Finite Group Databasebeta

← #13360978

Subgroups — #13360978

Order 1792.

1048 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#480C211yesmaximal, characteristic
2Dic7×#121C211yesmaximal, characteristic
3C2×C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
4#11729431C211yesmaximal, characteristic
5#11729307C211yesmaximal, characteristic, complemented
6#11728364C212yesmaximal, complemented
7#11728359C212yesmaximal, complemented
8#11727910C212yesmaximal, complemented
9#11727906C212yesmaximal, complemented
10#11727682C211yesmaximal, characteristic, complemented
11#11727660C211yesmaximal, characteristic, complemented
12C2×C7×#121C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C2×D4×Dic7—21no—
15C2×D4×Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4×Dic7—21no—
18C22×C4×Dic7C2211yescharacteristic
19C7×#480C418yescomplemented
20C2×C7×#107C2212yes—
21C2×C7×#102C2212yes—
22C2×C7×C22⋊C8C2211yescharacteristic
23#89993C2211yescharacteristic
24#89992—21no—
25C2×#6406C2212yes—
26#89984C2211yescharacteristic
27C2×C7⋊#95—21no—
28C2×C22⋊(C7⋊C8)C2211yescharacteristic
29C2×#6391C2212yes—
30C2×#6389C2212yes—
31#89757C2211yescharacteristic
32C4⋊C4×Dic7—21no—
33C4⋊C4×Dic7—21no—
34C4⋊C4×Dic7C2212yes—
35C4⋊C4×Dic7C2212yes—
36C4⋊C4×Dic7C2211yescharacteristic
37#89723—21no—
38C22⋊C4×Dic7—21no—
39#89346C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21578—71nomaximal, Sylow
45C22×C7×D4—21no—
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×C4×C7C2311yescharacteristic
49C23×Dic7—21no—
50C23×Dic7—21no—
51C23×Dic7D411yescharacteristic
52C23×Dic7D411yescharacteristic
53C22×C7×C8C2312yes—
54C7×#121—22no—
55C7×#121C2×C414yes—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4C2×C414yescomplemented
59C2×C7×C4⋊C4C2312yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C22⋊C4—21no—
62C2×C22⋊Dic7—21no—
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—21no—
65C2×C22⋊Dic7—21no—
66D4×Dic7—24no—
67D4×Dic7—44no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—24no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7C2311yescharacteristic
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2312yes—
74C2×C4×Dic7—24no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7—21no—
78C2×C4×Dic7—44no—
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7C2311yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C22×C7⋊C8C2312yes—
84C7×#107C2×C418yescomplemented
85C7×#102C2×C418yescomplemented
86C7×C22⋊C8C2×C414yescomplemented
87#6406—24no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93C22⋊(C7⋊C8)—22no—
94#6391—24no—
95#6389—24no—
96C2×#480D711yescharacteristic
97C4×#121—71no—
98C2×C4×C4⋊C4—71no—
99#1926—72no—
100#1921—72no—
101#1874—71no—
102#1853—71no—
103#1823—71no—
104#1776—72no—
105#1762—72no—
106#1749—71no—
107C24×C7—21no—
108C24×C7C2×D411yescharacteristic
109C2×C7×D4—24no—
110C2×C7×D4—22no—
111C2×C7×D4—22no—
112C2×C7×D4—21no—
113C2×C7×D4—42no—
114C2×C7×D4—42no—
115C2×C7×D4C22×C412yes—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—24no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—42no—
122C22×C4×C7—42no—
123C22×C4×C7C22×C412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7#4112yes—
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—24no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—21no—
135C22×Dic7—21no—
136C22×Dic7—41no—
137C22×Dic7—42no—
138C22×Dic7—42no—
139C22×Dic7—42no—
140C22×Dic7—42no—
141C22×Dic7—42no—
142C22×Dic7C2×D411yescharacteristic
143C22×Dic7C2×D411yescharacteristic
144C22×Dic7C2×D411yescharacteristic
145C22×Dic7C2×D411yescharacteristic
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8C22×C414yes—
