Finite Group Databasebeta

← #14219894

Subgroups — #14219894

Order 1792.

1746 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742912C211yesmaximal, characteristic
2C7×#2971C211yesmaximal, characteristic, complemented
3#11739965C211yesmaximal, characteristic, complemented
4#11737366C211yesmaximal, characteristic, complemented
5#11737365C211yesmaximal, characteristic, complemented
6Q8×#1131C211yesmaximal, characteristic, complemented
7C8×Q8×D7C211yesmaximal, characteristic, complemented
8#11736972C213yesmaximal, complemented
9#11736971C213yesmaximal, complemented
10#11736966C213yesmaximal, complemented
11#11736963C213yesmaximal, complemented
12#11736785C213yesmaximal, complemented
13#11736761C213yesmaximal, complemented
14#11735968C213yesmaximal, complemented
15#11735953C213yesmaximal, complemented
16C4×C7×#41C2211yescharacteristic
17#90522C411yescharacteristic
18#90522C411yescharacteristic
19Q8×D28C411yescharacteristic
20C4×Q8×D7C2211yescharacteristic
21C4×Q8×D7C2211yescharacteristic
22#90261C411yescharacteristic
23#90248C2213yes—
24#90248C2213yes—
25#90242C413yes—
26#90233C413yes—
27#90220C413yes—
28#90220C413yes—
29C4×#1157C2213yes—
30#90161C2213yes—
31C7×C8×Q8C2211yescharacteristic
32C7×C8×Q8C2211yescharacteristic
33C7×#436C2213yes—
34C7×#436C2213yes—
35C7×#432C2213yes—
36C4×C7×M16C2213yes—
37#89900C2213yes—
38#89900C2213yes—
39#89899C2213yes—
40#89899C2213yes—
41#89894C2216yes—
42#89891C2216yes—
43Q8×C7⋊C8C2211yescharacteristic
44Q8×C7⋊C8C2211yescharacteristic
45D4×C7⋊C8C2213yes—
46D4×C7⋊C8C2213yes—
47#89697C2213yes—
48C2×C4⋊(C7⋊C8)C2213yes—
49#89609C2213yes—
50#89607C2213yes—
51#89605C2213yes—
52D7×C4⋊C8C2213yes—
53#89603C2213yes—
54#89603C2213yes—
55#89602C2213yes—
56#89602C2213yes—
57#89502C2216yes—
58(C22⋊D7)⋊C8C2216yes—
59#89492C2216yes—
60#89492C2216yes—
61#89478C2211yescharacteristic
62#89478C2213yes—
63#89473C2213yes—
64#89473C2216yes—
65#89473C2213yes—
66C4×#1131C2213yes—
67#89457C2213yes—
68#89457C2213yes—
69C4×C8×D7C2213yes—
70#89451C2213yes—
71#89451C2213yes—
72C8×Dic14C2213yes—
73C8×Dic14C2213yes—
74#60335—71nomaximal, Sylow
75C2×C7×#41C2×C411yescharacteristic
76C2×Q8×D7C2×C411yescharacteristic
77C2×Q8×D7C2×C411yescharacteristic
78C2×#1157C2×C413yes—
79C2×C7×M16C2313yes—
80C2×C7×M16C2313yes—
81C4×C7×Q8C2×C411yescharacteristic
82C4×C7×Q8C2311yescharacteristic
83C4×C7×D4C2×C413yes—
84C4×C7×D4C2313yes—
85C7×#117C2×C413yes—
86C2×C42×C7C2313yes—
87Q8×Dic7C2×C411yescharacteristic
88Q8×Dic7C2×C411yescharacteristic
89#6501C2×C413yes—
90#6501C2×C413yes—
91#6499C2×C413yes—
92#6499C2×C413yes—
93C4×C22⋊D7C2×C413yes—
94C4×C22⋊D7C2×C413yes—
95C4×C22⋊D7C2313yes—
96C4×C22⋊D7C2313yes—
