Finite Group Databasebeta

← #13322063

Subgroups — #13322063

Order 1792.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3510C211yesmaximal, characteristic
2#11739964C211yesmaximal, characteristic, complemented
3#11739963C211yesmaximal, characteristic, complemented
4#11729577C211yesmaximal, characteristic, complemented
5#11729575C211yesmaximal, characteristic, complemented
6#11729503C212yesmaximal, complemented
7#11729502C212yesmaximal, complemented
8#11729501C212yesmaximal, complemented
9#11729500C212yesmaximal, complemented
10#11729472C211yesmaximal, characteristic, complemented
11#11729472C211yesmaximal, characteristic, complemented
12C7×#559—21no—
13C7×#559—21no—
14C7×#556—21no—
15C7×#556—21no—
16C7×#551—22no—
17C7×#551—22no—
18C7×#542C412yes—
19C7×#542C412yes—
20C7×#536C411yescharacteristic
21C7×#535C411yescharacteristic
22C7×#533C2211yescharacteristic
23C7×#529C2212yes—
24C7×#529C2212yes—
25C2×C7×#124C2211yescharacteristic
26C7×#519C411yescharacteristic
27C7×#518C411yescharacteristic
28C4×C7×#41C2211yescharacteristic
29#89796—21no—
30Q8×C7⋊C8—21no—
31#89784—22no—
32#89784—21no—
33#89783—22no—
34D4×C7⋊C8—21no—
35#89698C2211yescharacteristic
36#89698C2211yescharacteristic
37#89698C2212yescomplemented
38#89698C2212yescomplemented
39C4×#1136C2211yescharacteristic
40C4×#1136C2211yescharacteristic
41C4×#1136C2212yescomplemented
42C2×#6377C2212yescomplemented
43#89338C2211yescharacteristic, complemented
44#89338C2212yescomplemented
45#89338C2211yescharacteristic, complemented
46#89338C2212yescomplemented
47#89338C2211yescharacteristic, complemented
48#89338C2211yescharacteristic, complemented
49#89335C2212yes—
50#89335C2212yes—
51#89335C2212yes—
52#89335C2212yes—
53#17691—71nomaximal, Sylow
54C2×C7×#41C2×C411yescharacteristic
55C22×C7×Q8C2×C411yescharacteristic
56C22×C7×D4C2×C411yescharacteristic
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8C2×C411yescharacteristic
59C7×C4⋊Q8C2×C411yescharacteristic
60C7×#127—21no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#125C2311yescharacteristic
66C7×#125C2311yescharacteristic
67C7×#124—21no—
68C7×#124—21no—
69C7×#124—22no—
70C7×#124—22no—
71C7×#124—21no—
72C7×#124—21no—
73C7×#124C2×C412yes—
74C7×#124C2312yes—
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—21no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8C2×C412yes—
85C7×C22⋊Q8C2×C412yes—
86C7×#121C2×C412yes—
87C7×#121C2×C412yes—
88C7×#120—22no—
89C4×C7×Q8—21no—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8C2×C412yes—
94C4×C7×Q8C2×C412yes—
95C4×C7×Q8D411yescharacteristic
96C4×C7×Q8D411yescharacteristic
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4C2×C412yes—
102C4×C7×D4C2×C412yes—
103C4×C7×D4D411yescharacteristic
104C4×C7×D4D412yes—
105C4×C7×D4D412yes—
106C4×C7×D4D411yescharacteristic
107C7×#117—22no—
108C7×#117—22no—
109C7×#117—22no—
110C7×#117C2×C411yescharacteristic
111C7×#117C2×C411yescharacteristic
112C7×#117C2×C411yescharacteristic
113C7×#117C2×C412yes—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—22no—
