Finite Group Databasebeta

← #13390161

Subgroups — #13390161

Order 1792.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#89859C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic, complemented
4C2×C4×#1141C211yesmaximal, characteristic
5#11729546C212yesmaximal, complemented
6#11729541C211yesmaximal, characteristic, complemented
7#11729536C212yesmaximal, complemented
8#11729388C211yesmaximal, characteristic, complemented
9#11729367C212yesmaximal
10#11729360C212yesmaximal, complemented
11#11729358C211yesmaximal, characteristic, complemented
12C23×C7×Q8—21no—
13C7×D4×Q8—24no—
14C7×D4×Q8—24no—
15C2×C7×C4⋊Q8C2212yes—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C2×#6509—21no—
23C22×#1141C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6406C2212yes—
27#89859—24no—
28C4×#1141—24no—
29#89766—22no—
30#89759—21no—
31C2×#6383C2211yescharacteristic
32C2×#6383C2212yes—
33C2×#6381C2211yescharacteristic
34C2×#6381C2212yes—
35#89711C2212yes—
36#89707C2211yescharacteristic
37#89700C2212yes—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39C2×C4×C7⋊C8C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24326—71nomaximal, Sylow
46C22×C7×Q8—28no—
47C22×C7×Q8—22no—
48C22×C7×Q8—21no—
49C22×C7×Q8—44no—
50C22×C7×Q8—44no—
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—22no—
54C23×C4×C7—21no—
55C22×D28C2311yescharacteristic
56C22×C4×D7—21no—
57C7×C4⋊Q8—24no—
58C7×C4⋊Q8—24no—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8D414yes—
61C7×C22⋊Q8—28no—
62C7×C22⋊Q8—216no—
63C4×C7×Q8—22no—
64C4×C7×Q8—22no—
65C4×C7×D4—24no—
66C4×C7×D4—24no—
67C4×C7×D4—22no—
68C4×C7×D4D414yes—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D412yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7C2312yes—
78C2×C42×C7C2311yescharacteristic
79#6509—44no—
80C2×#1141—22no—
81C2×#1141—22no—
82C2×#1141Q814yescomplemented
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2312yes—
90C2×Dic7⋊C4—22no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2311yescharacteristic
94C4×D28—22no—
95C4×D28—22no—
96#6406—22no—
97#6406—24no—
98C7⋊#95—22no—
99#6383—24no—
100#6383—22no—
101#6381—24no—
102#6381—22no—
103C4⋊(C7⋊C8)—22no—
104C4×C7⋊C8—22no—
105C2×D4×Q8D711yescharacteristic, complemented
106C2×#474—71no—
107C2×C4×D8—71no—
108#2538—71no—
109#2057—72no—
110#2023—72no—
111#1990—71no—
112#1984—71no—
113#1879—72no—
114#1862—72no—
115#1802—71no—
116C24×C7—21no—
117C23×D7—21no—
118C2×C7×Q8—28no—
119C2×C7×Q8—22no—
120C2×C7×Q8—24no—
121C2×C7×Q8—22no—
122C2×C7×Q8—44no—
123C2×C7×Q8—432no—
124C2×C7×Q8—44no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×Q8D814yescomplemented
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4C2×D412yes—
132C2×C7×D4C2×Q812yes—
133C2×C7×D4#4112yes—
134C22×C4×C7—28no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—24no—
138C22×C4×C7—44no—
139C22×C4×C7—44no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—22no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28C2×Q812yescomplemented
155C2×C4×D7—42no—
156C2×C4×D7—42no—
157C2×C4×D7—42no—
158C7×C4⋊C4—28no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4C2×D414yes—
165C7×C4⋊C4C2×D412yes—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—24no—
168C42×C7—22no—
169C42×C7—22no—
170C42×C7C2×D412yes—
171C42×C7C2×D412yes—
172#1141—28no—
173(C2×D7)⋊C4—42no—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—82no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—42no—
181C4⋊Dic7—81no—
182Dic7⋊C4—44no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8C2×Q812yescomplemented
192C23×Q8—21no—
193C22×D8—71no—
194D4×Q8—24no—
195D4×Q8—24no—
196C2×C4⋊Q8C2×D712yes—
