Finite Group Databasebeta

← #13324869

Subgroups — #13324869

Order 1792.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737387C211yesmaximal, characteristic, complemented
2#11737099C211yesmaximal, characteristic
3#11736985C211yesmaximal, characteristic, complemented
4C7×#1592C211yesmaximal, characteristic, complemented
5#11728708C211yesmaximal, characteristic, complemented
6#11728650C211yesmaximal, characteristic, complemented
7#11728617C211yesmaximal, characteristic, complemented
8#11726329C211yesmaximal, characteristic, complemented
9#11726313C211yesmaximal, characteristic, complemented
10#11725996C211yesmaximal, characteristic, complemented
11#11725966C211yesmaximal, characteristic, complemented
12#11725921C211yesmaximal, characteristic, complemented
13#11725914C211yesmaximal, characteristic, complemented
14#11725831C211yesmaximal, characteristic, complemented
15#11725826C211yesmaximal, characteristic, complemented
16C2×D7×M16—21no—
17#90271C2211yescharacteristic
18C4×D4×D7C2211yescharacteristic
19C7×#446C2211yescharacteristic, complemented
20C4×C7×SD16C2211yescharacteristic
21C7×#436C2211yescharacteristic
22#89941C411yescharacteristic
23#89937C411yescharacteristic
24#89899—21no—
25C4×#1143C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic, complemented
27C4×#1141C2211yescharacteristic
28#89784C2211yescharacteristic
29#89643C411yescharacteristic
30#89635C411yescharacteristic
31#89612—21no—
32#89610—21no—
33#89610C2211yescharacteristic, complemented
34#89609C2211yescharacteristic
35#89606C2211yescharacteristic
36#89605C2211yescharacteristic
37#89591C411yescharacteristic
38#89581C411yescharacteristic
39#89545C411yescharacteristic
40#89538C411yescharacteristic
41#89500—21no—
42#89498—21no—
43#89480—21no—
44#89478—21no—
45C4×D56C2211yescharacteristic
46C4×#1131C2211yescharacteristic
47C8×D28C2211yescharacteristic
48C8×D28C2211yescharacteristic, complemented
49#89457C2211yescharacteristic
50#89457C2211yescharacteristic
51C7×#335C2211yescharacteristic, complemented
52C7×Q8⋊C8C2211yescharacteristic, complemented
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic, complemented
55Q8⋊(C7⋊C8)C2211yescharacteristic, complemented
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89278C2211yescharacteristic
58#89278C2211yescharacteristic, complemented
59#89275C2211yescharacteristic, complemented
60#89274C2211yescharacteristic, complemented
61#89254C2211yescharacteristic
62#89254C2211yescharacteristic, complemented
63#89252C2211yescharacteristic, complemented
64#89249C2211yescharacteristic, complemented
65#89249C2211yescharacteristic, complemented
66C8×C7⋊C8C2211yescharacteristic, complemented
67#18008—71nomaximal, Sylow
68C2×D4×D7C2×C411yescharacteristic
69C22×C4×D7—21no—
70C2×C7×SD16C2×C411yescharacteristic
71C2×C7×M16—21no—
72C4×C7×Q8C2311yescharacteristic
73C4×C7×D4C2311yescharacteristic
74#6510C2×C411yescharacteristic