117C2×C42×C7C2312yes—
118C2×C42×C7C2311yescharacteristic
119C2×#1136—21no—
120C2×#1136—21no—
121C2×#1136—22no—
122C22×C7⋊C8—22no—
123C22⋊(C7⋊C8)—21no—
124C22⋊(C7⋊C8)—21no—
125C22⋊(C7⋊C8)—22no—
126C22⋊(C7⋊C8)—22no—
127C4⋊(C7⋊C8)—21no—
128C4⋊(C7⋊C8)—21no—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—22no—
131#6377—22no—
132#6377C2311yescharacteristic
133#6377C2312yes—
134#6377C2311yescharacteristic
135#6377C2311yescharacteristic
136#6377C2311yescharacteristic
137#6377C2312yes—
138C4×C7⋊C8—21no—
139C4×C7⋊C8—21no—
140C4×C7⋊C8—22no—
141#3510D711yescharacteristic
142#2971—71no—
143#2970—71no—
144#1537—71no—
145#1535—72no—
146#1532—72no—
147#1529—72no—
148#1527—71no—
149#1527—71no—
150#1526—71no—
151#1524—72no—
152C24×C7—21no—
153C7×#41—42no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—42no—
158C2×C7×Q8C22⋊C411yescharacteristic
159C2×C7×Q8C22×C412yes—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—21no—
163C2×C7×D4—42no—
164C2×C7×D4C22⋊C411yescharacteristic
165C2×C7×D4C22⋊C412yes—
166C2×C7×D4C22×C412yes—
167C22×C4×C7—21no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7C22×C411yescharacteristic
171C22×C4×C7C22×C412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D412yes—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4C22⋊C411yescharacteristic
187C7×C4⋊C4C22⋊C411yescharacteristic
188C7×C4⋊C4C22⋊C412yes—
189C7×C4⋊C4C22⋊C412yes—
190C7×C4⋊C4C22×C411yescharacteristic
191C7×C4⋊C4C22×C411yescharacteristic
192C7×C4⋊C4C22×C411yescharacteristic
193C7×C4⋊C4C22×C411yescharacteristic
194C7×C22⋊C4—21no—
195C7×C22⋊C4—21no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4C22⋊C412yes—
207C7×C22⋊C4C22⋊C412yes—
208C7×C22⋊C4C22⋊C412yes—
209C42×C7—22no—
210C42×C7—22no—
211C42×C7—21no—
212C42×C7—21no—
213C42×C7C22×C412yes—
214C42×C7C22×C411yescharacteristic
215C42×C7C2×D412yes—
216C42×C7C2×D412yes—
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219C42×C7C2411yescharacteristic
220#1136—42no—
221#1136—42no—
222#1136—44no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—22no—
226C2×C7⋊C8—22no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—44no—
230#559—21no—
231#559—21no—
232#556—21no—
233#556—21no—
234#551—22no—
235#551—22no—
236#542Dic712yes—
237#542Dic712yes—
238#536Dic711yescharacteristic
239#535Dic711yescharacteristic
240#533C2×D711yescharacteristic
241#529C2×D712yes—
242#529C2×D712yes—
243C2×#124C2×D711yescharacteristic
244#519Dic711yescharacteristic
245#518Dic711yescharacteristic
246C4×#41C2×D711yescharacteristic
247#446—141no—
248C8×Q8—141no—
249#436—142no—
250#436—141no—
251#435—142no—
252C8×D4—141no—
253#432—71no—
254#432—71no—
255#432—72no—
256#432—72no—
257C4×M16—71no—
258C4×M16—71no—
259C4×M16—72no—
260C2×#97—72no—
261#330—72no—
262#330—72no—
263#330—71no—
264#330—71no—
265#330—71no—
266#330—71no—
267#329—72no—
268#329—72no—
269#329—72no—
270#329—72no—
271C23×C7—22no—
272C23×C7—42no—
273C23×C7—81no—
274C23×C7—81no—
275C23×C7C2×C22⋊C411yescharacteristic
276C23×C7C2×C22⋊C412yes—
277C7×Q8—22no—
278C7×Q8—42no—
279C7×Q8—42no—
