Finite Group Databasebeta

← #13322199

Subgroups — #13322199

Order 1792.

972 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
1D7×#533C211yesmaximal, characteristic
2#11736776C211yesmaximal, characteristic
3C2×#89605C211yesmaximal, characteristic
4C7×#1552C211yesmaximal, characteristic, complemented
5#11729585C211yesmaximal, characteristic, complemented
6#11728574C211yesmaximal, characteristic, complemented
7#11728193C211yesmaximal, characteristic, complemented
8#11726176C212yesmaximal, complemented
9#11726174C212yesmaximal, complemented
10#11726173C212yesmaximal, complemented
11#11726173C212yesmaximal, 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×#432C2211yescharacteristic
32C2×C7×C4⋊C8C2211yescharacteristic
33C2×(C2×D7)⋊C8—21no—
34C2×C8×Dic7—21no—
35#89698C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89609C2212yes—
38#89605—21no—
39#89605—21no—
40#89605C2212yes—
41#89605C2212yes—
42#89605C2212yes—
43#89498—21no—
44#89490—21no—
45#89479—21no—
46#89477—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
61C22×C7×C8—21no—
62C7×C4⋊Q8C2311yescharacteristic
63C7×C4⋊Q8C2311yescharacteristic
64C7×#125C2×C412yes—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8—21no—
67C4×C7×Q8—21no—
68C4×C7×Q8—21no—
69C7×#117C2×C412yes—
70C2×C42×C7C2311yescharacteristic
71#6509—21no—
72#6509—21no—
73Q8×Dic7—21no—
74Q8×Dic7—21no—
75#6507—21no—
76#6507—21no—
77#6489C2×C412yes—
78#6487—21no—
79#6487—21no—
80C2×C4×Dic7C2×C411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7D411yescharacteristic
83C22×C7⋊C8—21no—
84C2×#1131—22no—
85#6460—21no—
86#6460—21no—
87#6459—22no—
88#6459—22no—
89#6455C2×C412yes—
90#6455C2×C412yes—
91D7×C4⋊C4—22no—
92D7×C4⋊C4—22no—
93D7×C4⋊C4—21no—
94D7×C4⋊C4—21no—
95D7×C4⋊C4C2×C412yes—
96D7×C4⋊C4C2×C412yes—
97#6453C2×C412yes—
98#6453C2×C412yes—
99#6452—22no—
100#6452—22no—
101#6451—21no—
102#6451—21no—
103#6451C2×C412yes—
104#6451C2×C412yes—
105#6450—22no—
106#6450—22no—
107#6450—21no—
108#6450—21no—
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#6433C2×C412yes—
124#6432C2311yescharacteristic
125#6432C2311yescharacteristic
126C4×Dic14—21no—
127C4×Dic14—21no—
128C4×Dic14—22no—
129C4×Dic14—22no—
130C7×C4⋊C8—21no—
131C7×C4⋊C8C2312yes—
132C7×C4⋊C8C2312yes—
133C7×C22⋊C8—21no—
134C7×#97—21no—
135C22⋊(C7⋊C8)—21no—
136(C2×D7)⋊C8—22no—
137(C2×D7)⋊C8—22no—
138(C2×D7)⋊C8—22no—
139Dic7⋊C8—22no—
140C8×Dic7—22no—
141C8×Dic7—22no—
142C8×Dic7—22no—
143C4⋊(C7⋊C8)—21no—
144C4⋊(C7⋊C8)C2312yes—
145C4⋊(C7⋊C8)C2312yes—
146#6377—21no—
147C2×#533—71no—
148#2920—71no—
149C2×#431—71no—
150#1552—71no—
151#1552—71no—
152#1552—71no—
153#1552D711yescharacteristic, complemented
154#1549—72no—
155#1546—72no—
156#1544—72no—
157#1544—72no—
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—42no—
168Q8×D7—42no—
169Q8×D7—44no—
170C2×C4×D7—24no—
171C2×C4×D7—24no—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—41no—
175C2×C4×D7—41no—
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×D7C22×C411yescharacteristic
