Finite Group Databasebeta

← #13320441

Subgroups — #13320441

Order 1792.

988 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2#11736483C211yesmaximal, characteristic
3C2×#89457C211yesmaximal, characteristic
4C7×#1493C211yesmaximal, characteristic, complemented
5#11729580C211yesmaximal, characteristic, complemented
6#11728255C211yesmaximal, characteristic, complemented
7#11727492C211yesmaximal, characteristic, complemented
8#11725997C214yesmaximal, complemented
9#11725985C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11#90419C411yescharacteristic
12#90419C411yescharacteristic
13D7×C4⋊Q8C411yescharacteristic
14D7×C4⋊Q8C411yescharacteristic
15#90379—21no—
16D7×#125—21no—
17D7×C22⋊Q8—21no—
18D7×C22⋊Q8C412yes—
19#90313—21no—
20#90313C412yes—
21#90269—21no—
22#90269C2212yes—
23C4×Q8×D7—21no—
24C4×Q8×D7C2212yes—
25D7×#117—21no—
26#90208—21no—
27C2×C42×D7C2211yescharacteristic
28#90161C2211yescharacteristic
29C7×#423C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31#89898—21no—
32M16×Dic7—21no—
33C2×(C2×D7)⋊C8—21no—
34C2×Dic7⋊C8—21no—
35#89696C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89607—21no—
38#89607C2212yes—
39#89605—21no—
40#89605C2212yes—
41#89457—21no—
42#89457—21no—
43#89457C2212yes—
44#89457C2212yes—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17479—71nomaximal, Sylow
50C2×Q8×D7—21no—
51C2×Q8×D7C2×C412yes—
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C2×C7×M16—21no—
56C22×C7×C8—21no—
57C7×C4⋊Q8C2×C411yescharacteristic
58C7×C4⋊Q8C2×C411yescharacteristic
59C7×#125—21no—
60C7×C22⋊Q8—21no—
61C7×C22⋊Q8C2×C412yes—
62C4×C7×Q8—21no—
63C4×C7×Q8C2312yes—
64C7×#117—21no—
65C2×C42×C7C2311yescharacteristic
66#6509—21no—
67#6509C2×C412yes—
68Q8×Dic7—21no—
69Q8×Dic7C2×C412yes—
70#6507—21no—
71#6507C2×C412yes—
72#6489—21no—
73#6487—21no—
74#6487C2×C412yes—
75C2×C4×Dic7C2×C411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×C4×Dic7D411yescharacteristic
78C2×#1136—21no—
79C22×C7⋊C8—21no—
80#6460—21no—
81#6460C2×C412yes—
82#6459—22no—
83#6459—22no—
84#6455—21no—
85#6455—21no—
86D7×C4⋊C4—21no—
87D7×C4⋊C4—21no—
88D7×C4⋊C4—22no—
89D7×C4⋊C4—21no—
90D7×C4⋊C4—22no—
91D7×C4⋊C4C2×C412yes—
92#6453—21no—
93#6453—21no—
94#6452—22no—
95#6452—22no—
96#6451—21no—
97#6451—21no—
98#6451—21no—
99#6451C2×C412yes—
100#6450—22no—
101#6450—21no—
102#6450—22no—
103#6450C2×C412yes—
104D7×C22⋊C4—22no—
105#6441—22no—
106#6441—22no—
107#6440—22no—
108#6435—22no—
109C42×D7—22no—
110C42×D7C2×C411yescharacteristic
111C42×D7C2×C411yescharacteristic
112C42×D7D411yescharacteristic
113C42×D7D411yescharacteristic
114C42×D7D411yescharacteristic
115C42×D7D411yescharacteristic
116C42×D7C2311yescharacteristic
117C42×D7C2311yescharacteristic
118#6433—21no—
119#6432C2×C411yescharacteristic
120#6432C2×C411yescharacteristic
121C4×Dic14—21no—
122C4×Dic14—22no—
123C4×Dic14—22no—
124C4×Dic14C2312yes—
125C7×C4⋊C8—21no—
126C7×C4⋊C8C2312yes—
127C4×C7×C8—21no—
128C4×C7×C8C2312yes—
