Finite Group Databasebeta

← #13372081

Subgroups — #13372081

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×C4⋊C4C211yesmaximal, characteristic
2C2×C4×#1142C211yesmaximal, characteristic
3C2×#89790C212yesmaximal
4C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
5#11729360C212yesmaximal
6#11729358C211yesmaximal, characteristic
7#11727385C212yesmaximal, complemented
8#11727375C212yesmaximal, complemented
9#11727342C211yesmaximal, characteristic, complemented
10#11727331C211yesmaximal, characteristic, complemented
11#11727231C211yesmaximal, characteristic, complemented
12C2×C4×C7×D4C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C22×C7×C4⋊C4—21no—
15C22×#1142C2211yescharacteristic
16C2×C4×Dic14C2212yes—
17C2×C4×Dic14C2211yescharacteristic
18C7×#389C2212yes—
19C7×#385—22no—
20C7×#384C2211yescharacteristic
21C7×#380—21no—
22C4×C7×C4⋊C4C2211yescharacteristic
23C2×#6406C2212yes—
24C2×#6406C2211yescharacteristic
25#89790—28no—
26C4×#1142C418yescomplemented
27#89748—21no—
28C4⋊C4×Dic7—21no—
29C2×#6384C2212yes—
30C2×#6384C2211yescharacteristic
31C2×#6382C2211yescharacteristic
32C2×#6381C2212yes—
33C2×#6381C2212yes—
34#89707C2211yescharacteristic
35#89700C2212yes—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×#6377C2212yes—
38C2×C4×C7⋊C8C2211yescharacteristic
39#89419—22no—
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#22671—71nomaximal, Sylow
44C22×C7×D4C2311yescharacteristic
45C23×C4×C7—21no—
46C23×C4×C7—22no—
47C22×Dic14C2311yescharacteristic
48C4×C7×D4—24no—
49C4×C7×D4—22no—
50C4×C7×D4—42no—
51C4×C7×D4C2×C414yes—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—44no—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—22no—
62C2×C42×C7C2312yes—
63C2×C42×C7C2311yescharacteristic
64C2×#1142C2×C414yes—
65C2×#1142D414yes—
66C2×#1142Q814yes—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—22no—
73C2×Dic7⋊C4—22no—
74C2×C4×Dic7—21no—
75C2×C4×Dic7—22no—
76C22×C7⋊C8C2312yes—
77C22×C7⋊C8C2311yescharacteristic
78C22×C7⋊C8C2311yescharacteristic
79C4×Dic14—22no—
80C4×Dic14—24no—
81C4×Dic14—42no—
82C4×Dic14C2×C414yescomplemented
83C7×#95—22no—
84C7×#95D412yes—
85#6406—24no—
86#6406C2×C414yes—
87C7⋊#95—22no—
88#6384—24no—
89#6384C2×C414yescomplemented
90#6382C2×C414yescomplemented
91#6381—24no—
92#6381—42no—
93C4⋊(C7⋊C8)—22no—
94#6377—24no—
95C4×C7⋊C8C2×C414yescomplemented
96C2×#440—72no—
97C2×C4×SD16—71no—
98Q8×C4⋊C4—71no—
99D4×C4⋊C4D711yescharacteristic
100#1944—72no—
101#1863—72no—
102#1862—72no—
103#1845—71no—
104#1803—71no—
105#1802—71no—
106#1770—71no—
107C24×C7—21no—
108C2×C7×D4—21no—
109C2×C7×D4—21no—
110C2×C7×D4—21no—
111C2×C7×D4C22×C412yes—
112C2×C7×D4C2×D412yes—
113C2×C7×D4C2×Q812yes—
114C22×C4×C7—21no—
115C22×C4×C7—24no—
116C22×C4×C7—22no—
117C22×C4×C7—44no—
118C22×C4×C7—44no—
119C22×C4×C7—42no—
120C22×C4×C7—42no—
121C22×C4×C7—82no—
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—22no—
130C2×Dic14—21no—
131C2×Dic14—21no—
132C2×Dic14—21no—
133C2×Dic14C22×C412yes—
134C2×Dic14C2×D412yes—
135C2×Dic14C2×Q812yes—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—44no—
142C7×C4⋊C4—42no—
143C7×C4⋊C4—81no—
144C7×C4⋊C4C22×C412yes—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—44no—
147C42×C7—22no—
148C42×C7—22no—
149C42×C7C22×C412yes—
150C42×C7#4112yes—
151#1142C4⋊C4116yescomplemented
152C4⋊Dic7—21no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—42no—
156C4⋊Dic7—81no—
157C4⋊Dic7C22×C412yes—
158Dic7⋊C4—24no—
159Dic7⋊C4—44no—
160Dic7⋊C4—82no—
161C4×Dic7—24no—
