Finite Group Databasebeta

← #13354744

Subgroups — #13354744

Order 1792.

1502 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×#41C211yesmaximal, characteristic
2C2×#90005C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4C2×#89770C211yesmaximal, characteristic, complemented
5#11739514C211yesmaximal, characteristic, complemented
6#11729383C212yesmaximal, complemented
7#11729382C212yesmaximal, complemented
8#11729361C212yesmaximal, complemented
9#11729360C212yesmaximal, complemented
10#11729334C211yesmaximal, characteristic, complemented
11C2×#89319C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C4×C7×#41—24no—
14C4×C7×#41—24no—
15C2×C4×C7×Q8C2212yes—
16C2×C4×C7×D4—22no—
17C2×C4×C7×D4C2212yes—
18C2×C7×#117—21no—
19C2×C7×#117C2211yescharacteristic
20C22×C42×C7—21no—
21C22×C42×C7C2211yescharacteristic
22C2×#6489C2211yescharacteristic
23C2×#6489C2211yescharacteristic
24C23×C7⋊C8C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26#90005—24no—
27#90005C418yescomplemented
28C2×#6409C2212yes—
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6406C2212yes—
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89982—22no—
35#89770—24no—
36C2×#6382C2212yes—
37C2×#6381C2212yes—
38C4×Dic7⋊C4—21no—
39C4×Dic7⋊C4—21no—
40#89700C2212yes—
41#89700C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89319C2212yes—
45#89319C2212yes—
46#20872—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41C2×C414yes—
51C2×C7×#41D414yes—
52C2×C7×#41Q814yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—22no—
58C23×C4×C7D412yes—
59C23×C4×C7C2311yescharacteristic
60C4×C7×Q8—24no—
61C4×C7×Q8—24no—
62C4×C7×D4—216no—
63C4×C7×D4—24no—
64C4×C7×D4—24no—
65C7×#117—28no—
66C7×#117—22no—
67C7×#117D414yes—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—22no—
72C2×C42×C7—28no—
73C2×C42×C7—21no—
74C2×C42×C7—22no—
75C2×C42×C7—22no—
76C2×C42×C7—22no—
77C2×C42×C7D411yescharacteristic
78C2×C42×C7D411yescharacteristic
79C2×C42×C7C2312yes—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82#6489—22no—
83#6489—22no—
84#6489—22no—
85#6489C2×C414yescomplemented
86C2×C4⋊Dic7C2312yes—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—22no—
90C2×Dic7⋊C4—22no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C22×C7⋊C8—22no—
94C22×C7⋊C8—22no—
95C22×C7⋊C8—22no—
96C22×C7⋊C8—22no—
97C22×C7⋊C8—22no—
98C22×C7⋊C8C2×C414yescomplemented
99C22×C7⋊C8C2312yes—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6409—24no—
103#6409C2×C414yescomplemented
104#6409C2×C414yescomplemented
105#6406—24no—
106#6406C2×C414yescomplemented
107#6406C2×C414yescomplemented
108C7⋊#95—22no—
109C7⋊#95—22no—
110#6382—24no—
111#6381—24no—
112C2×C4×#41D711yescharacteristic
113C2×#427—71no—
114C2×#416—71no—
115C2×#416—71no—
116#2296—71no—
117#1865—72no—
118#1864—72no—
119#1863—72no—
120#1862—72no—
121#1735—71no—
122C2×#336—71no—
123C24×C7—21no—
124C24×C7C2×D411yescharacteristic
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41C4⋊C4116yescomplemented
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8C22×C412yes—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8C2×Q812yes—
135C2×C7×D4—24no—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C22×C412yes—
142C2×C7×D4C2×D412yes—
143C2×C7×D4C2×Q812yes—
144C22×C4×C7—21no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—28no—
149C22×C4×C7—24no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—42no—
153C22×C4×C7—44no—
154C22×C4×C7—44no—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×Q812yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—22no—
168C7×C4⋊C4—28no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4C2×D414yes—
171C7×C22⋊C4—28no—
172C7×C22⋊C4—24no—
173C42×C7—28no—
174C42×C7—22no—
175C42×C7—24no—
