Finite Group Databasebeta

← #11740261

Subgroups — #11740261

Order 896.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6509C211yesmaximal, characteristic, complemented
3C22×(C2×D7)⋊C4C211yesmaximal, characteristic
4C2×D7×C22⋊C4C211yesmaximal, characteristic
5#90003C211yesmaximal, characteristic, complemented
6#89982C211yesmaximal, characteristic, complemented
7#89766C211yesmaximal, characteristic, complemented
8#89766C211yesmaximal, characteristic, complemented
9#89764C211yesmaximal, characteristic, complemented
10#89761C211yesmaximal, characteristic, complemented
11#89761C211yesmaximal, characteristic, complemented
12#89759C211yesmaximal, characteristic, complemented
13#89727C211yesmaximal, characteristic
14#89722C211yesmaximal, characteristic, complemented
15#89714C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic, complemented
18C23×C4×C7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C7×C22⋊Q8—24no—
23C2×C7×C4⋊C4C2211yescharacteristic, complemented
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25C2×C7×C4⋊C4C2211yescharacteristic, complemented
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30#6509—24no—
31C2×(C2×D7)⋊C4—24no—
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×C4⋊Dic7C2211yescharacteristic, complemented
43C2×C4⋊Dic7C2211yescharacteristic, complemented
44C2×Dic7⋊C4C2211yescharacteristic, complemented
45C2×Dic7⋊C4C2211yescharacteristic, complemented
46C2×Dic7⋊C4C2211yescharacteristic, complemented
47C2×Dic7⋊C4C2211yescharacteristic, complemented
48D7×C22⋊C4—24no—
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2437—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—41no—
62C23×D7—41no—
63C23×D7Q814yescomplemented
64C23×D7C2311yescharacteristic
65C23×D7C2311yescharacteristic
66C2×C7×Q8—22no—
67C2×C7×Q8—22no—
68C2×C7×Q8—41no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—41no—
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C22×C4×C7C2311yescharacteristic
78C22×Dic7—22no—
79C22×Dic7—22no—
80C22×Dic7—41no—
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C2×C4×D7—22no—
88C2×C4×D7—22no—
89C2×C4×D7—22no—
90C2×C4×D7—22no—
91C2×C4×D7—41no—
92C2×C4×D7—41no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4D414yes—
98C22⋊Dic7—22no—
99C22⋊Dic7—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—22no—
105(C2×D7)⋊C4—22no—
106(C2×D7)⋊C4—22no—
107(C2×D7)⋊C4—41no—
108(C2×D7)⋊C4—41no—
109(C2×D7)⋊C4D414yes—
110C4⋊Dic7—22no—
111C4⋊Dic7—41no—
112Dic7⋊C4—22no—
113Dic7⋊C4—22no—
114Dic7⋊C4—41no—
115Dic7⋊C4—41no—
116C2×C22⋊Q8—71no—
117C2×C22⋊Q8D711yescharacteristic, complemented
118C22×C22⋊C4—71no—
119C22×C22⋊C4—71no—
120#399—71no—
121#396—71no—
122#396—71no—
123#393—71no—
124#385—71no—
125#385—71no—
126#385—71no—
127#385—71no—
128#380—71no—
129#380—71no—
130#379—71no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7C2×D412yes—
136C23×C7C2×Q812yes—
137C23×C7#4112yes—
138C23×C7C2411yescharacteristic
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—28no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—28no—
151C22×D7—44no—
152C22×D7—44no—
153C22×D7—44no—
154C22×D7—44no—
155C22×D7—44no—
156C22×D7C2×D412yes—
157C22×D7C2×Q812yes—
158C22×D7C2×Q812yes—
