Finite Group Databasebeta

← #13280874

Subgroups — #13280874

Order 1792.

1282 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C211yesmaximal, characteristic
2C2×#89757C211yesmaximal, characteristic
3C2×#89753C211yesmaximal, characteristic
4#11729323C211yesmaximal, characteristic
5#11729319C212yesmaximal
6#11729319C211yesmaximal, characteristic
7#11729317C212yesmaximal
8#11727173C211yesmaximal, characteristic
9#11727173C212yesmaximal
10#11727155C212yesmaximal
11Dic7×#95C211yesmaximal, characteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C4⋊Dic7C2212yes—
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2212yes—
21C7×#398C418yescomplemented
22C2×C7×#95C2211yescharacteristic
23C2×C7×#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36#89757—24no—
37#89753—24no—
38#14328—71nomaximal, Sylow
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×Dic7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—42no—
55C2×C7×C4⋊C4—42no—
56C2×C7×C4⋊C4C2×C414yes—
57C2×C7×C4⋊C4C2×C414yes—
58C2×C7×C4⋊C4C2×C418yes—
59C2×C4⋊Dic7—22no—
60C2×C4⋊Dic7—24no—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7—24no—
63C2×C4⋊Dic7—42no—
64C2×C4⋊Dic7—42no—
65C2×C4⋊Dic7—42no—
66C2×C4⋊Dic7—42no—
67C2×C4⋊Dic7—42no—
68C2×C4⋊Dic7—42no—
69C2×C4×Dic7—42no—
70C2×C4×Dic7—42no—
71C2×C4×Dic7—42no—
72C2×C4×Dic7D414yes—
73C2×C4×Dic7Q818yes—
74C7×#95C2×C418yes—
75C7×#95C2×C414yes—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95—24no—
79C7⋊#95—24no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95—24no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C2×#398D711yescharacteristic
89C2×#389—71no—
90C2×#387—71no—
91#1445—71no—
92#1444—71no—
93#1444—72no—
94#1441—71no—
95#1441—72no—
96C4⋊#95—72no—
97#1434—72no—
98C4×#95—71no—
99C24×C7C2411yescharacteristic
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7C22×C412yes—
123C22×C4×C7C22×C414yes—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C22×C412yes—
126C22×C4×C7C22×C414yes—
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×Q814yes—
129C22×C4×C7#4112yes—
130C22×C4×C7#4114yes—
131C22×Dic7—22no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—22no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—21no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C22×Dic7C2×D412yes—
162C22×Dic7C2×D412yes—
163C22×Dic7C2×Q814yes—
164C22×Dic7C2×Q814yes—
165C7×C4⋊C4—28no—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—48no—
168C4⋊Dic7—44no—
169C4⋊Dic7—48no—
170C4⋊Dic7—44no—
171C4⋊Dic7—48no—
172C4×Dic7—44no—
173C4×Dic7—48no—
174C22×C4⋊C4—72no—
175C22×C4⋊C4—72no—
176C22×C4⋊C4—71no—
177C22×C4⋊C4—71no—
178C22×C4⋊C4C2×D711yescharacteristic
179C22×C4⋊C4C2×D711yescharacteristic
180C22×C4⋊C4C2×D712yes—
181C22×C42—72no—
182C22×C42—71no—
183#398Dic718yescomplemented
184#389—144no—
185#387—144no—
186C2×#95—72no—
187C2×#95—71no—
188C2×#95—71no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95—71no—
192C2×#95—71no—
193C2×#95—71no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—72no—
198C2×#95C2×D711yescharacteristic
199C2×#95C2×D712yes—
200C23×C7C23×C411yescharacteristic