129(C2×D7)⋊C8—22no—
130(C2×D7)⋊C8—22no—
131(C2×D7)⋊C8—22no—
132(C2×D7)⋊C8—22no—
133#6389—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)C2312yes—
139C4×C7⋊C8—21no—
140C4×C7⋊C8C2312yes—
141C2×#533—71no—
142#2920—71no—
143C2×#431—71no—
144#1497—74no—
145#1493—71no—
146#1493—71no—
147#1493—71no—
148#1493D711yescharacteristic, complemented
149#1479—74no—
150C23×D7C22⋊C411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8C22×C412yes—
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×Dic7C22⋊C411yescharacteristic
157C22×Dic7C22⋊C411yescharacteristic
158C22×Dic7C22⋊C411yescharacteristic
159Q8×D7—28no—
160Q8×D7—44no—
161C2×C4×D7—22no—
162C2×C4×D7—22no—
163C2×C4×D7—24no—
164C2×C4×D7—22no—
165C2×C4×D7—22no—
166C2×C4×D7—42no—
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×D7C22×C411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7C2×D411yescharacteristic
179C2×Dic14—21no—
180C2×Dic14—22no—
181C2×Dic14—22no—
182C2×Dic14C22×C412yes—
183C7×M16—42no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—42no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4C22×C412yes—
193C7×C22⋊C4—22no—
194C42×C7—22no—
195C42×C7C22×C411yescharacteristic
196C42×C7C2×D411yescharacteristic
197C42×C7C2×D411yescharacteristic
198C42×C7C2411yescharacteristic
199C22⋊Dic7—22no—
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4—22no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7C22×C412yes—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216C4×Dic7—22no—
217C4×Dic7—22no—
218C4×Dic7—22no—
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22⋊C411yescharacteristic
225C4×Dic7C22×C411yescharacteristic
226C4×Dic7C22×C411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230C4×Dic7C2×D411yescharacteristic
231#1136—42no—
232C2×C7⋊C8—22no—
233C2×C7⋊C8—22no—
234C2×C7⋊C8—42no—
235#533—71no—
236#533—71no—
237#533—72no—
238#533—72no—
239#533—71no—
240#533—141no—
241#533—141no—
242#533—141no—
243#533—141no—
244#533C2×D711yescharacteristic
245C2×C4⋊Q8—71no—
246C2×C4⋊Q8—71no—
247C2×#125—141no—
248C2×C22⋊Q8—72no—
249C2×C22⋊Q8—141no—
250C2×C4×Q8—72no—
251C2×C4×Q8—141no—
252C2×#117—141no—
253C22×C42—71no—
254#432—72no—
255#432—141no—
256#431—72no—
257#431—72no—
258#431—72no—
259#431—141no—
260#431—141no—
261#431—141no—
262#423—71no—
263#423C2×D711yescharacteristic
264C2×C4⋊C8—141no—
265#407—141no—
266C2×C22⋊C8—141no—
267C4×M16—141no—
268C2×C4×C8—71no—
269C2×C4×C8C2×D711yescharacteristic
270Q8⋊C8—74no—
271Q8⋊C8—74no—
272Q8⋊C8—74no—
273Q8⋊C8C2×D714yescomplemented
274C23×C7C2×C22⋊C411yescharacteristic
275C22×D7—22no—
276C22×D7—22no—
277C22×D7—41no—
278C22×D7C2×C22⋊C411yescharacteristic
279C22×D7C2×C22⋊C411yescharacteristic
280C7×Q8—24no—
281C7×Q8—42no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7C2×C22⋊C411yescharacteristic
287C2×C4×C7C2×C22⋊C411yescharacteristic
288C2×C4×C7C2×C22⋊C411yescharacteristic
289C2×C4×C7#13712yes—
290C2×C4×C7#13712yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—41no—
