Finite Group Databasebeta

← #13363699

Subgroups — #13363699

Order 1792.

1194 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
1C2×#90271C211yesmaximal, characteristic
2C2×D7×C4⋊C8C211yesmaximal, characteristic, complemented
3#11731763C211yesmaximal, characteristic, complemented
4C7×#1803C211yesmaximal, characteristic, complemented
5#11729380C211yesmaximal, characteristic, complemented
6#11728233C211yesmaximal, characteristic, complemented
7#11728024C212yesmaximal, complemented
8#11727984C212yesmaximal, complemented
9#11727338C211yesmaximal, characteristic, complemented
10#11726637C212yesmaximal
11#11726555C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6510C2211yescharacteristic
14C2×#6509C2211yescharacteristic
15C22×C8×D7C2212yes—
16#90271—24no—
17#90271C418yescomplemented
18C2×#6461—21no—
19C2×#6458C2211yescharacteristic
20C2×#6457—21no—
21C2×#6455—21no—
22C2×#6453C2211yescharacteristic
23C2×C4×D28—21no—
24C2×C4×D28C2211yescharacteristic
25C2×#6435—21no—
26C2×C42×D7C2211yescharacteristic
27C2×C7×C4⋊C8C2211yescharacteristic
28C2×C7×#103C2212yes—
29C7×#384C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6395C2212yes—
32C2×Dic7⋊C8C2212yes—
33#89759—21no—
34#89759C2212yes—
35#89749—21no—
36C4⋊C4×Dic7—21no—
37C4⋊C4×Dic7C2212yes—
38C2×#6383C2212yes—
39#89707C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41D7×C4⋊C8—24no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21887—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×D28—21no—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C22×C4×D7D411yescharacteristic
52C22×C4×D7D411yescharacteristic
53C22×C4×D7C2311yescharacteristic
54C22×C7×C8C2312yes—
55C4×C7×Q8—22no—
56C4×C7×Q8C2×C414yes—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64#6510—22no—
65#6510C2×C414yes—
66#6509—22no—
67#6509C2×C414yes—
68C2×(C2×D7)⋊C4—21no—
69C2×(C2×D7)⋊C4—21no—
70C2×(C2×D7)⋊C4—21no—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—22no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7D411yescharacteristic
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7C2311yescharacteristic
85C22×C7⋊C8C2312yes—
86C2×C8×D7—24no—
87C2×C8×D7—42no—
88C2×C8×D7—42no—
89#6461—28no—
90#6458—22no—
91#6458C2×C414yes—
92#6457—28no—
93#6455—44no—
94#6453—22no—
95#6453C2×C414yes—
96C4×D28—28no—
97C4×D28—22no—
98C4×D28C2×C414yes—
99#6435—28no—
100C42×D7C2×C414yes—
101C42×D7D414yes—
102C7×C4⋊C8—22no—
103C7×#103—42no—
104#6409—42no—
105C7⋊#95—22no—
106C7⋊#95—22no—
107#6395—42no—
108Dic7⋊C8—24no—
109#6383—42no—
110C4⋊(C7⋊C8)—22no—
111C2×#529—71no—
112C22×C4⋊C8—71no—
113#2285—71no—
114#1989—72no—
115#1983—72no—
116#1893—72no—
117#1892—72no—
118#1803—71no—
119#1803D711yescharacteristic, complemented
120#1802—71no—
121#1802—71no—
122C23×D7—21no—
123C23×D7C2×D411yescharacteristic
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8C22×C412yes—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2411yescharacteristic
136C22×Dic7—21no—
137C22×Dic7—22no—
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C22×Dic7C2×D411yescharacteristic
141C2×D28—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—24no—
146C2×D28—21no—
147C2×D28—42no—
148C2×D28—42no—
149C2×D28C22×C412yes—
150C2×C4×D7—22no—
151C2×C4×D7—22no—
152C2×C4×D7—22no—
153C2×C4×D7—22no—
154C2×C4×D7—21no—
155C2×C4×D7—21no—
156C2×C4×D7—21no—
157C2×C4×D7—21no—
158C2×C4×D7—24no—
159C2×C4×D7—42no—
160C2×C4×D7—42no—
161C2×C4×D7C22⋊C414yes—
162C2×C4×D7C22⋊C414yes—
163C2×C4×D7C22×C412yes—
164C2×C4×D7C2×D412yes—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4C22×C412yes—
175C42×C7—24no—
176C42×C7C22×C412yes—
177C42×C7C2×D412yes—
178(C2×D7)⋊C4—24no—
179(C2×D7)⋊C4—24no—
180(C2×D7)⋊C4—24no—
181(C2×D7)⋊C4—24no—
182(C2×D7)⋊C4—24no—
183(C2×D7)⋊C4—44no—
184C4⋊Dic7—21no—
185C4⋊Dic7—24no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7—42no—
189C4⋊Dic7—42no—