197C2×C4⋊Q8C2×D711yescharacteristic
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8—22no—
200C2×C22⋊Q8—141no—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4—71no—
203C2×C4×D4C2×D712yes—
204C2×C4×D4C2×D711yescharacteristic
205#474—144no—
206C4×D8—144no—
207C2×#107—71no—
208C2×#107—72no—
209C2×C4⋊C8—71no—
210C2×#102—72no—
211C2×#102—71no—
212C2×#102—72no—
213C2×#102—71no—
214C2×C4×C8—71no—
215#396—142no—
216#390—72no—
217#389—72no—
218#384—71no—
219#380—141no—
220#340—72no—
221#340—72no—
222#340—72no—
223#336—72no—
224#336—72no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7C22×D411yescharacteristic
233C22×D7—22no—
234C22×D7—41no—
235C22×D7—41no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—41no—
239C22×D7—41no—
240C7×Q8—28no—
241C7×Q8—416no—
242C7×Q8—48no—
243C7×D4—22no—
244C7×D4—22no—
245C7×D4—22no—
246C7×D4C22⋊Q814yescomplemented
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—416no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—48no—
262C2×C4×C7#12012yes—
263C2×C4×C7#12012yes—
264C2×C4×C7C2×D814yes—
265C2×C4×C7C2×D812yes—
266C2×C4×C7AGL(1,ℤ/8)12yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×Q811yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C72−1+411yescharacteristic
272C2×C4×C72−1+411yescharacteristic
273C2×Dic7—41no—
274C2×Dic7—42no—
275C2×Dic7—41no—
276C2×Dic7—42no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279D28—22no—
280D28—22no—
281D28—41no—
282D28—41no—
283C4×D7—84no—
284C7⋊C8—22no—
285C7⋊C8—41no—
286C7⋊C8—42no—
287C22×Q8—28no—
288C22×Q8—22no—
289C22×Q8—21no—
290C22×Q8—44no—
291C22×Q8—44no—
292C22×Q8C22⋊D711yescharacteristic
293C22×D4—71no—
294C22×D4C22×D711yescharacteristic
295C23×C4—22no—
296C23×C4—21no—
297C23×C4—141no—
298C2×D8—74no—
299C2×D8—142no—
300C2×D8—142no—
301C22×C8—71no—
302C22×C8—71no—
303C22×C8—72no—
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8—22no—
307C4⋊Q8C22⋊D714yes—
308C22⋊Q8—28no—
309C22⋊Q8—216no—
310C22⋊Q8—284no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—22no—
316C4×D4—142no—
317C4×D4—142no—
318C4×D4C22⋊D714yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—71no—
322C2×C4⋊C4—72no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4C22⋊D711yescharacteristic
327C2×C4⋊C4C22⋊D712yes—
328C2×C4⋊C4C22×D712yes—
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—141no—
333C2×C22⋊C4—142no—
334C2×C22⋊C4—142no—
335C2×C42C22×D712yes—
336C2×C42C22×D711yescharacteristic
337#107—142no—
338#107—144no—
339C4⋊C8—142no—
340#102—144no—
341#102—142no—
342#102—142no—
343#102—144no—
344C4×C8—142no—
345#95—142no—
346C22×C7—24no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7C2×#12011yescharacteristic
354C22×C7#54811yescharacteristic
355C22×C7D4×Q811yescharacteristic
356C22×C7C22×D811yescharacteristic
357C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C7#57711yescharacteristic
359C22×C7#57811yescharacteristic
360C2×D7—42no—
361C2×D7—42no—
362C2×D7—42no—
363C2×D7—81no—
364C2×D7—81no—
365C2×D7—81no—
366C2×D7—81no—
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—44no—
370C4×C7—42no—
371C4×C7—42no—
372C4×C7#44714yes—
373C4×C7C2×C22⋊Q811yescharacteristic
374C4×C7#53611yescharacteristic
375C4×C7#54811yescharacteristic
376C4×C7D4×Q811yescharacteristic
377Dic7—82no—
378Dic7—81no—
379C24—21no—
380C24—141no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—44no—
386C2×Q8—432no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8#114114yescomplemented
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—72no—
395C2×D4—141no—
396C2×D4—141no—
397C2×D4—141no—
398C2×D4—141no—
399C2×D4—141no—
400C2×D4Q8×D712yes—
401C2×D4#116112yes—
