Finite Group Databasebeta

← #13408151

Subgroups — #13408151

Order 1792.

1032 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#542C211yesmaximal, characteristic
2C2×#89851C211yesmaximal, characteristic, complemented
3C2×#89827C211yesmaximal, characteristic, complemented
4#11739709C211yesmaximal, characteristic, complemented
5#11729542C211yesmaximal, characteristic, complemented
6#11729479C211yesmaximal, characteristic, complemented
7#11729448C212yesmaximal, complemented
8#11729434C212yesmaximal
9#11729383C212yesmaximal, complemented
10#11729361C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic, complemented
12C7×#542—24no—
13C7×#542—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—21no—
16C2×C7×#124—21no—
17C2×C7×#124C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117C2211yescharacteristic
25C2×#6432C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30#89984C2212yes—
31C2×C22⋊(C7⋊C8)C2212yes—
32#89851—24no—
33#89827—24no—
34#89758—21no—
35#89751—21no—
36C2×#6382C2212yes—
37C2×#6382C2212yes—
38#89721—21no—
39#89719—21no—
40#89708C2211yescharacteristic
41#89700C2212yes—
42C2×#6377C2211yescharacteristic
43C2×C4×C7⋊C8C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#25680—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×D4—21no—
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C22×Dic14—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8D414yes—
57C7×#126—28no—
58C7×#124—28no—
59C7×#124—22no—
60C7×#124D414yes—
61C7×#123—28no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C7×#121—22no—
66C7×#121—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C7×#117—22no—
72C7×#117—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4D411yescharacteristic
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C42×C7—21no—
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×Dic7⋊C4—22no—
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2312yes—
97#6432—22no—
98#6432—22no—
99#6409—24no—
100#6409—24no—
101#6406—24no—
102#6406—24no—
103C7⋊#95—22no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107C22⋊(C7⋊C8)—24no—
108#6382—24no—
109#6382—24no—
110#6377—22no—
111C4×C7⋊C8—22no—
112C2×#542D711yescharacteristic
113C2×#498—71no—
114C2×#489—71no—
115#2628—71no—
116#2035—72no—
117#2034—72no—
118#1993—71no—
119#1958—71no—
120#1865—72no—
121#1864—72no—
122#1750—71no—
123C24×C7—21no—
124C24×C7C2×D411yescharacteristic
125C2×C7×Q8—24no—
126C2×C7×Q8—21no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—82no—
130C2×C7×Q8C2×D412yes—
131C2×C7×D4—24no—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—82no—
136C2×C7×D4C2×D412yes—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—24no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—82no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—22no—
153C22×Dic7—22no—
154C2×Dic14—42no—
155C2×Dic14—82no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4C2×D412yes—
164C7×C4⋊C4C2×D412yes—
165C7×C22⋊C4—24no—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—24no—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—22no—
171C7×C22⋊C4—22no—
172C42×C7—24no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7C2×D412yes—
176C42×C7C2×D412yes—
177C22⋊Dic7—82no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—22no—
181C4⋊Dic7—22no—
182C4⋊Dic7—42no—
183Dic7⋊C4—82no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8#4114yes—
190#542—24no—
191#542—24no—
192C2×C4⋊Q8—71no—
193C2×C4⋊Q8C2×D711yescharacteristic
194C2×#126—21no—
195C2×#124—21no—
196C2×#124C2×D711yescharacteristic
197C2×#123—21no—
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8C2×D711yescharacteristic
200C2×#121C2×D711yescharacteristic
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4C2×D711yescharacteristic
203C2×#117C2×D711yescharacteristic
204#498—144no—
205#489—144no—
206C2×#106—72no—
207C2×#106—72no—
208C2×#103—72no—
209C2×#103—72no—
210C2×#102—72no—
211C2×#102—72no—
212C2×C22⋊C8—72no—
213C2×#97—71no—
214C2×C4×C8—71no—
215#400—141no—
216#399—141no—
217#395—141no—
218#393—141no—
219#389—72no—
220#382—71no—
221#380—72no—
222#340—72no—
223#340—72no—
224#336—72no—
225C23×C7—21no—
226C23×C7—22no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7C22×D411yescharacteristic
236C7×Q8—24no—
237C7×Q8—42no—
238C7×Q8—84no—
239C7×D4—24no—
240C7×D4—42no—
241C7×D4—84no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—21no—
245C2×C4×C7—22no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—84no—
261C2×C4×C7#12012yes—
262C2×C4×C7#12012yes—
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4112yes—
266C2×C4×C72−1+412yes—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273Dic14—84no—
