Finite Group Databasebeta

← #13363678

Subgroups — #13363678

Order 1792.

1162 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90313C211yesmaximal, characteristic
2C22×Dic7⋊C8C211yesmaximal, characteristic
3#11731309C211yesmaximal, characteristic
4C7×#1892C211yesmaximal, characteristic
5#11729726C211yesmaximal, characteristic
6#11729450C211yesmaximal, characteristic
7#11727983C212yesmaximal, complemented
8#11727975C212yesmaximal, complemented
9#11727268C211yesmaximal, characteristic
10#11726658C212yesmaximal, complemented
11#11726553C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C2×#6507C2211yescharacteristic
16C2×#6487C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19#90313—24no—
20#90313C418yescomplemented
21C2×#6452—21no—
22C2×#6451C2211yescharacteristic
23C2×#6450—21no—
24C2×#6450C2211yescharacteristic
25C2×#6441—21no—
26C2×#6440—21no—
27C2×#6440—21no—
28C2×C7×#103C2212yes—
29C7×#380C2211yescharacteristic
30C2×#6409C2212yes—
31#89984C2211yescharacteristic
32C2×#6390C2212yes—
33C2×Dic7⋊C8—24no—
34C2×Dic7⋊C8C2212yes—
35C2×Dic7⋊C8C2212yes—
36#89748C2212yes—
37C4⋊C4×Dic7C2212yes—
38C2×#6384C2212yes—
39#89714—21no—
40#89422—21no—
41#89421—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21885—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×Dic7D411yescharacteristic
50C23×Dic7D411yescharacteristic
51C22×Dic14—21no—
52C22×Dic14C2311yescharacteristic
53C22×C7×C8—22no—
54C22×C7×C8—42no—
55C22×C7×C8C2312yes—
56C7×C22⋊Q8—22no—
57C7×C22⋊Q8C2×C414yes—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2312yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—21no—
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—21no—
65Q8×Dic7—22no—
66Q8×Dic7C2×C414yes—
67#6507—22no—
68#6507C2×C414yes—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7C2312yes—
71C2×C4⋊Dic7C2311yescharacteristic
72#6487—22no—
73#6487C2×C414yes—
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—22no—
80C2×Dic7⋊C4—22no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7C2×C414yes—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D414yes—
88C2×C4×Dic7C2311yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8—22no—
91C22×C7⋊C8—42no—
92C22×C7⋊C8C2312yes—
93#6452—28no—
94#6451—22no—
95#6451C2×C414yes—
96#6450—28no—
97#6450—22no—
98#6450C2×C414yes—
99#6441—28no—
100#6440—28no—
101#6440—44no—
102C7×#103—42no—
103C7×#95—21no—
104#6409—42no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107#6390—42no—
108Dic7⋊C8—24no—
109Dic7⋊C8—24no—
110#6384—42no—
111C2×#533—71no—
112C22×C4⋊C8—71no—
113#2285—71no—
114#1987—72no—
115#1981—72no—
116#1892—71no—
117#1892—71no—
118#1892—71no—
119#1892D711yescharacteristic
120#1803—72no—
121#1803—72no—
122C24×C7C2×D411yescharacteristic
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8C22×C412yes—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2411yescharacteristic
136C22×Dic7—21no—
137C22×Dic7—21no—
138C22×Dic7—22no—
139C22×Dic7—42no—
140C22×Dic7C22⋊C414yes—
141C22×Dic7C22⋊C414yes—
142C22×Dic7SD1614yes—
143C22×Dic7Q1614yes—
144C22×Dic7C2×D411yescharacteristic
145C22×Dic7C2×D411yescharacteristic
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—21no—
151C2×Dic14—24no—
152C2×Dic14—44no—
153C2×Dic14C22×C412yes—
154C2×C7×C8—22no—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C2×C7×C8—44no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4C22×C412yes—
163C7×C22⋊C4—24no—