149C7×C4⋊C4—24no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4C22×C414yes—
153C7×C4⋊C4C22×C412yes—
154C7×C22⋊C4—24no—
155C22⋊Dic7—42no—
156C22⋊Dic7—42no—
157C22⋊Dic7—42no—
158C22⋊Dic7—42no—
159C4⋊Dic7—24no—
160C4⋊Dic7—22no—
161C4⋊Dic7—22no—
162C4⋊Dic7—22no—
163C4⋊Dic7—42no—
164C4⋊Dic7#4114yes—
165C4×Dic7—22no—
166C4×Dic7—22no—
167C4×Dic7—22no—
168C4×Dic7—44no—
169C4×Dic7—44no—
170C4×Dic7—42no—
171C4×Dic7#4114yes—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×#121C2×D711yescharacteristic
176C2×C4×D4—71no—
177C2×C4×D4—141no—
178C22×C4⋊C4—71no—
179C22×C4⋊C4C2×D711yescharacteristic
180C22×C42—71no—
181C22×C42—141no—
182#480Dic718yescomplemented
183C2×#107C2×D712yes—
184C2×#106—72no—
185C2×#102—72no—
186C2×#102C2×D712yes—
187C2×C22⋊C8—71no—
188C2×C22⋊C8C2×D711yescharacteristic
189C2×#97—72no—
190#390—71no—
191#389—71no—
192#388—141no—
193#386—141no—
194#380—71no—
195C4×C4⋊C4—72no—
196C4×C4⋊C4—72no—
197C4×C4⋊C4—71no—
198C4×C4⋊C4—141no—
199C4×C4⋊C4—141no—
200C4×C22⋊C4—141no—
201C2×#95—141no—
202#340—72no—
203#340—72no—
204#340—72no—
205#340—72no—
206#336—72no—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—21no—
211C23×C7—22no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7C4×D412yes—
219C23×C7AGL(1,ℤ/8)12yes—
220C23×C7AGL(1,ℤ/8)12yes—
221C23×C7C22×D411yescharacteristic
222C7×D4—24no—
223C7×D4—48no—
224C7×D4—42no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—22no—
230C2×C4×C7—24no—
231C2×C4×C7—42no—
232C2×C4×C7—41no—
233C2×C4×C7—42no—
234C2×C4×C7—41no—
235C2×C4×C7—48no—
236C2×C4×C7C4×D412yes—
237C2×C4×C7C23×C411yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C2×C4×C7C2×#4112yes—
240C2×C4×C7C2×#4112yes—
241C2×Dic7—28no—
242C2×Dic7—21no—
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—24no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—44no—
253C2×Dic7—48no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—84no—
263C2×Dic7#12312yes—
264C2×Dic7#12312yes—
265C2×Dic7#13712yes—
266C7×C8—24no—
267C7⋊C8—42no—
268C22×D4—21no—
269C22×D4C22×D711yescharacteristic
270C23×C4—21no—
271C23×C4—71no—
272C23×C4—71no—
273C23×C4—141no—
274C23×C4—141no—
275C23×C4C22×D711yescharacteristic
276C22×C8—72no—
277C22×C8C22×D712yes—
278#121—22no—
279#121C2×Dic714yes—
280C4×D4—144no—
281C4×D4—284no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—71no—
286C2×C4⋊C4—71no—
287C2×C4⋊C4—144no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4C2×Dic714yescomplemented
291C2×C4⋊C4C22×D712yes—
292C2×C4⋊C4C22×D711yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—141no—
296C2×C22⋊C4—141no—
297C2×C22⋊C4—141no—
298C2×C42—71no—
299C2×C42—71no—
300C2×C42—71no—
301C2×C42—71no—
302C2×C42—144no—
303C2×C42—141no—
304C2×C42—142no—
305C2×C42—142no—
306C2×C42—284no—
307#107C2×Dic718yescomplemented
308#106—144no—
309#102—144no—
310#102C2×Dic718yescomplemented
311C22⋊C8—142no—
312C22⋊C8C2×Dic714yescomplemented
313#97—144no—
314#95—142no—
315#95—142no—
316#95—141no—
317#95—141no—
318#95—141no—
319C22×C7—22no—
320C22×C7—22no—
321C22×C7—22no—
322C22×C7—22no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7—42no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7—81no—
329C22×C7—81no—
330C22×C7—81no—
331C22×C7—81no—
332C22×C7#44214yes—
333C22×C7C2×C4×D411yescharacteristic
334C22×C7C2×#12311yescharacteristic
335C22×C7#52511yescharacteristic
336C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
337C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
338C22×C7C2×#13711yescharacteristic
339C22×C7#57511yescharacteristic
340C4×C7—22no—
341C4×C7—44no—
342C4×C7—42no—
343C4×C7C4×#4112yes—
344C4×C7#52912yes—
345Dic7—22no—
346Dic7—22no—
347Dic7—22no—
348Dic7—41no—
349Dic7—44no—
350Dic7—81no—
351Dic7—81no—
352C24—21no—