190C4⋊Dic7C22×C412yes—
191Dic7⋊C4—24no—
192Dic7⋊C4—24no—
193C4×Dic7—44no—
194C4×Dic7C22⋊C414yes—
195C4×Dic7C22⋊C414yes—
196C4×Dic7C22×C412yes—
197C4×Dic7C2×D412yes—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C8×D7—48no—
202C23×C8—72no—
203#529—78no—
204#529—144no—
205C2×#126—141no—
206C2×#125—71no—
207C2×#124—71no—
208C2×#123—141no—
209C2×C22⋊Q8—71no—
210C2×#121—71no—
211C2×C4×Q8C2×D711yescharacteristic
212C2×C4×D4—71no—
213C2×C4×D4—141no—
214C2×#117—141no—
215C2×#117—141no—
216C22×C42—71no—
217C2×C4⋊C8—72no—
218C2×C4⋊C8—71no—
219C2×C4⋊C8—144no—
220C2×C4⋊C8C2×D711yescharacteristic
221C2×#103—72no—
222C2×#103C2×D712yes—
223C2×#102—72no—
224C2×#102—72no—
225#384—71no—
226#384C2×D711yescharacteristic
227#382—141no—
228#380—72no—
229#380—141no—
230C4×C4⋊C4—72no—
231C4×C4⋊C4—141no—
232#340—72no—
233#340—72no—
234#340—72no—
235#340C2×D712yes—
236C23×C7C22×D411yescharacteristic
237C22×D7—21no—
238C22×D7—21no—
239C22×D7—21no—
240C22×D7—21no—
241C22×D7—22no—
242C22×D7—41no—
243C22×D7—41no—
244C22×D7—41no—
245C22×D7—41no—
246C22×D7—41no—
247C22×D7—41no—
248C22×D7C2×C22⋊C412yes—
249C22×D7AGL(1,ℤ/8)12yes—
250C22×D7#13712yes—
251C7×Q8—22no—
252C7×Q8—22no—
253C7×Q8—41no—
254C7×Q8—41no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7C2×C22⋊C412yes—
267C2×C4×C7C2×C22⋊C412yes—
268C2×C4×C7#13512yes—
269C2×C4×C7#13512yes—
270C2×C4×C7C23×C411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7H(5,𝔽2)11yescharacteristic
273C2×C4×C7H(5,𝔽2)11yescharacteristic
274C2×C4×C72−1+411yescharacteristic
275C2×C4×C72−1+411yescharacteristic
276C2×Dic7—21no—
277C2×Dic7—21no—
278C2×Dic7—21no—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—22no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7C2×C22⋊C412yes—
289C2×Dic7C2×C22⋊C412yes—
290C2×Dic7C2×C22⋊C412yes—
291D28—22no—
292D28—22no—
293D28—48no—
294D28—41no—
295D28—41no—
296C4×D7—28no—
297C4×D7—22no—
298C4×D7—22no—
299C4×D7—22no—
300C4×D7—22no—
301C4×D7—28no—
302C4×D7—48no—
303C7×C8—42no—
304C7⋊C8—42no—
305C22×Q8C22×D711yescharacteristic
306C22×D4—71no—
307C22×D4—141no—
308C23×C4—71no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—141no—
312C22×C8—72no—
313C22×C8—144no—
314C22×C8—282no—
315C22×C8—282no—
316C22×C8C22×D712yes—
317#126—148no—
318#125—74no—
319#125—142no—
320#124—74no—
321#124—142no—
322#123—148no—
323C22⋊Q8—74no—
324C22⋊Q8—142no—
325#121—74no—
326#121—142no—
327C4×Q8—22no—
328C4×Q8C4×D714yes—
329C4×D4—74no—
330C4×D4—148no—
331C4×D4—142no—
332#117—148no—
333#117—284no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—72no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4C22×D712yes—
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—142no—
352C2×C42—21no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—74no—
357C2×C42—74no—
358C2×C42—142no—
359C2×C42C22×D711yescharacteristic
360C4⋊C8—22no—
361C4⋊C8—144no—
362C4⋊C8—142no—
363#103—42no—
364#103—282no—
365#102—282no—
366#102—282no—
367#95—142no—
368#95—142no—
369C22×C7C22×C22⋊C411yescharacteristic
370C22×C7#53711yescharacteristic
371C22×C7#53711yescharacteristic
372C22×C7C2×#13511yescharacteristic
373C22×C7C2×#13511yescharacteristic
374C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
375C22×C7C2×#13711yescharacteristic
376C2×D7—22no—
377C2×D7—22no—
378C2×D7—22no—
379C2×D7—22no—
380C2×D7—22no—
381C2×D7—22no—
382C2×D7—42no—
383C2×D7—42no—
384C2×D7—42no—
385C2×D7—81no—
386C2×D7—81no—
387C2×D7—81no—
388C2×D7—81no—
389C2×D7#41714yes—
390C4×C7—22no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—82no—
394C4×C7#41614yes—
395C4×C7#52011yescharacteristic
396C4×C7#52011yescharacteristic
397C4×C7#53511yescharacteristic
398C4×C7#53611yescharacteristic
399Dic7—22no—
400Dic7—22no—
401Dic7—22no—
402Dic7—42no—
403Dic7—82no—
404C24—71no—
405C24—141no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×C4×D712yes—
412C2×D4—72no—
413C2×D4—144no—
414C2×D4—141no—
