Finite Group Databasebeta

← #14219889

Subgroups — #14219889

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
1C7×#3482C211yesmaximal, characteristic
2#11740776C213yesmaximal, complemented
3#11740775C213yesmaximal, complemented
4#11740577C213yesmaximal, complemented
5#11740576C213yesmaximal, complemented
6#11740574C213yesmaximal, complemented
7#11740571C213yesmaximal, complemented
8#11739967C213yesmaximal, complemented
9#11739965C211yesmaximal, characteristic, complemented
10#11739964C211yesmaximal, characteristic, complemented
11#11739962C213yesmaximal, complemented
12#11739882C211yesmaximal, characteristic, complemented
13#11739880C211yesmaximal, characteristic, complemented
14Q8×#1136C211yesmaximal, characteristic, complemented
15C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
16C7×#557C411yescharacteristic
17C7×#556C413yes—
18C7×#552C413yes—
19C7×#552C413yes—
20C7×D4×Q8C411yescharacteristic
21C7×#548C411yescharacteristic
22C7×#548C411yescharacteristic
23C7×#547C413yes—
24C7×#533C2213yes—
25C7×#529C2213yes—
26C7×#529C2213yes—
27C4×C7×#41C2213yes—
28C4×C7×#41C2211yescharacteristic
29C2×C4×C7×Q8C2211yescharacteristic
30C2×C4×C7×Q8C2211yescharacteristic
31#89796C2213yes—
32#89796C2213yes—
33#89796C2216yes—
34#89796C2213yes—
35#89796C2213yes—
36#89796C2213yes—
37#89796C2213yes—
38Q8×C7⋊C8C2211yescharacteristic
39Q8×C7⋊C8C2211yescharacteristic
40Q8×C7⋊C8C2211yescharacteristic
41Q8×C7⋊C8C2211yescharacteristic
42Q8×C7⋊C8C2213yes—
43Q8×C7⋊C8C2213yes—
44Q8×C7⋊C8C2213yes—
45Q8×C7⋊C8C2213yes—
46#89784C2213yes—
47#89784C2213yes—
48#89784C2213yes—
49#89784C2213yes—
50#89783C2211yescharacteristic
51#89783C2213yes—
52#89783C2213yes—
53#89783C2213yes—
54#89783C2216yes—
55D4×C7⋊C8C2213yes—
56D4×C7⋊C8C2213yes—
57D4×C7⋊C8C2216yes—
58#89772C2216yes—
59#89772C2216yes—
60#89771C2216yes—
61#89769C2216yes—
62#89698C2213yes—
63#89698C2213yes—
64#89697C2213yes—
65#89697C2213yes—
66#89697C2213yes—
67#89697C2213yes—
68#89696C2213yes—
69C2×C4⋊(C7⋊C8)C2213yes—
70C2×C4⋊(C7⋊C8)C2213yes—
71C4×#1136C2213yes—
72C4×#1136C2213yes—
73C2×C4×C7⋊C8C2213yes—
74#60335—71nomaximal, Sylow
75C2×C7×#41C2×C411yescharacteristic
76C2×C7×#41C2×C413yes—
77C22×C7×Q8C2×C411yescharacteristic
78C22×C7×Q8C2×C411yescharacteristic
79C7×C4⋊Q8C2×C413yes—
80C7×C4⋊Q8C2×C413yes—
81C7×#126C2×C416yes—
82C7×#126C2×C416yes—
83C7×#125C2×C413yes—
84C7×#125C2×C413yes—