97#6489C2×C413yes—
98#6489C2316yes—
99#6487C2×C413yes—
100#6487C2×C413yes—
101#6487C2×C416yes—
102C22×C7⋊C8C2316yes—
103C2×#1131C2311yescharacteristic
104C2×#1131C2313yes—
105C2×C8×D7C2311yescharacteristic
106C2×C8×D7C2313yes—
107#6460C2×C413yes—
108#6460C2×C413yes—
109D7×C4⋊C4C2×C413yes—
110D7×C4⋊C4C2×C413yes—
111#6453C2×C413yes—
112#6453C2×C413yes—
113#6450C2×C413yes—
114#6450C2×C413yes—
115#6446C2×C416yes—
116#6446C2×C416yes—
117#6442C2×C416yes—
118#6442C2×C416yes—
119C4×D28C2×C413yes—
120C4×D28C2311yescharacteristic
121#6435C2×C413yes—
122#6435C2×C413yes—
123C42×D7C2313yes—
124C42×D7C2313yes—
125#6433C2×C416yes—
126#6432C2×C413yes—
127#6432C2×C413yes—
128C4×Dic14C2×C413yes—
129C4×Dic14C2313yes—
130C4×Dic14C2313yes—
131C4×Dic14C2313yes—
132C7×C4⋊C8C2313yes—
133C7×C4⋊C8C2313yes—
134C7×C22⋊C8C2316yes—
135C7×#97C2311yescharacteristic
136C7×#97C2313yes—
137C4×C7×C8C2313yes—
138C4×C7×C8C2313yes—
139C22⋊(C7⋊C8)C2313yes—
140C22⋊(C7⋊C8)C2313yes—
141(C2×D7)⋊C8C2313yes—
142(C2×D7)⋊C8C2313yes—
143(C2×D7)⋊C8C2311yescharacteristic
144(C2×D7)⋊C8C2311yescharacteristic
145#6389C2313yes—
146#6389C2316yes—
147#6389C2313yes—
148Dic7⋊C8C2313yes—
149Dic7⋊C8C2316yes—
150Dic7⋊C8C2313yes—
151Dic7⋊C8C2316yes—
152Dic7⋊C8C2313yes—
153Dic7⋊C8C2313yes—
154C8×Dic7C2313yes—
155C8×Dic7C2316yes—
156C8×Dic7C2313yes—
157C4⋊(C7⋊C8)C2313yes—
158C4⋊(C7⋊C8)C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4⋊(C7⋊C8)C2311yescharacteristic
161C4×C7⋊C8C2313yes—
162C4×C7⋊C8C2313yes—
163#3482—71no—
164#2994—73no—
165#2993—73no—
166#2990—73no—
167#2988—71no—
168#2985—73no—
169#2984—71no—
170#2982—71no—
171#2971D711yescharacteristic, complemented
172#2969—73no—
173#2966—73no—
174#2965—73no—
175Q8×M16—71no—
176#2962—73no—
177C2×C8×Q8—71no—
178C7×#41—24no—
179C2×C7×Q8C22×C411yescharacteristic
180C2×C7×D4C22×C413yes—
181C22×C4×C7C22×C413yes—
182C22×C4×C7C22×C413yes—
183C22×C4×C7C2413yes—
184C2×C22⋊D7C22×C413yes—
185C2×C22⋊D7C22×C413yes—
186Q8×D7—24no—
187Q8×D7—24no—
188#1157—46no—
189C2×D28C22×C411yescharacteristic
190C2×C4×D7C22×C413yes—
191C2×C4×D7C22×C413yes—
192C2×C4×D7C22×C413yes—
193C2×C4×D7C22×C413yes—
194C2×C4×D7C2411yescharacteristic
195C2×C4×D7C2411yescharacteristic
196C2×Dic14C22×C413yes—
197C2×Dic14C22×C413yes—
198C2×Dic14C22×C413yes—
199C7×M16—26no—
200C7×M16—43no—
201C2×C7×C8C2411yescharacteristic
202C2×C7×C8C2413yes—
203C2×C7×C8C2413yes—
204C2×C7×C8C2411yescharacteristic
205C7×C4⋊C4C22×C413yes—
206C7×C4⋊C4C22×C413yes—
207C7×C22⋊C4C22×C416yes—
208C42×C7C22×C413yes—
209C42×C7C22×C413yes—
210C42×C7C2413yes—
211C42×C7C2411yescharacteristic
212C22⋊Dic7C22×C413yes—
213C22⋊Dic7C22×C413yes—
214C22⋊Dic7C22×C416yes—
