Finite Group Databasebeta

← #13322341

Subgroups — #13322341

Order 1792.

952 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2#11736727C211yesmaximal, characteristic
3C2×#89607C211yesmaximal, characteristic
4C7×#1542C211yesmaximal, characteristic, complemented
5#11729582C211yesmaximal, characteristic, complemented
6#11728530C211yesmaximal, characteristic, complemented
7#11728191C211yesmaximal, characteristic, complemented
8#11726242C212yesmaximal, complemented
9#11726241C212yesmaximal, complemented
10#11726184C212yesmaximal, complemented
11#11726182C212yesmaximal, complemented
12C7×#533C2211yescharacteristic
13#90419C2211yescharacteristic
14#90419C2211yescharacteristic
15D7×C4⋊Q8C2211yescharacteristic
16D7×C4⋊Q8C2211yescharacteristic
17#90379—21no—
18D7×#125—21no—
19D7×C22⋊Q8—21no—
20D7×C22⋊Q8C412yes—
21#90313—21no—
22#90313C412yes—
23#90269—21no—
24#90269C412yes—
25C4×Q8×D7—21no—
26C4×Q8×D7C412yes—
27D7×#117—21no—
28#90208—21no—
29C2×C42×D7C2211yescharacteristic
30#90161C2211yescharacteristic
31C7×#431C2211yescharacteristic
32C2×C7×C4⋊C8C2211yescharacteristic
33C2×(C2×D7)⋊C8—21no—
34C2×#6389—21no—
35#89697C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89609C2212yes—
38#89607—21no—
39#89607—21no—
40#89607C2212yes—
41#89607C2212yes—
42#89607C2212yes—
43#89498—21no—
44#89491—21no—
45C4×#1131—21no—
46#89457—21no—
47C7×#333C2212yescomplemented
48C7×#332C2212yescomplemented
49#89342C2212yescomplemented
50#89341C2212yescomplemented
51#89281C2212yescomplemented
52#89277C2212yescomplemented
53#89276C2212yescomplemented
54#89276C2212yescomplemented
55#17734—71nomaximal, Sylow
56C2×Q8×D7—21no—
57C2×Q8×D7C2×C412yes—
58C22×C4×D7C2×C411yescharacteristic
59C22×C4×D7D411yescharacteristic
60C22×C4×D7D411yescharacteristic
61C22×C7×C8—21no—
62C7×C4⋊Q8C2311yescharacteristic
63C7×C4⋊Q8C2311yescharacteristic
64C7×#125—21no—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8C2×C412yes—
67C4×C7×Q8—21no—
68C4×C7×Q8C2×C412yes—
69C7×#117—21no—
70C2×C42×C7C2311yescharacteristic
71#6509—21no—
72#6509C2×C412yes—
73Q8×Dic7—21no—
74Q8×Dic7C2×C412yes—
75#6507—21no—
76#6507C2×C412yes—
77#6489—21no—
78#6487—21no—
79#6487C2×C412yes—
80C2×C4×Dic7C2×C411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7D411yescharacteristic
83C22×C7⋊C8—21no—
84C2×#1131—22no—
85#6460—21no—
86#6460C2×C412yes—
87#6459—22no—
88#6459—22no—
89#6455—21no—
90#6455—21no—
91D7×C4⋊C4—22no—
92D7×C4⋊C4—22no—
93D7×C4⋊C4—21no—
94D7×C4⋊C4—21no—
95D7×C4⋊C4—21no—
96D7×C4⋊C4C2×C412yes—
97#6453—21no—
98#6453—21no—
99#6452—22no—
100#6452—22no—
101#6451—21no—
102#6451—21no—
103#6451—21no—
104#6451C2×C412yes—
105#6450—22no—
106#6450—22no—
107#6450—21no—
108#6450C2×C412yes—
109D7×C22⋊C4—22no—
110#6441—22no—
111#6441—22no—
112#6440—22no—
113#6435—22no—
114C42×D7—22no—
115C42×D7C2×C411yescharacteristic
116C42×D7C2×C411yescharacteristic
117C42×D7D411yescharacteristic
118C42×D7D411yescharacteristic
119C42×D7D411yescharacteristic
120C42×D7D411yescharacteristic
121C42×D7C2311yescharacteristic
122C42×D7C2311yescharacteristic
123#6433—21no—
124#6432C2311yescharacteristic
125#6432C2311yescharacteristic
126C4×Dic14—21no—
127C4×Dic14—22no—
128C4×Dic14—22no—
129C4×Dic14C2×C412yes—
130C7×C4⋊C8—21no—
131C7×C4⋊C8C2312yes—
132C7×C4⋊C8C2312yes—
133C7×C22⋊C8—21no—
134C4×C7×C8—21no—
135C22⋊(C7⋊C8)—21no—
136(C2×D7)⋊C8—22no—
137(C2×D7)⋊C8—22no—
