Finite Group Databasebeta

← #13320732

Subgroups — #13320732

Order 1792.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#1157C211yesmaximal, characteristic
2#11736722C211yesmaximal, characteristic
3#11736485C211yesmaximal, characteristic
4C7×#1487C211yesmaximal, characteristic, complemented
5C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#11728522C211yesmaximal, characteristic, complemented
7#11728255C211yesmaximal, characteristic, complemented
8#11725997C212yesmaximal, complemented
9#11725993C212yesmaximal, complemented
10#11725989C212yesmaximal, complemented
11#11725985C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13Q8×C22⋊D7C412yes—
14#90270—21no—
15#90270C411yescharacteristic
16#90269—22no—
17#90269C2212yes—
18Q8×D28C411yescharacteristic
19C4×Q8×D7C2212yes—
20#90258—21no—
21#90258C411yescharacteristic
22Q8×Dic14C411yescharacteristic
23#90202—22no—
24#90202C412yes—
25C4×#1157—21no—
26C4×#1157C2211yescharacteristic
27#90161—21no—
28#90161C2211yescharacteristic
29C7×#431C2211yescharacteristic
30C7×#423C2211yescharacteristic
31#89901—21no—
32#89894—21no—
33C2×C4⋊(C7⋊C8)C2211yescharacteristic
34C2×C4×C7⋊C8C2211yescharacteristic
35#89612—22no—
36#89610—21no—
37#89607C2212yes—
38#89605C2212yes—
39#89603—21no—
40#89502—22no—
41C4×#1131C2212yes—
42#89457C2212yes—
43C7×Q8⋊C8C2212yes—
44C7×Q8⋊C8C2212yes—
45Q8⋊(C7⋊C8)C2211yescharacteristic
46Q8⋊(C7⋊C8)C2211yescharacteristic
47Q8⋊(C7⋊C8)C2211yescharacteristic
48Q8⋊(C7⋊C8)C2211yescharacteristic
49#89279C2212yes—
50#89279C2212yes—
51#89251C2212yes—
52#89251C2212yes—
53#17514—71nomaximal, Sylow
54C22×C7×Q8C2×C411yescharacteristic
55C2×Q8×D7C2×C412yes—
56C2×#1157D411yescharacteristic
57C2×C7×M16—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8—21no—
60C4×C7×Q8C2×C411yescharacteristic
61C4×C7×Q8C2×C411yescharacteristic
62C4×C7×Q8C2311yescharacteristic
63C4×C7×Q8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4C2×C411yescharacteristic
66C2×C42×C7—21no—
67C2×C42×C7C2311yescharacteristic
68#6509—22no—
69#6509C2×C412yes—
70Q8×Dic7C2×C412yes—
71#6507—22no—
72#6507C2×C412yes—
73C4×C22⋊D7—22no—
74C4×C22⋊D7—22no—
75C4×C22⋊D7C2×C412yes—
76C4×C22⋊D7D412yes—
77#6489—21no—
78#6489D411yescharacteristic
79#6487—21no—
80#6487—22no—
81#6487—22no—
82#6487C2×C412yes—
83C22×C7⋊C8—21no—
84C22×C7⋊C8—21no—
85C2×#1131—22no—
86#6460—22no—
87#6460C2×C412yes—
88#6459—22no—
89#6459—22no—
90#6459—22no—
91D7×C4⋊C4—22no—
92D7×C4⋊C4C2×C412yes—
93#6452—22no—
94#6452—22no—
95#6452—22no—
96#6451—22no—
97#6451C2×C412yes—
98#6450—22no—
99#6450C2×C412yes—
100C4×D28—21no—
101C4×D28—21no—
102C4×D28C2×C411yescharacteristic
103C4×D28D411yescharacteristic
104#6435—22no—
105#6435D412yes—
106C42×D7—22no—
107C42×D7C2312yes—
108#6433—21no—
109#6433—21no—
110#6433—21no—
111#6432—21no—
112#6432—21no—
113#6432C2×C411yescharacteristic
114#6432C2×C411yescharacteristic
115C4×Dic14—21no—
116C4×Dic14—21no—
117C4×Dic14—21no—
118C4×Dic14—22no—
119C4×Dic14—22no—
120C4×Dic14C2×C411yescharacteristic
121C4×Dic14D411yescharacteristic
122C4×Dic14C2312yes—
123C7×C4⋊C8—21no—
124C7×C4⋊C8—21no—
