Finite Group Databasebeta

← #13319973

Subgroups — #13319973

Order 1792.

848 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90392C211yesmaximal, characteristic
2#11736759C211yesmaximal, characteristic, complemented
3#11736709C211yesmaximal, characteristic, complemented
4C7×#1473C211yesmaximal, characteristic
5#11729499C211yesmaximal, characteristic
6#11728585C211yesmaximal, characteristic, complemented
7#11728541C211yesmaximal, characteristic, complemented
8#11725882C212yesmaximal, complemented
9#11725881C212yesmaximal, complemented
10#11725859C212yesmaximal
11#11725856C212yesmaximal
12C2×C7×#126C2211yescharacteristic
13#90392—22no—
14#90392—22no—
15#90392—22no—
16#90392—22no—
17C2×#6453C411yescharacteristic
18C2×#6452C411yescharacteristic
19C2×#6452C411yescharacteristic
20C2×#6452C2211yescharacteristic
21C2×#6451C411yescharacteristic
22C2×#6450C2211yescharacteristic
23C2×#6442C411yescharacteristic
24C2×#6442C411yescharacteristic
25C2×#6442C2211yescharacteristic
26C2×#6441C411yescharacteristic
27C2×#6441C411yescharacteristic
28C2×#6440C2211yescharacteristic
29C2×#6433C2211yescharacteristic
30C2×C4×Dic14C2211yescharacteristic
31C7×#432C2211yescharacteristic
32C7×#431C2211yescharacteristic
33#89698C2211yescharacteristic
34#89697C2211yescharacteristic
35#89603—22no—
36#89600—22no—
37#89491C2212yes—
38#89491C2212yes—
39#89491C2212yes—
40#89490C2212yes—
41#89473—22no—
42#89451—22no—
43C7×#324C2212yes—
44C7×#323C2212yes—
45#89323C2212yes—
46#89320C2212yes—
47#89264C2212yes—
48#89264C2212yes—
49#89264C2212yes—
50#89264C2212yes—
51#17410—71nomaximal, Sylow
52C23×Dic7C2×C411yescharacteristic
53C22×Dic14C2×C411yescharacteristic
54C22×Dic14C2×C411yescharacteristic
55C7×#126—21no—
56C7×#126—21no—
57C7×#126—21no—
58C7×#126—21no—
59C2×C7×C4⋊C4C2×C411yescharacteristic
60C2×C7×C4⋊C4C2×C411yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4C2×C411yescharacteristic
63C2×C7×C22⋊C4C2×C411yescharacteristic
64C2×C7×C22⋊C4C2311yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C22⋊Dic7C2×C411yescharacteristic
67C2×C22⋊Dic7C2×C411yescharacteristic
68C2×C22⋊Dic7C2×C411yescharacteristic
69C2×C4⋊Dic7C2×C411yescharacteristic
70C2×C4⋊Dic7C2×C411yescharacteristic
71C2×C4⋊Dic7C2×C411yescharacteristic
72C2×C4⋊Dic7C2×C411yescharacteristic
73C2×C4⋊Dic7C2×C411yescharacteristic
74C2×Dic7⋊C4C2×C411yescharacteristic
75C2×Dic7⋊C4C2×C411yescharacteristic
76C2×Dic7⋊C4C2×C411yescharacteristic
77C2×Dic7⋊C4C2×C411yescharacteristic
78C2×Dic7⋊C4C2×C411yescharacteristic
79C2×Dic7⋊C4C2×C411yescharacteristic
80C2×Dic7⋊C4C2×C411yescharacteristic
81C2×Dic7⋊C4C2×C411yescharacteristic
82C2×Dic7⋊C4C2×C411yescharacteristic
83C2×Dic7⋊C4C2×C411yescharacteristic
84C2×Dic7⋊C4C2311yescharacteristic
85C2×Dic7⋊C4C2311yescharacteristic
86C2×C4×Dic7C2×C411yescharacteristic
87C2×C4×Dic7C2×C411yescharacteristic
88C2×C4×Dic7C2×C411yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91#6453—22no—