176C42×C7—22no—
177C42×C7—22no—
178C42×C7—22no—
179C22⋊Dic7—24no—
180C22⋊Dic7—42no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185C4⋊Dic7C22×C412yes—
186C4⋊Dic7C22×C412yes—
187Dic7⋊C4—44no—
188Dic7⋊C4—44no—
189C4×Dic7—24no—
190C4×Dic7—42no—
191C2×C7⋊C8—28no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—44no—
200C2×C7⋊C8C22×C412yes—
201C2×C7⋊C8C22×C412yes—
202C22×#41C2×D711yescharacteristic
203C23×C8—71no—
204C23×C8—71no—
205C4×#41—24no—
206C4×#41—24no—
207C2×C4×Q8C2×D712yes—
208C2×C4×D4—22no—
209C2×C4×D4C2×D712yes—
210C2×#117—21no—
211C2×#117—71no—
212C2×#117—71no—
213C2×#117C2×D711yescharacteristic
214C22×C42—21no—
215C22×C42C2×D711yescharacteristic
216#427—144no—
217C2×#107—72no—
218C2×#106—72no—
219#416—78no—
220#416—144no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—72no—
224C2×#102—71no—
225C2×#102—71no—
226C2×#102—72no—
227#389—72no—
228#389—72no—
229#385—142no—
230C4×C4⋊C4—141no—
231C4×C4⋊C4—141no—
232#340—72no—
233#340—72no—
234#336—72no—
235#336—72no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7C4×D412yes—
244C23×C7#12012yes—
245C23×C7C22⋊Q812yes—
246C23×C7C22×D411yescharacteristic
247C7×Q8—22no—
248C7×Q8—22no—
249C7×Q8—22no—
250C7×Q8C2×C4⋊C414yes—
251C7×D4—216no—
252C7×D4—22no—
253C7×D4—22no—
254C7×D4—22no—
255C7×D4C2×C4⋊C414yes—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—216no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—48no—
267C2×C4×C7—416no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7C2×C4⋊C414yes—
271C2×C4×C7C4×D414yes—
272C2×C4×C7C4×D412yes—
273C2×C4×C7#12012yes—
274C2×C4×C7C22⋊Q814yes—
275C2×C4×C7C22⋊Q812yes—
276C2×C4×C7#12314yes—
277C2×C4×C7#13514yes—
278C2×C4×C7C23×C411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×Q811yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C7⋊C8—22no—
292C7⋊C8—22no—
293C7⋊C8—42no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41Dic1414yes—
298C2×#41C4×D714yes—
299C2×#41C22⋊D714yes—
300C22×Q8C22×D711yescharacteristic
301C22×D4—21no—
302C22×D4C22×D711yescharacteristic
303C23×C4—21no—
304C23×C4—22no—
305C23×C4C22⋊D712yes—
306C23×C4C22×D711yescharacteristic
307C22×C8—74no—
308C22×C8—71no—
309C22×C8—71no—
310C22×C8—72no—
311C22×C8—142no—
312C22×C8—142no—
313C22×C8—142no—
314C22×C8—142no—
315C22×C8—142no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×D4—216no—
319C4×D4—24no—
320C4×D4—24no—
321#117—28no—
322#117—22no—
323#117—74no—
324#117—142no—
325#117—142no—
326#117—142no—
327#117C22⋊D714yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—72no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4C22×D712yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—141no—
338C2×C22⋊C4—141no—
339C2×C42—28no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—141no—
345C2×C42—141no—
346C2×C42C22⋊D711yescharacteristic
347C2×C42C22⋊D711yescharacteristic
348C2×C42C22×D712yes—
349#107—144no—
350#106—144no—
351#103—74no—
352#103—74no—
353#103—144no—
354#102—74no—
355#102—74no—
356#102—144no—
357#95—142no—
358#95—142no—
359C22×C7—24no—
360C22×C7—22no—
361C22×C7—22no—
362C22×C7—24no—
363C22×C7—24no—
364C22×C7—44no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—44no—
368C22×C7#38514yes—
369C22×C7C2×C4×D411yescharacteristic
370C22×C7C2×#12011yescharacteristic
371C22×C7C2×C22⋊Q811yescharacteristic
372C22×C7C2×#13512yes—
373C22×C7#57512yes—
374C4×C7—24no—
375C4×C7—24no—
376C4×C7—42no—
377C4×C7—44no—
378C4×C7#38514yes—
379C4×C7C22×C4⋊C411yescharacteristic
380C4×C7C2×C4×D411yescharacteristic
381C4×C7C2×C22⋊Q811yescharacteristic
382C4×C7C2×#12311yescharacteristic
383Dic7—42no—
384Dic7—42no—
385Dic7—82no—
386C24—21no—
387C24C2×C22⋊D711yescharacteristic
388#41—28no—
389#41—28no—
390#41—28no—
391#41Dic7⋊C4116yescomplemented