75#6509C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77D4×Dic7C2×C411yescharacteristic
78C2×#1141C2×C411yescharacteristic
79C4×C22⋊D7—21no—
80C2×#1136—21no—
81D7×M16—44no—
82C2×D56C2×C411yescharacteristic
83C2×#1131—21no—
84C2×#1131—21no—
85C2×#1131D411yescharacteristic
86C2×#1131D411yescharacteristic
87C2×C8×D7—21no—
88C2×C8×D7—21no—
89C2×C8×D7—21no—
90#6461—21no—
91#6458C2×C411yescharacteristic
92#6457—21no—
93#6456C2×C411yescharacteristic
94D7×C4⋊C4C2×C411yescharacteristic
95#6453C2×C411yescharacteristic
96(C22⋊D7)⋊C4—21no—
97D7×C22⋊C4—21no—
98C4×D28—21no—
99C4×D28C2311yescharacteristic
100C4×D28C2311yescharacteristic
101#6435—21no—
102C42×D7C2311yescharacteristic
103C7×#106C2×C411yescharacteristic
104C7×C4⋊C8—21no—
105C7×C4⋊C8C2311yescharacteristic
106C7×C4⋊C8C2311yescharacteristic
107C7×#103C2×C411yescharacteristic
108C7×#102C2×C411yescharacteristic
109C7×C22⋊C8—21no—
110C7×#97—21no—
111C4×C7×C8C2311yescharacteristic
112C4×C7×C8C2311yescharacteristic
113C4×C7×C8C2311yescharacteristic
114#6409C2×C411yescharacteristic
115#6406C2×C411yescharacteristic
116C22⋊(C7⋊C8)—21no—
117#6395C2×C411yescharacteristic
118#6395C2×C411yescharacteristic
119(C2×D7)⋊C8—21no—
120(C2×D7)⋊C8—21no—
121(C2×D7)⋊C8—21no—
122(C2×D7)⋊C8—21no—
123(C2×D7)⋊C8—21no—
124(C2×D7)⋊C8—21no—
125(C2×D7)⋊C8C2×C411yescharacteristic
126(C2×D7)⋊C8C2×C411yescharacteristic
127#6392C2×C411yescharacteristic
128#6389D411yescharacteristic
129#6389D411yescharacteristic
130Dic7⋊C8—21no—
131Dic7⋊C8—21no—
132Dic7⋊C8C2×C411yescharacteristic
133Dic7⋊C8C2×C411yescharacteristic
134C8×Dic7—21no—
135#6383C2×C411yescharacteristic
136#6383C2×C411yescharacteristic
137#6382C2×C411yescharacteristic
138#6381C2×C411yescharacteristic
139C4⋊(C7⋊C8)—21no—
140C4⋊(C7⋊C8)C2311yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4×C7⋊C8—21no—
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145C4×C7⋊C8C2311yescharacteristic
146#3105—71no—
147#2969—71no—
148D4×M16—71no—
149#1592D711yescharacteristic, complemented
150#1589—71no—
151#1586—71no—
152#1585—71no—
153#1583—71no—
154#1581—71no—
155#1580—71no—
156C8×SD16—71no—
157#1574—71no—
158#1495—71no—
159#1478—71no—
160#1453—71no—
161C23×D7—21no—
162C2×C7×Q8C22×C411yescharacteristic
163C2×C7×D4C22×C411yescharacteristic
164C22×C4×C7—21no—
165C2×C22⋊D7—21no—
166C22×Dic7—21no—
167D4×D7—22no—
168D4×D7—41no—
169C2×D28—21no—
170C2×D28C22×C411yescharacteristic
171C2×D28C22×C411yescharacteristic
172C2×C4×D7—21no—
173C2×C4×D7—21no—
174C2×C4×D7—21no—
175C2×C4×D7—21no—
176C2×C4×D7—42no—
177C2×C4×D7—42no—
178C2×C4×D7C22×C411yescharacteristic
179C2×C4×D7C2×D411yescharacteristic
180C2×C4×D7#4111yescharacteristic
181C7×SD16—41no—
182C7×M16—42no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8C22×C411yescharacteristic
187C2×C7×C8C22×C411yescharacteristic
188C2×C7×C8C2×D411yescharacteristic
189C2×C7×C8C2×D411yescharacteristic
190C7×C4⋊C4—21no—