85C7×#125C2×C416yes—
86C7×#124C2×C413yes—
87C7×#124C2×C413yes—
88C7×#123C2×C416yes—
89C7×#123C2×C416yes—
90C7×C22⋊Q8C2×C413yes—
91C7×C22⋊Q8C2×C413yes—
92C7×C22⋊Q8C2×C416yes—
93C7×C22⋊Q8C2×C413yes—
94C7×C22⋊Q8C2×C413yes—
95C7×#121C2×C413yes—
96C7×#121C2×C413yes—
97C4×C7×Q8C2×C411yescharacteristic
98C4×C7×Q8C2×C411yescharacteristic
99C4×C7×Q8C2×C413yes—
100C4×C7×Q8C2×C413yes—
101C4×C7×Q8C2×C411yescharacteristic
102C4×C7×Q8C2×C413yes—
103C4×C7×Q8C2311yescharacteristic
104C4×C7×Q8C2313yes—
105C4×C7×Q8C2313yes—
106C4×C7×Q8C2313yes—
107C4×C7×D4C2×C413yes—
108C4×C7×D4C2×C413yes—
109C4×C7×D4C2×C413yes—
110C4×C7×D4C2×C413yes—
111C4×C7×D4C2311yescharacteristic
112C4×C7×D4C2313yes—
113C4×C7×D4C2313yes—
114C4×C7×D4C2313yes—
115C7×#117C2×C413yes—
116C7×#117C2×C413yes—
117C7×#117C2×C413yes—
118C7×#117C2×C413yes—
119C7×#117C2316yes—
120C2×C7×C4⋊C4C2×C413yes—
121C2×C7×C4⋊C4C2×C413yes—
122C2×C42×C7C2313yes—
123C2×C42×C7C2313yes—
124C2×C42×C7C2313yes—
125C2×#1136C2311yescharacteristic
126C2×#1136C2313yes—
127C2×#1136C2313yes—
128C2×#1136C2313yes—
129C22×C7⋊C8C2311yescharacteristic
130C22×C7⋊C8C2313yes—
131C22×C7⋊C8C2316yes—
132C22⋊(C7⋊C8)C2311yescharacteristic
133C22⋊(C7⋊C8)C2313yes—
134C22⋊(C7⋊C8)C2311yescharacteristic
135C22⋊(C7⋊C8)C2313yes—
136C22⋊(C7⋊C8)C2313yes—
137C22⋊(C7⋊C8)C2313yes—
138C22⋊(C7⋊C8)C2316yes—
139C4⋊(C7⋊C8)C2313yes—
140C4⋊(C7⋊C8)C2313yes—
141C4⋊(C7⋊C8)C2313yes—
142C4⋊(C7⋊C8)C2313yes—
143C4⋊(C7⋊C8)C2316yes—
144C4⋊(C7⋊C8)C2316yes—
145C4⋊(C7⋊C8)C2311yescharacteristic
146C4⋊(C7⋊C8)C2313yes—
147C4⋊(C7⋊C8)C2313yes—
148C4⋊(C7⋊C8)C2313yes—
149C4⋊(C7⋊C8)C2313yes—
150C4⋊(C7⋊C8)C2313yes—
151#6377C2311yescharacteristic
152#6377C2313yes—
153#6377C2313yes—
154#6377C2313yes—
155#6377C2316yes—
156C4×C7⋊C8C2313yes—
157C4×C7⋊C8C2313yes—
158C4×C7⋊C8C2313yes—
159C4×C7⋊C8C2313yes—
160C4×C7⋊C8C2313yes—
161C4×C7⋊C8C2313yes—
162C4×C7⋊C8C2316yes—
163#3482D711yescharacteristic
164#2994—73no—
165#2993—73no—
166#2990—73no—
167#2988—71no—
168#2985—73no—
169#2984—71no—
170#2982—71no—
171#2971—71no—
172#2969—73no—
173#2966—73no—
174#2965—73no—
175Q8×M16—71no—
176#2962—73no—
177C2×C8×Q8—71no—
178C7×#41—24no—
179C7×#41—46no—
180C2×C7×Q8—24no—
181C2×C7×Q8—24no—
182C2×C7×Q8C22×C413yes—
183C2×C7×Q8C22×C413yes—
184C2×C7×Q8C22×C413yes—
185C2×C7×Q8C22×C411yescharacteristic
186C2×C7×D4C22×C411yescharacteristic
187C2×C7×D4C22×C413yes—
