Finite Group Databasebeta

← #13322198

Subgroups — #13322198

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2#11736727C211yesmaximal, characteristic
3D7×#431C211yesmaximal, characteristic
4C7×#1544C211yesmaximal, characteristic, complemented
5#11729583C211yesmaximal, characteristic, complemented
6#11728531C211yesmaximal, characteristic, complemented
7#11728531C211yesmaximal, characteristic, complemented
8#11726179C212yesmaximal, complemented
9#11726178C212yesmaximal, complemented
10#11726176C212yesmaximal, complemented
11#11726174C212yesmaximal, complemented
12C7×#533C2211yescharacteristic
13#90419C2211yescharacteristic
14#90419C2211yescharacteristic
15D7×C4⋊Q8C2211yescharacteristic
16D7×C4⋊Q8C2211yescharacteristic
17#90379C412yes—
18D7×#125C412yes—
19D7×C22⋊Q8—21no—
20D7×C22⋊Q8—21no—
21#90313—21no—
22#90313—21no—
23#90269—21no—
24#90269—21no—
25C4×Q8×D7—21no—
26C4×Q8×D7—21no—
27D7×#117C412yes—
28#90208C412yes—
29C2×C42×D7C2211yescharacteristic
30#90161C2211yescharacteristic
31C7×#431C2211yescharacteristic
32C7×#431C2211yescharacteristic
33#89697C2211yescharacteristic
34#89697C2211yescharacteristic
35#89609C2212yes—
36#89607C2212yes—
37#89605C2212yes—
38D7×C4⋊C8C2212yes—
39#89498—21no—
40D7×C22⋊C8—21no—
41#89491—21no—
42#89490—21no—
43C4×#1131—21no—
44#89457—21no—
45#89457—21no—
46C4×C8×D7—21no—
47C7×#332C2212yescomplemented
48C7×#332C2212yescomplemented
49#89342C2212yescomplemented
50#89342C2212yescomplemented
51#89281C2212yescomplemented
52#89281C2212yescomplemented
53#89277C2212yescomplemented
54#89277C2212yescomplemented
55#17715—71nomaximal, Sylow
56C2×Q8×D7—21no—
57C2×Q8×D7—21no—
58C22×C4×D7C2×C411yescharacteristic
59C22×C4×D7D411yescharacteristic
60C22×C4×D7D411yescharacteristic
61C7×C4⋊Q8C2311yescharacteristic
62C7×C4⋊Q8C2311yescharacteristic
63C7×#125C2×C412yes—
64C7×C22⋊Q8—21no—
65C7×C22⋊Q8—21no—
66C4×C7×Q8—21no—
67C4×C7×Q8—21no—
68C7×#117C2×C412yes—
69C2×C42×C7C2311yescharacteristic
70#6509—21no—
71#6509—21no—
72Q8×Dic7—21no—
73Q8×Dic7—21no—
74#6507—21no—
75#6507—21no—
76#6489C2×C412yes—
77#6487—21no—
78#6487—21no—
79C2×C4×Dic7C2×C411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×#1131—22no—
83C2×C8×D7—22no—
84#6460—21no—
85#6460—21no—
86#6459—22no—
87#6459—22no—
88#6455C2×C412yes—
89#6455C2×C412yes—
90D7×C4⋊C4—22no—
91D7×C4⋊C4—22no—
92D7×C4⋊C4—21no—
93D7×C4⋊C4—21no—
94D7×C4⋊C4C2×C412yes—
95D7×C4⋊C4C2×C412yes—
96#6453C2×C412yes—
97#6453C2×C412yes—
98#6452—22no—
99#6452—22no—
100#6451—21no—
101#6451—21no—
102#6451C2×C412yes—
103#6451C2×C412yes—
104#6450—22no—
105#6450—22no—
106#6450—21no—
107#6450—21no—
108D7×C22⋊C4—22no—
109#6441—22no—
110#6441—22no—
111#6440—22no—
112#6435—22no—
113C42×D7—22no—
114C42×D7C2×C411yescharacteristic
115C42×D7C2×C411yescharacteristic
116C42×D7D411yescharacteristic
117C42×D7D411yescharacteristic
118C42×D7D411yescharacteristic
119C42×D7D411yescharacteristic
120C42×D7C2311yescharacteristic
121C42×D7C2311yescharacteristic
122#6433C2×C412yes—
123#6432C2311yescharacteristic
124#6432C2311yescharacteristic
125C4×Dic14—21no—
126C4×Dic14—21no—
127C4×Dic14—22no—