162C4×Dic7—44no—
163C2×C7⋊C8—21no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8C22×C412yes—
169C2×C7⋊C8C22×C412yes—
170C2×C7⋊C8C2×D412yes—
171C2×C7⋊C8C2×Q812yes—
172C22×SD16—71no—
173C2×C4×Q8—71no—
174C2×C4×Q8—72no—
175C2×C4×D4C2×D712yes—
176C2×C4×D4C2×D711yescharacteristic
177C22×C4⋊C4—21no—
178#440—148no—
179C4×SD16—78no—
180C2×#107—72no—
181C2×#107—72no—
182C2×#106—71no—
183C2×C4⋊C8—71no—
184C2×#103—71no—
185C2×#103—72no—
186C2×#102—71no—
187C2×#102—72no—
188C2×#97—72no—
189C2×C4×C8—71no—
190#389—72no—
191#389—142no—
192#389C2×D712yes—
193#385—22no—
194#384—71no—
195#384—141no—
196#384C2×D711yescharacteristic
197#380—21no—
198C4×C4⋊C4—141no—
199C4×C4⋊C4C2×D711yescharacteristic
200#340—72no—
201#340—72no—
202#336—72no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—41no—
207C23×C7—41no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7C22×D411yescharacteristic
211C7×D4—22no—
212C7×D4—22no—
213C7×D4—22no—
214C7×D4C2×C4⋊C414yes—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—48no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—88no—
231C2×C4×C7C4×D412yes—
232C2×C4×C7C4×D412yes—
233C2×C4×C7AGL(1,ℤ/8)12yes—
234C2×C4×C7#13712yes—
235C2×C4×C7C23×C411yescharacteristic
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C22×Q811yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C2×C4×C7H(5,𝔽2)11yescharacteristic
240C2×C4×C72−1+411yescharacteristic
241C2×Dic7—22no—
242C2×Dic7—22no—
243C2×Dic7—22no—
244C2×Dic7—22no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250Dic14—22no—
251Dic14—22no—
252Dic14—22no—
253Dic14C2×C4⋊C414yescomplemented
254C7⋊C8—22no—
255C7⋊C8—42no—
256C7⋊C8C2×C4⋊C414yescomplemented
257C22×Q8—71no—
258C22×D4C22×D711yescharacteristic
259C23×C4—21no—
260C23×C4—22no—
261C2×SD16—74no—
262C2×SD16—74no—
263C2×SD16—74no—
264C22×C8—71no—
265C22×C8—71no—
266C22×C8—72no—
267C4×Q8—74no—
268C4×Q8—144no—
269C4×Q8—142no—
270C4×Q8—282no—
271C4×D4—24no—
272C4×D4—22no—
273C4×D4—42no—
274C4×D4C4×D714yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—71no—
280C2×C4⋊C4—72no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4—142no—
283C2×C4⋊C4—141no—
284C2×C4⋊C4—142no—
285C2×C4⋊C4—141no—
286C2×C4⋊C4C22⋊D711yescharacteristic
287C2×C4⋊C4C22⋊D711yescharacteristic
288C2×C4⋊C4C22×D712yes—
289C2×C4⋊C4C22×D711yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C42—141no—
293C2×C42—142no—
294C2×C42C22×D712yes—
295C2×C42C22×D711yescharacteristic
296#107—144no—
297#107—282no—
298#106—74no—
299C4⋊C8—142no—
300#103—74no—
301#103—144no—
302#102—74no—
303#102—144no—
304#97—144no—
305C4×C8—74no—
306#95—22no—
307#95—142no—
308#95C22⋊D712yes—
309C22×C7—24no—
310C22×C7—42no—
311C22×C7—42no—
312C22×C7—42no—
313C22×C7—42no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7C2×C4×D411yescharacteristic
317C22×C7D4211yescharacteristic
318C22×C7D4×Q811yescharacteristic
319C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
320C22×C7C2×#13711yescharacteristic
321C22×C7#57611yescharacteristic
322C22×C7#57811yescharacteristic
323C4×C7—22no—
324C4×C7—42no—
325C4×C7—42no—
326C4×C7—42no—
327C4×C7—82no—
328C4×C7#44014yescomplemented
329C4×C7C22×C4⋊C411yescharacteristic
330C4×C7#52011yescharacteristic
331C4×C7#54711yescharacteristic
332C4×C7#55411yescharacteristic
333Dic7—24no—
334Dic7—42no—
335Dic7—82no—
336C24—21no—
337C2×Q8—72no—
338C2×Q8—72no—
339C2×Q8—72no—
340C2×Q8—141no—
341C2×Q8—141no—
342C2×Q8—141no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C4×D712yes—
347C2×D4D4×D712yes—