188C2×C7×D4C22×C413yes—
189C2×C7×D4C22×C413yes—
190C22×C4×C7C22×C413yes—
191C22×C4×C7C22×C413yes—
192C22×C4×C7C22×C413yes—
193C22×C4×C7C22×C413yes—
194C22×C4×C7C22×C413yes—
195C22×C4×C7C22×C413yes—
196C22×C4×C7C2411yescharacteristic
197C22×C4×C7C2413yes—
198C22×C4×C7C2411yescharacteristic
199C7×C4⋊C4C22×C413yes—
200C7×C4⋊C4C22×C413yes—
201C7×C4⋊C4C22×C413yes—
202C7×C4⋊C4C22×C416yes—
203C7×C4⋊C4C22×C416yes—
204C7×C4⋊C4C22×C411yescharacteristic
205C7×C4⋊C4C22×C413yes—
206C7×C4⋊C4C22×C413yes—
207C7×C4⋊C4C22×C413yes—
208C7×C4⋊C4C22×C413yes—
209C7×C4⋊C4C22×C413yes—
210C7×C4⋊C4C22×C413yes—
211C7×C4⋊C4C22×C416yes—
212C7×C4⋊C4C22×C416yes—
213C7×C22⋊C4C22×C411yescharacteristic
214C7×C22⋊C4C22×C411yescharacteristic
215C7×C22⋊C4C22×C413yes—
216C7×C22⋊C4C22×C413yes—
217C7×C22⋊C4C22×C416yes—
218C7×C22⋊C4C22×C413yes—
219C7×C22⋊C4C22×C413yes—
220C7×C22⋊C4C22×C416yes—
221C7×C22⋊C4C22×C416yes—
222C42×C7C22×C413yes—
223C42×C7C22×C413yes—
224C42×C7C22×C413yes—
225C42×C7C22×C413yes—
226C42×C7C22×C413yes—
227C42×C7C22×C413yes—
228C42×C7C2413yes—
229C42×C7C2411yescharacteristic
230C42×C7C2413yes—
231C42×C7C2413yes—
232#1136—22no—
233#1136—26no—
234#1136—43no—
235#1136—43no—
236C2×C7⋊C8—22no—
237C2×C7⋊C8—46no—
238C2×C7⋊C8—43no—
239C2×C7⋊C8C2411yescharacteristic
240C2×C7⋊C8C2411yescharacteristic
241C2×C7⋊C8C2413yes—
242C2×C7⋊C8C2413yes—
243C2×C7⋊C8C2413yes—
244C2×C7⋊C8C2413yes—
245C2×C7⋊C8C2411yescharacteristic
246C2×C7⋊C8C2411yescharacteristic
247#557Dic711yescharacteristic
248#556Dic713yes—
249#552Dic713yes—
250#552Dic713yes—
251D4×Q8Dic711yescharacteristic
252#548Dic711yescharacteristic
253#548Dic711yescharacteristic
254#547Dic713yes—
255#533C2×D713yes—
256#529C2×D713yes—
257#529C2×D713yes—
258C4×#41C2×D713yes—
259C4×#41C2×D711yescharacteristic
260C2×C4×Q8C2×D711yescharacteristic
261C2×C4×Q8C2×D711yescharacteristic
262#446—73no—
263#446—73no—
264#446—73no—
265#446—76no—
266#446—73no—
267#446—73no—
268#446—73no—
269C8×Q8—73no—
270C8×Q8—71no—
271C8×Q8—71no—
272C8×Q8—73no—
273C8×Q8—73no—
274C8×Q8—71no—
275C8×Q8—73no—
276C8×Q8—71no—
277#436—73no—
278#436—73no—
279#436—73no—
280#436—73no—
281#435—71no—
282#435—73no—
283#435—73no—
284#435—73no—
285#435—76no—
286C8×D4—73no—
287C8×D4—76no—
288C8×D4—73no—
289#433—76no—
290#433—76no—
291#432—73no—
292#432—73no—
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×M16—73no—
304C2×C4×C8—73no—
