Finite Group Databasebeta

← #13364977

Subgroups — #13364977

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6487C211yesmaximal, characteristic
2C2×#89701C211yesmaximal, characteristic
3C2×#89511C211yesmaximal, characteristic
4C4×C7×C22⋊C8C211yesmaximal, characteristic
5#11727717C211yesmaximal, characteristic
6#11727684C211yesmaximal, characteristic
7#11727681C211yesmaximal, characteristic
8#11727529C212yesmaximal, complemented
9#11727521C212yesmaximal, complemented
10#11726439C212yesmaximal, complemented
11#11726382C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6487C2211yescharacteristic
16C22×Dic28C2212yes—
17C2×C4×Dic14C2211yescharacteristic
18C2×C7×C22⋊C8C2211yescharacteristic
19C2×C7×C22⋊C8C2211yescharacteristic
20C2×C4×C7×C8C2212yes—
21#89984C2211yescharacteristic
22#89982—21no—
23C4×C22⋊Dic7—21no—
24C2×C7⋊#95—21no—
25C2×C7⋊#95—21no—
26C2×#6392C2212yes—
27C2×#6390C2212yes—
28C2×#6390C2212yes—
29C2×#6390C2212yes—
30C2×#6390C2212yes—
31C4×C4⋊Dic7C2211yescharacteristic
32#89705—21no—
33C4×Dic7⋊C4—21no—
34#89701—21no—
35#89701—21no—
36#89701C2212yes—
37#89701C2212yes—
38#89701C2211yescharacteristic
39#89700C2211yescharacteristic
40#89511C418yescomplemented
41C7×#336C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#22044—71nomaximal, Sylow
45C23×C4×C7D411yescharacteristic
46C23×C4×C7D411yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×Dic7—21no—
49C23×Dic7—21no—
50C22×Dic14—21no—
51C22×Dic14—22no—
52C22×Dic14—42no—
53C22×Dic14—42no—
54C22×Dic14C2×C414yes—
55C22×Dic14C2312yes—
56C22×Dic14C2311yescharacteristic
57C22×C7×C8C2312yes—
58C22×C7×C8C2312yes—
59C2×C42×C7—42no—
60C2×C42×C7D411yescharacteristic
61C2×C42×C7D411yescharacteristic
62C2×C42×C7C2311yescharacteristic
63C2×C42×C7C2311yescharacteristic
64C2×C22⋊Dic7—21no—
65C2×C22⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7—42no—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2311yescharacteristic
72#6487—22no—
73#6487C2×C414yes—
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×C4×Dic7—21no—
78C2×C4×Dic7—21no—
79C2×Dic28—42no—
80C2×Dic28—42no—
81C2×Dic28C2×C418yescomplemented
82C4×Dic14—42no—
83C7×C22⋊C8—22no—
84C7×C22⋊C8C2×C414yes—
85C4×C7×C8—42no—
86C7⋊#95—21no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95#6392—42no—
96#6390—42no—
97#6390—42no—
98#6390—42no—
99#6390C2×C418yescomplemented
100C2×#451—71no—
101C4×C22⋊Q8—71no—
102C2×#391—71no—
103#1969—72no—
104#1933—72no—
105#1928—72no—
106#1878—72no—
107#1872—71no—
108#1852—71no—
109#1787—71no—
110C4×C22⋊C8D711yescharacteristic
111C24×C7C2×D411yescharacteristic
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—41no—
117C22×C4×C7—41no—
118C22×C4×C7—41no—
119C22×C4×C7Q1614yescomplemented
120C22×C4×C7C22×C412yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7#4112yes—
127C22×C4×C7C2411yescharacteristic
128C22×Dic7—24no—
129C22×Dic7—22no—
130C22×Dic7—21no—
131C22×Dic7—21no—
132C22×Dic7—21no—
133C22×Dic7—22no—
134C22×Dic7—21no—
135C22×Dic7—42no—
136C22×Dic7—42no—
137C22×Dic7—42no—
138C22×Dic7—42no—
139C22×Dic7—42no—
140C2×Dic14—22no—
141C2×Dic14—22no—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C2×Dic14—22no—
145C2×Dic14—22no—
146C2×Dic14—24no—
147C2×Dic14—21no—
148C2×Dic14—22no—
149C2×Dic14—48no—
150C2×Dic14—42no—
151C2×Dic14—42no—
152C2×Dic14—48no—
153C2×Dic14C22×C412yes—
154C2×Dic14C22×C414yes—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8C22×C414yes—
161C42×C7—22no—
162C42×C7—22no—
163C42×C7—41no—
164C42×C7—41no—
165C22⋊Dic7—24no—
166C22⋊Dic7—42no—
167C4⋊Dic7—21no—
