Finite Group Databasebeta

← #13320340

Subgroups — #13320340

Order 1792.

988 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
3C2×#89605C211yesmaximal, characteristic
4C7×#1481C211yesmaximal, characteristic, complemented
5#11729578C211yesmaximal, characteristic, complemented
6#11728522C211yesmaximal, characteristic, complemented
7#11728183C211yesmaximal, characteristic, complemented
8#11725989C214yesmaximal, 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×#431C2211yescharacteristic
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×(C2×D7)⋊C8—21no—
32C2×C8×Dic7—21no—
33#89697C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35#89609—21no—
36#89607—21no—
37#89605—21no—
38#89605—21no—
39#89605C2212yes—
40#89605C2212yes—
41#89498—21no—
42#89491—21no—
43C4×#1131C2212yes—
44#89457C2212yes—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17464—71nomaximal, Sylow
50C2×Q8×D7—21no—
51C2×Q8×D7C2×C412yes—
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C22×C7×C8—21no—
56C7×C4⋊Q8C2×C411yescharacteristic
57C7×C4⋊Q8C2×C411yescharacteristic
58C7×#125—21no—
59C7×C22⋊Q8—21no—
60C7×C22⋊Q8C2×C412yes—
61C4×C7×Q8—21no—
62C4×C7×Q8C2312yes—
63C7×#117—21no—
64C2×C42×C7C2311yescharacteristic
65#6509—21no—
66#6509C2×C412yes—
67Q8×Dic7—21no—
68Q8×Dic7C2×C412yes—
69#6507—21no—
70#6507C2×C412yes—
71#6489—21no—
72#6487—21no—
73#6487C2×C412yes—
74C2×C4×Dic7C2×C411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C22×C7⋊C8—21no—
78C2×#1131—22no—
79#6460—21no—
80#6460C2×C412yes—
81#6459—22no—
82#6459—22no—
83#6455—21no—
84#6455—21no—
85D7×C4⋊C4—21no—
86D7×C4⋊C4—21no—
87D7×C4⋊C4—22no—
88D7×C4⋊C4—21no—
89D7×C4⋊C4—22no—
90D7×C4⋊C4C2×C412yes—
91#6453—21no—
92#6453—21no—
93#6452—22no—
94#6452—22no—
95#6451—21no—
96#6451—21no—
97#6451—21no—
98#6451C2×C412yes—
99#6450—22no—
100#6450—21no—
101#6450—22no—
102#6450C2×C412yes—
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#6433—21no—
118#6432C2×C411yescharacteristic
119#6432C2×C411yescharacteristic
120C4×Dic14—21no—
121C4×Dic14—22no—
122C4×Dic14—22no—
123C4×Dic14C2312yes—
124C7×C4⋊C8—21no—
125C7×C4⋊C8—21no—
126C7×C4⋊C8C2312yes—
127C7×C22⋊C8—21no—
128C4×C7×C8C2312yes—
129C22⋊(C7⋊C8)—21no—
130(C2×D7)⋊C8—22no—
131(C2×D7)⋊C8—22no—
132(C2×D7)⋊C8—22no—
133#6389—22no—
134Dic7⋊C8—22no—
135C8×Dic7—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2312yes—
140C4×C7⋊C8C2312yes—
141C2×#533—71no—
142#2920—71no—
143C2×#431—71no—
144#1485—74no—
145#1481—71no—
146#1481—71no—
147#1481—71no—
148#1481D711yescharacteristic, 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—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—42no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4C22×C412yes—
192C7×C22⋊C4—22no—
193C42×C7—22no—
194C42×C7C22×C411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2×D411yescharacteristic
197C42×C7C2411yescharacteristic
198C22⋊Dic7—22no—
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4—22no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—22no—
205C4⋊Dic7—22no—
206C4⋊Dic7C22×C412yes—
207Dic7⋊C4—22no—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215C4×Dic7—22no—
216C4×Dic7—22no—
217C4×Dic7—22no—
218C4×Dic7C22⋊C411yescharacteristic
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22×C411yescharacteristic
225C4×Dic7C22×C411yescharacteristic
226C4×Dic7C2×D411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230C2×C7⋊C8—22no—
231C2×C7⋊C8—22no—
232C2×C7⋊C8—42no—
233#1131—44no—
234#533—71no—