299C2×Dic7C2×C22⋊C411yescharacteristic
300C2×Dic7C2×C22⋊C411yescharacteristic
301C2×Dic7C2×C22⋊C411yescharacteristic
302C2×Dic7C2×C22⋊C411yescharacteristic
303C2×Dic7C2×C22⋊C411yescharacteristic
304C2×Dic7C2×C22⋊C411yescharacteristic
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—21no—
308C4×D7—21no—
309C4×D7—21no—
310C4×D7—21no—
311C4×D7—22no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—22no—
315C4×D7—22no—
316C4×D7—41no—
317C4×D7—41no—
318C4×D7—44no—
319C4×D7—44no—
320C4×D7#13114yes—
321Dic14—24no—
322Dic14—44no—
323Dic14—42no—
324Dic14—44no—
325C7×C8—42no—
326C7×C8—42no—
327C7⋊C8—42no—
328C7⋊C8—42no—
329C22×Q8—72no—
330C22×Q8—141no—
331C23×C4—71no—
332C23×C4—71no—
333C23×C4—71no—
334C2×M16—21no—
335C2×M16—141no—
336C22×C8—21no—
337C22×C8—141no—
338C4⋊Q8—72no—
339C4⋊Q8—72no—
340C4⋊Q8—71no—
341C4⋊Q8—71no—
342C4⋊Q8—141no—
343C4⋊Q8—141no—
344C4⋊Q8—141no—
345C4⋊Q8—141no—
346C4⋊Q8C4×D711yescharacteristic
347C4⋊Q8C4×D711yescharacteristic
348#125—21no—
349#125—142no—
350#125—142no—
351#125—141no—
352#125—141no—
353#125—141no—
354C22⋊Q8—21no—
355C22⋊Q8—72no—
356C22⋊Q8—72no—
357C22⋊Q8—72no—
358C22⋊Q8—142no—
359C22⋊Q8—142no—
360C22⋊Q8—142no—
361C22⋊Q8—142no—
362C22⋊Q8—141no—
363C22⋊Q8—141no—
364C22⋊Q8—141no—
365C22⋊Q8C4×D712yes—
366C4×Q8—21no—
367C4×Q8—72no—
368C4×Q8—72no—
369C4×Q8—72no—
370C4×Q8—142no—
371C4×Q8—142no—
372C4×Q8—142no—
373C4×Q8—141no—
374C4×Q8—142no—
375C4×Q8—141no—
376C4×Q8—141no—
377C4×Q8C22×D712yes—
378#117—21no—
379#117—142no—
380#117—142no—
381#117—141no—
382#117—141no—
383#117—141no—
384C2×C4⋊C4—72no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4—142no—
389C2×C4⋊C4—141no—
390C2×C22⋊C4—142no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—142no—
403C2×C42C22×D711yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—72no—
406C4⋊C8—142no—
407C4⋊C8—142no—
408C4⋊C8—141no—
409C4⋊C8C22×D712yes—
410C22⋊C8—142no—
411C22⋊C8—142no—
412C22⋊C8—142no—
413C22⋊C8—142no—
414#97—142no—
415C4×C8—21no—
416C4×C8—72no—
417C4×C8—142no—
418C4×C8—141no—
419C4×C8C22×D712yes—
420C22×C7—21no—
421C22×C7—21no—
422C22×C7#41912yes—
423C22×C7C22×C22⋊C411yescharacteristic
424C2×D7—21no—
425C2×D7—21no—
426C2×D7—21no—
427C2×D7—21no—
428C2×D7—21no—
429C2×D7—44no—
430C2×D7#40812yes—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7#41912yes—
436C4×C7#42112yes—
437C4×C7C2×#13112yes—
438C4×C7C2×#13711yescharacteristic
439C4×C7C2×#13711yescharacteristic
440Dic7—21no—
441Dic7—21no—
442Dic7—21no—
443Dic7—21no—
444Dic7—21no—
445Dic7—42no—
446Dic7—42no—
447Dic7#40812yes—
448C24—71no—
449C2×Q8—21no—
450C2×Q8—72no—
451C2×Q8—142no—
452C2×Q8—142no—
453C2×Q8—148no—
454C2×Q8—141no—
455C2×Q8—284no—
456C2×Q8C2×C4×D712yes—
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—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—144no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—282no—
478C22×C4C2×C4×D711yescharacteristic
479C22×C4D4×D711yescharacteristic