128C4×Dic14—22no—
129C7×C4⋊C8C2312yes—
130C7×C4⋊C8C2312yes—
131C7×C22⋊C8—21no—
132C7×C22⋊C8—21no—
133C4×C7×C8—21no—
134C4×C7×C8—21no—
135C22⋊(C7⋊C8)—21no—
136C22⋊(C7⋊C8)—21no—
137(C2×D7)⋊C8—22no—
138(C2×D7)⋊C8—22no—
139#6389—22no—
140Dic7⋊C8—22no—
141Dic7⋊C8—22no—
142C8×Dic7—22no—
143C4⋊(C7⋊C8)C2312yes—
144C4⋊(C7⋊C8)C2312yes—
145C4×C7⋊C8—21no—
146C4×C7⋊C8—21no—
147C2×#533—71no—
148#2920—71no—
149C2×#431—71no—
150#1552—72no—
151#1552—72no—
152#1549—72no—
153#1546—72no—
154#1544—71no—
155#1544—71no—
156#1544—71no—
157#1544D711yescharacteristic, complemented
158C23×D7C22⋊C411yescharacteristic
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
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—24no—
170C2×C4×D7—24no—
171C2×C4×D7—22no—
172C2×C4×D7—22no—
173C2×C4×D7—42no—
174C2×C4×D7C22⋊C411yescharacteristic
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×D7C2×D411yescharacteristic
183C2×C4×D7C2×D411yescharacteristic
184C2×C4×D7C2×D411yescharacteristic
185C2×C4×D7C2×D411yescharacteristic
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—22no—
189C2×Dic14—22no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4C22×C412yes—
197C7×C4⋊C4C22×C412yes—
198C7×C22⋊C4—22no—
199C42×C7—22no—
200C42×C7C22×C411yescharacteristic
201C42×C7C2×D411yescharacteristic
202C42×C7C2×D411yescharacteristic
203C42×C7C2411yescharacteristic
204C22⋊Dic7—22no—
205(C2×D7)⋊C4—22no—
206(C2×D7)⋊C4—22no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—22no—
210C4⋊Dic7—22no—
211C4⋊Dic7C22×C412yes—
212C4⋊Dic7C22×C412yes—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4—22no—
220Dic7⋊C4—22no—
221C4×Dic7—22no—
222C4×Dic7—22no—
223C4×Dic7—22no—
224C4×Dic7C22⋊C411yescharacteristic
225C4×Dic7C22⋊C411yescharacteristic
226C4×Dic7C22⋊C411yescharacteristic
227C4×Dic7C22⋊C411yescharacteristic
228C4×Dic7C22⋊C411yescharacteristic
229C4×Dic7C22⋊C411yescharacteristic
230C4×Dic7C22×C411yescharacteristic
231C4×Dic7C22×C411yescharacteristic
232C4×Dic7C2×D411yescharacteristic
233C4×Dic7C2×D411yescharacteristic
234C4×Dic7C2×D411yescharacteristic
235C4×Dic7C2×D411yescharacteristic
236C2×C7⋊C8—22no—
237C2×C7⋊C8—22no—
238#1131—44no—
239C8×D7—44no—
240#533—71no—
241#533—71no—
242#533—72no—
243#533—72no—
244#533—71no—
245#533—141no—
246#533—141no—
247#533—141no—
248#533—141no—
249#533C2×D711yescharacteristic
250C2×C4⋊Q8—71no—
251C2×C4⋊Q8—71no—
252C2×#125—72no—
253C2×C22⋊Q8—141no—
254C2×C22⋊Q8—141no—
255C2×C4×Q8—141no—
256C2×C4×Q8—141no—
257C2×#117—72no—
258C22×C42—71no—
259#432—72no—
260#432—141no—
261#431—71no—
262#431—71no—
263#431—72no—
264#431—141no—
265#431—141no—
266#431—141no—
267#431C2×D711yescharacteristic
268#431C2×D711yescharacteristic
269#423—72no—
270C2×C4⋊C8—72no—
271#407—141no—
272C2×C22⋊C8—141no—
273C4×M16—141no—
274C2×C4×C8—141no—
275#332—72no—
276#332—72no—
277#332—72no—
278#332—72no—
279#332—72no—
280#332—72no—
281#332C2×D712yescomplemented
282#332C2×D712yescomplemented
283C23×C7C2×C22⋊C411yescharacteristic
284C22×D7—22no—
285C22×D7—22no—
286C22×D7—41no—
287C22×D7C2×C22⋊C411yescharacteristic
288C22×D7C2×C22⋊C411yescharacteristic