402C2×D4C2×C22⋊D712yes—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—141no—
412C22×C4—142no—
413C22×C4—282no—
414C22×C4—282no—
415C22×C4—282no—
416C22×C4C2×C22⋊D712yes—
417C22×C4C2×C22⋊D712yes—
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C2×C22⋊D711yescharacteristic
420C22×C4C23×D711yescharacteristic
421D8—148no—
422C2×C8—72no—
423C2×C8—141no—
424C2×C8—141no—
425C2×C8—142no—
426C2×C8—141no—
427C2×C8—141no—
428C2×C8—142no—
429C2×C8—141no—
430C2×C8—142no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—284no—
442C4⋊C4—282no—
443C4⋊C4—561no—
444C4⋊C4C2×C22⋊D714yes—
445C4⋊C4C2×C22⋊D712yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—284no—
449C22⋊C4—282no—
450C22⋊C4—562no—
451C42—22no—
452C42—22no—
453C42C2×C22⋊D712yes—
454C42C2×C22⋊D712yes—
455C2×C7—44no—
456C2×C7#240911yescharacteristic
457C2×C7C2×#44711yescharacteristic
458C2×C7#301311yescharacteristic
459C2×C7#335811yescharacteristic
460C2×C7#336711yescharacteristic
461C2×C7Q8×D811yescharacteristic
462C2×C7#338411yescharacteristic
463D7—82no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—142no—
472C23—281no—
473C23—281no—
474C23—281no—
475C23—281no—
476C23—281no—
477C23—281no—
478C23C22×C22⋊D711yescharacteristic
479Q8—28no—
480Q8—416no—
481Q8—48no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—142no—
486D4—142no—
487D4—281no—
488D4—281no—
489D4#650914yescomplemented
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—282no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—281no—
511C2×C4—564no—
512C2×C4C2×#114114yes—
513C2×C4C2×#114112yes—
514C2×C4#649512yes—
515C2×C4#651612yes—
516C2×C4#651612yes—
517C2×C4C2×Q8×D711yescharacteristic
518C2×C4C2×#116111yescharacteristic
519C2×C4#655111yescharacteristic
520C2×C4#655111yescharacteristic
521C2×C4C22×C22⋊D711yescharacteristic
522C2×C4C22×C22⋊D711yescharacteristic
523C8—142no—
524C8—281no—
525C8—282no—
526C7#2432611yescharacteristic, complemented, Sylow
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—282no—
535C22—282no—
536C22—282no—
537C22—561no—
538C22—561no—
539C22—561no—
540C22—561no—
541C22C22×#114111yescharacteristic
542C22C2×#649511yescharacteristic
543C22Q8×C22⋊D711yescharacteristic
544C22#9050811yescharacteristic
545C22#9052811yescharacteristic
546C22#9052911yescharacteristic
547C22C2×#651611yescharacteristic
548C4—24no—
549C4—22no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—561no—
554C4—562no—
555C4#8998914yes—
556C4C2×#650911yescharacteristic
557C4Q8×C22⋊D711yescharacteristic
558C4#9050711yescharacteristic
559C4#9050811yescharacteristic
560C2—44no—
561C2—562no—
562C2Q8×#114111yescharacteristic
563C2#1174100711yescharacteristic
564C2#1174100811yescharacteristic
565C2#1174100911yescharacteristic
566C2C2×#8998911yescharacteristic
567C2#1174151311yescharacteristic
568C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×D4×Q8C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenterC2×D4×Q8D7
hypocenterC7#24326
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×D4×Q8C2semidirect
2. C2×#89859C2semidirect
3. #11739945C2semidirect
5. #11729546C2semidirect
6. #11729541C2semidirect
7. #11729536C2semidirect
8. #11729388C2semidirect
10. #11729360C2semidirect
11. #11729358C2semidirect
82. C2×#1141Q8semidirect
105. C2×D4×Q8D7semidirect
127. C2×C7×Q8D8semidirect
154. C2×D28C2×Q8semidirect
191. C2×C7⋊C8C2×Q8semidirect
246. C7×D4C22⋊Q8semidirect
390. C2×Q8#1141semidirect
489. D4#6509semidirect
526. C7#24326semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
C2×#898592
#117399452
C2×C4×#11412
#117295462
#117295412
#117295362
#117293882
#117293672
#117293602
#117293582
#243267