Finite Group Databasebeta

← #13411924

Subgroups — #13411924

Order 1792.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739782C211yesmaximal, characteristic
2#11731780C211yesmaximal, characteristic, complemented
3C7×#1802C211yesmaximal, characteristic
4C4×#6406C211yesmaximal, characteristic
5#11728222C211yesmaximal, characteristic, complemented
6#11728208C211yesmaximal, characteristic, complemented
7#11727933C212yesmaximal
8#11727917C212yesmaximal
9#11727328C211yesmaximal, characteristic, complemented
10#11726572C212yesmaximal, complemented
11#11726553C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6489—21no—
14C2×#6451—21no—
15C2×#6451C2212yes—
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×#102C2212yes—
18C7×#384C2211yescharacteristic
19#89993C2212yes—
20C2×#6406C2211yescharacteristic
21C2×#6406C2211yescharacteristic
22#89984—21no—
23C2×#6391C2212yes—
24C2×Dic7⋊C8C2212yes—
25#89751—22no—
26C4⋊C4×Dic7C2212yes—
27#89745—21no—
28#89745C2212yes—
29#89723—22no—
30#89707C2211yescharacteristic
31C4×C4⋊Dic7C2211yescharacteristic
32C4×C4⋊Dic7C2211yescharacteristic
33C2×C4×C7⋊C8C2211yescharacteristic
34C7×#340C2212yes—
35#89346C2212yes—
36#89345C2212yes—
37#89324C2212yes—
38#89324C2212yes—
39#89324C2211yescharacteristic
40#89324C2211yescharacteristic
41#26024—71nomaximal, Sylow
42C22×C7×D4C2311yescharacteristic
43C23×C4×C7—21no—
44C22×Dic14—22no—
45C22×C7×C8C2312yes—
46C4×C7×D4—22no—
47C4×C7×D4Q814yescomplemented
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4C2312yes—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C22⋊C4—21no—
53C2×C42×C7C2311yescharacteristic
54C2×C22⋊Dic7—22no—
55C2×C22⋊Dic7—22no—
56#6489—44no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7D411yescharacteristic
60C2×C4⋊Dic7D411yescharacteristic
61C2×C4⋊Dic7C2312yes—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2311yescharacteristic
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—22no—
66C2×C4×Dic7—21no—
67C2×C4×Dic7—22no—
68C2×C4×Dic7C2312yes—
69C22×C7⋊C8C2311yescharacteristic
70C22×C7⋊C8C2311yescharacteristic
71#6451—24no—
72#6451—42no—
73#6451—44no—
74C7×C4⋊C8—22no—
75C7×#102—42no—
76#6406—22no—
77#6406—22no—
78C7⋊#95—21no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95D412yes—
82#6391—24no—
83Dic7⋊C8—42no—
84C4×C7⋊C8—22no—
85#2551—71no—
86#2354—71no—
87#2065—72no—
88#2023—72no—
89#1987—72no—
90#1948—71no—
91#1939—71no—
92#1920—72no—
93#1834—71no—
94#1802D711yescharacteristic
95C4×#102—71no—
96C24×C7—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4C2×Q812yes—
103C22×C4×C7—21no—
104C22×C4×C7—24no—
105C22×C4×C7—22no—
106C22×C4×C7—42no—
107C22×C4×C7—42no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—21no—
112C22×Dic7—21no—
113C22×Dic7—22no—
114C22×Dic7C2×D412yes—
115C22×Dic7C2×D412yes—
116C2×Dic14—44no—
117C2×Dic14—44no—
118C2×Dic14—82no—
119C2×C7×C8—22no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—42no—
126C7×C4⋊C4—81no—
127C7×C4⋊C4C2×Q812yes—
128C7×C22⋊C4—24no—
129C42×C7C2×Q812yes—
130C42×C7#4112yes—
131C22⋊Dic7—44no—
132C22⋊Dic7—82no—
133C4⋊Dic7—22no—
134C4⋊Dic7—21no—
135C4⋊Dic7—21no—
136C4⋊Dic7—22no—
137C4⋊Dic7—22no—
138C4⋊Dic7—42no—
139C4⋊Dic7—81no—
140C4⋊Dic7SD1614yes—
141C4⋊Dic7#4112yes—
142C4⋊Dic7#4112yes—
143Dic7⋊C4—44no—