191C7×C4⋊C4C22×C411yescharacteristic
192C7×C4⋊C4C22×C411yescharacteristic
193C7×C22⋊C4—21no—
194C42×C7—21no—
195C42×C7C2411yescharacteristic
196C22⋊Dic7—21no—
197#1143—22no—
198#1141—41no—
199(C2×D7)⋊C4—21no—
200(C2×D7)⋊C4—21no—
201(C2×D7)⋊C4—21no—
202(C2×D7)⋊C4—21no—
203(C2×D7)⋊C4—21no—
204(C2×D7)⋊C4—21no—
205(C2×D7)⋊C4—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7C22×C411yescharacteristic
208C4⋊Dic7C22×C411yescharacteristic
209Dic7⋊C4—21no—
210Dic7⋊C4—21no—
211Dic7⋊C4—21no—
212C4×Dic7C22×C411yescharacteristic
213C4×Dic7C2×D411yescharacteristic
214C4×Dic7#4111yescharacteristic
215#1136—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C22×C411yescharacteristic
220C2×C7⋊C8C22×C411yescharacteristic
221C2×C7⋊C8C2×D411yescharacteristic
222C2×C7⋊C8C2×D411yescharacteristic
223D56—22no—
224#1131—22no—
225#1131—41no—
226#1131—42no—
227#1131—42no—
228C8×D7—42no—
229C8×D7—42no—
230C8×D7—42no—
231C22×M16—141no—
232#529—71no—
233C2×C4×D4—71no—
234#484—71no—
235#481—71no—
236#478—71no—
237#478—71no—
238#463—71no—
239#463—71no—
240#450—71no—
241#447—71no—
242#446C2×D711yescharacteristic, complemented
243C8×Q8—71no—
244C4×SD16—71no—
245C4×SD16C2×D711yescharacteristic
246C4×D8—71no—
247C4×D8—71no—
248#436—71no—
249#436—71no—
250#436—141no—
251#436C2×D711yescharacteristic
252#435—141no—
253#435—141no—
254#435—141no—
255C8×D4—71no—
256C8×D4—71no—
257C8×D4—71no—
258#433—141no—
259#431—71no—
260#431—71no—
261#431—71no—
262#423—71no—
263#407—141no—
264#405—141no—
265C4×M16—71no—
266#335—71no—
267#335C2×D711yescharacteristic, complemented
268#334—71no—
269#334—71no—
270Q8⋊C8—71no—
271Q8⋊C8C2×D711yescharacteristic, complemented
272D4⋊C8—71no—
273D4⋊C8—71no—
274D4⋊C8—71no—
275D4⋊C8—71no—
276D4⋊C8—71no—
277D4⋊C8C2×D711yescharacteristic
278#322—71no—
279#322—71no—
280#322C2×D711yescharacteristic, complemented
281C82—71no—
282C23×C7—21no—
283C22×D7—21no—
284C22×D7—21no—
285C22×D7—42no—
286C22×D7—42no—
287C22×D7—81no—
288C22×D7C4×D411yescharacteristic
289C7×Q8—21no—
290C7×Q8—21no—
291C7×D4—21no—
292C7×D4—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C22⋊D7—42no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—42no—
305C2×Dic7C4×D411yescharacteristic
306C2×Dic7C4×D411yescharacteristic
307C2×Dic7C4×D411yescharacteristic
308D28—21no—
309D28—21no—
310D28—21no—
311D28—81no—
312D28#13112yes—
313C4×D7—21no—
314C4×D7—21no—
315C4×D7—21no—
316C4×D7—21no—
317C4×D7—41no—
318C4×D7—42no—
319C4×D7—42no—
320C4×D7—41no—
321C4×D7—41no—
322C4×D7—41no—
323C4×D7—41no—
324C7×C8—21no—
325C7×C8—21no—
326C7×C8—21no—
327C7×C8—21no—
328C7×C8—41no—
329C7×C8—41no—
330C7×C8—41no—
331C7⋊C8—21no—
332C7⋊C8—21no—
333C7⋊C8—21no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C7⋊C8—41no—
337C7⋊C8AGL(1,ℤ/8)12yes—
338C22×D4—71no—
339C23×C4—141no—
340C2×SD16—71no—
341C2×SD16C4×D711yescharacteristic
342C2×D8—71no—
343C2×D8—71no—
344C2×M16—21no—
345C2×M16—71no—