305C23×C7C23×C413yes—
306C23×C7C23×C411yescharacteristic
307C23×C7C23×C411yescharacteristic
308C7×Q8—26no—
309C7×Q8—26no—
310C7×Q8—43no—
311C7×Q8#13114yes—
312C7×D4—26no—
313C7×D4—22no—
314C7×D4—26no—
315C7×D4—43no—
316C2×C4×C7—26no—
317C2×C4×C7—26no—
318C2×C4×C7—26no—
319C2×C4×C7—26no—
320C2×C4×C7—21no—
321C2×C4×C7—23no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—23no—
325C2×C4×C7—21no—
326C2×C4×C7—43no—
327C2×C4×C7—43no—
328C2×C4×C7C23×C411yescharacteristic
329C2×C4×C7C23×C411yescharacteristic
330C2×C4×C7C23×C411yescharacteristic
331C2×C4×C7C23×C413yes—
332C2×C4×C7C23×C411yescharacteristic
333C2×C4×C7C23×C413yes—
334C2×C4×C7C23×C413yes—
335C2×C4×C7C23×C413yes—
336C2×C4×C7C23×C413yes—
337C2×C4×C7C23×C413yes—
338C2×C4×C7C23×C413yes—
339C2×C4×C7C2511yescharacteristic
340C7⋊C8—21no—
341C7⋊C8—21no—
342C7⋊C8—23no—
343C7⋊C8—23no—
344C7⋊C8—23no—
345C7⋊C8—23no—
346C7⋊C8—21no—
347C7⋊C82−1+412yes—
348C2×#41C2×Dic711yescharacteristic
349C2×#41C2×Dic713yes—
350C22×Q8C2×Dic711yescharacteristic
351C22×Q8C2×Dic711yescharacteristic
352C2×M16—71no—
353C2×M16—73no—
354C2×M16—73no—
355C2×M16—73no—
356C22×C8—71no—
357C22×C8—73no—
358C22×C8—76no—
359C4⋊Q8C2×Dic713yes—
360C4⋊Q8C2×Dic713yes—
361#126C2×Dic716yes—
362#126C2×Dic716yes—
363#125C2×Dic713yes—
364#125C2×Dic713yes—
365#125C2×Dic716yes—
366#124C2×Dic713yes—
367#124C2×Dic713yes—
368#123C2×Dic716yes—
369#123C2×Dic716yes—
370C22⋊Q8C2×Dic713yes—
371C22⋊Q8C2×Dic713yes—
372C22⋊Q8C2×Dic716yes—
373C22⋊Q8C2×Dic713yes—
374C22⋊Q8C2×Dic713yes—
375#121C2×Dic713yes—
376#121C2×Dic713yes—
377C4×Q8C2×Dic711yescharacteristic
378C4×Q8C2×Dic711yescharacteristic
379C4×Q8C2×Dic713yes—
380C4×Q8C2×Dic713yes—
381C4×Q8C2×Dic711yescharacteristic
382C4×Q8C2×Dic713yes—
383C4×Q8C22×D711yescharacteristic
384C4×Q8C22×D713yes—
385C4×Q8C22×D713yes—
386C4×Q8C22×D713yes—
387C4×D4C2×Dic713yes—
388C4×D4C2×Dic713yes—
389C4×D4C2×Dic713yes—
390C4×D4C2×Dic713yes—
391C4×D4C22×D711yescharacteristic
392C4×D4C22×D713yes—
393C4×D4C22×D713yes—
394C4×D4C22×D713yes—
395#117C2×Dic713yes—
396#117C2×Dic713yes—
397#117C2×Dic713yes—
398#117C2×Dic713yes—
399#117C22×D716yes—
400C2×C4⋊C4C2×Dic713yes—
401C2×C4⋊C4C2×Dic713yes—
402C2×C42C22×D713yes—
403C2×C42C22×D713yes—
404C2×C42C22×D713yes—
405C4⋊C8—73no—
406C4⋊C8—73no—
407C4⋊C8—73no—
408C4⋊C8—73no—
409C4⋊C8—76no—
410C4⋊C8—76no—
411C4⋊C8—73no—
412C4⋊C8—73no—
413C4⋊C8—71no—