138(C2×D7)⋊C8—22no—
139#6389—22no—
140#6389—22no—
141#6389—22no—
142Dic7⋊C8—22no—
143C4⋊(C7⋊C8)—21no—
144C4⋊(C7⋊C8)C2312yes—
145C4⋊(C7⋊C8)C2312yes—
146C4×C7⋊C8—21no—
147C2×#533—71no—
148#2920—71no—
149C2×#432—71no—
150#1559—72no—
151#1559—72no—
152#1558—72no—
153#1556—72no—
154#1542—71no—
155#1542—71no—
156#1542—71no—
157#1542D711yescharacteristic, complemented
158C23×D7C22⋊C411yescharacteristic
159C2×C7×Q8—21no—
160C2×C7×Q8C22×C412yes—
161C22×C4×C7C22×C411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×Dic7C22⋊C411yescharacteristic
165C22×Dic7C22⋊C411yescharacteristic
166C22×Dic7C22⋊C411yescharacteristic
167Q8×D7—44no—
168Q8×D7—44no—
169C2×C4×D7—22no—
170C2×C4×D7—22no—
171C2×C4×D7—22no—
172C2×C4×D7—22no—
173C2×C4×D7—42no—
174C2×C4×D7—42no—
175C2×C4×D7C22⋊C411yescharacteristic
176C2×C4×D7C22⋊C411yescharacteristic
177C2×C4×D7C22⋊C411yescharacteristic
178C2×C4×D7C22⋊C411yescharacteristic
179C2×C4×D7C22⋊C411yescharacteristic
180C2×C4×D7C22⋊C411yescharacteristic
181C2×C4×D7C22×C411yescharacteristic
182C2×C4×D7C22×C411yescharacteristic
183C2×C4×D7C2×D411yescharacteristic
184C2×C4×D7C2×D411yescharacteristic
185C2×C4×D7C2×D411yescharacteristic
186C2×C4×D7C2×D411yescharacteristic
187C2×Dic14—21no—
188C2×Dic14—22no—
189C2×Dic14—22no—
190C2×Dic14C22×C412yes—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C2×C7×C8—42no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4C22×C412yes—
200C7×C22⋊C4—22no—
201C42×C7—22no—
202C42×C7C22×C411yescharacteristic
203C42×C7C2×D411yescharacteristic
204C42×C7C2×D411yescharacteristic
205C42×C7C2411yescharacteristic
206C22⋊Dic7—22no—
207(C2×D7)⋊C4—22no—
208(C2×D7)⋊C4—22no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—22no—
213C4⋊Dic7—22no—
214C4⋊Dic7C22×C412yes—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4—22no—
220Dic7⋊C4—22no—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223C4×Dic7—22no—
224C4×Dic7—22no—
225C4×Dic7—22no—
226C4×Dic7C22⋊C411yescharacteristic
227C4×Dic7C22⋊C411yescharacteristic
228C4×Dic7C22⋊C411yescharacteristic
229C4×Dic7C22⋊C411yescharacteristic
230C4×Dic7C22⋊C411yescharacteristic
231C4×Dic7C22⋊C411yescharacteristic
232C4×Dic7C22×C411yescharacteristic
233C4×Dic7C22×C411yescharacteristic
234C4×Dic7C2×D411yescharacteristic
235C4×Dic7C2×D411yescharacteristic
236C4×Dic7C2×D411yescharacteristic
237C4×Dic7C2×D411yescharacteristic
238C2×C7⋊C8—22no—
239C2×C7⋊C8—22no—
240C2×C7⋊C8—42no—
241#1131—44no—
242#533—71no—
243#533—71no—
244#533—72no—
245#533—72no—
246#533—71no—
247#533—141no—
248#533—141no—
249#533—141no—
250#533—141no—
251#533C2×D711yescharacteristic
252C2×C4⋊Q8—71no—
253C2×C4⋊Q8—71no—
254C2×#125—141no—
255C2×C22⋊Q8—72no—
256C2×C22⋊Q8—141no—
257C2×C4×Q8—72no—
258C2×C4×Q8—141no—
259C2×#117—141no—
260C22×C42—71no—
261#432—72no—
262#432—72no—
263#432—72no—
264#432—141no—
265#432—141no—
266#432—141no—
267#431—71no—
268#431—141no—
269#431C2×D711yescharacteristic
270#423—72no—
271C2×C4⋊C8—71no—
272C2×C4⋊C8C2×D711yescharacteristic
273#407—141no—
274C2×C22⋊C8—141no—
275C4×M16—141no—
276C2×#97—141no—
277#333—72no—
278#333—72no—
279#333—72no—
280#333C2×D712yescomplemented
281#332—72no—
282#332—72no—
283#332—72no—
284#332C2×D712yescomplemented
285C23×C7C2×C22⋊C411yescharacteristic
286C22×D7—22no—
287C22×D7—22no—
288C22×D7—41no—
289C22×D7C2×C22⋊C411yescharacteristic
290C22×D7C2×C22⋊C411yescharacteristic
