Finite Group Databasebeta

← #13390169

Subgroups — #13390169

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×Q8×D28C211yesmaximal, characteristic, complemented
2C7×#2538C211yesmaximal, characteristic, complemented
3C2×#89779C211yesmaximal, characteristic, complemented
4C2×C4×#1141C211yesmaximal, characteristic
5#11729388C211yesmaximal, characteristic, complemented
6#11727411C212yesmaximal
7#11727392C212yesmaximal, complemented
8#11727379C212yesmaximal, complemented
9#11727358C212yesmaximal, complemented
10#11727338C211yesmaximal, characteristic, complemented
11#11727336C211yesmaximal, characteristic, complemented
12C22×Q8×D7—21no—
13C2×C7×C22⋊Q8—21no—
14C2×C4×C7×Q8C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C22×#1141C2211yescharacteristic
17Q8×D28—24no—
18Q8×D28—24no—
19C2×#6460—21no—
20C2×#6460—22no—
21C2×C4×D28C2212yes—
22C2×C4×D28C2211yescharacteristic
23C2×#6432C2212yes—
24C2×#6432C2211yescharacteristic
25C7×#396—22no—
26C7×#390C2212yes—
27C7×#389C2212yes—
28C7×#384C2211yescharacteristic
29C7×#380—21no—
30C2×#6406C2211yescharacteristic
31C2×#6406C2212yes—
32C4×#1141—24no—
33#89779—24no—
34C2×#6383C2211yescharacteristic
35C2×#6383C2212yes—
36C2×#6381C2212yes—
37C2×#6381C2211yescharacteristic
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×Q8D411yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C2×Q8×D7—28no—
50C2×Q8×D7—44no—
51C2×Q8×D7—44no—
52C22×D28C2311yescharacteristic
53C22×C4×D7—22no—
54C22×C4×D7—21no—
55C22×Dic14—22no—
56C22×Dic14—21no—
57C7×C22⋊Q8—44no—
58C4×C7×Q8—22no—
59C4×C7×Q8—22no—
60C4×C7×D4—22no—
61C4×C7×D4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C4⋊C4D412yes—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2312yes—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C42×C7C2312yes—
73C2×C42×C7C2311yescharacteristic
74C2×#1141—22no—
75C2×#1141—22no—
76C2×#1141Q814yescomplemented
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—21no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2312yes—
83C22×C7⋊C8C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85C22×C7⋊C8C2311yescharacteristic
86#6460—28no—
87#6460—216no—
88C4×D28—24no—
89C4×D28—22no—
90C4×D28—24no—
91C4×D28D414yes—
92#6432—22no—
93#6432—24no—
94#6432—24no—
95#6432D414yes—
96C7×#95—22no—
97#6406—24no—
98#6406—22no—
99#6383—22no—
100#6383—24no—
101#6381—24no—
102#6381—22no—
103C4⋊(C7⋊C8)—22no—
104C4×C7⋊C8—22no—
105C2×D4×Q8—71no—
106C2×#474—71no—
107C2×C4×D8—71no—
108#2538D711yescharacteristic, complemented
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—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8D814yescomplemented
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4C2×Q812yescomplemented
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—42no—
130C22×C4×C7—42no—
131C22×C4×C7—42no—
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C22×Dic7—21no—
138C22×Dic7—22no—
139Q8×D7—432no—
140C2×D28—21no—
141C2×D28—21no—
142C2×D28—21no—
143C2×D28C2×D412yes—
144C2×D28C2×Q812yes—
145C2×D28#4112yes—
146C2×C4×D7—28no—
147C2×C4×D7—24no—
148C2×C4×D7—44no—
149C2×C4×D7—44no—
150C2×C4×D7—42no—
151C2×C4×D7—42no—
152C2×Dic14—28no—
153C2×Dic14—24no—
154C2×Dic14—44no—
155C2×Dic14—44no—
156C2×Dic14—42no—
157C2×Dic14—42no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—42no—
165C7×C4⋊C4—44no—
166C7×C4⋊C4—81no—
167C7×C22⋊C4—42no—
168C7×C22⋊C4—44no—
169C7×C22⋊C4—82no—
170C42×C7—22no—
171C42×C7—22no—
172C42×C7C2×D412yes—
173C42×C7C2×D412yes—
174#1141—28no—
175(C2×D7)⋊C4—28no—