168C4⋊Dic7—22no—
169C4⋊Dic7—22no—
170C4⋊Dic7—22no—
171C4⋊Dic7—22no—
172C4⋊Dic7—42no—
173C4⋊Dic7C22×C412yes—
174Dic7⋊C4—24no—
175Dic7⋊C4—42no—
176Dic7⋊C4—42no—
177C4×Dic7—42no—
178C4×Dic7—42no—
179Dic28—48no—
180C23×Q8—71no—
181C22×Q16—72no—
182C2×C22⋊Q8—71no—
183C2×C4×Q8—71no—
184C22×C4⋊C4—71no—
185C22×C42C2×D711yescharacteristic
186#451—78no—
187C2×#107—72no—
188C2×#103—72no—
189C2×#103—72no—
190C2×#103—72no—
191C2×#103—72no—
192C2×C22⋊C8C2×D711yescharacteristic
193C2×C22⋊C8C2×D711yescharacteristic
194C2×C4×C8C2×D712yes—
195#391—72no—
196#391—72no—
197#391—71no—
198#391—141no—
199#391—141no—
200#389—71no—
201#387—141no—
202#385—141no—
203#380—71no—
204C4×C4⋊C4—71no—
205C4×C4⋊C4—141no—
206C4×C22⋊C4—141no—
207C2×#95—141no—
208C2×#95—141no—
209#340—72no—
210#340—72no—
211#336C2×D712yes—
212C23×C7—21no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7C4×D412yes—
217C23×C7C2×Q1612yes—
218C23×C7#13512yes—
219C23×C7C22×D411yescharacteristic
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—44no—
234C2×C4×C7—44no—
235C2×C4×C7C4×D412yes—
236C2×C4×C7C2×Q1612yes—
237C2×C4×C7C2×Q1612yes—
238C2×C4×C7#13712yes—
239C2×C4×C7C23×C411yescharacteristic
240C2×C4×C7C2×#4111yescharacteristic
241C2×C4×C7H(5,𝔽2)12yes—
242C2×C4×C7H(5,𝔽2)12yes—
243C2×Dic7—22no—
244C2×Dic7—22no—
245C2×Dic7—24no—
246C2×Dic7—48no—
247C2×Dic7—41no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—41no—
251C2×Dic7—41no—
252C2×Dic7—41no—
253C2×Dic7—41no—
254C2×Dic7—42no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—42no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—42no—
262C2×Dic7—84no—
263Dic14—24no—
264Dic14—24no—
265Dic14—24no—
266Dic14—48no—
267Dic14—42no—
268Dic14—42no—
269Dic14—42no—
270C7×C8—24no—
271C7×C8—42no—
272C22×Q8—74no—
273C22×Q8—71no—
274C22×Q8—72no—
275C22×Q8—141no—
276C22×Q8—142no—
277C22×Q8—282no—
278C22×Q8—282no—
279C23×C4—141no—
280C23×C4—141no—
281C23×C4D2811yescharacteristic
282C23×C4D2811yescharacteristic
283C23×C4C22×D711yescharacteristic
284C2×Q16—78no—
285C2×Q16—282no—
286C2×Q16—282no—
287C22×C8C22×D712yes—
288C22×C8C22×D712yes—
289C22⋊Q8—74no—
290C22⋊Q8—142no—
291C4×Q8—282no—
292C2×C4⋊C4—71no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—71no—
295C2×C4⋊C4—71no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—141no—
299C2×C4⋊C4—141no—
300C2×C4⋊C4—282no—
301C2×C22⋊C4—141no—
302C2×C22⋊C4—141no—
303C2×C42—42no—
304C2×C42—141no—
305C2×C42—141no—
306C2×C42D2811yescharacteristic
307C2×C42D2811yescharacteristic
308C2×C42C22×D711yescharacteristic
309C2×C42C22×D711yescharacteristic
310#107—282no—
311#103—78no—
312#103—282no—
313#103—282no—
314#103—282no—
315C22⋊C8—22no—
316C22⋊C8C4×D714yes—
317C4×C8—42no—
318#95—142no—
319#95—142no—
320#95—142no—
321#95—142no—
322#95—141no—
323#95—141no—
324#95—141no—
325#95—141no—
326#95—141no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—22no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—22no—
333C22×C7C4×Q1614yes—
334C22×C7C2×C4×D411yescharacteristic
335C22×C7#53411yescharacteristic
336C22×C7#53511yescharacteristic
337C22×C7C22×Q1611yescharacteristic
338C22×C7C2×#13511yescharacteristic
339C22×C7C2×#13711yescharacteristic
340C22×C7#57511yescharacteristic
341C4×C7—22no—
342C4×C7—22no—
343C4×C7—22no—
344C4×C7—22no—
345C4×C7—41no—
346C4×C7#51812yes—
347C4×C7#53812yes—
348Dic7—44no—
349Dic7—42no—
350Dic7—42no—
351Dic7—81no—
352Dic7—81no—
353Dic7—82no—
354C24C2×D2811yescharacteristic
355C2×Q8—74no—
356C2×Q8—72no—
357C2×Q8—144no—
358C2×Q8—141no—
359C2×Q8—142no—
360C2×Q8—142no—
