Finite Group Databasebeta

← #13320815

Subgroups — #13320815

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742911C211yesmaximal, characteristic
2C7×#2991C211yesmaximal, characteristic, complemented
3#11739958C211yesmaximal, characteristic, complemented
4#11728515C211yesmaximal, characteristic, complemented
5#11728513C212yesmaximal, complemented
6#11728510C212yesmaximal, complemented
7#11728508C211yesmaximal, characteristic, complemented
8#11728253C212yesmaximal, complemented
9#11728252C211yesmaximal, characteristic, complemented
10#11728251C211yesmaximal, characteristic, complemented
11#11728250C212yesmaximal, complemented
12C4×C7×#41C2211yescharacteristic
13#90520—21no—
14#90276C411yescharacteristic
15#90271—21no—
16#90270C411yescharacteristic
17#90256C412yes—
18#90241C412yes—
19D4×D28C411yescharacteristic
20#90238—22no—
21C4×D4×D7—21no—
22#90234C411yescharacteristic
23#90221—21no—
24#90218—22no—
25#90170C2212yes—
26#90168C2211yescharacteristic
27C4×#1157C2211yescharacteristic
28C2×C4×D28C2212yes—
29C7×C8×Q8—21no—
30C7×#436—21no—
31C7×#436—22no—
32C7×#431C2211yescharacteristic, complemented
33C7×#431C2212yescomplemented
34C7×#423C2212yescomplemented
35C7×#423C2211yescharacteristic, complemented
36#90006—21no—
37#89772—22no—
38#89771—21no—
39#89697C2212yescomplemented
40#89697C2212yescomplemented
41C4×#1136C2211yescharacteristic, complemented
42C2×C4×C7⋊C8C2211yescharacteristic, complemented
43#89279C2212yescomplemented
44#89279C2211yescharacteristic, complemented
45#89279C2211yescharacteristic, complemented
46#89278C2211yescharacteristic
47#89278C2212yes—
48#89278C2211yescharacteristic
49#89278C2212yescomplemented
50#89278C2212yescomplemented
51#89278C2212yescomplemented
52#89278C2212yescomplemented
53#17522—71nomaximal, Sylow
54C2×C7×#41D411yescharacteristic
55C2×D4×D7—21no—
56C2×#1157—21no—
57C2×#1157C2×C411yescharacteristic
58C22×D28C2×C412yes—
59C22×C4×D7—22no—
60C2×C7×M16—22no—
61C2×C7×M16—21no—
62C4×C7×Q8C2×C411yescharacteristic
63C4×C7×Q8D411yescharacteristic
64C4×C7×D4C2×C411yescharacteristic
65C4×C7×D4C2×C412yes—
66C4×C7×D4D411yescharacteristic
67C4×C7×D4D412yes—
68C7×#117D412yes—
69C7×#117D411yescharacteristic
70C2×C42×C7C2312yes—
71C2×C42×C7C2311yescharacteristic
72#6510—21no—
73#6509—21no—
74#6502—22no—
75#6500—22no—
76D4×Dic7—21no—
77#6493—21no—
78#6493C2×C412yes—
79#6493C2×C412yes—
80#6492—22no—
81C4×C22⋊D7—21no—
82C4×C22⋊D7—21no—
83C4×C22⋊D7—22no—
84C2×(C2×D7)⋊C4—22no—
85#6489C2×C411yescharacteristic
86C2×C4⋊Dic7C2×C412yes—
87#6487C2×C412yes—
88C2×#1136—21no—
89C22×C7⋊C8—21no—
90#6461—22no—
91#6459—22no—
92#6458—21no—
93#6457—22no—
94#6457—22no—
95#6456—21no—
96D7×C4⋊C4—21no—
97#6453—21no—
98#6447—22no—
99#6446—22no—
100#6445—22no—
101(C22⋊D7)⋊C4—22no—
102D7×C22⋊C4—22no—
103#6441—22no—
104#6439—22no—
105#6438—21no—