291C7×Q8—42no—
292C7×Q8—42no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—41no—
297C2×C4×C7C2×C22⋊C411yescharacteristic
298C2×C4×C7C2×C22⋊C411yescharacteristic
299C2×C4×C7C2×C22⋊C411yescharacteristic
300C2×C4×C7#13712yes—
301C2×C4×C7#13712yes—
302C2×C4×C7C23×C411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D411yescharacteristic
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7C2×C22⋊C411yescharacteristic
311C2×Dic7C2×C22⋊C411yescharacteristic
312C2×Dic7C2×C22⋊C411yescharacteristic
313C2×Dic7C2×C22⋊C411yescharacteristic
314C2×Dic7C2×C22⋊C411yescharacteristic
315C2×Dic7C2×C22⋊C411yescharacteristic
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—21no—
319C4×D7—21no—
320C4×D7—21no—
321C4×D7—21no—
322C4×D7—22no—
323C4×D7—22no—
324C4×D7—22no—
325C4×D7—22no—
326C4×D7—41no—
327C4×D7—41no—
328C4×D7—41no—
329C4×D7—41no—
330C4×D7—44no—
331C4×D7—44no—
332Dic14—44no—
333Dic14—44no—
334Dic14—42no—
335Dic14—42no—
336C7×C8—42no—
337C7×C8—42no—
338C7⋊C8—42no—
339C7⋊C8—42no—
340C22×Q8—72no—
341C22×Q8—141no—
342C23×C4—71no—
343C23×C4—71no—
344C23×C4—71no—
345C2×M16—142no—
346C22×C8—21no—
347C22×C8—141no—
348C4⋊Q8—72no—
349C4⋊Q8—72no—
350C4⋊Q8—71no—
351C4⋊Q8—71no—
352C4⋊Q8—141no—
353C4⋊Q8—141no—
354C4⋊Q8—141no—
355C4⋊Q8—141no—
356C4⋊Q8C22×D711yescharacteristic
357C4⋊Q8C22×D711yescharacteristic
358#125—21no—
359#125—142no—
360#125—142no—
361#125—141no—
362#125—141no—
363#125—141no—
364C22⋊Q8—21no—
365C22⋊Q8—72no—
366C22⋊Q8—72no—
367C22⋊Q8—72no—
368C22⋊Q8—142no—
369C22⋊Q8—142no—
370C22⋊Q8—142no—
371C22⋊Q8—142no—
372C22⋊Q8—141no—
373C22⋊Q8—141no—
374C22⋊Q8—141no—
375C22⋊Q8C4×D712yes—
376C4×Q8—21no—
377C4×Q8—72no—
378C4×Q8—72no—
379C4×Q8—72no—
380C4×Q8—142no—
381C4×Q8—142no—
382C4×Q8—142no—
383C4×Q8—142no—
384C4×Q8—141no—
385C4×Q8—141no—
386C4×Q8—141no—
387C4×Q8C4×D712yes—
388#117—21no—
389#117—142no—
390#117—142no—
391#117—141no—
392#117—141no—
393#117—141no—
394C2×C4⋊C4—72no—
395C2×C4⋊C4—141no—
396C2×C4⋊C4—141no—
397C2×C4⋊C4—142no—
398C2×C4⋊C4—142no—
399C2×C4⋊C4—141no—
400C2×C22⋊C4—142no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—71no—
406C2×C42—71no—
407C2×C42—71no—
408C2×C42—71no—
409C2×C42—71no—
410C2×C42—71no—
411C2×C42—71no—
412C2×C42—142no—
413C2×C42C22×D711yescharacteristic
414C4⋊C8—21no—
415C4⋊C8—72no—
416C4⋊C8—72no—
417C4⋊C8—142no—
418C4⋊C8—141no—
419C4⋊C8C22×D712yes—
420C4⋊C8C22×D712yes—
421C22⋊C8—21no—
422C22⋊C8—142no—
423C22⋊C8—142no—
424C22⋊C8—142no—
425C22⋊C8—141no—
426#97—142no—
427#97—142no—
428#97—142no—
429C4×C8—21no—
430C4×C8—141no—
431C22×C7—21no—
432C22×C7—21no—
433C22×C7#41912yes—
434C22×C7C22×C22⋊C411yescharacteristic
435C2×D7—21no—
436C2×D7—21no—
437C2×D7—21no—
438C2×D7—21no—
439C2×D7—21no—
440C2×D7—44no—
441C2×D7#41312yes—
442C4×C7—21no—
443C4×C7—21no—
444C4×C7—21no—
445C4×C7—42no—
446C4×C7—42no—
447C4×C7#41912yes—
448C4×C7C2×#13711yescharacteristic
449C4×C7C2×#13711yescharacteristic
450C4×C7#57512yes—
451Dic7—21no—
452Dic7—21no—
453Dic7—21no—
454Dic7—21no—
455Dic7—21no—
456Dic7—21no—
457Dic7—42no—
458Dic7—42no—
459C24—71no—
460C2×Q8—21no—
461C2×Q8—72no—
462C2×Q8—142no—
463C2×Q8—142no—
464C2×Q8—141no—
465C2×Q8—284no—
466C2×Q8—284no—
467C2×Q8C2×C4×D712yes—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—71no—
479C22×C4—71no—
480C22×C4—71no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4C2×C4×D711yescharacteristic