184C2×C4×D7C2×D411yescharacteristic
185C2×C4×D7C2×D411yescharacteristic
186C2×C4×D7C2×D411yescharacteristic
187C2×C4×D7C2×D411yescharacteristic
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—22no—
191C2×Dic14—22no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C2×C7×C8—42no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4C22×C412yes—
200C7×C4⋊C4C22×C412yes—
201C7×C22⋊C4—22no—
202C42×C7—22no—
203C42×C7C22×C411yescharacteristic
204C42×C7C2×D411yescharacteristic
205C42×C7C2×D411yescharacteristic
206C42×C7C2411yescharacteristic
207C22⋊Dic7—22no—
208(C2×D7)⋊C4—22no—
209(C2×D7)⋊C4—22no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—22no—
213C4⋊Dic7—22no—
214C4⋊Dic7C22×C412yes—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4—22no—
220Dic7⋊C4—22no—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224C4×Dic7—22no—
225C4×Dic7—22no—
226C4×Dic7—22no—
227C4×Dic7C22⋊C411yescharacteristic
228C4×Dic7C22⋊C411yescharacteristic
229C4×Dic7C22⋊C411yescharacteristic
230C4×Dic7C22⋊C411yescharacteristic
231C4×Dic7C22⋊C411yescharacteristic
232C4×Dic7C22⋊C411yescharacteristic
233C4×Dic7C22×C411yescharacteristic
234C4×Dic7C22×C411yescharacteristic
235C4×Dic7C2×D411yescharacteristic
236C4×Dic7C2×D411yescharacteristic
237C4×Dic7C2×D411yescharacteristic
238C4×Dic7C2×D411yescharacteristic
239C2×C7⋊C8—22no—
240C2×C7⋊C8—22no—
241C2×C7⋊C8—42no—
242#1131—44no—
243#533—71no—
244#533—71no—
245#533—72no—
246#533—72no—
247#533—71no—
248#533—141no—
249#533—141no—
250#533—141no—
251#533—141no—
252#533C2×D711yescharacteristic
253C2×C4⋊Q8—71no—
254C2×C4⋊Q8—71no—
255C2×#125—72no—
256C2×C22⋊Q8—141no—
257C2×C22⋊Q8—141no—
258C2×C4×Q8—141no—
259C2×C4×Q8—141no—
260C2×#117—72no—
261C22×C42—71no—
262#432—71no—
263#432—141no—
264#432C2×D711yescharacteristic
265#431—72no—
266#431—72no—
267#431—72no—
268#431—141no—
269#431—141no—
270#431—141no—
271#423—72no—
272C2×C4⋊C8—71no—
273C2×C4⋊C8C2×D711yescharacteristic
274#407—141no—
275C2×C22⋊C8—141no—
276C4×M16—141no—
277C2×C4×C8—141no—
278#332—72no—
279#332—72no—
280#332—72no—
281#332—72no—
282#332—72no—
283#332—72no—
284#332C2×D712yescomplemented
285#332C2×D712yescomplemented
286C23×C7C2×C22⋊C411yescharacteristic
287C22×D7—22no—
288C22×D7—22no—
289C22×D7—41no—
290C22×D7C2×C22⋊C411yescharacteristic
291C22×D7C2×C22⋊C411yescharacteristic
292C7×Q8—42no—
293C7×Q8—41no—
294C7×Q8—41no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7C2×C22⋊C411yescharacteristic
302C2×C4×C7C2×C22⋊C411yescharacteristic
303C2×C4×C7C2×C22⋊C411yescharacteristic
304C2×C4×C7C23×C411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—41no—
311C2×Dic7#10114yes—
312C2×Dic7C2×C22⋊C411yescharacteristic
313C2×Dic7C2×C22⋊C411yescharacteristic
314C2×Dic7C2×C22⋊C411yescharacteristic
315C2×Dic7C2×C22⋊C411yescharacteristic
316C2×Dic7C2×C22⋊C411yescharacteristic
317C2×Dic7C2×C22⋊C411yescharacteristic
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—21no—
321C4×D7—21no—
322C4×D7—21no—
323C4×D7—21no—
324C4×D7—22no—
325C4×D7—22no—
326C4×D7—22no—