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×Dic1412yes—
396C2×Q8C2×C4×D712yes—
397C2×Q8C2×C22⋊D712yes—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4C2×Dic1412yes—
405C2×D4C2×C4×D712yes—
406C2×D4C2×C22⋊D712yes—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—142no—
421C22×C4C2×Dic1412yes—
422C22×C4C2×C4×D712yes—
423C22×C4#115712yes—
424C22×C4#115712yes—
425C22×C4C2×C22⋊D712yes—
426C22×C4C2×C22⋊D712yes—
427C22×C4C2×C22⋊D712yes—
428C22×C4C2×C22⋊D712yes—
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—72no—
432C2×C8—72no—
433C2×C8—148no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—142no—
439C2×C8—142no—
440C2×C8—142no—
441C2×C8—284no—
442C4⋊C4—28no—
443C4⋊C4—22no—
444C4⋊C4—72no—
445C4⋊C4—72no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—142no—
449C4⋊C4—142no—
450C4⋊C4—284no—
451C4⋊C4—284no—
452C4⋊C4C2×C22⋊D714yes—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—144no—
456C22⋊C4—282no—
457C42—28no—
458C42—22no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—144no—
464C42—282no—
465C2×C7—24no—
466C2×C7—44no—
467C2×C7C2×#38511yescharacteristic
468C2×C7C4×#13511yescharacteristic
469C2×C7#294311yescharacteristic
470C2×C7#300212yes—
471C2×C7#307612yes—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—42no—
478C23—42no—
479C23#648712yes—
480C23C4×C22⋊D712yes—
481C23#651612yes—
482C23C22×C22⋊D711yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8C2×Dic7⋊C414yes—
487D4—216no—
488D4—22no—
489D4—22no—
490D4—22no—
491D4C2×Dic7⋊C414yes—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—216no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—48no—
503C2×C4—416no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—142no—
507C2×C4—142no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4C2×Dic7⋊C414yes—
514C2×C4#648714yes—
515C2×C4#648712yes—
516C2×C4C4×C22⋊D714yes—
517C2×C4C4×C22⋊D712yes—
518C2×C4#649214yes—
519C2×C4#651314yes—
520C2×C4#651612yes—
521C2×C4C22×Dic1411yescharacteristic
522C2×C4C22×C4×D711yescharacteristic
523C2×C4C2×#115711yescharacteristic
524C2×C4C2×#115711yescharacteristic
525C2×C4C22×C22⋊D711yescharacteristic
526C2×C4C22×C22⋊D711yescharacteristic
527C8—142no—
528C8—142no—
529C8—282no—
530C7#2087211yescharacteristic, complemented, Sylow
531C22—22no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22#8998214yes—
541C22C2×#648711yescharacteristic
542C22C2×C4×C22⋊D711yescharacteristic
543C22C2×#651312yes—
544C22#9051312yes—
545C22C2×#651611yescharacteristic
546C4—24no—
547C4—24no—
548C4—42no—
549C4—44no—
550C4—282no—
551C4—282no—
552C4—562no—
553C4#8998214yes—
554C4C22×Dic7⋊C411yescharacteristic
555C4C2×#648711yescharacteristic
556C4C2×C4×C22⋊D711yescharacteristic
557C4C2×#649211yescharacteristic
558C2—24no—
559C2—44no—
560C2C4×#651311yescharacteristic
561C2#1173995111yescharacteristic
562C2#1173995212yes—
563C2C2×#8998211yescharacteristic
564C2#1174152512yes—
subgroupG/H
centerC22×C4C2×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×#41C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×C4×#41D7
hypocenterC7#20872
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
4. C2×#89770C2semidirect
5. #11739514C2semidirect
6. #11729383C2semidirect
7. #11729382C2semidirect
8. #11729361C2semidirect
9. #11729360C2semidirect
10. #11729334C2semidirect
11. C2×#89319C2semidirect
27. #90005C4semidirect
85. #6489C2×C4semidirect
98. C22×C7⋊C8C2×C4semidirect
103. #6409C2×C4semidirect
104. #6409C2×C4semidirect
106. #6406C2×C4semidirect
107. #6406C2×C4semidirect
128. C7×#41C4⋊C4semidirect
391. #41Dic7⋊C4semidirect
530. C7#20872semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×#412
C2×#900052
C2×#897702
#117395142
#117293832
#117293822
#117293612
#117293602
#117293342
C2×#893192
#208727