490C22×C4D4×D711yescharacteristic
491C22×C4D4×D711yescharacteristic
492M16—284no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—42no—
496C2×C8—142no—
497C2×C8—142no—
498C2×C8—282no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—72no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—142no—
508C4⋊C4—142no—
509C4⋊C4—142no—
510C4⋊C4—142no—
511C4⋊C4—142no—
512C4⋊C4—142no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4C2×C4×D712yes—
519C22⋊C4—22no—
520C22⋊C4—142no—
521C22⋊C4—142no—
522C22⋊C4—142no—
523C42—22no—
524C42—71no—
525C42—71no—
526C42—71no—
527C42—71no—
528C42—71no—
529C42—71no—
530C42—71no—
531C42—71no—
532C42—71no—
533C42—71no—
534C42—71no—
535C42—71no—
536C42—142no—
537C42—142no—
538C42—142no—
539C42C2×C4×D711yescharacteristic
540C42D4×D711yescharacteristic
541C42D4×D711yescharacteristic
542C42C23×D711yescharacteristic
543C2×C7—22no—
544C2×C7C2×#41311yescharacteristic
545C2×C7C2×#41911yescharacteristic
546C2×C7#289611yescharacteristic
547D7—22no—
548D7—41no—
549C23—71no—
550C23—71no—
551C23—142no—
552C23—142no—
553C23—281no—
554C23D7×C22⋊C411yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—282no—
558Q8—284no—
559Q8—284no—
560Q8—282no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—71no—
566C2×C4—71no—
567C2×C4—71no—
568C2×C4—71no—
569C2×C4—71no—
570C2×C4—71no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—141no—
575C2×C4—141no—
576C2×C4—141no—
577C2×C4—141no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—284no—
589C2×C4—284no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4D7×C22⋊C411yescharacteristic
593C2×C4D7×C22⋊C411yescharacteristic
594C2×C4D7×C22⋊C411yescharacteristic
595C2×C4#647812yes—
596C2×C4#648112yes—
597C2×C4C22×C4×D711yescharacteristic
598C2×C4C2×D4×D711yescharacteristic
599C2×C4C2×D4×D711yescharacteristic
600C8—42no—
601C8—42no—
602C8—282no—
603C8—282no—
604C7#1773411yescharacteristic, complemented, Sylow
605C22—21no—
606C22—21no—
607C22—72no—
608C22—141no—
609C22—141no—
610C22—141no—
611C22—141no—
612C22—141no—
613C22—284no—
614C22#8957412yes—
615C22C2×D7×C22⋊C411yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—42no—
621C4—141no—
622C4—141no—
623C4—141no—
624C4—141no—
625C4—141no—
626C4—141no—
627C4—282no—
628C4—282no—
629C4#8957412yes—
630C4C2×#647811yescharacteristic
631C4C2×#648111yescharacteristic
632C4#9045912yes—
633C2—22no—
634C2—142no—
635C2—281no—
636C2D7×#41311yescharacteristic
637C2C2×#8957411yescharacteristic
638C2#1173627511yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1542C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1542D7
hypocenterC7#17734
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
4. C7×#1542C2semidirect
5. #11729582C2semidirect
6. #11728530C2semidirect
7. #11728191C2semidirect
8. #11726242C2semidirect
9. #11726241C2semidirect
10. #11726184C2semidirect
11. #11726182C2semidirect
47. C7×#333C22semidirect
48. C7×#332C22semidirect
49. #89342C22semidirect
50. #89341C22semidirect
51. #89281C22semidirect
52. #89277C22semidirect
53. #89276C22semidirect
54. #89276C22semidirect
157. #1542D7semidirect
280. #333C2×D7semidirect
284. #332C2×D7semidirect
604. C7#17734semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117367272
C2×#896072
C7×#15422
#117295822
#117285302
#117281912
#117262422
#117262412
#117261842
#117261822
#177347