235#533—71no—
236#533—72no—
237#533—72no—
238#533—71no—
239#533—141no—
240#533—141no—
241#533—141no—
242#533—141no—
243#533C2×D711yescharacteristic
244C2×C4⋊Q8—71no—
245C2×C4⋊Q8—71no—
246C2×#125—141no—
247C2×C22⋊Q8—72no—
248C2×C22⋊Q8—141no—
249C2×C4×Q8—72no—
250C2×C4×Q8—141no—
251C2×#117—141no—
252C22×C42—71no—
253#432—141no—
254#432—141no—
255#431—71no—
256#431—72no—
257#431—72no—
258#431—72no—
259#431—141no—
260#431—141no—
261#431C2×D711yescharacteristic
262#423—141no—
263C2×C4⋊C8—71no—
264C2×C4⋊C8C2×D711yescharacteristic
265#407—141no—
266C2×C22⋊C8—141no—
267C4×M16—72no—
268C2×C4×C8—141no—
269Q8⋊C8—74no—
270Q8⋊C8—74no—
271Q8⋊C8—74no—
272Q8⋊C8C2×D714yescomplemented
273C23×C7C2×C22⋊C411yescharacteristic
274C22×D7—22no—
275C22×D7—22no—
276C22×D7—41no—
277C22×D7C2×C22⋊C411yescharacteristic
278C22×D7C2×C22⋊C411yescharacteristic
279C7×Q8—24no—
280C7×Q8—42no—
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×C7#13712yes—
289C2×C4×C7#13712yes—
290C2×C4×C7C23×C411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7C2×C22⋊C411yescharacteristic
299C2×Dic7C2×C22⋊C411yescharacteristic
300C2×Dic7C2×C22⋊C411yescharacteristic
301C2×Dic7C2×C22⋊C411yescharacteristic
302C2×Dic7C2×C22⋊C411yescharacteristic
303C2×Dic7C2×C22⋊C411yescharacteristic
304C4×D7—22no—
305C4×D7—22no—
306C4×D7—21no—
307C4×D7—21no—
308C4×D7—21no—
309C4×D7—21no—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—22no—
315C4×D7—41no—
316C4×D7—41no—
317C4×D7—44no—
318C4×D7—44no—
319C4×D7#13114yes—
320Dic14—24no—
321Dic14—42no—
322Dic14—44no—
323Dic14—44no—
324C7×C8—42no—
325C7×C8—42no—
326C7⋊C8—42no—
327C7⋊C8—42no—
328C22×Q8—72no—
329C22×Q8—141no—
330C23×C4—71no—
331C23×C4—71no—
332C23×C4—71no—
333C2×M16—142no—
334C22×C8—21no—
335C22×C8—141no—
336C4⋊Q8—72no—
337C4⋊Q8—72no—
338C4⋊Q8—71no—
339C4⋊Q8—71no—
340C4⋊Q8—141no—
341C4⋊Q8—141no—
342C4⋊Q8—141no—
343C4⋊Q8—141no—
344C4⋊Q8C4×D711yescharacteristic
345C4⋊Q8C4×D711yescharacteristic
346#125—21no—
347#125—142no—
348#125—142no—
349#125—141no—
350#125—141no—
351#125—141no—
352C22⋊Q8—21no—
353C22⋊Q8—72no—
354C22⋊Q8—72no—
355C22⋊Q8—72no—
356C22⋊Q8—142no—
357C22⋊Q8—142no—
358C22⋊Q8—142no—
359C22⋊Q8—142no—
360C22⋊Q8—141no—
361C22⋊Q8—141no—
362C22⋊Q8—141no—
363C22⋊Q8C4×D712yes—
364C4×Q8—21no—
365C4×Q8—72no—
366C4×Q8—72no—
367C4×Q8—72no—
368C4×Q8—142no—
369C4×Q8—142no—
370C4×Q8—142no—
371C4×Q8—141no—
372C4×Q8—142no—
373C4×Q8—141no—
374C4×Q8—141no—
375C4×Q8C22×D712yes—
376#117—21no—
377#117—142no—
378#117—142no—
379#117—141no—
380#117—141no—
381#117—141no—
382C2×C4⋊C4—72no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—142no—
386C2×C4⋊C4—142no—
387C2×C4⋊C4—141no—
388C2×C22⋊C4—142no—
389C2×C42—71no—
390C2×C42—71no—
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—142no—
401C2×C42C22×D711yescharacteristic
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—72no—
405C4⋊C8—142no—
406C4⋊C8—141no—
407C4⋊C8—141no—
408C4⋊C8C22×D712yes—
409C22⋊C8—21no—
410C22⋊C8—142no—
411C22⋊C8—142no—
412C22⋊C8—142no—
413C22⋊C8—141no—
414#97—142no—
415C4×C8—72no—
416C4×C8—142no—
417C4×C8—142no—
418C4×C8C22×D712yes—
419C22×C7—21no—
420C22×C7—21no—
421C22×C7#41912yes—
422C22×C7C22×C22⋊C411yescharacteristic
423C2×D7—21no—
424C2×D7—21no—
425C2×D7—21no—
426C2×D7—21no—
427C2×D7—21no—
428C2×D7—44no—
429C2×D7#40812yes—
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7#41912yes—
436C4×C7C2×#13112yes—
437C4×C7C2×#13711yescharacteristic
438C4×C7C2×#13711yescharacteristic
439Dic7—21no—
