Finite Group Databasebeta

← #13408118

Subgroups — #13408118

Order 1792.

1038 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×#533C211yesmaximal, characteristic
2C2×#89852C211yesmaximal, characteristic, complemented
3C2×#89854C211yesmaximal, characteristic, complemented
4#11739519C211yesmaximal, characteristic
5#11729542C211yesmaximal, characteristic, complemented
6#11729540C211yesmaximal, characteristic, complemented
7#11729449C212yesmaximal
8#11729448C212yesmaximal
9#11729383C212yesmaximal, complemented
10#11729382C212yesmaximal, complemented
11#11729283C211yesmaximal, characteristic
12C7×#533—24no—
13C7×#533—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2212yes—
21C2×C7×#117—21no—
22C22×C42×C7C2211yescharacteristic
23C2×#6433C2211yescharacteristic
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28#89984C2211yescharacteristic
29#89984C2211yescharacteristic
30#89982—21no—
31#89982—21no—
32C2×C22⋊(C7⋊C8)C2212yes—
33#89854—24no—
34#89852—24no—
35C2×#6382C2212yes—
36C2×#6381C2212yes—
37#89708C2211yescharacteristic
38#89705—21no—
39#89705—21no—
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×#6377C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#25677—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8C2312yes—
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8D414yes—
56C7×C4⋊Q8D414yes—
57C7×#125—28no—
58C7×C22⋊Q8—28no—
59C7×C22⋊Q8—24no—
60C7×C22⋊Q8—24no—
61C4×C7×Q8—28no—
62C4×C7×Q8—24no—
63C4×C7×Q8—24no—
64C7×#117—28no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—22no—
72C2×C42×C7—22no—
73C2×C42×C7—22no—
74C2×C42×C7—22no—
75C2×C42×C7D411yescharacteristic
76C2×C42×C7D411yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C22×C7⋊C8C2312yes—
90C22×C7⋊C8C2312yes—
91#6433—22no—
92#6433—22no—
93#6409—24no—
94#6409—24no—
95#6409—24no—
96#6409—24no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C22⋊(C7⋊C8)—24no—
104#6382—24no—
105#6381—24no—
106#6377—22no—
107#6377—22no—
108C2×#533D711yescharacteristic
109C2×#501—71no—
110C2×#491—71no—
111#2851—71no—
112#2036—72no—
113#2035—72no—
114#1993—71no—
115#1960—71no—
116#1865—72no—
117#1863—72no—
118#1749—71no—
119C24×C7C2×D411yescharacteristic
120C2×C7×Q8—24no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—42no—
127C2×C7×Q8—42no—
128C2×C7×Q8C2×D414yes—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C7×C4⋊C4—28no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—24no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4C2×D414yes—
158C7×C22⋊C4—28no—
159C42×C7—28no—
160C42×C7—22no—
161C42×C7—22no—
162C42×C7—21no—
163C42×C7—21no—
164C42×C7C2×D412yes—
165C42×C7C2×D412yes—
166C22⋊Dic7—42no—
167C22⋊Dic7—42no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—22no—
173C4⋊Dic7—22no—
174Dic7⋊C4—42no—
175Dic7⋊C4—42no—
176Dic7⋊C4—42no—
177Dic7⋊C4—42no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8#4114yes—
184#533—24no—
185#533—24no—
186C2×C4⋊Q8C2×D711yescharacteristic
187C2×C4⋊Q8C2×D711yescharacteristic
188C2×#125—21no—
189C2×#125—71no—
190C2×C22⋊Q8—21no—
191C2×C22⋊Q8C2×D712yes—
192C2×C4×Q8—21no—
193C2×C4×Q8C2×D712yes—
194C2×#117—21no—
195C22×C42C2×D711yescharacteristic
196#501—144no—
197#491—144no—
198C2×#107—72no—
199C2×#106—72no—
200C2×#103—72no—
201C2×#103—72no—
202C2×#103—72no—
203C2×#103—72no—
204C2×C22⋊C8—72no—
205C2×#97—71no—
206C2×#97—71no—
207#389—71no—
208#389—71no—
209#387—141no—
210#387—141no—
211#385—141no—
212#385—141no—
213#382—71no—
214#380—71no—
215#380—71no—
216#340—72no—
217#340—72no—
218#336—72no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7#13712yes—
226C23×C7C22×D411yescharacteristic
227C7×Q8—24no—
228C7×Q8—24no—
229C7×Q8—48no—
230C7×Q8—42no—
231C7×Q8—42no—
232C2×C4×C7—28no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—44no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7#12012yes—
255C2×C4×C7#12012yes—
256C2×C4×C7#13712yes—
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C2×#4112yes—
260C2×C4×C72−1+412yes—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C7⋊C8—24no—
274C7⋊C8—42no—
275C22×Q8—21no—
276C22×Q8C22×D712yes—
277C23×C4C22⋊D711yescharacteristic
278C23×C4C22⋊D711yescharacteristic
279C23×C4C22×D711yescharacteristic
280C22×C8—72no—
281C22×C8—72no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8C22⋊D714yes—
285C4⋊Q8C22⋊D714yes—
286#125—28no—
287#125—142no—