125C7×C4⋊C8C2312yes—
126C7×C22⋊C8—21no—
127C4×C7×C8C2312yes—
128(C2×D7)⋊C8—22no—
129(C2×D7)⋊C8—22no—
130(C2×D7)⋊C8—22no—
131#6389—22no—
132#6389—22no—
133Dic7⋊C8—22no—
134C8×Dic7—22no—
135C4⋊(C7⋊C8)—21no—
136C4⋊(C7⋊C8)C2311yescharacteristic
137C4⋊(C7⋊C8)C2311yescharacteristic
138C4×C7⋊C8—21no—
139C4×C7⋊C8C2311yescharacteristic
140C4×C7⋊C8C2311yescharacteristic
141Q8×#41—71no—
142#2965—71no—
143#2925—71no—
144#1497—72no—
145#1493—71no—
146#1491—72no—
147#1487D711yescharacteristic, complemented
148#1485—72no—
149#1481—71no—
150#1479—72no—
151C2×Q8⋊C8—71no—
152C2×C7×Q8—21no—
153C2×C7×Q8—22no—
154C2×C7×Q8—41no—
155C2×C7×Q8C22×C411yescharacteristic
156C2×C7×Q8C22×C411yescharacteristic
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7C22×C411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C2×C22⋊D7C22⋊C412yes—
163Q8×D7—24no—
164Q8×D7—42no—
165#1157—22no—
166#1157—22no—
167C2×D28C22⋊C411yescharacteristic
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7C22×C412yes—
171C2×C4×D7C2×D412yes—
172C2×C4×D7C2×D412yes—
173C2×Dic14—22no—
174C2×Dic14—21no—
175C2×Dic14—22no—
176C2×Dic14C22⋊C411yescharacteristic
177C2×Dic14C22×C412yes—
178C7×M16—42no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4C22×C411yescharacteristic
187C7×C4⋊C4C22×C411yescharacteristic
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7C22×C411yescharacteristic
193C42×C7C2×D411yescharacteristic
194C42×C7C2×D411yescharacteristic
195C42×C7C2411yescharacteristic
196C22⋊Dic7—22no—
197C22⋊Dic7C22⋊C412yes—
198(C2×D7)⋊C4—22no—
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4C22⋊C412yes—
201(C2×D7)⋊C4C22⋊C412yes—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7C22⋊C411yescharacteristic
208C4⋊Dic7C22⋊C411yescharacteristic
209C4⋊Dic7C22×C412yes—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4C22⋊C412yes—
219Dic7⋊C4C22⋊C412yes—
220C4×Dic7—22no—
221C4×Dic7—22no—
222C4×Dic7—22no—
223C4×Dic7C22×C412yes—
224C4×Dic7C2×D412yes—
225C4×Dic7C2×D412yes—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—42no—
231C2×C7⋊C8—42no—
232#1131—44no—
233Q82—71no—
234#557—71no—
235#557—141no—
236#554—71no—
237#554—72no—
238#554—142no—
239#554—141no—
240D4×Q8—72no—
241D4×Q8—71no—
242#533—72no—
243#533—71no—
244#533—141no—
245#533—142no—
246C4×#41—71no—
247C4×#41—141no—
248C2×C4×Q8—72no—
249C2×C4×Q8C2×D711yescharacteristic
250#446—141no—
251#436—141no—
252#435—142no—
253#433—142no—
254#432—72no—
255#431—72no—
256#431—72no—
257#431C2×D711yescharacteristic
258#423C2×D711yescharacteristic
259C2×C4⋊C8—71no—
260#408—141no—
261#405—141no—
262C4×M16—72no—
263C2×C4×C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8—71no—
268Q8⋊C8—72no—
269Q8⋊C8—72no—
270Q8⋊C8—72no—
271Q8⋊C8—72no—
272Q8⋊C8C2×D712yes—
273Q8⋊C8C2×D712yes—
274C23×C7C2×C22⋊C411yescharacteristic
275C22×D7C2×C22⋊C412yes—
276C7×Q8—21no—
277C7×Q8—21no—
278C7×Q8—21no—
279C7×Q8—42no—
280C7×Q8#13112yes—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—81no—
290C2×C4×C7C2×C22⋊C411yescharacteristic
291C2×C4×C7C2×C22⋊C411yescharacteristic
292C2×C4×C7C2×C22⋊C411yescharacteristic