106#6438C2×C412yes—
107#6437C2×C411yescharacteristic
108C4×D28—22no—
109C4×D28—22no—
110C4×D28—21no—
111C4×D28C2×C412yes—
112C4×D28C2×C412yes—
113C4×D28C2×C411yescharacteristic
114C4×D28C2311yescharacteristic
115C4×D28C2312yes—
116#6435—21no—
117#6435—22no—
118C42×D7—21no—
119#6433C2×C412yes—
120#6432C2×C411yescharacteristic
121C4×Dic14C2×C411yescharacteristic
122C4×Dic14C2311yescharacteristic
123C7×C4⋊C8—21no—
124C7×C4⋊C8—22no—
125C7×C4⋊C8C2311yescharacteristic
126C7×C4⋊C8C2312yes—
127C7×C4⋊C8C2311yescharacteristic
128C7×C22⋊C8—22no—
129C7×C22⋊C8—21no—
130C4×C7×C8—21no—
131C4×C7×C8—22no—
132C22⋊(C7⋊C8)—22no—
133C22⋊(C7⋊C8)—22no—
134C4⋊(C7⋊C8)—22no—
135C4⋊(C7⋊C8)—22no—
136#6377—21no—
137C4×C7⋊C8—21no—
138C4×C7⋊C8C2311yescharacteristic
139C4×C7⋊C8C2311yescharacteristic
140C4×C7⋊C8C2312yes—
141#3481—71no—
142#2991D711yescharacteristic, complemented
143#2922—71no—
144#1495—72no—
145#1492—71no—
146#1489—71no—
147#1486—72no—
148#1484—71no—
149#1482—72no—
150#1478—72no—
151#1476—71no—
152C23×D7—22no—
153C7×#41—42no—
154C2×C7×Q8C22⋊C411yescharacteristic
155C2×C7×D4C22⋊C411yescharacteristic
156C2×C7×D4C22⋊C412yes—
157C22×C4×C7C22×C412yes—
158C22×C4×C7C22×C411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C2×C22⋊D7—22no—
164C2×C22⋊D7—21no—
165C2×C22⋊D7—22no—
166C22×Dic7—22no—
167D4×D7—44no—
168#1157—22no—
169#1157—41no—
170#1157—81no—
171#1157—81no—
172C2×D28—24no—
173C2×D28—22no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28—42no—
177C2×D28C22×C412yes—
178C2×D28C22×C411yescharacteristic
179C2×C4×D7—21no—
180C2×C4×D7—21no—
181C2×C4×D7—21no—
182C2×C4×D7—22no—
183C2×C4×D7—22no—
184C2×C4×D7—22no—
185C2×C4×D7—82no—
186C2×C4×D7—82no—
187C2×Dic14—21no—
188C2×Dic14C22×C411yescharacteristic
189C7×M16—82no—
190C7×M16—81no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—22no—
194C7×C4⋊C4C22⋊C411yescharacteristic
195C7×C4⋊C4C22⋊C411yescharacteristic
196C7×C4⋊C4C22⋊C412yes—
197C7×C4⋊C4C22⋊C412yes—
198C7×C22⋊C4C22⋊C412yes—
199C7×C22⋊C4C22⋊C412yes—
200C7×C22⋊C4C22⋊C412yes—
201C42×C7C22×C411yescharacteristic
202C42×C7C22×C412yes—
203C42×C7C2×D411yescharacteristic
204C42×C7C2×D412yes—
205C42×C7C2×D411yescharacteristic
206C42×C7C2×D412yes—
207C42×C7C2411yescharacteristic
208C22⋊Dic7—22no—
209C22⋊Dic7—21no—
210(C2×D7)⋊C4—21no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214(C2×D7)⋊C4—22no—
215(C2×D7)⋊C4—22no—
216(C2×D7)⋊C4—22no—
217(C2×D7)⋊C4—21no—
218(C2×D7)⋊C4—22no—
219C4⋊Dic7—21no—
220C4⋊Dic7—22no—
221C4⋊Dic7C22×C411yescharacteristic
222C4⋊Dic7C22×C411yescharacteristic
223C4⋊Dic7C22×C412yes—
224Dic7⋊C4—21no—
225Dic7⋊C4—22no—
226Dic7⋊C4—22no—
227Dic7⋊C4—22no—
228Dic7⋊C4—21no—
229C4×Dic7—21no—
230C4×Dic7—21no—
231C4×Dic7—21no—
232#1136—42no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—22no—
236C2×C7⋊C8—42no—
237#559—71no—
238#554—71no—