176(C2×D7)⋊C4—24no—
177C4⋊Dic7—28no—
178C4⋊Dic7—24no—
179C4⋊Dic7—22no—
180C4⋊Dic7—21no—
181C4⋊Dic7C2×D412yes—
182C4⋊Dic7C2×D414yes—
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—141no—
193C22×D8—71no—
194D4×Q8—144no—
195D4×Q8—144no—
196C2×C4⋊Q8—71no—
197C2×C4⋊Q8—72no—
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8—141no—
200C2×C22⋊Q8—142no—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4—71no—
203C2×C4×D4—72no—
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—22no—
216#390C2×D712yes—
217#389C2×D712yes—
218#384C2×D711yescharacteristic
219#380—21no—
220#340—72no—
221#340—72no—
222#340—72no—
223#336—72no—
224#336—72no—
225C23×C7—22no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7C22×D411yescharacteristic
233C22×D7—22no—
234C22×D7—22no—
235C22×D7—22no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—41no—
239C22×D7—41no—
240C7×Q8—28no—
241C7×D4—22no—
242C7×D4—22no—
243C7×D4—41no—
244C7×D4—41no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—84no—
259C2×C4×C7#12012yes—
260C2×C4×C7#12012yes—
261C2×C4×C7C2×D814yes—
262C2×C4×C7C2×D812yes—
263C2×C4×C7AGL(1,ℤ/8)12yes—
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×Q811yescharacteristic
267C2×C4×C7C2×#4111yescharacteristic
268C2×C4×C72−1+411yescharacteristic
269C2×C4×C72−1+411yescharacteristic
270C2×Dic7—22no—
271C2×Dic7—24no—
272C2×Dic7—41no—
273C2×Dic7—42no—
274C2×Dic7—41no—
275C2×Dic7—42no—
276D28—22no—
277D28—22no—
278D28—22no—
279D28C22⋊Q814yescomplemented
280C4×D7—416no—
281C4×D7—48no—
282Dic14—416no—
283Dic14—48no—
284C7⋊C8—22no—
285C7⋊C8—41no—
286C7⋊C8—42no—
287C22×Q8—148no—
288C22×Q8—142no—
289C22×Q8—141no—
290C22×Q8—284no—
291C22×Q8—284no—
292C22×Q8C22⋊D711yescharacteristic
293C22×D4—71no—
294C22×D4C22×D711yescharacteristic
295C23×C4—21no—
296C23×C4—142no—
297C23×C4—141no—
298C2×D8—74no—
299C2×D8—142no—
300C2×D8—142no—
301C22×C8—71no—
302C22×C8—71no—
303C22×C8—72no—
304C4⋊Q8—74no—
305C4⋊Q8—144no—
306C4⋊Q8—144no—
307C4⋊Q8—142no—
308C22⋊Q8—44no—
309C22⋊Q8—148no—
310C22⋊Q8—1416no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—74no—
316C4×D4—144no—
317C4×D4—144no—
318C4×D4—142no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—72no—
324C2×C4⋊C4—142no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4C22⋊D711yescharacteristic
327C2×C4⋊C4C22⋊D712yes—
328C2×C4⋊C4C22×D711yescharacteristic
329C2×C4⋊C4C22×D712yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—142no—
334C2×C22⋊C4—141no—
335C2×C42C22×D712yes—
336C2×C42C22×D711yescharacteristic
337#107—142no—
338#107—144no—
339C4⋊C8—142no—
340#102—142no—
341#102—144no—
342#102—142no—
343#102—144no—
344C4×C8—142no—
345#95—22no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
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—24no—
361C2×D7—42no—
362C2×D7—42no—
363C2×D7—42no—
364C2×D7—42no—
365C2×D7—42no—
366C2×D7—42no—
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—42no—
370C4×C7—82no—
371C4×C7—81no—
372C4×C7#44714yes—
373C4×C7C2×C22⋊Q811yescharacteristic
374C4×C7#53611yescharacteristic
375C4×C7#54811yescharacteristic
376C4×C7D4×Q811yescharacteristic
377Dic7—42no—
378Dic7—44no—
379C24—21no—
380C24—141no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—148no—
384C2×Q8—144no—
385C2×Q8—284no—
386C2×Q8—2832no—
387C2×Q8—284no—
388C2×Q8—282no—
389C2×Q8—282no—
390C2×Q8#114114yescomplemented
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—72no—
397C2×D4—72no—
398C2×D4—72no—
399C2×D4—141no—
400C2×D4—141no—
401C2×D4—141no—
402C2×D4Q8×D712yescomplemented