201C23×C7C22×D411yescharacteristic
202C23×C7C22×Q812yes—
203C23×C7C2×#4111yescharacteristic
204C23×C7C2×#4112yes—
205C23×C7H(5,𝔽2)11yescharacteristic
206C23×C7H(5,𝔽2)11yescharacteristic
207C23×C7H(5,𝔽2)12yes—
208C23×C7H(5,𝔽2)11yescharacteristic
209C23×C72−1+412yes—
210C23×C72−1+411yescharacteristic
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—22no—
215C2×C4×C7—22no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—24no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—24no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—22no—
228C2×C4×C7—24no—
229C2×C4×C7—41no—
230C2×C4×C7—42no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—41no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—41no—
240C2×C4×C7C4×D414yes—
241C2×C4×C7C4×Q818yes—
242C2×Dic7—22no—
243C2×Dic7—22no—
244C2×Dic7—24no—
245C2×Dic7—24no—
246C2×Dic7—22no—
247C2×Dic7—24no—
248C2×Dic7—22no—
249C2×Dic7—22no—
250C2×Dic7—24no—
251C2×Dic7—24no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—42no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7C4⋊Q814yes—
276C23×C4—71no—
277C23×C4—71no—
278C23×C4—72no—
279C23×C4—71no—
280C23×C4—72no—
281C23×C4—71no—
282C23×C4C22×D712yes—
283C23×C4C22×D711yescharacteristic
284C23×C4C22×D711yescharacteristic
285C23×C4C22×D711yescharacteristic
286C23×C4C22×D712yes—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—142no—
294C2×C4⋊C4—144no—
295C2×C4⋊C4—142no—
296C2×C4⋊C4—144no—
297C2×C4⋊C4—282no—
298C2×C4⋊C4—282no—
299C2×C4⋊C4—282no—
300C2×C4⋊C4—282no—
301C2×C4⋊C4—282no—
302C2×C4⋊C4—282no—
303C2×C4⋊C4C2×Dic714yes—
304C2×C4⋊C4C2×Dic714yes—
305C2×C4⋊C4C2×Dic718yes—
306C2×C42—74no—
307C2×C42—78no—
308C2×C42—282no—
309C2×C42—282no—
310C2×C42—282no—
311#95—142no—
312#95—142no—
313#95—142no—
314#95—144no—
315#95—142no—
316#95—142no—
317#95—144no—
318#95—144no—
319#95—282no—
320#95—282no—
321#95—282no—
322#95—282no—
323#95C2×Dic718yes—
324#95C2×Dic714yes—
325C22×C7C2×C4×D411yescharacteristic
326C22×C7C2×C4×Q812yes—
327C22×C7#51811yescharacteristic
328C22×C7#52012yes—
329C22×C7#52011yescharacteristic
330C22×C7C2×C4⋊Q811yescharacteristic
331C22×C7#53211yescharacteristic
332C22×C7#53312yes—
333C22×C7#53411yescharacteristic
334C22×C7#53511yescharacteristic
335C22×C7#53611yescharacteristic
336C22×C7#53711yescharacteristic
337C22×C7#53812yes—
338C22×C7#53912yes—
339C22×C7#53912yes—
340C22×C7#54012yes—
341C22×C7#54011yescharacteristic
342C22×C7#54112yes—
343C22×C7#54211yescharacteristic
344C22×C7#54312yes—
345C22×C7#54412yes—
346C22×C7#54412yes—
347C22×C7#54412yes—
348C4×C7—24no—
349C4×C7—28no—
350C4×C7—42no—
351C4×C7—42no—
352C4×C7—44no—
353Dic7—24no—
354Dic7—42no—
355Dic7—44no—
356Dic7—82no—
357Dic7—81no—
358Dic7—81no—
359C24C23×D711yescharacteristic
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—72no—
383C22×C4—74no—
384C22×C4—72no—
385C22×C4—74no—
386C22×C4—141no—
387C22×C4—142no—
388C22×C4—141no—
389C22×C4—142no—
390C22×C4—142no—
391C22×C4—141no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—141no—
398C22×C4—142no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—142no—
402C22×C4—141no—
403C22×C4—142no—