144Dic7⋊C4—44no—
145C4×Dic7—22no—
146C4×Dic7—22no—
147C4×Dic7—42no—
148C4×Dic7—44no—
149C2×C7⋊C8—21no—
150C2×C7⋊C8—21no—
151C2×C7⋊C8—21no—
152C2×C7⋊C8—21no—
153C2×C7⋊C8—21no—
154C2×C7⋊C8—21no—
155C2×C4⋊Q8—72no—
156C2×C4⋊Q8—141no—
157C2×C4×D4C2×D711yescharacteristic
158C2×#117—141no—
159C2×#106—72no—
160C2×C4⋊C8—72no—
161C2×C4⋊C8C2×D711yescharacteristic
162C2×#102—71no—
163C2×#102—71no—
164C2×#102C2×D712yes—
165C2×C4×C8—71no—
166#395—142no—
167#390—72no—
168#389—72no—
169#389—141no—
170#388—142no—
171#384—71no—
172#384C2×D711yescharacteristic
173#380—141no—
174C4×C4⋊C4—71no—
175C4×C4⋊C4—72no—
176C4×C4⋊C4—71no—
177#340—72no—
178#340—72no—
179#340—72no—
180#340—71no—
181#340—71no—
182#340C2×D712yes—
183#336—72no—
184C23×C7—22no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7—41no—
191C23×C7C22×D411yescharacteristic
192C7×D4—22no—
193C7×D4—22no—
194C7×D4—41no—
195C7×D4—41no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—22no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—48no—
202C2×C4×C7—42no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7C22⋊Q812yes—
206C2×C4×C7C22⋊Q812yes—
207C2×C4×C7C2×SD1612yes—
208C2×C4×C7AGL(1,ℤ/8)12yes—
209C2×C4×C7C22×Q811yescharacteristic
210C2×C4×C7C2×#4111yescharacteristic
211C2×C4×C7C2×#4111yescharacteristic
212C2×C4×C7C2×#4111yescharacteristic
213C2×C4×C7H(5,𝔽2)11yescharacteristic
214C2×C4×C72−1+411yescharacteristic
215C2×Dic7—22no—
216C2×Dic7—22no—
217C2×Dic7—22no—
218C2×Dic7—22no—
219C2×Dic7—22no—
220C2×Dic7—22no—
221C2×Dic7—22no—
222C2×Dic7—41no—
223C2×Dic7—41no—
224C2×Dic7—41no—
225C2×Dic7—41no—
226C2×Dic7—42no—
227C2×Dic7—42no—
228C2×Dic7—42no—
229C2×Dic7C2×SD1614yes—
230Dic14—88no—
231C7×C8—42no—
232C7⋊C8—41no—
233C7⋊C8—41no—
234C22×Q8—142no—
235C22×D4C22×D711yescharacteristic
236C23×C4—21no—
237C22×C8—71no—
238C22×C8—71no—
239C22×C8C22×D712yes—
240C4⋊Q8—144no—
241C4⋊Q8—282no—
242C4⋊Q8—284no—
243C4×D4—22no—
244C4×D4Dic1414yescomplemented
245#117—284no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—71no—
249C2×C4⋊C4—71no—
250C2×C4⋊C4—71no—
251C2×C4⋊C4—71no—
252C2×C4⋊C4—72no—
253C2×C4⋊C4—142no—
254C2×C4⋊C4—142no—
255C2×C4⋊C4—141no—
256C2×C4⋊C4—141no—
257C2×C4⋊C4C22×D712yes—
258C2×C4⋊C4C22×D711yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—142no—
261C2×C22⋊C4—142no—
262C2×C42—72no—
263C2×C42—142no—
264C2×C42—141no—
265C2×C42C22×D711yescharacteristic
266#106—144no—
267C4⋊C8—22no—
268C4⋊C8—282no—
269#102—42no—
270#102—142no—
271#102—142no—
272C4×C8—142no—
273#95—72no—
274#95—142no—
275#95—142no—
276#95—141no—
277C22×C7—42no—
278C22×C7—42no—
279C22×C7—42no—
280C22×C7—81no—
281C22×C7—81no—
282C22×C7—81no—
283C22×C7—81no—
284C22×C7C2×C22⋊Q811yescharacteristic
285C22×C7#54611yescharacteristic
286C22×C7#54811yescharacteristic
287C22×C7C22×SD1611yescharacteristic
288C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C7#57611yescharacteristic
290C22×C7#57811yescharacteristic
291C4×C7—22no—
292C4×C7—42no—
293C4×C7—82no—
294C4×C7#47614yes—
295C4×C7#53311yescharacteristic
296C4×C7#54411yescharacteristic
297C4×C7#55511yescharacteristic
298C4×C7#55611yescharacteristic
299Dic7—24no—
300Dic7—42no—
301Dic7—42no—
302Dic7—42no—
303Dic7—82no—
304C24—21no—
305C2×Q8—284no—
306C2×Q8—284no—