480C22×C4D4×D711yescharacteristic
481M16—42no—
482M16—282no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—42no—
486C2×C8—142no—
487C2×C8—142no—
488C2×C8—282no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—72no—
495C4⋊C4—142no—
496C4⋊C4—142no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—142no—
508C4⋊C4C2×C4×D712yes—
509C22⋊C4—22no—
510C22⋊C4—142no—
511C22⋊C4—142no—
512C22⋊C4—142no—
513C42—22no—
514C42—71no—
515C42—71no—
516C42—71no—
517C42—71no—
518C42—71no—
519C42—71no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—71no—
525C42—71no—
526C42—142no—
527C42—142no—
528C42—142no—
529C42C2×C4×D711yescharacteristic
530C42D4×D711yescharacteristic
531C42D4×D711yescharacteristic
532C42C23×D711yescharacteristic
533C2×C7—22no—
534C2×C7C2×#40811yescharacteristic
535C2×C7C2×#41911yescharacteristic
536C2×C7C2×#42111yescharacteristic
537D7—22no—
538D7—41no—
539C23—71no—
540C23—71no—
541C23—142no—
542C23—142no—
543C23—281no—
544C23D7×C22⋊C411yescharacteristic
545Q8—24no—
546Q8—42no—
547Q8—144no—
548Q8—284no—
549Q8—284no—
550Q8—282no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—71no—
556C2×C4—74no—
557C2×C4—71no—
558C2×C4—71no—
559C2×C4—71no—
560C2×C4—71no—
561C2×C4—71no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—141no—
568C2×C4—141no—
569C2×C4—141no—
570C2×C4—141no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—284no—
580C2×C4—284no—
581C2×C4—281no—
582C2×C4D7×C22⋊C411yescharacteristic
583C2×C4D7×C22⋊C411yescharacteristic
584C2×C4D7×C22⋊C411yescharacteristic
585C2×C4#647812yes—
586C2×C4#648112yes—
587C2×C4C22×C4×D711yescharacteristic
588C2×C4C2×D4×D711yescharacteristic
589C2×C4C2×D4×D711yescharacteristic
590C8—42no—
591C8—42no—
592C8—282no—
593C8—282no—
594C7#1747911yescharacteristic, complemented, Sylow
595C22—21no—
596C22—21no—
597C22—72no—
598C22—141no—
599C22—141no—
600C22—141no—
601C22—141no—
602C22—141no—
603C22—284no—
604C22#8957412yes—
605C22C2×D7×C22⋊C411yescharacteristic
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—72no—
611C4—141no—
612C4—141no—
613C4—141no—
614C4—141no—
615C4—141no—
616C4—282no—
617C4—282no—
618C4#8957412yes—
619C4#8959312yes—
620C4D7×#13112yes—
621C4C2×#647811yescharacteristic
622C4C2×#648111yescharacteristic
623C2—22no—
624C2—142no—
625C2—281no—
626C2D7×#40811yescharacteristic
627C2C2×#8957411yescharacteristic
628C2C2×#8959311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1493C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1493D7
hypocenterC7#17479
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
4. C7×#1493C2semidirect
5. #11729580C2semidirect
6. #11728255C2semidirect
7. #11727492C2semidirect
8. #11725997C2semidirect
9. #11725985C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
148. #1493D7semidirect
273. Q8⋊C8C2×D7semidirect
594. C7#17479semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117364832
C2×#894572
C7×#14932
#117295802
#117282552
#117274922
#117259972
#117259852
#174797