159C22×D7#4112yes—
160C7×Q8—44no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—44no—
177C2×C4×C7C2×D412yes—
178C2×C4×C7C2×D412yes—
179C2×C4×C7C2×D412yes—
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C2×Dic7—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—44no—
198C2×Dic7C2×D412yes—
199C4×D7—44no—
200C4×D7—44no—
201C25—71no—
202C22×Q8C2×D711yescharacteristic, complemented
203C23×C4—71no—
204C23×C4—71no—
205C23×C4—71no—
206C23×C4C2×D711yescharacteristic
207C22⋊Q8—24no—
208C22⋊Q8—144no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4—71no—
211C2×C4⋊C4—71no—
212C2×C4⋊C4—71no—
213C2×C4⋊C4—71no—
214C2×C4⋊C4—71no—
215C2×C4⋊C4C2×D711yescharacteristic, complemented
216C2×C4⋊C4C2×D711yescharacteristic, complemented
217C2×C4⋊C4C2×D711yescharacteristic, complemented
218C2×C22⋊C4—71no—
219C2×C22⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4—71no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4—71no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4—71no—
227C2×C22⋊C4—71no—
228C2×C22⋊C4—71no—
229C2×C22⋊C4—71no—
230C2×C22⋊C4—144no—
231C2×C22⋊C4—144no—
232C2×C22⋊C4C2×D711yescharacteristic
233C2×C22⋊C4C2×D711yescharacteristic
234#95—71no—
235#95—71no—
236#95—71no—
237#95—71no—
238#95—71no—
239#95—71no—
240C22×C7—22no—
241C22×C7—22no—
242C22×C7—22no—
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7C22⋊Q814yescomplemented
247C22×C7C22×D411yescharacteristic
248C22×C7C22×D411yescharacteristic
249C22×C7C22×Q811yescharacteristic
250C22×C7C2×#4111yescharacteristic
251C22×C7C2×#4111yescharacteristic
252C22×C7H(5,𝔽2)11yescharacteristic
253C22×C7H(5,𝔽2)11yescharacteristic
254C2×D7—22no—
255C2×D7—22no—
256C2×D7—22no—
257C2×D7—22no—
258C2×D7—22no—
259C2×D7—22no—
260C2×D7—22no—
261C2×D7—22no—
262C2×D7—22no—
263C2×D7—416no—
264C2×D7—41no—
265C2×D7—41no—
266C2×D7—41no—
267C2×D7—41no—
268C2×D7C22⋊Q814yescomplemented
269C4×C7—22no—
270C4×C7—22no—
271C4×C7—22no—
272C4×C7—41no—
273C4×C7—41no—
274C4×C7—41no—
275Dic7—22no—
276Dic7—41no—
277Dic7—41no—
278Dic7—41no—
279Dic7—41no—
280Dic7—41no—
281C24—71no—
282C24—71no—
283C24—74no—
284C24—142no—
285C24—142no—
286C24—142no—
287C24—142no—
288C24—281no—
289C24—281no—
290C24C22×D711yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—41no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4—71no—
300C22×C4—71no—
301C22×C4—71no—
302C22×C4—71no—
303C22×C4—142no—
304C22×C4—142no—
305C22×C4—142no—
306C22×C4—142no—
307C22×C4—142no—
308C22×C4—142no—
309C22×C4—281no—
310C22×C4—281no—
311C22×C4—281no—
312C22×C4C22×D711yescharacteristic
313C22×C4C22×D711yescharacteristic
314C22×C4C22×D711yescharacteristic
315C22×C4C22×D711yescharacteristic
316C22×C4C22×D711yescharacteristic
317C22×C4C22×D711yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—142no—
322C4⋊C4—142no—
323C4⋊C4—142no—
324C4⋊C4—281no—
325C4⋊C4—281no—
326C4⋊C4—281no—
327C22⋊C4—22no—
328C22⋊C4—74no—
329C22⋊C4—142no—
330C22⋊C4—142no—
331C22⋊C4—142no—
332C22⋊C4—142no—
333C22⋊C4—142no—
334C22⋊C4—142no—
335C22⋊C4—142no—
336C22⋊C4—142no—
337C22⋊C4—142no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C22⋊C4C22⋊D714yes—
341C2×C7—24no—
342C2×C7C2×C22⋊Q811yescharacteristic
343C2×C7C2×C22⋊Q811yescharacteristic