327C4×D7—22no—
328C4×D7—22no—
329C4×D7—41no—
330C4×D7—41no—
331C4×D7—44no—
332C4×D7—44no—
333C4×D7#13514yes—
334Dic14—44no—
335Dic14—42no—
336Dic14—41no—
337Dic14—41no—
338Dic14—44no—
339C7×C8—42no—
340C7×C8—42no—
341C7⋊C8—42no—
342C7⋊C8—42no—
343C22×Q8—141no—
344C22×Q8—141no—
345C23×C4—71no—
346C23×C4—71no—
347C23×C4—71no—
348C2×M16—142no—
349C22×C8—21no—
350C22×C8—141no—
351C4⋊Q8—72no—
352C4⋊Q8—72no—
353C4⋊Q8—71no—
354C4⋊Q8—71no—
355C4⋊Q8—141no—
356C4⋊Q8—141no—
357C4⋊Q8—141no—
358C4⋊Q8—141no—
359C4⋊Q8C22×D711yescharacteristic
360C4⋊Q8C22×D711yescharacteristic
361#125—72no—
362#125—72no—
363#125—72no—
364#125—142no—
365#125—142no—
366#125C4×D712yes—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—142no—
370C22⋊Q8—142no—
371C22⋊Q8—142no—
372C22⋊Q8—142no—
373C22⋊Q8—141no—
374C22⋊Q8—141no—
375C22⋊Q8—141no—
376C22⋊Q8—141no—
377C22⋊Q8—141no—
378C22⋊Q8—141no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—142no—
382C4×Q8—142no—
383C4×Q8—142no—
384C4×Q8—142no—
385C4×Q8—141no—
386C4×Q8—141no—
387C4×Q8—141no—
388C4×Q8—141no—
389C4×Q8—141no—
390C4×Q8—141no—
391#117—72no—
392#117—72no—
393#117—72no—
394#117—142no—
395#117—142no—
396#117C4×D712yes—
397C2×C4⋊C4—72no—
398C2×C4⋊C4—72no—
399C2×C4⋊C4—142no—
400C2×C4⋊C4—142no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C22⋊C4—142no—
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—71no—
413C2×C42—71no—
414C2×C42—71no—
415C2×C42—142no—
416C2×C42C22×D711yescharacteristic
417C4⋊C8—21no—
418C4⋊C8—72no—
419C4⋊C8—72no—
420C4⋊C8—142no—
421C4⋊C8—141no—
422C4⋊C8C22×D712yes—
423C4⋊C8C22×D712yes—
424C22⋊C8—21no—
425C22⋊C8—142no—
426C22⋊C8—142no—
427C22⋊C8—142no—
428C22⋊C8—141no—
429#97—21no—
430#97—141no—
431C4×C8—142no—
432C4×C8—142no—
433C4×C8—142no—
434C22×C7—21no—
435C22×C7—21no—
436C22×C7C2×#10112yes—
437C22×C7C22×C22⋊C411yescharacteristic
438C2×D7—21no—
439C2×D7—21no—
440C2×D7—21no—
441C2×D7—21no—
442C2×D7—21no—
443C2×D7—44no—
444C2×D7#41612yes—
445C4×C7—21no—
446C4×C7—21no—
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—42no—
450C4×C7—42no—
451C4×C7C2×#13512yes—
452C4×C7#57511yescharacteristic
453C4×C7#57511yescharacteristic
454Dic7—21no—
455Dic7—21no—
456Dic7—21no—
457Dic7—42no—
458Dic7—42no—
459Dic7C2×#10112yes—
460Dic7C2×#10112yes—
461Dic7#41612yes—
462C24—71no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—142no—
466C2×Q8—142no—
467C2×Q8—141no—
468C2×Q8—141no—
469C2×Q8—284no—
470C2×Q8—282no—
471C2×Q8—282no—
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—71no—
484C22×C4—71no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—144no—
488C22×C4—144no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—281no—
492C22×C4—281no—
493C22×C4C2×C4×D711yescharacteristic
494C22×C4D4×D711yescharacteristic
495C22×C4D4×D711yescharacteristic
496M16—284no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—42no—
500C2×C8—142no—
501C2×C8—142no—
502C2×C8—282no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—72no—
508C4⋊C4—72no—
509C4⋊C4—142no—
510C4⋊C4—142no—