293C2×C4×C7#13712yes—
294C2×C4×C7#13712yes—
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C22⋊D7—24no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7C2×C22⋊C412yes—
302C2×Dic7C2×C22⋊C412yes—
303C2×Dic7C2×C22⋊C412yes—
304D28—22no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—42no—
310C4×D7—44no—
311C4×D7—82no—
312Dic14—22no—
313Dic14—22no—
314Dic14—22no—
315Dic14—44no—
316Dic14—82no—
317Dic14—81no—
318C7×C8—42no—
319C7×C8—42no—
320C7⋊C8—22no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C7⋊C8—41no—
324C2×#41—71no—
325C22×Q8—72no—
326C22×Q8C4×D711yescharacteristic
327C2×M16—21no—
328C2×M16—142no—
329C22×C8—141no—
330C22×C8—141no—
331C4⋊Q8—72no—
332C4⋊Q8—72no—
333C4⋊Q8—71no—
334C4⋊Q8—71no—
335C4⋊Q8—141no—
336C4⋊Q8—141no—
337C4⋊Q8—142no—
338C4⋊Q8—142no—
339#125—141no—
340#125—141no—
341#125—142no—
342#125—142no—
343#125—142no—
344#125—141no—
345C22⋊Q8—72no—
346C22⋊Q8—72no—
347C22⋊Q8—72no—
348C22⋊Q8—142no—
349C22⋊Q8—142no—
350C22⋊Q8—142no—
351C22⋊Q8—142no—
352C22⋊Q8—142no—
353C22⋊Q8—142no—
354C22⋊Q8—142no—
355C22⋊Q8—141no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—71no—
359C4×Q8—71no—
360C4×Q8—72no—
361C4×Q8—72no—
362C4×Q8—72no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—141no—
366C4×Q8—141no—
367C4×Q8—142no—
368C4×Q8—141no—
369C4×Q8C4×D711yescharacteristic
370C4×Q8C4×D711yescharacteristic
371C4×Q8C22×D711yescharacteristic
372C4×Q8C22×D711yescharacteristic
373C4×D4—72no—
374C4×D4—72no—
375C4×D4—71no—
376C4×D4—71no—
377C4×D4—141no—
378C4×D4—141no—
379C4×D4—142no—
380C4×D4—142no—
381#117—71no—
382#117—72no—
383#117—142no—
384#117—141no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—72no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4C4×D711yescharacteristic
389C2×C42—21no—
390C2×C42—72no—
391C2×C42—142no—
392C2×C42C22×D711yescharacteristic
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—71no—
396C4⋊C8—71no—
397C4⋊C8—142no—
398C4⋊C8—141no—
399C4⋊C8C22×D712yes—
400C22⋊C8—21no—
401C22⋊C8—142no—
402C22⋊C8—142no—
403C22⋊C8—142no—
404#97—142no—
405#97—142no—
406C4×C8—71no—
407C4×C8—71no—
408C4×C8—142no—
409C4×C8—141no—
410C4×C8C22×D712yes—
411C22×C7—21no—
412C22×C7—21no—
413C22×C7#41912yes—
414C22×C7C22×C22⋊C411yescharacteristic
415C2×D7—22no—
416C2×D7—22no—
417C2×D7—22no—
418C4×C7—22no—
419C4×C7—21no—
420C4×C7—21no—
421C4×C7—21no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7#41912yes—
426C4×C7C2×#13111yescharacteristic
427C4×C7#56811yescharacteristic
428C4×C7C2×#13711yescharacteristic
429C4×C7C2×#13711yescharacteristic
430Dic7—22no—
431Dic7—22no—
432Dic7—22no—
433Dic7—42no—
434Dic7—42no—
435#41—142no—
436#41—142no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—41no—
440C2×Q8—71no—
441C2×Q8—72no—
442C2×Q8—142no—
443C2×Q8—142no—
444C2×Q8—141no—
445C2×Q8—144no—
446C2×Q8—282no—
447C2×Q8C2×C4×D711yescharacteristic
448C2×Q8C2×C4×D711yescharacteristic
449C2×D4—72no—
450C2×D4—71no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—72no—
454C22×C4—72no—
455C22×C4—72no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4C2×C4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4D4×D711yescharacteristic