280C7×Q8—81no—
281C7×Q8—81no—
282C7×D4—22no—
283C7×D4—42no—
284C7×D4—42no—
285C7×D4—42no—
286C7×D4—82no—
287C7×D4—82no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—82no—
301C2×C4×C7—82no—
302C2×C4×C7C2×C22⋊C412yes—
303C2×C4×C7C2×C22⋊C412yes—
304C2×C4×C7C2×C22⋊C411yescharacteristic
305C2×C4×C7C2×C22⋊C411yescharacteristic
306C2×C4×C7C2×C22⋊C412yes—
307C2×C4×C7C2×C22⋊C411yescharacteristic
308C2×C4×C7C23×C411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C7⋊C8—22no—
312C7⋊C8—22no—
313C7⋊C8—42no—
314C7⋊C8—41no—
315C7⋊C8—42no—
316C7⋊C8—41no—
317C2×#41C2×Dic711yescharacteristic
318C22×Q8C2×Dic711yescharacteristic
319C22×D4C2×Dic711yescharacteristic
320C2×M16—141no—
321C2×M16—141no—
322C2×M16—142no—
323C22×C8—142no—
324C4⋊Q8—21no—
325C4⋊Q8C2×Dic711yescharacteristic
326C4⋊Q8C2×Dic711yescharacteristic
327#127—21no—
328#126—22no—
329#126—22no—
330#126—22no—
331#126—22no—
332#125C22×D711yescharacteristic
333#125C22×D711yescharacteristic
334#124—21no—
335#124—21no—
336#124—22no—
337#124—22no—
338#124—21no—
339#124—21no—
340#124C2×Dic712yes—
341#124C22×D712yes—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8C2×Dic712yes—
352C22⋊Q8C2×Dic712yes—
353#121C2×Dic712yes—
354#121C2×Dic712yes—
355#120—22no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8C2×Dic712yes—
361C4×Q8C2×Dic712yes—
362C4×Q8C22⋊D711yescharacteristic
363C4×Q8C22⋊D711yescharacteristic
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4C2×Dic712yes—
369C4×D4C2×Dic712yes—
370C4×D4C22⋊D712yes—
371C4×D4C22⋊D712yes—
372C4×D4C22⋊D711yescharacteristic
373C4×D4C22⋊D711yescharacteristic
374#117—22no—
375#117—22no—
376#117—22no—
377#117C2×Dic711yescharacteristic
378#117C2×Dic711yescharacteristic
379#117C2×Dic711yescharacteristic
380#117C2×Dic712yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C42C22×D711yescharacteristic
385C2×C42C22×D712yes—
386C4⋊C8—141no—
387C4⋊C8—142no—
388C4⋊C8—142no—
389C4⋊C8—141no—
390C22⋊C8—142no—
391C22⋊C8—141no—
392C22⋊C8—141no—
393C22⋊C8—142no—
394#97—71no—
395#97—72no—
396#97—71no—
397#97—72no—
398#97—71no—
399#97—71no—
400#97—142no—
401C4×C8—141no—
402C4×C8—142no—
403C4×C8—141no—
404C22×C7—21no—
405C22×C7—22no—
406C22×C7—22no—
407C22×C7—22no—
408C22×C7—21no—
409C22×C7—21no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7—162no—
414C22×C7C22×C22⋊C411yescharacteristic
415C4×C7—21no—
416C4×C7—22no—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—22no—
420C4×C7—22no—
421C4×C7—21no—
422C4×C7—21no—
423C4×C7—42no—
424C4×C7—41no—
425C4×C7—41no—
426C4×C7—41no—
427C4×C7—41no—
428C4×C7#41312yes—
429C24—21no—
430#41—42no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—42no—
435C2×Q8C22⋊Dic711yescharacteristic
436C2×Q8C22×Dic712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—42no—
441C2×D4C22⋊Dic711yescharacteristic
442C2×D4C22⋊Dic712yes—
443C2×D4C22×Dic712yes—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4C22×Dic711yescharacteristic
448C22×C4C22×Dic712yes—
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D712yes—
452C22×C4C2×C22⋊D712yes—