353C24D4×D711yescharacteristic
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4C22×Dic712yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—71no—
369C22×C4—71no—
370C22×C4—71no—
371C22×C4—71no—
372C22×C4—142no—
373C22×C4—142no—
374C22×C4—142no—
375C22×C4—142no—
376C22×C4—144no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—141no—
380C22×C4—141no—
381C22×C4—282no—
382C22×C4—281no—
383C22×C4—282no—
384C22×C4—282no—
385C22×C4—282no—
386C22×C4—282no—
387C22×C4D4×D711yescharacteristic
388C22×C4#115912yes—
389C22×C4C22×Dic712yes—
390C22×C4C23×D711yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—142no—
394C2×C8—142no—
395C2×C8—142no—
396C2×C8C22×Dic714yes—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—74no—
401C4⋊C4—144no—
402C4⋊C4—142no—
403C4⋊C4—142no—
404C4⋊C4—142no—
405C4⋊C4—282no—
406C4⋊C4C22×Dic714yes—
407C4⋊C4C22×Dic712yes—
408C22⋊C4—24no—
409C22⋊C4—282no—
410C22⋊C4—282no—
411C22⋊C4—282no—
412C22⋊C4—282no—
413C42—74no—
414C42—142no—
415C42—142no—
416C42—142no—
417C42—284no—
418C42—284no—
419C42—282no—
420C2×C7—24no—
421C2×C7—82no—
422C2×C7C4×#12311yescharacteristic
423C2×C7C2×#44211yescharacteristic
424C2×C7C4×#13711yescharacteristic
425C2×C7#309211yescharacteristic
426C2×C7#309411yescharacteristic
427C2×C7#311211yescharacteristic
428C2×C7#311411yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23#647412yes—
441C23#647412yes—
442C23D4×Dic712yes—
443C23C2×D4×D711yescharacteristic
444D4—24no—
445D4—48no—
446D4—42no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—72no—
459C2×C4—72no—
460C2×C4—72no—
461C2×C4—148no—
462C2×C4—141no—
463C2×C4—141no—
464C2×C4—141no—
465C2×C4—141no—
466C2×C4—141no—
467C2×C4—141no—
468C2×C4—141no—
469C2×C4—141no—
470C2×C4—141no—
471C2×C4—144no—
472C2×C4—284no—
473C2×C4—288no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4—281no—
480C2×C4—281no—
481C2×C4—281no—
482C2×C4—564no—
483C2×C4D4×Dic712yes—
484C2×C4C2×#115911yescharacteristic
485C2×C4D7×#4112yes—
486C2×C4D7×#4112yes—
487C2×C4C23×Dic711yescharacteristic
488C8—24no—
489C8—282no—
490C7#2157811yescharacteristic, complemented, Sylow
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#8992414yes—
505C22D7×#12311yescharacteristic
506C22#9034711yescharacteristic
507C22C2×#647411yescharacteristic
508C22C2×#647411yescharacteristic
509C22D7×#13711yescharacteristic
510C22#9046811yescharacteristic
511C22C2×D4×Dic711yescharacteristic
512C4—22no—
513C4—44no—
514C4—42no—
515C4—142no—
516C4—142no—
517C4—142no—
518C4—284no—
519C4—281no—
520C4—561no—
521C4—561no—
522C4#9034012yes—
523C4#41×Dic712yes—
524C2—24no—
525C2—82no—
526C2#1173871511yescharacteristic
527C2#1173871611yescharacteristic
528C2#1173872711yescharacteristic
529C2#1173872811yescharacteristic
530C2Dic7×#12311yescharacteristic
531C2C2×#8992411yescharacteristic
532C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#480C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
cosocleC22×C4×C7C24
hypercenterC2×#480D7
hypocenterC7#21578
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×Dic7C2semidirect
5. #11729307C2semidirect
6. #11728364C2semidirect
7. #11728359C2semidirect
8. #11727910C2semidirect
9. #11727906C2semidirect
10. #11727682C2semidirect
11. #11727660C2semidirect
19. C7×#480C4semidirect
58. C2×C7×C4⋊C4C2×C4semidirect
84. C7×#107C2×C4semidirect
85. C7×#102C2×C4semidirect
86. C7×C22⋊C8C2×C4semidirect
182. #480Dic7semidirect
290. C2×C4⋊C4C2×Dic7semidirect
307. #107C2×Dic7semidirect
310. #102C2×Dic7semidirect
312. C22⋊C8C2×Dic7semidirect
490. C7#21578semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4802
Dic7×#1212
C2×C4⋊C4×Dic72
#117294312
#117293072
#117283642
#117283592
#117279102
#117279062
#117276822
#117276602
#215787