414C4⋊C8—73no—
415C4⋊C8—73no—
416C4⋊C8—73no—
417C22⋊C8—71no—
418C22⋊C8—73no—
419C22⋊C8—71no—
420C22⋊C8—73no—
421C22⋊C8—73no—
422C22⋊C8—73no—
423C22⋊C8—76no—
424#97—71no—
425#97—73no—
426#97—73no—
427#97—73no—
428#97—76no—
429C4×C8—73no—
430C4×C8—73no—
431C4×C8—73no—
432C4×C8—73no—
433C4×C8—73no—
434C4×C8—73no—
435C4×C8—76no—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7—23no—
439C22×C7—21no—
440C22×C7—21no—
441C22×C7—23no—
442C22×C7—21no—
443C22×C7—23no—
444C22×C7#56812yes—
445C22×C7C24×C411yescharacteristic
446C4×C7—21no—
447C4×C7—23no—
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—23no—
451C4×C7—23no—
452C4×C7—23no—
453C4×C7—23no—
454C4×C7—21no—
455C4×C7—23no—
456C4×C7C2×#13116yes—
457C4×C7C2×2−1+411yescharacteristic
458C4×C7#58511yescharacteristic
459#41—24no—
460#41—46no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8C22×Dic713yes—
464C2×Q8C22×Dic713yes—
465C2×Q8C22×Dic713yes—
466C2×Q8C22×Dic711yescharacteristic
467C2×D4C22×Dic711yescharacteristic
468C2×D4C22×Dic713yes—
469C2×D4C22×Dic713yes—
470C2×D4C22×Dic713yes—
471C22×C4C22×Dic713yes—
472C22×C4C22×Dic713yes—
473C22×C4C22×Dic713yes—
474C22×C4C22×Dic713yes—
475C22×C4C22×Dic713yes—
476C22×C4C22×Dic713yes—
477C22×C4C23×D711yescharacteristic
478C22×C4C23×D711yescharacteristic
479C22×C4C23×D713yes—
480M16—142no—
481M16—146no—
482M16—283no—
483M16—283no—
484C2×C8—71no—
485C2×C8—71no—
486C2×C8—73no—
487C2×C8—71no—
488C2×C8—73no—
489C2×C8—71no—
490C2×C8—73no—
491C2×C8—73no—
492C2×C8—142no—
493C2×C8—286no—
494C2×C8—283no—
495C4⋊C4C22×Dic713yes—
496C4⋊C4C22×Dic713yes—
497C4⋊C4C22×Dic713yes—
498C4⋊C4C22×Dic716yes—
499C4⋊C4C22×Dic716yes—
500C4⋊C4C22×Dic711yescharacteristic
501C4⋊C4C22×Dic713yes—
502C4⋊C4C22×Dic713yes—
503C4⋊C4C22×Dic713yes—
504C4⋊C4C22×Dic713yes—
505C4⋊C4C22×Dic713yes—
506C4⋊C4C22×Dic713yes—
507C4⋊C4C22×Dic716yes—
508C4⋊C4C22×Dic716yes—
509C22⋊C4C22×Dic711yescharacteristic
510C22⋊C4C22×Dic711yescharacteristic
511C22⋊C4C22×Dic713yes—
512C22⋊C4C22×Dic713yes—
513C22⋊C4C22×Dic716yes—
514C22⋊C4C22×Dic713yes—
515C22⋊C4C22×Dic713yes—
516C22⋊C4C22×Dic716yes—
517C22⋊C4C22×Dic716yes—
518C42C22×Dic713yes—
519C42C22×Dic713yes—
520C42C22×Dic713yes—
521C42C22×Dic713yes—
522C42C22×Dic713yes—
523C42C22×Dic713yes—
524C42C23×D713yes—
525C42C23×D713yes—
526C42C23×D713yes—
527C42C23×D711yescharacteristic
528C2×C7—22no—
529C2×C7—41no—