239#554—71no—
240#553—72no—
241#553—141no—
242#551—142no—
243#546—72no—
244#546—141no—
245D42—71no—
246#532—71no—
247#529—72no—
248#529—141no—
249#525—142no—
250C4×#41—71no—
251C4×#41C2×D711yescharacteristic
252C2×C4×D4—72no—
253C2×C4×D4—141no—
254C8×Q8—21no—
255#436—21no—
256#436—22no—
257#433—142no—
258#431—72no—
259#431—72no—
260#431C2×D711yescharacteristic, complemented
261#431C2×D712yescomplemented
262#424—141no—
263#423C2×D712yescomplemented
264#423C2×D711yescharacteristic, complemented
265#408—141no—
266C4×M16—71no—
267C2×C4×C8—71no—
268Q8⋊C8—71no—
269Q8⋊C8—71no—
270Q8⋊C8—72no—
271D4⋊C8—71no—
272D4⋊C8—71no—
273D4⋊C8—72no—
274D4⋊C8—72no—
275D4⋊C8—72no—
276D4⋊C8—72no—
277D4⋊C8—72no—
278C23×C7C2×C22⋊C412yes—
279C23×C7C2×C22⋊C411yescharacteristic
280C22×D7—21no—
281C22×D7—22no—
282C22×D7—44no—
283C22×D7—44no—
284C22×D7—82no—
285C7×Q8—22no—
286C7×D4—22no—
287C7×D4—42no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—42no—
298C2×C4×C7C2×C22⋊C411yescharacteristic
299C2×C4×C7C2×C22⋊C412yes—
300C2×C4×C7C2×C22⋊C412yes—
301C2×C4×C7C2×C22⋊C411yescharacteristic
302C2×C4×C7C2×C22⋊C412yes—
303C2×C4×C7C2×C22⋊C411yescharacteristic
304C2×C4×C7C23×C411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C22⋊D7—42no—
308C22⋊D7—44no—
309C22⋊D7—82no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—22no—
314C2×Dic7—82no—
315D28—22no—
316D28—21no—
317D28—22no—
318D28—44no—
319D28—42no—
320D28—81no—
321D28#13112yes—
322C4×D7—42no—
323C4×D7—41no—
324C4×D7—42no—
325C4×D7—41no—
326C4×D7—42no—
327C4×D7—42no—
328C4×D7—82no—
329C4×D7—82no—
330Dic14—21no—
331Dic14—21no—
332Dic14—81no—
333C7×C8—41no—
334C7×C8—42no—
335C7×C8—41no—
336C7⋊C8—41no—
337C7⋊C8—41no—
338C7⋊C8—42no—
339C2×#41—71no—
340C2×#41—141no—
341C2×#41D2811yescharacteristic
342C22×D4—72no—
343C22×D4—141no—
344C23×C4—142no—
345C2×M16—22no—
346C2×M16—21no—
347C2×M16—141no—
348C22×C8—141no—
349C4⋊Q8—71no—
350#127—71no—
351#126—142no—
352#126—142no—
353#125—72no—
354#125—141no—
355#124—72no—
356#124—141no—
357#124—141no—
358#123—142no—
359#123—142no—
360#123—142no—
361#123—142no—
362C22⋊Q8—72no—
363C22⋊Q8—141no—
364C22⋊Q8—142no—
365C22⋊Q8—142no—
366#121—72no—
367#121—72no—
368#121—142no—
369#121—141no—
370#121—141no—
371#121—142no—
372#120—142no—
373#120—142no—
374C4×Q8—71no—
375C4×Q8—71no—
376C4×Q8C4×D711yescharacteristic
377C4×Q8C22⋊D711yescharacteristic
378C4×D4—72no—
379C4×D4—72no—
380C4×D4—71no—
381C4×D4—71no—
382C4×D4—72no—
383C4×D4—141no—
384C4×D4—141no—
385C4×D4—142no—
386C4×D4—141no—
387C4×D4—141no—
388C4×D4—142no—
389C4×D4—142no—
390C4×D4—142no—
391C4×D4—141no—
392C4×D4C4×D711yescharacteristic
393C4×D4C4×D712yes—
394C4×D4C22⋊D711yescharacteristic
395C4×D4C22⋊D712yes—
396#117—71no—
397#117—141no—
398#117—142no—
399#117D2812yes—
400#117D2811yescharacteristic
401C2×C4⋊C4—72no—
402C2×C4⋊C4—141no—