440Dic7—21no—
441Dic7—21no—
442Dic7—21no—
443Dic7—42no—
444Dic7—42no—
445Dic7#40812yes—
446Dic7#42112yes—
447C24—71no—
448C2×Q8—21no—
449C2×Q8—72no—
450C2×Q8—142no—
451C2×Q8—142no—
452C2×Q8—148no—
453C2×Q8—141no—
454C2×Q8—284no—
455C2×Q8C2×C4×D712yes—
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—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—144no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—282no—
477C22×C4C2×C4×D711yescharacteristic
478C22×C4D4×D711yescharacteristic
479C22×C4D4×D711yescharacteristic
480M16—284no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—42no—
484C2×C8—142no—
485C2×C8—142no—
486C2×C8—282no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—72no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—142no—
496C4⋊C4—142no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—141no—
505C4⋊C4—142no—
506C4⋊C4C2×C4×D712yes—
507C22⋊C4—22no—
508C22⋊C4—142no—
509C22⋊C4—142no—
510C22⋊C4—142no—
511C42—22no—
512C42—71no—
513C42—71no—
514C42—71no—
515C42—71no—
516C42—71no—
517C42—71no—
518C42—71no—
519C42—71no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—142no—
525C42—142no—
526C42—142no—
527C42C2×C4×D711yescharacteristic
528C42D4×D711yescharacteristic
529C42D4×D711yescharacteristic
530C42C23×D711yescharacteristic
531C2×C7—22no—
532C2×C7C2×#40811yescharacteristic
533C2×C7C2×#41911yescharacteristic
534C2×C7C2×#42111yescharacteristic
535D7—22no—
536D7—41no—
537C23—71no—
538C23—71no—
539C23—142no—
540C23—142no—
541C23—281no—
542C23D7×C22⋊C411yescharacteristic
543Q8—24no—
544Q8—42no—
545Q8—144no—
546Q8—282no—
547Q8—284no—
548Q8—284no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—71no—
554C2×C4—71no—
555C2×C4—74no—
556C2×C4—71no—
557C2×C4—71no—
558C2×C4—71no—
559C2×C4—71no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—141no—
569C2×C4—141no—
570C2×C4—141no—
571C2×C4—141no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—284no—
579C2×C4—284no—
580C2×C4D7×C22⋊C411yescharacteristic
581C2×C4D7×C22⋊C411yescharacteristic
582C2×C4D7×C22⋊C411yescharacteristic
583C2×C4#647812yes—
584C2×C4#648112yes—
585C2×C4C22×C4×D711yescharacteristic
586C2×C4C2×D4×D711yescharacteristic
587C2×C4C2×D4×D711yescharacteristic
588C8—42no—
589C8—42no—
590C8—282no—
591C8—282no—
592C7#1746411yescharacteristic, complemented, Sylow
593C22—21no—
594C22—21no—
595C22—72no—
596C22—141no—
597C22—141no—
598C22—141no—
599C22—141no—
600C22—141no—
601C22—284no—
602C22#8957412yes—
603C22C2×D7×C22⋊C411yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—42no—
609C4—72no—
610C4—72no—
611C4—141no—
612C4—141no—
613C4—141no—
614C4—141no—
615C4—282no—
616C4—282no—
617C4#8957412yes—
618C4D7×#13112yes—
619C4C2×#647811yescharacteristic
620C4C2×#648111yescharacteristic
621C2—22no—
622C2—142no—
623C2—281no—
624C2D7×#40811yescharacteristic
625C2C2×#8957411yescharacteristic
626C2D7×#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1481C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1481D7
hypocenterC7#17464
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
4. C7×#1481C2semidirect
5. #11729578C2semidirect
6. #11728522C2semidirect
7. #11728183C2semidirect
8. #11725989C2semidirect
9. #11725985C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
148. #1481D7semidirect
272. Q8⋊C8C2×D7semidirect
592. C7#17464semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117367272
C2×#896052
C7×#14812
#117295782
#117285222
#117281832
#117259892
#117259852
#174647