Finite Group Databasebeta

← #13321898

Subgroups — #13321898

Order 1792.

952 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2#11736776C211yesmaximal, characteristic
3C2×#89477C211yesmaximal, characteristic
4C7×#1531C211yesmaximal, characteristic, complemented
5#11729576C211yesmaximal, characteristic, complemented
6#11728567C211yesmaximal, characteristic, complemented
7#11727555C211yesmaximal, characteristic, complemented
8#11726123C214yesmaximal, complemented
9#11726117C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11#90419C411yescharacteristic
12#90419C411yescharacteristic
13D7×C4⋊Q8C411yescharacteristic
14D7×C4⋊Q8C411yescharacteristic
15#90379C2212yes—
16D7×#125C2212yes—
17D7×C22⋊Q8—21no—
18D7×C22⋊Q8—21no—
19#90313—21no—
20#90313—21no—
21#90269—21no—
22#90269—21no—
23C4×Q8×D7—21no—
24C4×Q8×D7—21no—
25D7×#117C412yes—
26#90208C412yes—
27C2×C42×D7C2211yescharacteristic
28#90161C2211yescharacteristic
29C7×#432C2211yescharacteristic
30C2×C7×#97C2211yescharacteristic
31C2×(C2×D7)⋊C8—21no—
32C2×Dic7⋊C8—21no—
33#89698C2211yescharacteristic
34C2×#6377C2211yescharacteristic
35#89609—21no—
36#89605—21no—
37#89498—21no—
38#89490—21no—
39#89479C2212yes—
40#89477—21no—
41#89477—21no—
42#89477C2212yes—
43#89477C2212yes—
44#89477C2212yes—
45C7×#330C2214yescomplemented
46#89338C2214yescomplemented
47#89260C2214yescomplemented
48#89260C2214yescomplemented
49#17672—71nomaximal, Sylow
50C2×Q8×D7—21no—
51C2×Q8×D7—21no—
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C22×C7×C8—21no—
56C7×C4⋊Q8C2×C411yescharacteristic
57C7×C4⋊Q8C2×C411yescharacteristic
58C7×#125C2312yes—
59C7×C22⋊Q8—21no—
60C7×C22⋊Q8—21no—
61C4×C7×Q8—21no—
62C4×C7×Q8—21no—
63C7×#117C2×C412yes—
64C2×C42×C7C2311yescharacteristic
65#6509—21no—
66#6509—21no—
67Q8×Dic7—21no—
68Q8×Dic7—21no—
69#6507—21no—
70#6507—21no—
71#6489C2×C412yes—
72#6487—21no—
73#6487—21no—
74C2×C4×Dic7C2×C411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C22×C7⋊C8—21no—
78C2×#1131—22no—
79#6460—21no—
80#6460—21no—
81#6459—22no—
82#6459—22no—
83#6455C2×C412yes—
84#6455C2×C412yes—
85D7×C4⋊C4—21no—
86D7×C4⋊C4—21no—
87D7×C4⋊C4—22no—
88D7×C4⋊C4—22no—
89D7×C4⋊C4C2×C412yes—
90D7×C4⋊C4C2×C412yes—
91#6453C2×C412yes—
92#6453C2×C412yes—
93#6452—22no—
94#6452—22no—
95#6451—21no—
96#6451—21no—
97#6451C2×C412yes—
98#6451C2×C412yes—
99#6450—21no—
100#6450—21no—
101#6450—22no—
102#6450—22no—
103D7×C22⋊C4—22no—
104#6441—22no—
105#6441—22no—
106#6440—22no—
107#6435—22no—
108C42×D7—22no—
109C42×D7C2×C411yescharacteristic
110C42×D7C2×C411yescharacteristic
111C42×D7D411yescharacteristic
112C42×D7D411yescharacteristic
113C42×D7D411yescharacteristic
114C42×D7D411yescharacteristic
115C42×D7C2311yescharacteristic
116C42×D7C2311yescharacteristic
117#6433C2312yes—
118#6432C2×C411yescharacteristic
119#6432C2×C411yescharacteristic
120C4×Dic14—21no—
121C4×Dic14—21no—
122C4×Dic14—22no—
123C4×Dic14—22no—
124C7×C4⋊C8—21no—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97C2312yes—
128C7×#97C2312yes—
129C22⋊(C7⋊C8)—21no—