307C2×Q8—562no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4C2×Dic1412yes—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—72no—
320C22×C4—72no—
321C22×C4—141no—
322C22×C4—141no—
323C22×C4—142no—
324C22×C4D4×D711yescharacteristic
325C22×C4D4×D711yescharacteristic
326C22×C4C23×D711yescharacteristic
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—141no—
331C2×C8—141no—
332C2×C8—141no—
333C2×C8—141no—
334C2×C8—141no—
335C2×C8—141no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—74no—
341C4⋊C4—72no—
342C4⋊C4—72no—
343C4⋊C4—81no—
344C4⋊C4—142no—
345C4⋊C4—141no—
346C4⋊C4—141no—
347C4⋊C4—142no—
348C4⋊C4—142no—
349C4⋊C4—284no—
350C4⋊C4—284no—
351C4⋊C4—282no—
352C4⋊C4—561no—
353C4⋊C4C2×Dic1412yes—
354C22⋊C4—24no—
355C22⋊C4—284no—
356C22⋊C4—562no—
357C42—142no—
358C42—142no—
359C42—284no—
360C42—282no—
361C42C2×Dic1412yes—
362C42#115912yes—
363C2×C7—82no—
364C2×C7#243911yescharacteristic
365C2×C7C2×#47611yescharacteristic
366C2×C7#308311yescharacteristic
367C2×C7#329811yescharacteristic
368C2×C7#332511yescharacteristic
369C2×C7#336211yescharacteristic
370C2×C7#336711yescharacteristic
371C23—22no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23C2×D4×D711yescharacteristic
379Q8—568no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—74no—
394C2×C4—142no—
395C2×C4—142no—
396C2×C4—142no—
397C2×C4—142no—
398C2×C4—142no—
399C2×C4—142no—
400C2×C4—142no—
401C2×C4—281no—
402C2×C4—281no—
403C2×C4—281no—
404C2×C4—281no—
405C2×C4—282no—
406C2×C4—282no—
407C2×C4—282no—
408C2×C4#644112yes—
409C2×C4#644112yes—
410C2×C4#647412yes—
411C2×C4D7×SD1612yes—
412C2×C4C22×Dic1411yescharacteristic
413C2×C4C2×#115911yescharacteristic
414C2×C4#654811yescharacteristic
415C2×C4#655111yescharacteristic
416C2×C4D7×#4111yescharacteristic
417C2×C4D7×#4111yescharacteristic
418C8—42no—
419C8—281no—
420C8—281no—
421C7#2602411yescharacteristic, complemented, Sylow
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C2×#644111yescharacteristic
430C22#9030311yescharacteristic
431C22#9032411yescharacteristic
432C22C2×#647411yescharacteristic
433C22C2×D7×SD1611yescharacteristic
434C22#9046511yescharacteristic
435C22#9047011yescharacteristic
436C4—22no—
437C4—42no—
438C4—82no—
439C4—144no—
440C4—282no—
441C4—282no—
442C4—282no—
443C4—562no—
444C4#8953314yes—
445C4#9019111yescharacteristic
446C4#9020811yescharacteristic
447C4#9030711yescharacteristic
448C4#9033211yescharacteristic
449C2—82no—
450C2#1173134311yescharacteristic
451C2C2×#8953311yescharacteristic
452C2#1173631711yescharacteristic
453C2#1173780111yescharacteristic
454C2#1173789511yescharacteristic
455C2#1173843811yescharacteristic
456C2#1173853511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1802C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#1802D7
hypocenterC7#26024
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #11731780C2semidirect
5. #11728222C2semidirect
6. #11728208C2semidirect
9. #11727328C2semidirect
10. #11726572C2semidirect
11. #11726553C2semidirect
47. C4×C7×D4Q8semidirect
244. C4×D4Dic14semidirect
421. C7#26024semidirect

Maximal subgroups

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

subgroupindex
#117397822
#117317802
C7×#18022
C4×#64062
#117282222
#117282082
#117279332
#117279172
#117273282
#117265722
#117265532
#260247