289C7×Q8—42no—
290C7×Q8—42no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7C2×C22⋊C411yescharacteristic
297C2×C4×C7C2×C22⋊C411yescharacteristic
298C2×C4×C7C2×C22⋊C411yescharacteristic
299C2×C4×C7C23×C411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—41no—
307C2×Dic7C2×C22⋊C411yescharacteristic
308C2×Dic7C2×C22⋊C411yescharacteristic
309C2×Dic7C2×C22⋊C411yescharacteristic
310C2×Dic7C2×C22⋊C411yescharacteristic
311C2×Dic7C2×C22⋊C411yescharacteristic
312C2×Dic7C2×C22⋊C411yescharacteristic
313C4×D7—22no—
314C4×D7—22no—
315C4×D7—22no—
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—41no—
322C4×D7—41no—
323C4×D7—44no—
324C4×D7—44no—
325C4×D7#10114yes—
326C4×D7#13514yes—
327Dic14—44no—
328Dic14—42no—
329Dic14—42no—
330Dic14—44no—
331C7×C8—42no—
332C7×C8—42no—
333C7⋊C8—42no—
334C7⋊C8—42no—
335C22×Q8—141no—
336C22×Q8—141no—
337C23×C4—71no—
338C23×C4—71no—
339C23×C4—71no—
340C2×M16—142no—
341C22×C8—142no—
342C4⋊Q8—72no—
343C4⋊Q8—72no—
344C4⋊Q8—71no—
345C4⋊Q8—71no—
346C4⋊Q8—141no—
347C4⋊Q8—141no—
348C4⋊Q8—141no—
349C4⋊Q8—141no—
350C4⋊Q8C22×D711yescharacteristic
351C4⋊Q8C22×D711yescharacteristic
352#125—72no—
353#125—72no—
354#125—72no—
355#125—142no—
356#125—142no—
357#125C4×D712yes—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—142no—
361C22⋊Q8—142no—
362C22⋊Q8—142no—
363C22⋊Q8—142no—
364C22⋊Q8—141no—
365C22⋊Q8—141no—
366C22⋊Q8—141no—
367C22⋊Q8—141no—
368C22⋊Q8—141no—
369C22⋊Q8—141no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—142no—
373C4×Q8—142no—
374C4×Q8—142no—
375C4×Q8—142no—
376C4×Q8—141no—
377C4×Q8—141no—
378C4×Q8—141no—
379C4×Q8—141no—
380C4×Q8—141no—
381C4×Q8—141no—
382#117—72no—
383#117—72no—
384#117—72no—
385#117—142no—
386#117—142no—
387#117C4×D712yes—
388C2×C4⋊C4—72no—
389C2×C4⋊C4—72no—
390C2×C4⋊C4—142no—
391C2×C4⋊C4—142no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—141no—
394C2×C22⋊C4—142no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—71no—
406C2×C42—142no—
407C2×C42C22×D711yescharacteristic
408C4⋊C8—72no—
409C4⋊C8—72no—
410C4⋊C8—142no—
411C4⋊C8—142no—
412C4⋊C8C22×D712yes—
413C4⋊C8C22×D712yes—
414C22⋊C8—21no—
415C22⋊C8—21no—
416C22⋊C8—142no—
417C22⋊C8—142no—
418C22⋊C8—141no—
419C22⋊C8—141no—
420#97—142no—
421C4×C8—21no—
422C4×C8—21no—
423C4×C8—142no—
424C4×C8—141no—
425C4×C8—141no—
426C22×C7—21no—
427C22×C7—21no—
428C22×C7—21no—
429C22×C7C22×C22⋊C411yescharacteristic
430C2×D7—21no—
431C2×D7—21no—
432C2×D7—21no—
433C2×D7—21no—
434C2×D7—44no—
435C2×D7C2×#10112yes—
436C2×D7#41612yes—
437C4×C7—21no—
438C4×C7—21no—
439C4×C7—21no—
440C4×C7—42no—
441C4×C7—42no—
442C4×C7C2×#10112yes—
443C4×C7C2×#13512yes—
444C4×C7#57512yes—
445Dic7—21no—
446Dic7—21no—
447Dic7—21no—
448Dic7—21no—
449Dic7—42no—
450Dic7—42no—
451Dic7C2×#10112yes—
452Dic7#41612yes—
453C24—71no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—142no—
457C2×Q8—142no—
458C2×Q8—141no—
459C2×Q8—141no—
460C2×Q8—284no—
461C2×Q8—284no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—71no—
467C22×C4—71no—
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—144no—