404C22×C4—141no—
405C22×C4—142no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—141no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—141no—
415C22×C4—142no—
416C22×C4D4×D712yes—
417C22×C4#115912yes—
418C22×C4Q8×D714yes—
419C22×C4#116114yes—
420C22×C4C22×Dic712yes—
421C22×C4C22×Dic714yes—
422C22×C4C22×Dic712yes—
423C22×C4C22×Dic712yes—
424C22×C4C22×Dic714yes—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—48no—
428C4⋊C4—288no—
429C4⋊C4—284no—
430C4⋊C4—288no—
431C4⋊C4—284no—
432C42—288no—
433C42—284no—
434C2×C7C4×C4⋊Q811yescharacteristic
435C2×C7#232211yescharacteristic
436C2×C7#232411yescharacteristic
437C2×C7#232912yes—
438C2×C7#233012yes—
439C2×C7#265911yescharacteristic
440C2×C7#266011yescharacteristic
441C2×C7#266111yescharacteristic
442C2×C7#266411yescharacteristic
443C2×C7#266612yes—
444C2×C7#266712yes—
445C23C2×D4×D711yescharacteristic
446C23C2×#115911yescharacteristic
447C23#654811yescharacteristic
448C23#654811yescharacteristic
449C23C2×Q8×D712yes—
450C23C2×#116112yes—
451C23#655311yescharacteristic
452C23#655312yes—
453C23#655411yescharacteristic
454C23#655412yes—
455C23C23×Dic711yescharacteristic
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—74no—
486C2×C4—144no—
487C2×C4—142no—
488C2×C4—144no—
489C2×C4—142no—
490C2×C4—144no—
491C2×C4—142no—
492C2×C4—144no—
493C2×C4—142no—
494C2×C4—144no—
495C2×C4—142no—
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—281no—
507C2×C4—282no—
508C2×C4—281no—
509C2×C4—282no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—281no—
517C2×C4—282no—
518C2×C4—281no—
519C2×C4D4×Dic714yes—
520C2×C4Q8×Dic718yes—
521C7#1432811yescharacteristic, complemented, Sylow
522C22#9029411yescharacteristic
523C22#9029811yescharacteristic
524C22#9029911yescharacteristic
525C22#9031011yescharacteristic
526C22#9031412yes—
527C22#9032512yes—
528C22#9033512yes—
529C22#9033612yes—
530C22#9033911yescharacteristic
531C22#9034411yescharacteristic
532C22#9035612yes—
533C22#9038012yes—
534C22#9038012yes—
535C22#9038812yes—
536C22#9038912yes—
537C22D7×C4⋊Q811yescharacteristic
538C22#9041611yescharacteristic
539C22#9041912yes—
540C22C2×D4×Dic711yescharacteristic
541C22#9048911yescharacteristic
542C22C2×Q8×Dic712yes—
543C22#9051812yes—
544C22#9051811yescharacteristic
545C4—28no—
546C4—24no—
547C4—42no—
548C4—42no—
549C4—44no—
550C4—144no—
551C4—282no—
552C4—284no—
553C4—562no—
554C4—561no—
555C4—561no—
556C2#1173528211yescharacteristic
557C2#1173528311yescharacteristic
558C2#1173528511yescharacteristic
559C2#1173528812yes—
560C2#1173529012yes—
561C2#1173541111yescharacteristic
562C2#1174007811yescharacteristic
563C2#1174019012yes—
564C2#1174028111yescharacteristic
565C2#1174046812yes—
566C2Dic7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#398C2
socle, prime radical, cosocleC24×C7C24
hypercenterC2×#398D7
hypocenterC7#14328
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
21. C7×#398C4semidirect
183. #398Dic7semidirect
521. C7#14328semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3982
C2×#897572
C2×#897532
#117293232
#117293192
#117293172
#117271732
#117271552
Dic7×#952
#143287