348C2×D4Q8×D712yes—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—82no—
357C22×C4—141no—
358C22×C4—141no—
359C22×C4—142no—
360C22×C4C2×C22⋊D712yes—
361C22×C4C2×C22⋊D712yes—
362C22×C4C2×C22⋊D711yescharacteristic
363C22×C4C2×C22⋊D711yescharacteristic
364C22×C4C23×D711yescharacteristic
365SD16—716no—
366C2×C8—72no—
367C2×C8—72no—
368C2×C8—72no—
369C2×C8—72no—
370C2×C8—141no—
371C2×C8—141no—
372C2×C8—142no—
373C2×C8—142no—
374C2×C8—142no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—72no—
383C4⋊C4—81no—
384C4⋊C4—144no—
385C4⋊C4—141no—
386C4⋊C4—142no—
387C4⋊C4—142no—
388C4⋊C4—284no—
389C4⋊C4—282no—
390C4⋊C4—562no—
391C4⋊C4—561no—
392C4⋊C4C2×C4×D712yes—
393C22⋊C4—24no—
394C22⋊C4—44no—
395C42—22no—
396C42—22no—
397C42—144no—
398C42—284no—
399C42C2×C4×D712yes—
400C42#115712yes—
401C2×C7—44no—
402C2×C7D4×C4⋊C411yescharacteristic
403C2×C7C2×#44011yescharacteristic
404C2×C7#295111yescharacteristic
405C2×C7#327411yescharacteristic
406C2×C7#327611yescharacteristic
407C2×C7#338511yescharacteristic
408C2×C7#338611yescharacteristic
409C23—22no—
410C23—22no—
411C23—22no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C22×C22⋊D711yescharacteristic
417Q8—74no—
418Q8—142no—
419Q8—142no—
420Q8—142no—
421D4—22no—
422D4—22no—
423D4—22no—
424D4D7×C4⋊C414yes—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—88no—
441C2×C4—142no—
442C2×C4—142no—
443C2×C4—142no—
444C2×C4—142no—
445C2×C4—281no—
446C2×C4—281no—
447C2×C4—282no—
448C2×C4—282no—
449C2×C4—282no—
450C2×C4C4×C22⋊D712yes—
451C2×C4C4×C22⋊D712yes—
452C2×C4#649512yes—
453C2×C4#651412yes—
454C2×C4C22×C4×D711yescharacteristic
455C2×C4C2×#115711yescharacteristic
456C2×C4C2×D4×D711yescharacteristic
457C2×C4#654811yescharacteristic
458C2×C4C2×Q8×D711yescharacteristic
459C2×C4#655111yescharacteristic
460C8—74no—
461C8—142no—
462C8—282no—
463C7#2267111yescharacteristic, complemented, Sylow
464C22—24no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×C4×C22⋊D711yescharacteristic
472C22C2×#649511yescharacteristic
473C22D4×C22⋊D711yescharacteristic
474C22Q8×C22⋊D711yescharacteristic
475C22C2×#651411yescharacteristic
476C22#9052611yescharacteristic
477C22#9052911yescharacteristic
478C4—22no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—144no—
484C4—282no—
485C4—562no—
486C4#8979014yescomplemented
487C4C2×D7×C4⋊C411yescharacteristic
488C4#9019511yescharacteristic
489C4#9019811yescharacteristic
490C4#9020211yescharacteristic
491C2—44no—
492C2C4⋊C4×C22⋊D711yescharacteristic
493C2C2×#8979011yescharacteristic
494C2#1174000211yescharacteristic
495C2#1174093711yescharacteristic
496C2#1174097411yescharacteristic
497C2#1174100311yescharacteristic
498C2#1174107611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×D4×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterD4×C4⋊C4D7
hypocenterC7#22671
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic14C2semidirect
7. #11727385C2semidirect
8. #11727375C2semidirect
9. #11727342C2semidirect
10. #11727331C2semidirect
11. #11727231C2semidirect
26. C4×#1142C4semidirect
82. C4×Dic14C2×C4semidirect
89. #6384C2×C4semidirect
90. #6382C2×C4semidirect
95. C4×C7⋊C8C2×C4semidirect
151. #1142C4⋊C4semidirect
253. Dic14C2×C4⋊C4semidirect
256. C7⋊C8C2×C4⋊C4semidirect
328. C4×C7#440semidirect
463. C7#22671semidirect
486. C4#89790semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C4⋊C42
C2×C4×#11422
C2×#897902
C4⋊C4×Dic142
#117293602
#117293582
#117273852
#117273752
#117273422
#117273312
#117272312
#226717