478C22×C4—144no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—282no—
482C22×C4C2×C4×D711yescharacteristic
483C22×C4D4×D711yescharacteristic
484C22×C4D4×D711yescharacteristic
485M16—284no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—142no—
489C2×C8—142no—
490C2×C8—284no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—72no—
496C4⋊C4—72no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4C2×C4×D712yes—
510C4⋊C4C2×C4×D712yes—
511C22⋊C4—22no—
512C22⋊C4—142no—
513C22⋊C4—142no—
514C22⋊C4—142no—
515C42—22no—
516C42—71no—
517C42—71no—
518C42—71no—
519C42—71no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—71no—
525C42—71no—
526C42—71no—
527C42—71no—
528C42—142no—
529C42—142no—
530C42—142no—
531C42C2×C4×D711yescharacteristic
532C42D4×D711yescharacteristic
533C42D4×D711yescharacteristic
534C42C23×D711yescharacteristic
535C2×C7—41no—
536C2×C7C22×#10111yescharacteristic
537C2×C7C2×#41611yescharacteristic
538C2×C7#289611yescharacteristic
539D7—22no—
540D7—22no—
541C23—71no—
542C23—71no—
543C23—142no—
544C23—142no—
545C23—281no—
546C23D7×C22⋊C411yescharacteristic
547Q8—42no—
548Q8—42no—
549Q8—284no—
550Q8—284no—
551Q8—282no—
552Q8—282no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—71no—
559C2×C4—71no—
560C2×C4—74no—
561C2×C4—74no—
562C2×C4—71no—
563C2×C4—71no—
564C2×C4—71no—
565C2×C4—71no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—284no—
581C2×C4—284no—
582C2×C4—281no—
583C2×C4D7×C22⋊C411yescharacteristic
584C2×C4D7×C22⋊C411yescharacteristic
585C2×C4D7×C22⋊C411yescharacteristic
586C2×C4C22×C4×D711yescharacteristic
587C2×C4C2×D4×D711yescharacteristic
588C2×C4C2×D4×D711yescharacteristic
589C8—42no—
590C8—42no—
591C8—282no—
592C8—282no—
593C7#1771511yescharacteristic, complemented, Sylow
594C22—21no—
595C22—21no—
596C22—21no—
597C22—72no—
598C22—72no—
599C22—141no—
600C22—141no—
601C22—141no—
602C22—141no—
603C22—284no—
604C22C2×D7×C22⋊C411yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—72no—
611C4—72no—
612C4—141no—
613C4—141no—
614C4—141no—
615C4—141no—
616C4—282no—
617C4—282no—
618C4D7×#10112yes—
619C4D7×#13512yes—
620C4#9045912yes—
621C2—41no—
622C2—142no—
623C2—142no—
624C2C2×D7×#10111yescharacteristic
625C2D7×#41611yescharacteristic
626C2#1173627511yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1544C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1544D7
hypocenterC7#17715
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
4. C7×#1544C2semidirect
5. #11729583C2semidirect
6. #11728531C2semidirect
7. #11728531C2semidirect
8. #11726179C2semidirect
9. #11726178C2semidirect
10. #11726176C2semidirect
11. #11726174C2semidirect
47. C7×#332C22semidirect
48. C7×#332C22semidirect
49. #89342C22semidirect
50. #89342C22semidirect
51. #89281C22semidirect
52. #89281C22semidirect
53. #89277C22semidirect
54. #89277C22semidirect
157. #1544D7semidirect
281. #332C2×D7semidirect
282. #332C2×D7semidirect
593. C7#17715semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117367272
D7×#4312
C7×#15442
#117295832
#117285312
#117261792
#117261782
#117261762
#117261742
#177157