415C2×D4—141no—
416C2×D4—141no—
417C2×D4—141no—
418C2×D4—141no—
419C2×D4—282no—
420C2×D4—282no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—71no—
425C22×C4—71no—
426C22×C4—71no—
427C22×C4—74no—
428C22×C4—74no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—144no—
441C22×C4—142no—
442C22×C4—282no—
443C22×C4—282no—
444C22×C4D4×D711yescharacteristic
445C22×C4D4×D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—142no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—288no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—72no—
461C4⋊C4—144no—
462C4⋊C4—144no—
463C4⋊C4—144no—
464C4⋊C4—141no—
465C4⋊C4—142no—
466C4⋊C4—142no—
467C4⋊C4—282no—
468C4⋊C4—282no—
469C4⋊C4C2×C4×D712yes—
470C22⋊C4—144no—
471C22⋊C4—144no—
472C22⋊C4—144no—
473C22⋊C4—144no—
474C22⋊C4—144no—
475C22⋊C4—284no—
476C42—24no—
477C42—74no—
478C42—74no—
479C42—72no—
480C42—72no—
481C42—284no—
482C42C2×C4×D712yes—
483C42D4×D712yes—
484C2×C7#231111yescharacteristic
485C2×C7C2×#41611yescharacteristic
486C2×C7C2×#41711yescharacteristic
487C2×C7#324711yescharacteristic
488C2×C7#324711yescharacteristic
489C2×C7#324811yescharacteristic
490C2×C7#324811yescharacteristic
491D7—24no—
492D7—82no—
493C23—72no—
494C23—72no—
495C23—72no—
496C23—141no—
497C23—141no—
498C23—141no—
499C23—141no—
500C23—142no—
501C23—281no—
502C23—281no—
503C23—281no—
504C23—281no—
505C23—281no—
506C23—281no—
507C23C2×D4×D711yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512D4—142no—
513D4—142no—
514D4—288no—
515D4—281no—
516D4—281no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—72no—
529C2×C4—72no—
530C2×C4—72no—
531C2×C4—148no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—148no—
543C2×C4—142no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—288no—
547C2×C4—282no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4D7×C22⋊C412yes—
551C2×C4D7×C22⋊C412yes—
552C2×C4#647912yes—
553C2×C4#648212yes—
554C2×C4C22×C4×D711yescharacteristic
555C2×C4C2×D4×D711yescharacteristic
556C2×C4#654811yescharacteristic
557C2×C4#655111yescharacteristic
558C2×C4#655311yescharacteristic
559C2×C4#655411yescharacteristic
560C8—42no—
561C8—282no—
562C7#2188711yescharacteristic, complemented, Sylow
563C22—74no—
564C22—142no—
565C22—142no—
566C22—142no—
567C22—142no—
568C22—142no—
569C22—142no—
570C22—282no—
571C22—282no—
572C22—282no—
573C22—561no—
574C22—561no—
575C22—561no—
576C22—561no—
577C22C2×D7×C22⋊C411yescharacteristic
578C22#9029911yescharacteristic
579C22#9032011yescharacteristic
580C22C2×#647911yescharacteristic
581C22C2×#648211yescharacteristic
582C22D7×AGL(1,ℤ/8)11yescharacteristic
583C22D7×#13711yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—142no—
589C4—142no—
590C4—142no—
591C4—282no—
592C4—562no—
593C4#8957614yes—
594C4#9019511yescharacteristic
595C4#9021411yescharacteristic
596C4#9029811yescharacteristic
597C4#9031911yescharacteristic
598C2—144no—
599C2—562no—
600C2#1173144811yescharacteristic
601C2C2×#8957611yescharacteristic
602C2D7×#41711yescharacteristic
603C2#1173784611yescharacteristic
604C2#1173794211yescharacteristic
605C2#1173847911yescharacteristic
606C2#1173857611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1803C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
cosocleC22×C4×C7C24
hypercenter#1803D7
hypocenterC7#21887
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×D7×C4⋊C8C2semidirect
3. #11731763C2semidirect
4. C7×#1803C2semidirect
5. #11729380C2semidirect
6. #11728233C2semidirect
7. #11728024C2semidirect
8. #11727984C2semidirect
9. #11727338C2semidirect
11. #11726555C2semidirect
17. #90271C4semidirect
119. #1803D7semidirect
562. C7#21887semidirect

Maximal subgroups

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

subgroupindex
C2×#902712
C2×D7×C4⋊C82
#117317632
C7×#18032
#117293802
#117282332
#117280242
#117279842
#117273382
#117266372
#117265552
#218877