164C7×C22⋊C4—42no—
165C22⋊Dic7—24no—
166C22⋊Dic7—42no—
167C4⋊Dic7—21no—
168C4⋊Dic7—24no—
169C4⋊Dic7—22no—
170C4⋊Dic7—22no—
171C4⋊Dic7C22×C412yes—
172Dic7⋊C4—24no—
173Dic7⋊C4—24no—
174Dic7⋊C4—24no—
175Dic7⋊C4—24no—
176Dic7⋊C4—24no—
177Dic7⋊C4—44no—
178Dic7⋊C4—44no—
179C4×Dic7—24no—
180C4×Dic7—24no—
181C4×Dic7—44no—
182C4×Dic7C22⋊C414yes—
183C4×Dic7C22⋊C414yes—
184C4×Dic7C22×C412yes—
185C4×Dic7C22×C412yes—
186C4×Dic7C2×D412yes—
187C4×Dic7C2×D412yes—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—44no—
192C23×C8—71no—
193C23×C8C2×D711yescharacteristic
194#533—78no—
195#533—144no—
196C2×C4⋊Q8—71no—
197C2×C4⋊Q8—71no—
198C2×#125—141no—
199C2×C22⋊Q8—71no—
200C2×C22⋊Q8—141no—
201C2×C22⋊Q8C2×D711yescharacteristic
202C2×C4×Q8—71no—
203C2×C4×Q8—71no—
204C2×C4×Q8—141no—
205C2×#117—141no—
206C2×#117—141no—
207C22×C42—71no—
208C2×C4⋊C8—72no—
209C2×C4⋊C8—72no—
210C2×C4⋊C8—144no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103—72no—
214C2×#103C2×D712yes—
215#384—72no—
216#382—141no—
217#380—71no—
218#380—141no—
219#380C2×D711yescharacteristic
220C4×C4⋊C4—72no—
221C4×C4⋊C4—141no—
222#340—72no—
223#340—72no—
224#340—72no—
225#340C2×D712yes—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7C2×C22⋊C412yes—
229C23×C7C2×SD1612yes—
230C23×C7C2×Q1612yes—
231C23×C7C22×D411yescharacteristic
232C7×Q8—24no—
233C7×Q8—41no—
234C7×Q8—41no—
235C2×C4×C7—22no—
236C2×C4×C7—24no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7C2×C22⋊C412yes—
247C2×C4×C7C23×C411yescharacteristic
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7H(5,𝔽2)12yes—
250C2×C4×C72−1+412yes—
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—44no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7#103116yescomplemented
262C2×Dic7C2×C22⋊C412yes—
263C2×Dic7C2×C22⋊C412yes—
264C2×Dic7C2×C22⋊C412yes—
265C2×Dic7C2×C22⋊C412yes—
266C2×Dic7C2×SD1612yes—
267C2×Dic7C2×SD1612yes—
268C2×Dic7C2×Q1612yes—
269C2×Dic7C2×Q1612yes—
270Dic14—24no—
271Dic14—48no—
272Dic14—41no—
273Dic14—41no—
274C7×C8—42no—
275C7⋊C8—42no—
276C22×Q8—71no—
277C22×Q8—141no—
278C22×Q8C22×D711yescharacteristic
279C23×C4—71no—
280C23×C4—71no—
281C23×C4C22×D711yescharacteristic
282C22×C8—22no—
283C22×C8—42no—
284C22×C8—72no—
285C22×C8—142no—
286C22×C8—282no—
287C22×C8C22×D712yes—
288C4⋊Q8—74no—
289C4⋊Q8—74no—
290C4⋊Q8—142no—
291C4⋊Q8—142no—
292#125—148no—
293C22⋊Q8—22no—
294C22⋊Q8—74no—
295C22⋊Q8—148no—
296C22⋊Q8—142no—
297C22⋊Q8C4×D714yes—
298C4×Q8—74no—
299C4×Q8—74no—
300C4×Q8—142no—
301C4×Q8—148no—
302C4×Q8—142no—
303#117—148no—
304#117—284no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—72no—
308C2×C4⋊C4—141no—
309C2×C4⋊C4—141no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4C22×D712yes—
317C2×C4⋊C4C22×D711yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—141no—
321C2×C22⋊C4—141no—
322C2×C42—74no—
323C2×C42—71no—
324C2×C42—71no—
325C2×C42—74no—
326C2×C42—71no—
327C2×C42—71no—
328C2×C42—141no—
329C2×C42—141no—
330C2×C42—142no—
331C4⋊C8—144no—
332C4⋊C8—144no—
333#103—42no—
334#103—282no—
335#103—282no—
336#103—282no—
337#95—21no—
338#95—142no—
339#95—141no—
340C22×C7—22no—
341C22×C7—22no—
342C22×C7—24no—
343C22×C7—24no—
344C22×C7C2×#10314yes—
345C22×C7C22×C22⋊C411yescharacteristic
346C22×C7#53712yes—
347C22×C7C22×SD1611yescharacteristic
348C22×C7C22×Q1611yescharacteristic
349C22×C7#57512yes—
350C4×C7—22no—
351C4×C7—42no—
352C4×C7—42no—
353C4×C7—82no—
354C4×C7#52012yes—