403C2×C22⋊C4—142no—
404C2×C22⋊C4—142no—
405C2×C42—141no—
406C2×C42C22×D712yes—
407C2×C42C22×D711yescharacteristic
408C4⋊C8—21no—
409C4⋊C8—22no—
410C4⋊C8—142no—
411C4⋊C8—142no—
412C4⋊C8C22×D711yescharacteristic
413C4⋊C8C22×D712yes—
414C4⋊C8C22×D711yescharacteristic
415C22⋊C8—22no—
416C22⋊C8—21no—
417C22⋊C8—142no—
418C22⋊C8—142no—
419#97—141no—
420C4×C8—21no—
421C4×C8—22no—
422C4×C8—71no—
423C4×C8—71no—
424C4×C8—72no—
425C4×C8—141no—
426C22×C7—21no—
427C22×C7—22no—
428C22×C7—22no—
429C22×C7—21no—
430C22×C7—22no—
431C22×C7—21no—
432C22×C7C22×C22⋊C411yescharacteristic
433C2×D7—22no—
434C2×D7—41no—
435C2×D7—41no—
436C2×D7—42no—
437C2×D7—42no—
438C2×D7—42no—
439C2×D7—88no—
440C4×C7—21no—
441C4×C7—22no—
442C4×C7—21no—
443C4×C7—22no—
444C4×C7—21no—
445C4×C7—22no—
446C4×C7#42112yes—
447C4×C7C2×#13111yescharacteristic
448C4×C7#56811yescharacteristic
449C4×C7#57611yescharacteristic
450C4×C7#57711yescharacteristic
451Dic7—41no—
452Dic7—42no—
453Dic7—41no—
454Dic7—41no—
455C24—142no—
456#41—42no—
457#41—142no—
458#41—281no—
459#41—561no—
460#41—561no—
461C2×Q8—71no—
462C2×Q8—141no—
463C2×Q8(C2×D7)⋊C411yescharacteristic
464C2×D4—71no—
465C2×D4—72no—
466C2×D4—144no—
467C2×D4—141no—
468C2×D4—142no—
469C2×D4—142no—
470C2×D4—142no—
471C2×D4—141no—
472C2×D4—141no—
473C2×D4—282no—
474C2×D4—284no—
475C2×D4(C2×D7)⋊C411yescharacteristic
476C2×D4(C2×D7)⋊C412yes—
477C22×C4—142no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—562no—
485C22×C4—562no—
486C22×C4C2×C4×D712yes—
487C22×C4C2×C4×D711yescharacteristic
488C22×C4C2×D2811yescharacteristic
489C22×C4C2×D2812yes—
490C22×C4C2×C22⋊D712yes—
491C22×C4C2×C22⋊D711yescharacteristic
492M16—82no—
493M16—81no—
494M16—282no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—142no—
501C2×C8—282no—
502C4⋊C4—71no—
503C4⋊C4—72no—
504C4⋊C4—71no—
505C4⋊C4—142no—
506C4⋊C4—141no—
507C4⋊C4—142no—
508C4⋊C4—142no—
509C4⋊C4—142no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4(C2×D7)⋊C411yescharacteristic
513C4⋊C4(C2×D7)⋊C411yescharacteristic
514C4⋊C4(C2×D7)⋊C412yes—
515C4⋊C4(C2×D7)⋊C412yes—
516C22⋊C4—142no—
517C22⋊C4—141no—
518C22⋊C4—142no—
519C22⋊C4—142no—
520C22⋊C4—142no—
521C22⋊C4—142no—
522C22⋊C4—141no—
523C22⋊C4—142no—
524C22⋊C4—142no—
525C22⋊C4—141no—
526C22⋊C4—142no—
527C22⋊C4(C2×D7)⋊C412yes—
528C22⋊C4(C2×D7)⋊C412yes—
529C22⋊C4(C2×D7)⋊C412yes—
530C42—141no—
531C42—141no—
532C42—141no—
533C42C2×C4×D712yes—
534C42C2×C4×D711yescharacteristic
535C42C2×D2811yescharacteristic
536C42C2×D2812yes—
537C42C2×C22⋊D711yescharacteristic
538C42C2×C22⋊D712yes—
539C42C23×D711yescharacteristic
540C2×C7—42no—
541C2×C7—41no—
542C2×C7#288011yescharacteristic
543C2×C7#289711yescharacteristic
544C2×C7C2×#42111yescharacteristic
545D7—42no—
546D7—82no—
547C23—141no—
548C23—142no—
549C23—284no—
550C23—284no—
551C23—562no—
552C23C2×(C2×D7)⋊C412yes—