92#6453—22no—
93#6452—22no—
94#6452—22no—
95#6452—22no—
96#6452—22no—
97#6452—22no—
98#6452—22no—
99#6451—22no—
100#6451—22no—
101#6450—22no—
102#6450—22no—
103#6442—22no—
104#6442—22no—
105#6442—22no—
106#6442—22no—
107#6442—22no—
108#6442—22no—
109#6441—22no—
110#6441—22no—
111#6441—22no—
112#6441—22no—
113#6440—22no—
114#6440—22no—
115#6433—21no—
116#6433—21no—
117#6433—21no—
118#6433—21no—
119C4×Dic14D412yes—
120C4×Dic14D412yes—
121C4×Dic14D412yes—
122C4×Dic14D412yes—
123C7×C4⋊C8—21no—
124C7×C4⋊C8—21no—
125C7×C22⋊C8C2312yes—
126C7×C22⋊C8C2312yes—
127C7×#97—21no—
128C4×C7×C8—21no—
129C22⋊(C7⋊C8)C2312yes—
130C22⋊(C7⋊C8)C2312yes—
131#6389—22no—
132#6389—22no—
133#6389—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136Dic7⋊C8—22no—
137Dic7⋊C8—22no—
138C8×Dic7—22no—
139C4⋊(C7⋊C8)—21no—
140C4⋊(C7⋊C8)—21no—
141#6377—21no—
142C4×C7⋊C8—21no—
143C2×#541—71no—
144#2983—71no—
145#2983—71no—
146#1473—72no—
147#1473—71no—
148#1473D711yescharacteristic
149#1470—71no—
150#1470—71no—
151#1470—72no—
152#1467—72no—
153#1464—72no—
154C24×C7C22×C411yescharacteristic
155C22×C4×C7C22×C411yescharacteristic
156C22×C4×C7C22×C411yescharacteristic
157C22×C4×C7C22×C411yescharacteristic
158C22×C4×C7C22×C411yescharacteristic
159C22×C4×C7C22×C411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—22no—
162C22×Dic7—42no—
163C22×Dic7C22×C411yescharacteristic
164C22×Dic7C22×C411yescharacteristic
165C22×Dic7C22×C411yescharacteristic
166C22×Dic7C22×C411yescharacteristic
167C22×Dic7C22×C411yescharacteristic
168C22×Dic7C22×C411yescharacteristic
169C22×Dic7C22×C411yescharacteristic
170C22×Dic7C22×C411yescharacteristic
171C2×Dic14—22no—
172C2×Dic14—42no—
173C2×Dic14—42no—
174C2×Dic14C22⋊C412yes—
175C2×Dic14C22⋊C412yes—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C22⋊C4—21no—
185C7×C22⋊C4—21no—
186C7×C22⋊C4—21no—
187C7×C22⋊C4—21no—
188C7×C22⋊C4—21no—
189C7×C22⋊C4—21no—
190C42×C7C2×D411yescharacteristic
191C42×C7C2×D411yescharacteristic
192C42×C7C2×D411yescharacteristic
193C42×C7C2×D411yescharacteristic
194C22⋊Dic7—22no—
195C22⋊Dic7—22no—
196C22⋊Dic7—22no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—22no—
202C4⋊Dic7—22no—
203C4⋊Dic7C22⋊C412yes—
204C4⋊Dic7C22⋊C412yes—
205Dic7⋊C4—21no—
206Dic7⋊C4—21no—
207Dic7⋊C4—21no—
208Dic7⋊C4—21no—
209Dic7⋊C4—21no—
210Dic7⋊C4—21no—
211Dic7⋊C4—21no—
212Dic7⋊C4—21no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4C22⋊C412yes—
220Dic7⋊C4C22⋊C412yes—
221Dic7⋊C4C22⋊C412yes—
222Dic7⋊C4C22⋊C412yes—
223C4×Dic7—22no—
224C4×Dic7—22no—
225C4×Dic7—22no—
226C4×Dic7C2×D412yes—
227C4×Dic7C2×D412yes—
228C4×Dic7C2×D412yes—
229C4×Dic7C2×D412yes—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—22no—
232#541—142no—
233#541—142no—
234#541—142no—
235#541—142no—
236C2×C4⋊Q8—71no—
237C2×#126—71no—
238C2×#126—71no—
239C2×#126—71no—
240C2×#126C2×D711yescharacteristic
241C2×#125—71no—