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—148no—
409C22×C4—142no—
410C22×C4—141no—
411C22×C4—144no—
412C22×C4—284no—
413C22×C4—284no—
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—141no—
426C2×C8—142no—
427C2×C8—141no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8—141no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—72no—
440C4⋊C4—74no—
441C4⋊C4—81no—
442C4⋊C4—148no—
443C4⋊C4—144no—
444C4⋊C4—142no—
445C4⋊C4—141no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—82no—
449C22⋊C4—148no—
450C22⋊C4—144no—
451C42—22no—
452C42—22no—
453C42D4×D712yes—
454C42D4×D712yes—
455C2×C7—82no—
456C2×C7#240911yescharacteristic
457C2×C7C2×#44711yescharacteristic
458C2×C7#301311yescharacteristic
459C2×C7#335811yescharacteristic
460C2×C7#336711yescharacteristic
461C2×C7Q8×D811yescharacteristic
462C2×C7#338411yescharacteristic
463D7—44no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—142no—
472C23—142no—
473C23—142no—
474C23—281no—
475C23—281no—
476C23—281no—
477C23—281no—
478C23C22×C22⋊D711yescharacteristic
479Q8—28no—
480Q8—2816no—
481Q8—288no—
482D4—22no—
483D4—22no—
484D4—41no—
485D4—41no—
486D4—74no—
487D4—142no—
488D4—142no—
489D4—142no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—144no—
505C2×C4—142no—
506C2×C4—2816no—
507C2×C4—282no—
508C2×C4—281no—
509C2×C4—282no—
510C2×C4—281no—
511C2×C4—288no—
512C2×C4C2×#114114yes—
513C2×C4C2×#114112yes—
514C2×C4#650012yes—
515C2×C4#650012yes—
516C2×C4#651212yes—
517C2×C4C2×D4×D711yescharacteristic
518C2×C4C2×D4×D711yescharacteristic
519C2×C4C2×Q8×D711yescharacteristic
520C2×C4#655111yescharacteristic
521C2×C4D7×#4111yescharacteristic
522C2×C4#655411yescharacteristic
523C8—142no—
524C8—281no—
525C8—282no—
526C7#2432611yescharacteristic, complemented, Sylow
527C22—42no—
528C22—42no—
529C22—42no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—144no—
535C22—282no—
536C22—282no—
537C22—282no—
538C22—282no—
539C22—282no—
540C22—282no—
541C22C22×#114111yescharacteristic
542C22C2×#650011yescharacteristic
543C22Q8×C22⋊D711yescharacteristic
544C22C2×#651211yescharacteristic
545C22#9052211yescharacteristic
546C22#9052711yescharacteristic
547C22#9052911yescharacteristic
548C4—24no—
549C4—22no—
550C4—42no—
551C4—82no—
552C4—81no—
553C4—284no—
554C4—282no—
555C4#8980814yes—
556C4D7×C22⋊Q811yescharacteristic
557C4#9031911yescharacteristic
558C4#9032311yescharacteristic
559C4#9032411yescharacteristic
560C2—82no—
561C2—284no—
562C2C2×#8980811yescharacteristic
563C2#1174019611yescharacteristic
564C2#1174087211yescharacteristic
565C2Q8×#114111yescharacteristic
566C2#1174102511yescharacteristic
567C2#1174107511yescharacteristic
568C2#1174111311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2538C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenter#2538D7
hypocenterC7#24326
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D28C2semidirect
2. C7×#2538C2semidirect
3. C2×#89779C2semidirect
5. #11729388C2semidirect
7. #11727392C2semidirect
8. #11727379C2semidirect
9. #11727358C2semidirect
10. #11727338C2semidirect
11. #11727336C2semidirect
76. C2×#1141Q8semidirect
108. #2538D7semidirect
120. C2×C7×Q8D8semidirect
126. C2×C7×D4C2×Q8semidirect
191. C2×C7⋊C8C2×Q8semidirect
279. D28C22⋊Q8semidirect
390. C2×Q8#1141semidirect
402. C2×D4Q8×D7semidirect
526. C7#24326semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D282
C7×#25382
C2×#897792
C2×C4×#11412
#117293882
#117274112
#117273922
#117273792
#117273582
#117273382
#117273362
#243267