346C2×M16—71no—
347C2×M16—141no—
348C2×M16—141no—
349C2×M16—141no—
350C2×M16—284no—
351C22×C8—141no—
352C22×C8—141no—
353C22×C8—141no—
354#126—141no—
355#125—71no—
356#124—71no—
357#123—141no—
358C22⋊Q8—71no—
359#121—71no—
360C4×Q8C22×D711yescharacteristic
361C4×D4—71no—
362C4×D4—71no—
363C4×D4—71no—
364C4×D4—71no—
365C4×D4—141no—
366C4×D4—141no—
367C4×D4—141no—
368C4×D4C22×D711yescharacteristic
369#117—141no—
370C2×C4⋊C4—71no—
371C2×C22⋊C4—141no—
372C2×C42—71no—
373#107—71no—
374#107—71no—
375#106—71no—
376#106C4×D711yescharacteristic
377C4⋊C8—21no—
378C4⋊C8—71no—
379C4⋊C8—71no—
380C4⋊C8—71no—
381C4⋊C8—71no—
382C4⋊C8—141no—
383C4⋊C8—141no—
384C4⋊C8—141no—
385C4⋊C8C22×D711yescharacteristic
386C4⋊C8C22×D711yescharacteristic
387#103—71no—
388#103C4×D711yescharacteristic
389#102—71no—
390#102—71no—
391#102—71no—
392#102—71no—
393#102—71no—
394#102C4×D711yescharacteristic
395C22⋊C8—21no—
396C22⋊C8—71no—
397C22⋊C8—71no—
398C22⋊C8—141no—
399C22⋊C8—141no—
400C22⋊C8—141no—
401C22⋊C8—141no—
402C22⋊C8—141no—
403C22⋊C8—141no—
404C22⋊C8—141no—
405#97—21no—
406#97—71no—
407#97—71no—
408C4×C8—71no—
409C4×C8—71no—
410C4×C8—71no—
411C4×C8—141no—
412C4×C8—141no—
413C4×C8C22×D711yescharacteristic
414C4×C8C22×D711yescharacteristic
415C4×C8C22×D711yescharacteristic
416C22×C7—41no—
417C22×C7—41no—
418C22×C7—41no—
419C22×C7C2×C4×D411yescharacteristic
420C2×D7—21no—
421C2×D7—22no—
422C2×D7—21no—
423C2×D7—21no—
424C2×D7—41no—
425C2×D7—41no—
426C2×D7—41no—
427C2×D7—41no—
428C2×D7—41no—
429C2×D7—84no—
430C4×C7—21no—
431C4×C7—41no—
432C4×C7—41no—
433C4×C7—41no—
434C4×C7C2×#13111yescharacteristic
435C4×C7#56811yescharacteristic
436C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
437C4×C7#57511yescharacteristic
438Dic7—21no—
439Dic7—21no—
440Dic7—21no—
441Dic7—41no—
442Dic7—41no—
443C24—141no—
444C2×Q8C2×C4×D711yescharacteristic
445C2×D4—71no—
446C2×D4—71no—
447C2×D4—141no—
448C2×D4—141no—
449C2×D4—142no—
450C2×D4—281no—
451C2×D4C2×C4×D711yescharacteristic
452C22×C4—21no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—141no—
461C22×C4—282no—
462C22×C4—282no—
463SD16—41no—
464SD16—142no—
465D8—142no—
466D8—281no—
467M16—42no—
468M16—142no—
469M16—281no—
470M16—282no—
471M16—282no—
472M16—282no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—71no—
477C2×C8—71no—
478C2×C8—71no—
479C2×C8—71no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—282no—
484C2×C8—282no—
485C2×C8—282no—
486C2×C8C2×C4×D711yescharacteristic
487C2×C8C2×C4×D711yescharacteristic
488C2×C8D4×D711yescharacteristic
489C2×C8D4×D711yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—71no—
492C4⋊C4—71no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4C2×C4×D711yescharacteristic
498C4⋊C4C2×C4×D711yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—141no—
501C22⋊C4—141no—
502C22⋊C4—141no—
503C22⋊C4—141no—
504C22⋊C4—141no—
505C22⋊C4—141no—
506C22⋊C4—141no—
507C22⋊C4—141no—