553C23C2×(C2×D7)⋊C411yescharacteristic
554Q8—22no—
555Q8—141no—
556Q8—141no—
557Q8—561no—
558D4—22no—
559D4—42no—
560D4—72no—
561D4—142no—
562D4—141no—
563D4—142no—
564D4—284no—
565D4—282no—
566D4—284no—
567D4—282no—
568D4—561no—
569D4—562no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—42no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—142no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—281no—
587C2×C4—282no—
588C2×C4—281no—
589C2×C4—282no—
590C2×C4—562no—
591C2×C4—562no—
592C2×C4—562no—
593C2×C4C2×(C2×D7)⋊C411yescharacteristic
594C2×C4C2×(C2×D7)⋊C412yes—
595C2×C4C2×(C2×D7)⋊C411yescharacteristic
596C2×C4C2×(C2×D7)⋊C412yes—
597C2×C4C2×(C2×D7)⋊C412yes—
598C2×C4C2×(C2×D7)⋊C411yescharacteristic
599C2×C4C22×C4×D711yescharacteristic
600C2×C4C22×D2811yescharacteristic
601C2×C4C22×C22⋊D711yescharacteristic
602C8—41no—
603C8—42no—
604C8—41no—
605C8—281no—
606C8—281no—
607C8—282no—
608C7#1752211yescharacteristic, complemented, Sylow
609C22—21no—
610C22—22no—
611C22—21no—
612C22—22no—
613C22—22no—
614C22—21no—
615C22—142no—
616C22—281no—
617C22—281no—
618C22—282no—
619C22—282no—
620C22—282no—
621C22—568no—
622C22C22×(C2×D7)⋊C411yescharacteristic
623C4—21no—
624C4—22no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—21no—
629C4—281no—
630C4—281no—
631C4—281no—
632C4—282no—
633C4#8991112yes—
634C4D7×#13111yescharacteristic
635C4#9044111yescharacteristic
636C4#9052611yescharacteristic
637C4#9052811yescharacteristic
638C2—42no—
639C2—41no—
640C2—282no—
641C2—562no—
642C2#1174000811yescharacteristic
643C2C2×#8991111yescharacteristic
644C2#1174123811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#2991C2
socleC22×C7C22×C22⋊C4
prime radical#11742911C2
cosocleC42×C7C24
hypercenter#2991D7
hypocenterC7#17522
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. C7×#2991C2semidirect
3. #11739958C2semidirect
4. #11728515C2semidirect
5. #11728513C2semidirect
6. #11728510C2semidirect
7. #11728508C2semidirect
8. #11728253C2semidirect
9. #11728252C2semidirect
10. #11728251C2semidirect
11. #11728250C2semidirect
32. C7×#431C22semidirect
33. C7×#431C22semidirect
34. C7×#423C22semidirect
35. C7×#423C22semidirect
39. #89697C22semidirect
40. #89697C22semidirect
41. C4×#1136C22semidirect
42. C2×C4×C7⋊C8C22semidirect
43. #89279C22semidirect
44. #89279C22semidirect
45. #89279C22semidirect
49. #89278C22semidirect
50. #89278C22semidirect
51. #89278C22semidirect
52. #89278C22semidirect
142. #2991D7semidirect
260. #431C2×D7semidirect
261. #431C2×D7semidirect
263. #423C2×D7semidirect
264. #423C2×D7semidirect
608. C7#17522semidirect

Maximal subgroups

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

subgroupindex
#117429112
C7×#29912
#117399582
#117285152
#117285132
#117285102
#117285082
#117282532
#117282522
#117282512
#117282502
#175227