130(C2×D7)⋊C8—22no—
131(C2×D7)⋊C8—22no—
132(C2×D7)⋊C8—22no—
133Dic7⋊C8—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—21no—
138#6377—21no—
139#6377C2312yes—
140#6377C2312yes—
141C2×#533—71no—
142#2920—71no—
143C2×#432—71no—
144#1537—74no—
145#1531—71no—
146#1531—71no—
147#1531—71no—
148#1531D711yescharacteristic, complemented
149#1528—74no—
150C23×D7C22⋊C411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×Dic7C22⋊C411yescharacteristic
157C22×Dic7C22⋊C411yescharacteristic
158C22×Dic7C22⋊C411yescharacteristic
159Q8×D7—44no—
160Q8×D7—44no—
161C2×C4×D7—24no—
162C2×C4×D7—22no—
163C2×C4×D7—22no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7C22⋊C411yescharacteristic
167C2×C4×D7C22⋊C411yescharacteristic
168C2×C4×D7C22⋊C411yescharacteristic
169C2×C4×D7C22⋊C411yescharacteristic
170C2×C4×D7C22⋊C411yescharacteristic
171C2×C4×D7C22⋊C411yescharacteristic
172C2×C4×D7C22×C411yescharacteristic
173C2×C4×D7C22×C411yescharacteristic
174C2×C4×D7C2×D411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—22no—
181C2×Dic14—22no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—42no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4C22×C412yes—
190C7×C4⋊C4C22×C412yes—
191C7×C22⋊C4—22no—
192C42×C7—22no—
193C42×C7C22×C411yescharacteristic
194C42×C7C2×D411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2411yescharacteristic
197C22⋊Dic7—22no—
198(C2×D7)⋊C4—22no—
199(C2×D7)⋊C4—22no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—22no—
203C4⋊Dic7—22no—
204C4⋊Dic7C22×C412yes—
205C4⋊Dic7C22×C412yes—
206Dic7⋊C4—22no—
207Dic7⋊C4—22no—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214C4×Dic7—22no—
215C4×Dic7—22no—
216C4×Dic7—22no—
217C4×Dic7C22⋊C411yescharacteristic
218C4×Dic7C22⋊C411yescharacteristic
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22×C411yescharacteristic
224C4×Dic7C22×C411yescharacteristic
225C4×Dic7C2×D411yescharacteristic
226C4×Dic7C2×D411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C2×C7⋊C8—22no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—42no—
232#1131—44no—
233#533—71no—
234#533—71no—
235#533—72no—
236#533—72no—
237#533—71no—
238#533—141no—
239#533—141no—
240#533—141no—
241#533—141no—
242#533C2×D711yescharacteristic
243C2×C4⋊Q8—71no—
244C2×C4⋊Q8—71no—
245C2×#125—72no—
246C2×C22⋊Q8—141no—
247C2×C22⋊Q8—141no—
248C2×C4×Q8—141no—
249C2×C4×Q8—141no—
250C2×#117—72no—
251C22×C42—71no—
252#432—71no—
253#432—72no—
254#432—72no—
255#432—72no—
256#432—141no—
257#432—141no—
258#432C2×D711yescharacteristic
259#431—141no—
260#431—141no—
261#423—141no—
262C2×C4⋊C8—141no—
263#407—141no—
264C2×C22⋊C8—141no—
265C4×M16—72no—
266C2×#97—71no—
267C2×#97C2×D711yescharacteristic
268#330—74no—
269#330—74no—
270#330—74no—
271#330C2×D714yescomplemented
272C23×C7C2×C22⋊C411yescharacteristic
273C22×D7—22no—
274C22×D7—22no—
275C22×D7—41no—
276C22×D7C2×C22⋊C411yescharacteristic
277C22×D7C2×C22⋊C411yescharacteristic
278C7×Q8—42no—
279C7×Q8—42no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—41no—