461M16—42no—
462M16—284no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—282no—
470C2×C8—282no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—72no—
477C4⋊C4—72no—
478C4⋊C4—71no—
479C4⋊C4—71no—
480C4⋊C4—72no—
481C4⋊C4—142no—
482C4⋊C4—142no—
483C4⋊C4—142no—
484C4⋊C4—142no—
485C4⋊C4—142no—
486C4⋊C4—142no—
487C4⋊C4—142no—
488C4⋊C4—142no—
489C4⋊C4—142no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—142no—
494C4⋊C4C2×C4×D711yescharacteristic
495C4⋊C4C2×C4×D711yescharacteristic
496C22⋊C4—72no—
497C22⋊C4—72no—
498C22⋊C4—72no—
499C22⋊C4—142no—
500C22⋊C4—142no—
501C22⋊C4—142no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—72no—
507C42—72no—
508C42—72no—
509C42—142no—
510C42—142no—
511C42—142no—
512C42C2×C4×D711yescharacteristic
513C42D4×D711yescharacteristic
514C42D4×D711yescharacteristic
515C42C23×D711yescharacteristic
516C2×C7—22no—
517C2×C7#288011yescharacteristic
518C2×C7C2×#41911yescharacteristic
519C2×C7#290111yescharacteristic
520D7—42no—
521C23—72no—
522C23D7×C22⋊C411yescharacteristic
523Q8—21no—
524Q8—21no—
525Q8—21no—
526Q8—42no—
527Q8—142no—
528Q8—142no—
529Q8—142no—
530Q8—284no—
531Q8—561no—
532Q8—562no—
533Q8#647012yes—
534D4—144no—
535D4—142no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—72no—
547C2×C4—81no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—282no—
555C2×C4—284no—
556C2×C4—562no—
557C2×C4D7×C22⋊C411yescharacteristic
558C2×C4D7×C22⋊C411yescharacteristic
559C2×C4D7×C22⋊C411yescharacteristic
560C2×C4#647812yes—
561C2×C4#648112yes—
562C2×C4C22×C4×D711yescharacteristic
563C2×C4C2×D4×D711yescharacteristic
564C2×C4C2×D4×D711yescharacteristic
565C8—42no—
566C8—42no—
567C8—142no—
568C8—281no—
569C8—281no—
570C8—281no—
571C7#1751411yescharacteristic, complemented, Sylow
572C22—21no—
573C22—21no—
574C22—142no—
575C22—142no—
576C22—142no—
577C22#8957412yes—
578C22C2×D7×C22⋊C411yescharacteristic
579C4—21no—
580C4—22no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—142no—
587C4—142no—
588C4—142no—
589C4—282no—
590C4—282no—
591C4#8957412yes—
592C4C2×#647011yescharacteristic
593C4#9043611yescharacteristic
594C4C2×#647811yescharacteristic
595C4C2×#648111yescharacteristic
596C2—22no—
597C2—282no—
598C2#1173607611yescharacteristic
599C2C2×#8957411yescharacteristic
600C2#1173643711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1487C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#1157D4
cosocleC42×C7C24
hypercenter#1487D7
hypocenterC7#17514
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. C7×#1487C2semidirect
5. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11728522C2semidirect
7. #11728255C2semidirect
8. #11725997C2semidirect
9. #11725993C2semidirect
10. #11725989C2semidirect
11. #11725985C2semidirect
147. #1487D7semidirect
571. C7#17514semidirect

Maximal subgroups

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

subgroupindex
Q8×#11572
#117367222
#117364852
C7×#14872
C2×Q8⋊(C7⋊C8)2
#117285222
#117282552
#117259972
#117259932
#117259892
#117259852
#175147