344C2×C7#54311yescharacteristic
345C2×C7D4211yescharacteristic
346C2×C7#54611yescharacteristic
347C2×C7#54611yescharacteristic
348C2×C7#55111yescharacteristic
349D7—24no—
350D7—42no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—72no—
356C23—72no—
357C23—72no—
358C23—72no—
359C23—148no—
360C23—141no—
361C23—141no—
362C23—141no—
363C23—141no—
364C23—148no—
365C23—141no—
366C23—141no—
367C23—141no—
368C23—141no—
369C23—141no—
370C23—141no—
371C23—284no—
372C23—284no—
373C23—284no—
374C23—284no—
375C23—284no—
376C23Q8×D712yes—
377C23#116112yes—
378C23C2×C22⋊D712yes—
379C23C23×D711yescharacteristic
380Q8—44no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—44no—
397C2×C4—72no—
398C2×C4—141no—
399C2×C4—141no—
400C2×C4—141no—
401C2×C4—141no—
402C2×C4—141no—
403C2×C4—141no—
404C2×C4—141no—
405C2×C4—141no—
406C2×C4—141no—
407C2×C4—141no—
408C2×C4—141no—
409C2×C4—141no—
410C2×C4—141no—
411C2×C4—141no—
412C2×C4—141no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—284no—
416C2×C4—284no—
417C2×C4—284no—
418C2×C4D4×D712yes—
419C2×C4C2×C22⋊D712yes—
420C2×C4C2×C22⋊D712yes—
421C7#243711yescharacteristic, complemented, Sylow
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—74no—
429C22—142no—
430C22—142no—
431C22—142no—
432C22—142no—
433C22—142no—
434C22—142no—
435C22—142no—
436C22—142no—
437C22—142no—
438C22—2816no—
439C22—281no—
440C22—281no—
441C22—281no—
442C22—281no—
443C22#650914yescomplemented
444C22C2×D4×D711yescharacteristic
445C22#654811yescharacteristic
446C22C2×Q8×D711yescharacteristic
447C22C2×#116111yescharacteristic
448C22D7×#4111yescharacteristic
449C22#655311yescharacteristic
450C22C22×C22⋊D711yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—142no—
458C4—281no—
459C4—281no—
460C4—281no—
461C4—281no—
462C4—281no—
463C2—24no—
464C2—144no—
465C2—282no—
466C2D7×C22⋊Q811yescharacteristic
467C2#9032111yescharacteristic
468C2#9032511yescharacteristic
469C2#9032811yescharacteristic
470C2D4×C22⋊D711yescharacteristic
471C2C2×#650911yescharacteristic
472C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC24×D7C22
hypercenterC2×C22⋊Q8D7
hypocenterC7#2437
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6509C2semidirect
5. #90003C2semidirect
6. #89982C2semidirect
7. #89766C2semidirect
8. #89766C2semidirect
9. #89764C2semidirect
10. #89761C2semidirect
11. #89761C2semidirect
12. #89759C2semidirect
14. #89722C2semidirect
15. #89714C2semidirect
17. C22×C7×Q8C22semidirect
23. C2×C7×C4⋊C4C22semidirect
24. C2×C7×C4⋊C4C22semidirect
25. C2×C7×C4⋊C4C22semidirect
42. C2×C4⋊Dic7C22semidirect
43. C2×C4⋊Dic7C22semidirect
44. C2×Dic7⋊C4C22semidirect
45. C2×Dic7⋊C4C22semidirect
46. C2×Dic7⋊C4C22semidirect
47. C2×Dic7⋊C4C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
63. C23×D7Q8semidirect
117. C2×C22⋊Q8D7semidirect
202. C22×Q8C2×D7semidirect
215. C2×C4⋊C4C2×D7semidirect
216. C2×C4⋊C4C2×D7semidirect
217. C2×C4⋊C4C2×D7semidirect
246. C22×C7C22⋊Q8semidirect
268. C2×D7C22⋊Q8semidirect
421. C7#2437semidirect
443. C22#6509semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65092
C22×(C2×D7)⋊C42
C2×D7×C22⋊C42
#900032
#899822
#897662
#897642
#897612
#897592
#897272
#897222
#897142
#24377