453M16—282no—
454M16—284no—
455M16—282no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—142no—
460C2×C8—141no—
461C2×C8—142no—
462C2×C8—284no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4C22⋊Dic711yescharacteristic
474C4⋊C4C22⋊Dic711yescharacteristic
475C4⋊C4C22⋊Dic712yes—
476C4⋊C4C22⋊Dic712yes—
477C4⋊C4C22×Dic711yescharacteristic
478C4⋊C4C22×Dic711yescharacteristic
479C4⋊C4C22×Dic711yescharacteristic
480C4⋊C4C22×Dic711yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4C22⋊Dic712yes—
494C22⋊C4C22⋊Dic712yes—
495C22⋊C4C22⋊Dic712yes—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—21no—
500C42C22×Dic712yes—
501C42C22×Dic711yescharacteristic
502C42C2×C22⋊D712yes—
503C42C2×C22⋊D711yescharacteristic
504C42C2×C22⋊D711yescharacteristic
505C42C2×C22⋊D712yes—
506C42C23×D711yescharacteristic
507C2×C7—42no—
508C2×C7—41no—
509C2×C7—82no—
510C2×C7C2×#41311yescharacteristic
511C2×C7#290111yescharacteristic
512C2×C7#290111yescharacteristic
513C23—22no—
514C23—42no—
515C23—81no—
516C23—81no—
517C23C2×C22⋊Dic711yescharacteristic
518C23C2×C22⋊Dic712yes—
519Q8—22no—
520Q8—42no—
521Q8—42no—
522Q8—81no—
523Q8—81no—
524D4—22no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—82no—
529D4—82no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4C2×C22⋊Dic712yes—
545C2×C4C2×C22⋊Dic712yes—
546C2×C4C2×C22⋊Dic711yescharacteristic
547C2×C4C2×C22⋊Dic711yescharacteristic
548C2×C4C2×C22⋊Dic712yes—
549C2×C4C2×C22⋊Dic711yescharacteristic
550C2×C4C23×Dic711yescharacteristic
551C2×C4C22×C22⋊D711yescharacteristic
552C2×C4C22×C22⋊D711yescharacteristic
553C8—142no—
554C8—142no—
555C8—282no—
556C8—281no—
557C8—282no—
558C8—281no—
559C7#1769111yescharacteristic, complemented, Sylow
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—162no—
570C22C22×C22⋊Dic711yescharacteristic
571C4—21no—
572C4—22no—
573C4—21no—
574C4—21no—
575C4—22no—
576C4—22no—
577C4—21no—
578C4—21no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4#9000912yes—
585C2—41no—
586C2—42no—
587C2—82no—
588C2C2×#9000911yescharacteristic
589C2#1174156511yescharacteristic
590C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#3510C2
socleC22×C7C22×C22⋊C4
prime radicalC7×#535C4
cosocleC42×C7C24
hypercenter#3510D7
hypocenterC7#17691
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. #11739964C2semidirect
3. #11739963C2semidirect
4. #11729577C2semidirect
5. #11729575C2semidirect
6. #11729503C2semidirect
7. #11729502C2semidirect
8. #11729501C2semidirect
9. #11729500C2semidirect
10. #11729472C2semidirect
11. #11729472C2semidirect
37. #89698C22semidirect
38. #89698C22semidirect
41. C4×#1136C22semidirect
42. C2×#6377C22semidirect
43. #89338C22semidirect
44. #89338C22semidirect
45. #89338C22semidirect
46. #89338C22semidirect
47. #89338C22semidirect
48. #89338C22semidirect
559. C7#17691semidirect

Maximal subgroups

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

subgroupindex
C7×#35102
#117399642
#117399632
#117295772
#117295752
#117295032
#117295022
#117295012
#117295002
#117294722
#176917