508C42—21no—
509C42—71no—
510C42—71no—
511C42—71no—
512C42C23×D711yescharacteristic
513C2×C7—81no—
514C2×C7#293311yescharacteristic
515C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
516C2×C7#295711yescharacteristic
517D7—41no—
518D7—42no—
519D7—81no—
520C23—21no—
521C23—71no—
522C23—141no—
523C23—141no—
524C23—282no—
525C23—282no—
526C23—561no—
527Q8—21no—
528Q8—21no—
529D4—21no—
530D4—21no—
531D4—72no—
532D4—141no—
533D4—141no—
534D4—141no—
535D4—282no—
536D4—561no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—71no—
543C2×C4—71no—
544C2×C4—71no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—281no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4C22×C4×D711yescharacteristic
560C2×C4C2×D4×D711yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—72no—
570C8—141no—
571C8—141no—
572C8—141no—
573C8—281no—
574C8—281no—
575C8—281no—
576C7#1800811yescharacteristic, complemented, Sylow
577C22—41no—
578C22—41no—
579C22—41no—
580C22—141no—
581C22—141no—
582C22—141no—
583C22—142no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22—281no—
588C22—281no—
589C22—564no—
590C22C4×D4×D711yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—141no—
596C4—141no—
597C4—141no—
598C4—281no—
599C4—281no—
600C4#9043611yescharacteristic
601C4D7×#13111yescharacteristic
602C4D7×AGL(1,ℤ/8)11yescharacteristic
603C4#9046811yescharacteristic
604C2—81no—
605C2—281no—
606C2—282no—
607C2—561no—
608C2#1173698411yescharacteristic
609C2#1173721811yescharacteristic
610C2#1173734411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1592C2
socleC22×C7C2×C4×D4
prime radical#89545C4
cosocleC42×C7C24
hypercenter#1592D7
hypocenterC7#18008
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
1. #11737387C2semidirect
3. #11736985C2semidirect
4. C7×#1592C2semidirect
5. #11728708C2semidirect
6. #11728650C2semidirect
7. #11728617C2semidirect
8. #11726329C2semidirect
9. #11726313C2semidirect
10. #11725996C2semidirect
11. #11725966C2semidirect
12. #11725921C2semidirect
13. #11725914C2semidirect
14. #11725831C2semidirect
15. #11725826C2semidirect
19. C7×#446C22semidirect
26. Q8×C7⋊C8C22semidirect
33. #89610C22semidirect
48. C8×D28C22semidirect
51. C7×#335C22semidirect
52. C7×Q8⋊C8C22semidirect
54. C7×#322C22semidirect
55. Q8⋊(C7⋊C8)C22semidirect
58. #89278C22semidirect
59. #89275C22semidirect
60. #89274C22semidirect
62. #89254C22semidirect
63. #89252C22semidirect
64. #89249C22semidirect
65. #89249C22semidirect
66. C8×C7⋊C8C22semidirect
149. #1592D7semidirect
242. #446C2×D7semidirect
267. #335C2×D7semidirect
271. Q8⋊C8C2×D7semidirect
280. #322C2×D7semidirect
576. C7#18008semidirect

Maximal subgroups

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

subgroupindex
#117373872
#117370992
#117369852
C7×#15922
#117287082
#117286502
#117286172
#117263292
#117263132
#117259962
#117259662
#117259212
#117259142
#117258312
#117258262
#180087