#1174067612yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#395C2
socle, prime radical, cosocleC24×C7C24
hypercenterC2×#395D7
hypocenterC7#14626
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
18. C7×#395C4semidirect
177. #395Dic7semidirect
517. C7#14626semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3952
C2×#900022
C2×#897532
#117298882
#117293232
#117293192
#117271702
#117271662
#117271552
#146267