511C4⋊C4—142no—
512C4⋊C4—142no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4—142no—
516C4⋊C4—142no—
517C4⋊C4—142no—
518C4⋊C4—142no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4C2×C4×D712yes—
522C4⋊C4C2×C4×D712yes—
523C22⋊C4—22no—
524C22⋊C4—142no—
525C22⋊C4—142no—
526C22⋊C4—142no—
527C42—22no—
528C42—71no—
529C42—71no—
530C42—71no—
531C42—71no—
532C42—71no—
533C42—71no—
534C42—71no—
535C42—71no—
536C42—71no—
537C42—71no—
538C42—71no—
539C42—71no—
540C42—142no—
541C42—142no—
542C42—142no—
543C42C2×C4×D711yescharacteristic
544C42D4×D711yescharacteristic
545C42D4×D711yescharacteristic
546C42C23×D711yescharacteristic
547C2×C7—22no—
548C2×C7C22×#10111yescharacteristic
549C2×C7C2×#41611yescharacteristic
550C2×C7#289611yescharacteristic
551D7—22no—
552D7—41no—
553C23—71no—
554C23—71no—
555C23—142no—
556C23—142no—
557C23—281no—
558C23D7×C22⋊C411yescharacteristic
559Q8—42no—
560Q8—41no—
561Q8—41no—
562Q8—284no—
563Q8—284no—
564Q8—282no—
565Q8—281no—
566Q8—281no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—71no—
574C2×C4—71no—
575C2×C4—71no—
576C2×C4—74no—
577C2×C4—74no—
578C2×C4—71no—
579C2×C4—71no—
580C2×C4—71no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—142no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—284no—
598C2×C4—284no—
599C2×C4—281no—
600C2×C4D7×C22⋊C411yescharacteristic
601C2×C4D7×C22⋊C411yescharacteristic
602C2×C4D7×C22⋊C411yescharacteristic
603C2×C4C22×C4×D711yescharacteristic
604C2×C4C2×D4×D711yescharacteristic
605C2×C4C2×D4×D711yescharacteristic
606C8—42no—
607C8—42no—
608C8—282no—
609C8—282no—
610C7#1771511yescharacteristic, complemented, Sylow
611C22—21no—
612C22—21no—
613C22—72no—
614C22—141no—
615C22—141no—
616C22—141no—
617C22—141no—
618C22—141no—
619C22—284no—
620C22D7×#10112yes—
621C22C2×D7×C22⋊C411yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—42no—
627C4—42no—
628C4—72no—
629C4—72no—
630C4—72no—
631C4—141no—
632C4—141no—
633C4—141no—
634C4—282no—
635C4—282no—
636C4D7×#13512yes—
637C4#9046411yescharacteristic
638C4#9046811yescharacteristic
639C2—22no—
640C2—142no—
641C2—281no—
642C2C2×D7×#10111yescharacteristic
643C2D7×#41611yescharacteristic
644C2#1173632611yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1552C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1552D7
hypocenterC7#17715
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
4. C7×#1552C2semidirect
5. #11729585C2semidirect
6. #11728574C2semidirect
7. #11728193C2semidirect
8. #11726176C2semidirect
9. #11726174C2semidirect
10. #11726173C2semidirect
11. #11726173C2semidirect
47. C7×#332C22semidirect
48. C7×#332C22semidirect
49. #89342C22semidirect
50. #89342C22semidirect
51. #89281C22semidirect
52. #89281C22semidirect
53. #89277C22semidirect
54. #89277C22semidirect
153. #1552D7semidirect
284. #332C2×D7semidirect
285. #332C2×D7semidirect
610. C7#17715semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117367762
C2×#896052
C7×#15522
#117295852
#117285742
#117281932
#117261762
#117261742
#117261732
#177157