285C2×C4×C7C2×C22⋊C411yescharacteristic
286C2×C4×C7C2×C22⋊C411yescharacteristic
287C2×C4×C7C2×C22⋊C411yescharacteristic
288C2×C4×C7C23×C411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7#10114yes—
296C2×Dic7C2×C22⋊C411yescharacteristic
297C2×Dic7C2×C22⋊C411yescharacteristic
298C2×Dic7C2×C22⋊C411yescharacteristic
299C2×Dic7C2×C22⋊C411yescharacteristic
300C2×Dic7C2×C22⋊C411yescharacteristic
301C2×Dic7C2×C22⋊C411yescharacteristic
302C4×D7—22no—
303C4×D7—22no—
304C4×D7—22no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—41no—
311C4×D7—41no—
312C4×D7—41no—
313C4×D7—41no—
314C4×D7—44no—
315C4×D7—44no—
316Dic14—44no—
317Dic14—44no—
318Dic14—42no—
319Dic14—42no—
320C7×C8—42no—
321C7×C8—42no—
322C7⋊C8—42no—
323C7⋊C8—42no—
324C22×Q8—141no—
325C22×Q8—141no—
326C23×C4—71no—
327C23×C4—71no—
328C23×C4—71no—
329C2×M16—142no—
330C22×C8—21no—
331C22×C8—141no—
332C4⋊Q8—72no—
333C4⋊Q8—72no—
334C4⋊Q8—71no—
335C4⋊Q8—71no—
336C4⋊Q8—141no—
337C4⋊Q8—141no—
338C4⋊Q8—141no—
339C4⋊Q8—141no—
340C4⋊Q8C4×D711yescharacteristic
341C4⋊Q8C4×D711yescharacteristic
342#125—72no—
343#125—72no—
344#125—72no—
345#125—142no—
346#125—142no—
347#125C22×D712yes—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—142no—
351C22⋊Q8—142no—
352C22⋊Q8—142no—
353C22⋊Q8—142no—
354C22⋊Q8—141no—
355C22⋊Q8—141no—
356C22⋊Q8—141no—
357C22⋊Q8—141no—
358C22⋊Q8—141no—
359C22⋊Q8—141no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—141no—
365C4×Q8—141no—
366C4×Q8—142no—
367C4×Q8—142no—
368C4×Q8—141no—
369C4×Q8—141no—
370C4×Q8—141no—
371C4×Q8—141no—
372#117—72no—
373#117—72no—
374#117—72no—
375#117—142no—
376#117—142no—
377#117C4×D712yes—
378C2×C4⋊C4—72no—
379C2×C4⋊C4—72no—
380C2×C4⋊C4—142no—
381C2×C4⋊C4—142no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C22⋊C4—142no—
385C2×C42—71no—
386C2×C42—71no—
387C2×C42—71no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—142no—
397C2×C42C22×D711yescharacteristic
398C4⋊C8—21no—
399C4⋊C8—142no—
400C4⋊C8—142no—
401C4⋊C8—142no—
402C4⋊C8—141no—
403C22⋊C8—21no—
404C22⋊C8—142no—
405C22⋊C8—142no—
406C22⋊C8—142no—
407C22⋊C8—141no—
408#97—21no—
409#97—72no—
410#97—72no—
411#97—141no—
412#97C22×D712yes—
413#97C22×D712yes—
414C4×C8—142no—
415C22×C7—21no—
416C22×C7—21no—
417C22×C7C2×#10112yes—
418C22×C7C22×C22⋊C411yescharacteristic
419C2×D7—21no—
420C2×D7—21no—
421C2×D7—21no—
422C2×D7—21no—
423C2×D7—21no—
424C2×D7—44no—
425C2×D7#42112yes—
426C4×C7—21no—
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—21no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7#42112yes—
434Dic7—21no—
435Dic7—21no—
436Dic7—21no—
437Dic7—21no—
438Dic7—42no—
439Dic7—42no—
440Dic7C2×#10112yes—
441Dic7C2×#10112yes—
442C24—71no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—142no—
446C2×Q8—142no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—284no—
450C2×Q8—284no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—71no—
457C22×C4—71no—
458C22×C4—71no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—144no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—282no—