242C2×#125—71no—
243C2×#125—71no—
244C2×#125—71no—
245C2×#125—71no—
246C2×C22⋊Q8—71no—
247C2×C22⋊Q8—71no—
248C2×C4×Q8—71no—
249C2×C4×Q8—71no—
250C2×#117—71no—
251#446—142no—
252#446—142no—
253#446—142no—
254#446—142no—
255#432—71no—
256#432—72no—
257#432—72no—
258#432—72no—
259#432C2×D711yescharacteristic
260#431—71no—
261#431—72no—
262#431C2×D711yescharacteristic
263#324—72no—
264#324—72no—
265#324—72no—
266#324—72no—
267#324—72no—
268#324C2×D712yes—
269#323—72no—
270#323C2×D712yes—
271C23×C7—21no—
272C23×C7—22no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7C23×C411yescharacteristic
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7C2×C22⋊C411yescharacteristic
286C2×C4×C7C2×C22⋊C411yescharacteristic
287C2×C4×C7C2×C22⋊C411yescharacteristic
288C2×C4×C7C2×C22⋊C411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—44no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7C2×C22⋊C412yes—
305C2×Dic7C2×C22⋊C412yes—
306C2×Dic7C2×C22⋊C412yes—
307C2×Dic7C2×C22⋊C412yes—
308Dic14—42no—
309Dic14—42no—
310Dic14—44no—
311C7×C8—42no—
312C7×C8—42no—
313C7⋊C8—42no—
314C7⋊C8—42no—
315C22×Q8—71no—
316C22×Q8—71no—
317C23×C4—71no—
318C4⋊Q8—142no—
319C4⋊Q8—142no—
320#126—21no—
321#126—21no—
322#126—21no—
323#126—21no—
324#126—142no—
325#126—142no—
326#126—142no—
327#126—142no—
328#126—142no—
329#126—142no—
330#125—142no—
331#125—142no—
332#125—142no—
333#125—142no—
334#125—142no—
335#125—142no—
336#125—142no—
337#125—142no—
338#125—141no—
339#125—141no—
340#125—141no—
341#125—141no—
342C22⋊Q8—142no—
343C22⋊Q8—142no—
344C22⋊Q8—142no—
345C22⋊Q8—142no—
346C4×Q8—72no—
347C4×Q8—72no—
348C4×Q8—72no—
349C4×Q8—72no—
350C4×Q8—142no—
351C4×Q8—142no—
352#117—142no—
353#117—142no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4C4×D711yescharacteristic
372C2×C4⋊C4C4×D711yescharacteristic
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C22⋊C4—71no—
375C2×C22⋊C4—71no—
376C2×C22⋊C4—71no—
377C2×C22⋊C4C4×D711yescharacteristic
378C2×C22⋊C4C4×D711yescharacteristic
379C2×C22⋊C4C22×D711yescharacteristic
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—71no—
383C2×C42—71no—
384C2×C42—71no—
385C2×C42C22×D711yescharacteristic
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—142no—
389C4⋊C8—142no—
390C4⋊C8—142no—
391C4⋊C8—142no—
392C4⋊C8—141no—
393C4⋊C8—141no—
394C22⋊C8—72no—
395C22⋊C8—72no—
396C22⋊C8C22×D712yes—
397C22⋊C8C22×D712yes—
398#97—21no—
399#97—142no—
400#97—142no—
401#97—142no—
402#97—141no—
403C4×C8—21no—
404C4×C8—142no—
405C4×C8—141no—
406C22×C7—21no—
407C22×C7—21no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7—41no—
411C22×C7—82no—
412C22×C7C22×C22⋊C411yescharacteristic
413C22×C7#56812yes—
414C4×C7—21no—
415C4×C7—21no—
416C4×C7—21no—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—21no—
420C4×C7—41no—
421C4×C7—41no—
422C4×C7—41no—
423Dic7—22no—
424Dic7—22no—
425Dic7—22no—
426Dic7—22no—
427Dic7—41no—