361C2×Q8—142no—
362C2×Q8—142no—
363C2×Q8—142no—
364C2×Q8—142no—
365C2×Q8—142no—
366C2×Q8—288no—
367C2×Q8—282no—
368C2×Q8—282no—
369C2×Q8—288no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—144no—
378C22×C4—142no—
379C22×C4—141no—
380C22×C4—141no—
381C22×C4—141no—
382C22×C4—142no—
383C22×C4—141no—
384C22×C4—282no—
385C22×C4—282no—
386C22×C4—282no—
387C22×C4—282no—
388C22×C4—282no—
389C22×C4Dic2814yescomplemented
390C22×C4C2×C4×D712yes—
391C22×C4C2×D2811yescharacteristic
392C22×C4C2×D2811yescharacteristic
393C22×C4C2×D2811yescharacteristic
394C22×C4C2×D2811yescharacteristic
395C22×C4C2×D2811yescharacteristic
396C22×C4#115712yes—
397C22×C4C23×D711yescharacteristic
398Q16—288no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8C2×C4×D714yes—
405C4⋊C4—72no—
406C4⋊C4—144no—
407C4⋊C4—141no—
408C4⋊C4—142no—
409C4⋊C4—142no—
410C4⋊C4—142no—
411C4⋊C4—142no—
412C4⋊C4—282no—
413C4⋊C4—282no—
414C4⋊C4—282no—
415C22⋊C4—144no—
416C22⋊C4—282no—
417C42—22no—
418C42—22no—
419C42—41no—
420C42—41no—
421C42—282no—
422C42—282no—
423C2×C7—24no—
424C2×C7#232111yescharacteristic
425C2×C7C2×C4×Q1611yescharacteristic
426C2×C7C4×#13711yescharacteristic
427C2×C7#318911yescharacteristic
428C2×C7#319011yescharacteristic
429C2×C7#320911yescharacteristic
430C2×C7#321111yescharacteristic
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23C4×D2812yes—
436C23#646712yes—
437C23C2×Dic2812yes—
438C23C22×D2811yescharacteristic
439Q8—144no—
440Q8—144no—
441Q8—144no—
442Q8—288no—
443Q8—282no—
444Q8—282no—
445Q8—282no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—142no—
462C2×C4—142no—
463C2×C4—144no—
464C2×C4—288no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—281no—
468C2×C4—282no—
469C2×C4—282no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4—282no—
474C2×C4—281no—
475C2×C4—281no—
476C2×C4—282no—
477C2×C4—281no—
478C2×C4—282no—
479C2×C4—281no—
480C2×C4—564no—
481C2×C4C4×D2812yes—
482C2×C4C2×Dic2812yes—
483C2×C4C2×Dic2812yes—
484C2×C4#647212yes—
485C2×C4C22×C4×D711yescharacteristic
486C2×C4C2×#115711yescharacteristic
487C2×C4#654812yes—
488C2×C4#655312yes—
489C8—24no—
490C8—42no—
491C7#2204411yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22C4×Dic2814yes—
499C22C2×C4×D2811yescharacteristic
500C22#9018411yescharacteristic
501C22#9021611yescharacteristic
502C22C2×#646711yescharacteristic
503C22C22×Dic2811yescharacteristic
504C22C2×#647211yescharacteristic
505C22#9043911yescharacteristic
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—284no—
512C4—282no—
513C4—282no—
514C4—562no—
515C4—561no—
516C4—561no—
517C4#9023612yes—
518C4#9025112yes—
519C2—24no—
520C2#1173101411yescharacteristic
521C2C2×C4×Dic2811yescharacteristic
522C2C4×#647211yescharacteristic
523C2#1173606211yescharacteristic
524C2#1173606311yescharacteristic
525C2#1173675211yescharacteristic
526C2#1173675811yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×C22⋊C8C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC4×C22⋊C8D7
hypocenterC7#22044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
8. #11727529C2semidirect
9. #11727521C2semidirect
10. #11726439C2semidirect
11. #11726382C2semidirect
40. #89511C4semidirect
81. C2×Dic28C2×C4semidirect
99. #6390C2×C4semidirect
119. C22×C4×C7Q16semidirect
389. C22×C4Dic28semidirect
491. C7#22044semidirect

Maximal subgroups

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

subgroupindex
C4×#64872
C2×#897012
C2×#895112
C4×C7×C22⋊C82
#117277172
#117276842
#117276812
#117275292
#117275212
#117264392
#117263822
#220447