215C22⋊Dic7C22×C416yes—
216(C2×D7)⋊C4C22×C411yescharacteristic
217(C2×D7)⋊C4C22×C411yescharacteristic
218(C2×D7)⋊C4C22×C413yes—
219(C2×D7)⋊C4C22×C413yes—
220C4⋊Dic7C22×C411yescharacteristic
221C4⋊Dic7C22×C413yes—
222C4⋊Dic7C22×C413yes—
223C4⋊Dic7C22×C413yes—
224C4⋊Dic7C22×C416yes—
225C4⋊Dic7C22×C416yes—
226Dic7⋊C4C22×C413yes—
227Dic7⋊C4C22×C413yes—
228Dic7⋊C4C22×C413yes—
229Dic7⋊C4C22×C413yes—
230Dic7⋊C4C22×C416yes—
231Dic7⋊C4C22×C416yes—
232C4×Dic7C22×C413yes—
233C4×Dic7C22×C413yes—
234C4×Dic7C22×C413yes—
235C4×Dic7C22×C413yes—
236C4×Dic7C2413yes—
237C4×Dic7C2413yes—
238C2×C7⋊C8—46no—
239C2×C7⋊C8C2411yescharacteristic
240C2×C7⋊C8C2411yescharacteristic
241C2×C7⋊C8C2413yes—
242C2×C7⋊C8C2413yes—
243#1131—22no—
244#1131—43no—
245C8×D7—22no—
246C8×D7—43no—
247#557—71no—
248#556—73no—
249#552—73no—
250#552—73no—
251D4×Q8—71no—
252#548—71no—
253#548—71no—
254#547—73no—
255#533—73no—
256#529—73no—
257#529—73no—
258C4×#41—73no—
259C4×#41C2×D711yescharacteristic
260C2×C4×Q8—71no—
261C2×C4×Q8—71no—
262#446—73no—
263#446—73no—
264#446—73no—
265#446—76no—
266#446—73no—
267#446—73no—
268#446—73no—
269C8×Q8—71no—
270C8×Q8—71no—
271C8×Q8—73no—
272C8×Q8—73no—
273C8×Q8—73no—
274C8×Q8—73no—
275C8×Q8C2×D711yescharacteristic
276C8×Q8C2×D711yescharacteristic
277#436—73no—
278#436—73no—
279#436C2×D713yes—
280#436C2×D713yes—
281#435—73no—
282#435—73no—
283#435—76no—
284#435—71no—
285#435—73no—
286C8×D4—73no—
287C8×D4—73no—
288C8×D4—76no—
289#433—76no—
290#433—76no—
291#432—73no—
292#432C2×D713yes—
293#431—73no—
294#431—73no—
295#431—73no—
296#431—73no—
297#424—76no—
298#423—73no—
299C2×C4⋊C8—73no—
300C2×C4⋊C8—73no—
301#405—76no—
302C4×M16—73no—
303C4×M16C2×D713yes—
304C2×C4×C8—73no—
305C23×C7C23×C413yes—
306C22×D7C23×C411yescharacteristic
307C22×D7C23×C411yescharacteristic
308C7×Q8#13114yes—
309C7×D4—26no—
310C2×C4×C7—26no—
311C2×C4×C7—23no—
312C2×C4×C7—23no—
313C2×C4×C7—43no—
314C2×C4×C7C23×C413yes—
315C2×C4×C7C23×C411yescharacteristic
316C2×C4×C7C23×C413yes—
317C2×C4×C7C23×C411yescharacteristic
318C2×C4×C7C23×C413yes—
319C2×C4×C7C2511yescharacteristic
320C22⋊D7—26no—
321C22⋊D7—43no—
322C2×Dic7C23×C411yescharacteristic
323C2×Dic7C23×C411yescharacteristic
324C2×Dic7C23×C413yes—
325C2×Dic7C23×C413yes—
326C2×Dic7C23×C413yes—
327C2×Dic7C23×C413yes—
328D28—22no—
329C4×D7—26no—
330C4×D7—26no—
331C4×D7—26no—
332C4×D7—21no—
333C4×D7—21no—
334C4×D7—21no—
335C4×D7—21no—
336C4×D7—43no—
337Dic14—26no—
338Dic14—26no—
339Dic14—43no—
340C7×C8—23no—
341C7×C8—23no—
342C7×C8—21no—
343C7×C82−1+412yes—
344C7⋊C8—21no—
345C7⋊C8—21no—
346C7⋊C8—23no—
347C7⋊C8—23no—
348C2×#41—73no—
349C2×#41C4×D711yescharacteristic
350C22×Q8—71no—