428Dic7—41no—
429Dic7—41no—
430Dic7—41no—
431C24C2×C4×D711yescharacteristic
432C2×Q8—72no—
433C2×Q8—72no—
434C2×Q8—142no—
435C2×Q8—282no—
436C2×Q8—282no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—142no—
446C22×C4—282no—
447C22×C4C2×C4×D711yescharacteristic
448C22×C4C2×C4×D711yescharacteristic
449C22×C4C2×C4×D711yescharacteristic
450C22×C4C2×C4×D711yescharacteristic
451C22×C4C2×C4×D711yescharacteristic
452C22×C4C23×D711yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—142no—
456C2×C8—142no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—72no—
464C4⋊C4—72no—
465C4⋊C4—72no—
466C4⋊C4—72no—
467C4⋊C4—72no—
468C4⋊C4—72no—
469C4⋊C4—142no—
470C4⋊C4—142no—
471C4⋊C4—142no—
472C4⋊C4—142no—
473C4⋊C4—142no—
474C4⋊C4—142no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—142no—
496C22⋊C4—142no—
497C22⋊C4—142no—
498C42—72no—
499C42—72no—
500C42—72no—
501C42—72no—
502C42—142no—
503C42—142no—
504C42—142no—
505C42D4×D711yescharacteristic
506C42D4×D711yescharacteristic
507C42D4×D711yescharacteristic
508C42D4×D711yescharacteristic
509C2×C7—22no—
510C2×C7—81no—
511C2×C7#287911yescharacteristic
512C2×C7#288411yescharacteristic
513C2×C7#288911yescharacteristic
514C23—21no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23C22×C4×D711yescharacteristic
519Q8—282no—
520Q8—282no—
521Q8—284no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—72no—
534C2×C4—72no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—284no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4D7×C22⋊C411yescharacteristic
549C2×C4D7×C22⋊C411yescharacteristic
550C2×C4D7×C22⋊C411yescharacteristic
551C2×C4D7×C22⋊C411yescharacteristic
552C2×C4C2×D4×D711yescharacteristic
553C2×C4C2×D4×D711yescharacteristic
554C8—42no—
555C8—42no—
556C8—282no—
557C8—282no—
558C7#1741011yescharacteristic, complemented, Sylow
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—82no—
565C22C2×D7×C22⋊C411yescharacteristic
566C22#9044112yes—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—142no—
577C4—142no—
578C4—142no—
579C4—142no—
580C4—281no—
581C4—281no—
582C4—281no—
583C4—281no—
584C2—22no—
585C2—81no—
586C2#1173610711yescharacteristic
587C2#1173613411yescharacteristic
588C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1473C2
socleC22×C7C22×C22⋊C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1473D7
hypocenterC7#17410
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1473

Complements

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

NH ≅ G/Ntype
2. #11736759C2semidirect
3. #11736709C2semidirect
6. #11728585C2semidirect
7. #11728541C2semidirect
8. #11725882C2semidirect
9. #11725881C2semidirect
558. C7#17410semidirect

Maximal subgroups

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

subgroupindex
C2×#903922
#117367592
#117367092
C7×#14732
#117294992
#117285852
#117285412
#117258822
#117258812
#117258592
#117258562
#174107