355C4×C7#53511yescharacteristic
356C4×C7#53611yescharacteristic
357Dic7—22no—
358Dic7—22no—
359Dic7—42no—
360Dic7—42no—
361Dic7—82no—
362Dic7C2×#10314yes—
363Dic7C2×#10314yes—
364C24D4×D711yescharacteristic
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—72no—
369C2×Q8—144no—
370C2×Q8—141no—
371C2×Q8—142no—
372C2×Q8—142no—
373C2×Q8—284no—
374C2×Q8C2×C4×D712yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—74no—
381C22×C4—74no—
382C22×C4—74no—
383C22×C4—71no—
384C22×C4—71no—
385C22×C4—71no—
386C22×C4—71no—
387C22×C4—74no—
388C22×C4—141no—
389C22×C4—141no—
390C22×C4—142no—
391C22×C4—282no—
392C22×C4C2×C4×D712yes—
393C22×C4D4×D711yescharacteristic
394C22×C4D4×D712yes—
395C22×C4C23×D711yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—44no—
400C2×C8—142no—
401C2×C8—142no—
402C2×C8—142no—
403C2×C8—284no—
404C4⋊C4—24no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—72no—
409C4⋊C4—144no—
410C4⋊C4—144no—
411C4⋊C4—144no—
412C4⋊C4—144no—
413C4⋊C4—144no—
414C4⋊C4—144no—
415C4⋊C4—141no—
416C4⋊C4—142no—
417C4⋊C4—142no—
418C4⋊C4—284no—
419C4⋊C4—284no—
420C4⋊C4C2×C4×D712yes—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—144no—
424C22⋊C4—282no—
425C42—74no—
426C42—74no—
427C42—72no—
428C42—72no—
429C42—72no—
430C42—72no—
431C42—144no—
432C42—144no—
433C42—284no—
434C2×C7—24no—
435C2×C7#231111yescharacteristic
436C2×C7C22×#10311yescharacteristic
437C2×C7#289611yescharacteristic
438C2×C7#324612yes—
439C2×C7#324912yes—
440C23—22no—
441C23—22no—
442C23D7×C22⋊C412yes—
443C23D7×SD1612yes—
444C23D7×Q1612yes—
445C23C2×D4×D711yescharacteristic
446Q8—24no—
447Q8—41no—
448Q8—41no—
449Q8—144no—
450Q8—288no—
451Q8—281no—
452Q8—281no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—72no—
465C2×C4—72no—
466C2×C4—72no—
467C2×C4—72no—
468C2×C4—72no—
469C2×C4—72no—
470C2×C4—716no—
471C2×C4—72no—
472C2×C4—72no—
473C2×C4—142no—
474C2×C4—142no—
475C2×C4—142no—
476C2×C4—142no—
477C2×C4—284no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4—282no—
482C2×C4—282no—
483C2×C4D7×C22⋊C412yes—
484C2×C4C22×C4×D711yescharacteristic
485C2×C4C2×D4×D711yescharacteristic
486C2×C4#654812yes—
487C2×C4#655412yes—
488C8—42no—
489C8—282no—
490C7#2188511yescharacteristic, complemented, Sylow
491C22—24no—
492C22—22no—
493C22—22no—
494C22—24no—
495C22D7×#10314yes—
496C22C2×D7×C22⋊C411yescharacteristic
497C22#9028812yes—
498C22C2×D7×SD1611yescharacteristic
499C22C2×D7×Q1611yescharacteristic
500C22#9045912yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—74no—
505C4—74no—
506C4—82no—
507C4—142no—
508C4—142no—
509C4—282no—
510C4—282no—
511C4—562no—
512C4#9023712yes—
513C4#9028311yescharacteristic
514C4#9033911yescharacteristic
515C2—24no—
516C2#1173210711yescharacteristic
517C2C2×D7×#10311yescharacteristic
518C2#1173627511yescharacteristic
519C2#1173808512yes—
520C2#1173819212yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1892C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1892D7
hypocenterC7#21885
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
7. #11727983C2semidirect
8. #11727975C2semidirect
10. #11726658C2semidirect
11. #11726553C2semidirect
20. #90313C4semidirect
261. C2×Dic7#103semidirect
490. C7#21885semidirect

Maximal subgroups

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

subgroupindex
C2×#903132
C22×Dic7⋊C82
#117313092
C7×#18922
#117297262
#117294502
#117279832
#117279752
#117272682
#117266582
#117265532
#218857