288#125—142no—
289C22⋊Q8—28no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8—24no—
295#117—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—71no—
302C2×C4⋊C4—71no—
303C2×C4⋊C4—71no—
304C2×C4⋊C4—71no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—141no—
308C2×C4⋊C4—141no—
309C2×C4⋊C4C22×D712yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—141no—
312C2×C22⋊C4—141no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C22⋊D711yescharacteristic
317C2×C42C22⋊D711yescharacteristic
318C2×C42C22×D711yescharacteristic
319C2×C42C22×D711yescharacteristic
320#107—144no—
321#106—144no—
322#103—144no—
323#103—144no—
324#103—144no—
325#103—144no—
326C22⋊C8—144no—
327#97—142no—
328#97—142no—
329#95—141no—
330#95—141no—
331#95—141no—
332#95—141no—
333#95—141no—
334#95—141no—
335C22×C7—22no—
336C22×C7—22no—
337C22×C7—22no—
338C22×C7—41no—
339C22×C7—41no—
340C22×C7—41no—
341C22×C7—41no—
342C22×C7C2×#12011yescharacteristic
343C22×C7#53211yescharacteristic
344C22×C7#53611yescharacteristic
345C22×C7C2×#13711yescharacteristic
346C22×C7C2×#13711yescharacteristic
347C22×C7#57511yescharacteristic
348C22×C7#57511yescharacteristic
349C4×C7—22no—
350C4×C7—22no—
351C4×C7—22no—
352C4×C7—41no—
353C4×C7—44no—
354C4×C7—41no—
355C4×C7—44no—
356C4×C7#54712yes—
357C4×C7#54812yes—
358Dic7—81no—
359Dic7—81no—
360Dic7—81no—
361Dic7—81no—
362C24C2×C22⋊D711yescharacteristic
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8C2×C22⋊D714yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—141no—
385C22×C4—141no—
386C22×C4—141no—
387C22×C4—141no—
388C22×C4C2×C22⋊D711yescharacteristic
389C22×C4C2×C22⋊D711yescharacteristic
390C22×C4C2×C22⋊D711yescharacteristic
391C22×C4C2×C22⋊D711yescharacteristic
392C22×C4C2×C22⋊D711yescharacteristic
393C22×C4C23×D711yescharacteristic
394C2×C8—74no—
395C2×C8—142no—
396C2×C8—142no—
397C2×C8—142no—
398C2×C8—142no—
399C2×C8—142no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—142no—
407C4⋊C4—142no—
408C4⋊C4—141no—
409C4⋊C4—141no—
410C4⋊C4—141no—
411C4⋊C4—141no—
412C4⋊C4—282no—
413C4⋊C4—282no—
414C4⋊C4—282no—
415C4⋊C4—282no—
416C4⋊C4C2×C22⋊D714yes—
417C22⋊C4—28no—
418C22⋊C4—282no—
419C22⋊C4—282no—
420C42—28no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42C2×C22⋊D712yes—
426C42C2×C22⋊D712yes—
427C2×C7—42no—
428C2×C7#243211yescharacteristic
429C2×C7#300411yescharacteristic
430C2×C7#300511yescharacteristic
431C2×C7#317211yescharacteristic
432C2×C7#318111yescharacteristic
433C2×C7#323611yescharacteristic
434C2×C7#323711yescharacteristic
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23#650512yes—
442C23C22×C22⋊D711yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—48no—
446Q8—42no—
447Q8—42no—
448C2×C4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4—281no—
474C2×C4—281no—
475C2×C4—281no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4—281no—
480C2×C4—281no—
481C2×C4—281no—
482C2×C4#650512yes—
483C2×C4#651612yes—
484C2×C4#651612yes—
485C2×C4D7×#4112yes—
486C2×C4#655412yes—
487C2×C4C22×C22⋊D711yescharacteristic
488C2×C4C22×C22⋊D711yescharacteristic
489C8—144no—
490C8—282no—
491C7#2567711yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22C2×#650511yescharacteristic
500C22C2×#650511yescharacteristic
501C22#9051311yescharacteristic
502C22#9051311yescharacteristic
503C22#9051611yescharacteristic
504C22#9052311yescharacteristic
505C22C2×#651611yescharacteristic
506C4—22no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—44no—
511C4—41no—
512C4—44no—
513C4—561no—
514C4—561no—
515C4—561no—
516C4—561no—
517C4#9051512yes—
518C4#9052212yes—
519C2—42no—
520C2#1174077111yescharacteristic
521C2#1174077211yescharacteristic
522C2#1174079011yescharacteristic
523C2#1174079111yescharacteristic
524C2#1174150411yescharacteristic
525C2#1174152611yescharacteristic
526C2#1174153811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#533C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#533D7
hypocenterC7#25677
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. C2×#89852C2semidirect
3. C2×#89854C2semidirect
5. #11729542C2semidirect
6. #11729540C2semidirect
9. #11729383C2semidirect
10. #11729382C2semidirect
491. C7#25677semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×#898522
C2×#898542
#117395192
#117295422
#117295402
#117294492
#117294482
#117293832
#117293822
#117292832
#256777