274C7⋊C8—24no—
275C7⋊C8—42no—
276C22×Q8—21no—
277C22×Q8—141no—
278C22×Q8C22×D711yescharacteristic
279C22×D4—21no—
280C22×D4C22×D711yescharacteristic
281C23×C4—21no—
282C23×C4C22×D711yescharacteristic
283C22×C8—72no—
284C22×C8—72no—
285C4⋊Q8—22no—
286C4⋊Q8—142no—
287C4⋊Q8—142no—
288C4⋊Q8C22⋊D714yes—
289#126—28no—
290#124—28no—
291#124—22no—
292#124C22⋊D714yes—
293#123—28no—
294C22⋊Q8—28no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297#121—22no—
298#121—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×D4—22no—
302C4×D4—22no—
303#117—22no—
304#117—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—72no—
309C2×C4⋊C4—72no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—142no—
312C2×C4⋊C4C22⋊D711yescharacteristic
313C2×C4⋊C4C22⋊D711yescharacteristic
314C2×C4⋊C4C22×D711yescharacteristic
315C2×C4⋊C4C22×D711yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—142no—
322C2×C22⋊C4C22⋊D711yescharacteristic
323C2×C22⋊C4C22⋊D711yescharacteristic
324C2×C42—21no—
325C2×C42C22×D711yescharacteristic
326C2×C42C22×D711yescharacteristic
327#106—144no—
328#106—144no—
329#103—144no—
330#103—144no—
331#102—144no—
332#102—144no—
333C22⋊C8—144no—
334#97—142no—
335C4×C8—142no—
336#95—142no—
337#95—141no—
338#95—141no—
339#95—142no—
340C22×C7—42no—
341C22×C7—41no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—41no—
346C22×C7—41no—
347C22×C7—42no—
348C22×C7—82no—
349C22×C7—82no—
350C22×C7C2×#12011yescharacteristic
351C22×C7#53211yescharacteristic
352C22×C7#53611yescharacteristic
353C22×C7#57712yes—
354C22×C7#57712yes—
355C4×C7—22no—
356C4×C7—22no—
357C4×C7—41no—
358C4×C7—42no—
359C4×C7—41no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—41no—
363C4×C7#54712yes—
364C4×C7#54812yes—
365Dic7—82no—
366Dic7—82no—
367C24—21no—
368C24C2×C22⋊D711yescharacteristic
369C2×Q8—24no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—82no—
374C2×Q8—282no—
375C2×Q8—562no—
376C2×Q8C2×C22⋊D712yes—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—82no—
382C2×D4C2×C22⋊D712yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—82no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4C2×C22⋊D711yescharacteristic
395C22×C4C2×C22⋊D711yescharacteristic
396C22×C4C2×C22⋊D711yescharacteristic
397C22×C4C2×C22⋊D711yescharacteristic
398C22×C4C2×C22⋊D711yescharacteristic
399C22×C4C23×D711yescharacteristic
400C2×C8—74no—
401C2×C8—142no—
402C2×C8—142no—
403C2×C8—142no—
404C2×C8—142no—
405C2×C8—142no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—142no—
414C4⋊C4—142no—
415C4⋊C4—142no—
416C4⋊C4—142no—
417C4⋊C4—282no—
418C4⋊C4—562no—
419C4⋊C4C2×C22⋊D712yes—
420C4⋊C4C2×C22⋊D712yes—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—562no—
429C42—24no—
430C42—21no—
431C42—21no—
432C42C2×C22⋊D712yes—
433C42C2×C22⋊D712yes—
434C2×C7—81no—
435C2×C7—82no—
436C2×C7#243211yescharacteristic
437C2×C7#300911yescharacteristic
438C2×C7#301011yescharacteristic
439C2×C7#317312yes—
440C2×C7#324412yes—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—22no—
445C23—21no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23C22×C22⋊D711yescharacteristic
452Q8—24no—
453Q8—42no—
454Q8—84no—
455Q8—564no—
456D4—24no—
457D4—42no—
458D4—84no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4—282no—
482C2×C4—282no—
483C2×C4—282no—
484C2×C4#651612yes—
485C2×C4#651612yes—
486C2×C4D7×#4112yes—
487C2×C4#655412yes—
488C2×C4C22×C22⋊D711yescharacteristic
489C2×C4C22×C22⋊D711yescharacteristic
490C8—144no—
491C8—282no—
492C7#2568011yescharacteristic, complemented, Sylow
493C22—41no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22#9051611yescharacteristic
504C22#9052311yescharacteristic
505C22#9052712yes—
506C22#9052812yes—
507C22C2×#651611yescharacteristic
508C4—22no—
509C4—22no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—562no—
517C4—562no—
518C4#9051512yes—
519C4#9052212yes—
520C2—81no—
521C2—82no—
522C2#1174090012yes—
523C2#1174090112yes—
524C2#1174153811yescharacteristic
525C2#1174155011yescharacteristic
526C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#542C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC2×#542D7
hypocenterC7#25680
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
2. C2×#89851C2semidirect
3. C2×#89827C2semidirect
4. #11739709C2semidirect
5. #11729542C2semidirect
6. #11729479C2semidirect
7. #11729448C2semidirect
9. #11729383C2semidirect
10. #11729361C2semidirect
11. #11729336C2semidirect
492. C7#25680semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5422
C2×#898512
C2×#898272
#117397092
#117295422
#117294792
#117294482
#117294342
#117293832
#117293612
#117293362
#256807