351C22×Q8—71no—
352C2×M16—71no—
353C2×M16—73no—
354C2×M16C22×D713yes—
355C2×M16C22×D713yes—
356C22×C8—76no—
357C22×C8—71no—
358C22×C8—73no—
359C4⋊Q8—73no—
360C4⋊Q8—73no—
361#126—76no—
362#126—76no—
363#125—73no—
364#125—73no—
365#125—76no—
366#124—73no—
367#124—73no—
368#123—76no—
369#123—76no—
370C22⋊Q8—76no—
371C22⋊Q8—73no—
372C22⋊Q8—73no—
373C22⋊Q8—73no—
374C22⋊Q8—73no—
375#121—73no—
376#121—73no—
377C4×Q8—71no—
378C4×Q8—71no—
379C4×Q8—73no—
380C4×Q8—73no—
381C4×Q8—73no—
382C4×Q8—73no—
383C4×Q8—73no—
384C4×Q8—73no—
385C4×Q8C4×D711yescharacteristic
386C4×Q8C22×D711yescharacteristic
387C4×D4—73no—
388C4×D4—73no—
389C4×D4—73no—
390C4×D4—73no—
391C4×D4—73no—
392C4×D4—71no—
393C4×D4C4×D713yes—
394C4×D4C22×D713yes—
395#117—73no—
396#117—76no—
397#117—73no—
398#117—73no—
399#117C4×D713yes—
400C2×C4⋊C4—73no—
401C2×C4⋊C4—73no—
402C2×C42—73no—
403C2×C42—73no—
404C2×C42C22×D713yes—
405C4⋊C8—76no—
406C4⋊C8—76no—
407C4⋊C8—73no—
408C4⋊C8—73no—
409C4⋊C8—73no—
410C4⋊C8—73no—
411C4⋊C8—73no—
412C4⋊C8—73no—
413C4⋊C8—73no—
414C4⋊C8—71no—
415C4⋊C8C22×D713yes—
416C4⋊C8C22×D713yes—
417C22⋊C8—73no—
418C22⋊C8—73no—
419C22⋊C8—71no—
420C22⋊C8—73no—
421C22⋊C8—71no—
422C22⋊C8—73no—
423C22⋊C8C22×D716yes—
424#97—73no—
425#97—73no—
426#97—76no—
427#97C22×D711yescharacteristic
428#97C22×D713yes—
429C4×C8—73no—
430C4×C8—73no—
431C4×C8—76no—
432C4×C8—73no—
433C4×C8—73no—
434C4×C8C22×D713yes—
435C4×C8C22×D713yes—
436C22×C7—23no—
437C22×C7—23no—
438C22×C7—23no—
439C22×C7C24×C411yescharacteristic
440C2×D7—21no—
441C2×D7—21no—
442C2×D7—21no—
443C2×D7—21no—
444C2×D7—21no—
445C2×D7#56812yes—
446C4×C7—23no—
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—23no—
450C4×C7C2×#13116yes—
451C4×C7C2×2−1+411yescharacteristic
452C4×C7#58511yescharacteristic
453Dic7—21no—
454Dic7—21no—
455Dic7—23no—
456Dic7—23no—
457Dic7—23no—
458Dic7—23no—
459#41—24no—
460#41—286no—
461C2×Q8—73no—
462C2×Q8—73no—
463C2×Q8—73no—
464C2×Q8—144no—
465C2×Q8—144no—
466C2×Q8C2×C4×D711yescharacteristic
467C2×D4—73no—
468C2×D4—73no—
469C2×D4—71no—
470C2×D4C2×C4×D713yes—
471C22×C4—73no—
472C22×C4—73no—
473C22×C4—73no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—73no—
477C22×C4C2×C4×D713yes—
478C22×C4C2×C4×D713yes—
479C22×C4C23×D713yes—
480M16—26no—
481M16—43no—
482M16—142no—
483M16—283no—
484C2×C8—71no—
485C2×C8—73no—
486C2×C8—71no—
487C2×C8—73no—
488C2×C8—142no—
489C2×C8—283no—
490C2×C8—286no—
491C2×C8C23×D713yes—
492C2×C8C23×D713yes—
493C2×C8C23×D711yescharacteristic
494C2×C8C23×D711yescharacteristic
495C4⋊C4—76no—
496C4⋊C4—76no—
497C4⋊C4—73no—
498C4⋊C4—73no—
499C4⋊C4—73no—
500C4⋊C4—73no—
501C4⋊C4—73no—
502C4⋊C4—73no—
503C4⋊C4—71no—
504C4⋊C4—73no—