470C22×C4—282no—
471C22×C4C2×C4×D711yescharacteristic
472C22×C4D4×D711yescharacteristic
473C22×C4D4×D711yescharacteristic
474M16—284no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—42no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8—282no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—72no—
486C4⋊C4—72no—
487C4⋊C4—142no—
488C4⋊C4—142no—
489C4⋊C4—142no—
490C4⋊C4—142no—
491C4⋊C4—142no—
492C4⋊C4—142no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4C2×C4×D712yes—
500C4⋊C4C2×C4×D712yes—
501C22⋊C4—22no—
502C22⋊C4—142no—
503C22⋊C4—142no—
504C22⋊C4—142no—
505C42—22no—
506C42—71no—
507C42—71no—
508C42—71no—
509C42—71no—
510C42—71no—
511C42—71no—
512C42—71no—
513C42—71no—
514C42—71no—
515C42—71no—
516C42—71no—
517C42—71no—
518C42—142no—
519C42—142no—
520C42—142no—
521C42C2×C4×D711yescharacteristic
522C42D4×D711yescharacteristic
523C42D4×D711yescharacteristic
524C42C23×D711yescharacteristic
525C2×C7—22no—
526C2×C7C22×#10111yescharacteristic
527C2×C7C2×#42111yescharacteristic
528C2×C7C2×#42111yescharacteristic
529D7—22no—
530D7—41no—
531C23—71no—
532C23—71no—
533C23—142no—
534C23—142no—
535C23—281no—
536C23D7×C22⋊C411yescharacteristic
537Q8—42no—
538Q8—42no—
539Q8—282no—
540Q8—284no—
541Q8—284no—
542Q8—282no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—71no—
549C2×C4—71no—
550C2×C4—74no—
551C2×C4—71no—
552C2×C4—71no—
553C2×C4—71no—
554C2×C4—71no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—284no—
570C2×C4—284no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4D7×C22⋊C411yescharacteristic
574C2×C4D7×C22⋊C411yescharacteristic
575C2×C4D7×C22⋊C411yescharacteristic
576C2×C4C22×C4×D711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4C2×D4×D711yescharacteristic
579C8—42no—
580C8—42no—
581C8—282no—
582C8—282no—
583C7#1767211yescharacteristic, complemented, Sylow
584C22—21no—
585C22—21no—
586C22—72no—
587C22—141no—
588C22—141no—
589C22—141no—
590C22—141no—
591C22—141no—
592C22—284no—
593C22D7×#10112yes—
594C22C2×D7×C22⋊C411yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—42no—
602C4—72no—
603C4—72no—
604C4—141no—
605C4—141no—
606C4—141no—
607C4—141no—
608C4—282no—
609C4—282no—
610C4#8959312yes—
611C2—22no—
612C2—142no—
613C2—281no—
614C2C2×D7×#10111yescharacteristic
615C2C2×#8959311yescharacteristic
616C2D7×#42111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1531C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1531D7
hypocenterC7#17672
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
4. C7×#1531C2semidirect
5. #11729576C2semidirect
6. #11728567C2semidirect
7. #11727555C2semidirect
8. #11726123C2semidirect
9. #11726117C2semidirect
45. C7×#330C22semidirect
46. #89338C22semidirect
47. #89260C22semidirect
48. #89260C22semidirect
148. #1531D7semidirect
271. #330C2×D7semidirect
583. C7#17672semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117367762
C2×#894772
C7×#15312
#117295762
#117285672
#117275552
#117261232
#117261172
#176727