530C2×C7—43no—
531C2×C7C4×2−1+411yescharacteristic
532C2×C7C22×#13111yescharacteristic
533C2×C7C2×#56811yescharacteristic
534C23C23×Dic713yes—
535C23C23×Dic711yescharacteristic
536C23C23×Dic711yescharacteristic
537Q8—26no—
538Q8—26no—
539Q8—43no—
540Q8#651114yes—
541D4—26no—
542D4—22no—
543D4—26no—
544D4—43no—
545C2×C4—26no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—23no—
552C2×C4—21no—
553C2×C4—23no—
554C2×C4—21no—
555C2×C4—43no—
556C2×C4—43no—
557C2×C4C23×Dic711yescharacteristic
558C2×C4C23×Dic711yescharacteristic
559C2×C4C23×Dic711yescharacteristic
560C2×C4C23×Dic713yes—
561C2×C4C23×Dic711yescharacteristic
562C2×C4C23×Dic713yes—
563C2×C4C23×Dic713yes—
564C2×C4C23×Dic713yes—
565C2×C4C23×Dic713yes—
566C2×C4C23×Dic713yes—
567C2×C4C23×Dic713yes—
568C2×C4C24×D711yescharacteristic
569C8—72no—
570C8—141no—
571C8—141no—
572C8—143no—
573C8—143no—
574C8—143no—
575C8—143no—
576C8—141no—
577C7#6033511yescharacteristic, complemented, Sylow
578C22—21no—
579C22—21no—
580C22—21no—
581C22—23no—
582C22—21no—
583C22—23no—
584C22—23no—
585C22—21no—
586C22#9051012yes—
587C22C24×Dic711yescharacteristic
588C4—21no—
589C4—23no—
590C4—23no—
591C4—23no—
592C4—23no—
593C4—21no—
594C4—23no—
595C4—21no—
596C4—23no—
597C4—21no—
598C4C2×#651116yes—
599C4D7×2−1+411yescharacteristic
600C4#9062011yescharacteristic
601C2—22no—
602C2—41no—
603C2—43no—
604C2C22×#651111yescharacteristic
605C2C2×#9051011yescharacteristic
606C2Dic7×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
Fitting, prime radicalC7×#3482C2
cosocleC2×C4×C7C25
hypercenter#3482D7
hypocenterC7#60335
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #3482

Complements

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

NH ≅ G/Ntype
2. #11740776C2semidirect
3. #11740775C2semidirect
4. #11740577C2semidirect
5. #11740576C2semidirect
6. #11740574C2semidirect
7. #11740571C2semidirect
8. #11739967C2semidirect
9. #11739965C2semidirect
10. #11739964C2semidirect
11. #11739962C2semidirect
12. #11739882C2semidirect
13. #11739880C2semidirect
14. Q8×#1136C2semidirect
15. C2×Q8×C7⋊C8C2semidirect
577. C7#60335semidirect

Maximal subgroups

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

subgroupindex
C7×#34822
#117407762
#117407752
#117405772
#117405762
#117405742
#117405712
#117399672
#117399652
#117399642
#117399622
#117398822
#117398802
Q8×#11362
C2×Q8×C7⋊C82
#603357