505C4⋊C4—76no—
506C4⋊C4—76no—
507C4⋊C4C2×C4×D713yes—
508C4⋊C4C2×C4×D713yes—
509C22⋊C4—73no—
510C22⋊C4—73no—
511C22⋊C4—76no—
512C22⋊C4—76no—
513C22⋊C4—71no—
514C22⋊C4—71no—
515C22⋊C4—73no—
516C22⋊C4—73no—
517C22⋊C4C2×C4×D716yes—
518C42—73no—
519C42—73no—
520C42—73no—
521C42—73no—
522C42—73no—
523C42—73no—
524C42C2×C4×D713yes—
525C42C2×C4×D713yes—
526C42C23×D713yes—
527C42C23×D711yescharacteristic
528C2×C7—43no—
529C2×C7C4×2−1+411yescharacteristic
530C2×C7C22×#13111yescharacteristic
531C2×C7C2×#56811yescharacteristic
532D7—22no—
533D7—41no—
534C23—71no—
535C23—71no—
536C23C22×C4×D713yes—
537Q8—146no—
538Q8—146no—
539Q8—283no—
540Q8#647014yes—
541D4—26no—
542D4—142no—
543D4—146no—
544D4—283no—
545C2×C4—26no—
546C2×C4—23no—
547C2×C4—23no—
548C2×C4—43no—
549C2×C4—71no—
550C2×C4—71no—
551C2×C4—73no—
552C2×C4—73no—
553C2×C4—73no—
554C2×C4—73no—
555C2×C4—146no—
556C2×C4—146no—
557C2×C4—146no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—283no—
563C2×C4C22×C4×D713yes—
564C2×C4C22×C4×D711yescharacteristic
565C2×C4C22×C4×D711yescharacteristic
566C2×C4C22×C4×D713yes—
567C2×C4C22×C4×D713yes—
568C2×C4C24×D711yescharacteristic
569C8—23no—
570C8—23no—
571C8—21no—
572C8—141no—
573C8—141no—
574C8—143no—
575C8—143no—
576C8#655412yes—
577C7#6033511yescharacteristic, complemented, Sylow
578C22—23no—
579C22—23no—
580C22—23no—
581C22—72no—
582C22—141no—
583C22—141no—
584C22—141no—
585C22—141no—
586C22—141no—
587C22C23×C4×D711yescharacteristic
588C4—23no—
589C4—21no—
590C4—21no—
591C4—23no—
592C4—141no—
593C4—141no—
594C4—143no—
595C4—143no—
596C4—143no—
597C4—143no—
598C4C2×#647016yes—
599C4C2×#655411yescharacteristic
600C4#9061611yescharacteristic
601C2—43no—
602C2—142no—
603C2—281no—
604C2C4×#655411yescharacteristic
605C2C22×#647011yescharacteristic
606C2D7×#56811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2971C2
prime radical#11742912C2
cosocleC2×C4×C7C25
hypercenter#2971D7
hypocenterC7#60335
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. C7×#2971C2semidirect
3. #11739965C2semidirect
4. #11737366C2semidirect
5. #11737365C2semidirect
6. Q8×#1131C2semidirect
7. C8×Q8×D7C2semidirect
8. #11736972C2semidirect
9. #11736971C2semidirect
10. #11736966C2semidirect
11. #11736963C2semidirect
12. #11736785C2semidirect
13. #11736761C2semidirect
14. #11735968C2semidirect
15. #11735953C2semidirect
171. #2971D7semidirect
577. C7#60335semidirect

Maximal subgroups

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

subgroupindex
#117429122
C7×#29712
#117399652
#117373662
#117373652
Q8×#11312
C8×Q8×D72
#117369722
#117369712
#117369662
#117369632
#117367852
#117367612
#117359682
#117359532
#603357