Finite Group Databasebeta

← #11734556

Subgroups — #11734556

Order 896.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6500C211yesmaximal, characteristic
2C2×#6498C211yesmaximal, characteristic, complemented
3C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#394C211yesmaximal, characteristic, complemented
6#89764C211yesmaximal, characteristic, complemented
7#89761C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89730C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89727C211yesmaximal, characteristic, complemented
12#89724C211yesmaximal, characteristic, complemented
13#89717C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89439C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic, complemented
21C22×C4×D7C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic, complemented
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic, complemented
27C2×C7×C22⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic, complemented
32#6500—24no—
33#6498—24no—
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×Dic7⋊C4C2211yescharacteristic, complemented
43C2×Dic7⋊C4C2211yescharacteristic, complemented
44C2×Dic7⋊C4C2211yescharacteristic, complemented
45C2×Dic7⋊C4C2211yescharacteristic, complemented
46C2×C4×Dic7C2211yescharacteristic, complemented
47(C22⋊D7)⋊C4—24no—
48D7×C22⋊C4—24no—
49C7×#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2418—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—22no—
63C23×D7—41no—
64C23×D7—41no—
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—22no—
69C2×C7×D4—41no—
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C2×C22⋊D7—22no—
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C2×C22⋊D7—41no—
80C2×C22⋊D7D414yes—
81C22×Dic7—22no—
82C22×Dic7—22no—
83C22×Dic7—41no—
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C2×C4×D7—22no—
91C2×C4×D7—22no—
92C2×C4×D7—22no—
93C2×C4×D7—22no—
94C2×C4×D7—41no—
95C2×C4×D7—41no—
96C7×C4⋊C4—41no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103C22⋊Dic7—22no—
104C22⋊Dic7D414yescomplemented
105(C2×D7)⋊C4—22no—
106(C2×D7)⋊C4—22no—
107(C2×D7)⋊C4—22no—
108(C2×D7)⋊C4—22no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—22no—
112(C2×D7)⋊C4—41no—
113Dic7⋊C4—22no—
114Dic7⋊C4—22no—
115Dic7⋊C4—22no—
116Dic7⋊C4—41no—
117C4×Dic7—22no—
118C2×#123—71no—
119C2×#120—71no—
120C2×C4×D4—71no—
121C22×C22⋊C4—71no—
122#399—71no—
123#397—71no—
124#394—71no—
125#394D711yescharacteristic, complemented
126#388—71no—
127#388—71no—
128#386—71no—
129#386—71no—
130#385—71no—
131#380—71no—
132#379—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7C2×D412yes—
146C23×C7C2×D412yes—
147C23×C7C2411yescharacteristic
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—28no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—44no—
160C22×D7—44no—
161C22×D7—44no—
162C22×D7—44no—
163C22×D7—44no—
164C22×D7—44no—
165C22×D7C2×D412yes—
166C22×D7#4112yes—
167C22×D7#4112yes—
168C22×D7#4112yes—
169C7×D4—44no—
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—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C22⋊D7—28no—
186C22⋊D7—44no—
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—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202C2×Dic7—44no—
203C2×Dic7C2×D412yes—
204C2×Dic7C2×D412yes—
205C2×Dic7C2×D412yes—
206C4×D7—44no—
207C4×D7—44no—
208C25—71no—
209C22×D4—71no—
210C22×D4—71no—
211C22×D4C2×D711yescharacteristic
212C23×C4—71no—
213C23×C4—71no—
214C23×C4—71no—
215#123—144no—
216#120—144no—
217C4×D4—144no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4—71no—
222C2×C4⋊C4C2×D711yescharacteristic, complemented
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—71no—
231C2×C22⋊C4—71no—
232C2×C22⋊C4—71no—
233C2×C22⋊C4—71no—
234C2×C22⋊C4—71no—
235C2×C22⋊C4—144no—
236C2×C22⋊C4C2×D711yescharacteristic
237C2×C22⋊C4C2×D711yescharacteristic
238C2×C22⋊C4C2×D711yescharacteristic, complemented
239C2×C22⋊C4C2×D711yescharacteristic, complemented
240C2×C42—71no—
241#95—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95—71no—
246#95C2×D711yescharacteristic, complemented
247C22×C7—22no—
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7C22×D411yescharacteristic
262C22×C7C22×D411yescharacteristic
263C22×C7C2×#4111yescharacteristic
264C22×C7C2×#4111yescharacteristic
265C22×C7C2×#4111yescharacteristic
266C22×C7H(5,𝔽2)11yescharacteristic
267C22×C7H(5,𝔽2)11yescharacteristic
268C2×D7—22no—
269C2×D7—22no—
270C2×D7—22no—
271C2×D7—22no—
272C2×D7—22no—
273C2×D7—22no—
274C2×D7—22no—
275C2×D7—22no—
276C2×D7—22no—
277C2×D7—416no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C2×D7#12314yescomplemented
283C4×C7—41no—
284C4×C7—41no—
285C4×C7—41no—
286C4×C7—41no—
287C4×C7—41no—
288Dic7—22no—
289Dic7—22no—
290Dic7—22no—
291Dic7—41no—
292Dic7—41no—
293Dic7—41no—
294C24—71no—
295C24—71no—
296C24—142no—
297C24—142no—
298C24—142no—
299C24—142no—
300C24—142no—
301C24—281no—
302C24—281no—
303C24C22×D711yescharacteristic
304C24C22×D711yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—74no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—281no—
314C22×C4—71no—
315C22×C4—71no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—71no—
320C22×C4—142no—
321C22×C4—142no—
322C22×C4—142no—
323C22×C4—142no—
324C22×C4—142no—
325C22×C4—142no—
326C22×C4—281no—
327C22×C4—281no—
328C22×C4—281no—
329C22×C4C22×D711yescharacteristic
330C22×C4C22×D711yescharacteristic
331C22×C4C22×D711yescharacteristic
332C22×C4C22×D711yescharacteristic
333C22×C4C22×D711yescharacteristic
334C4⋊C4—41no—
335C4⋊C4—142no—
336C4⋊C4—142no—
337C4⋊C4—142no—
338C4⋊C4—281no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—74no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4—142no—
349C22⋊C4—142no—
350C22⋊C4—142no—
351C22⋊C4—142no—
352C22⋊C4—142no—
353C22⋊C4—142no—
354C22⋊C4—281no—
355C42—142no—
356C2×C7—42no—
357C2×C7—42no—
358C2×C7C2×#12311yescharacteristic
359C2×C7#52511yescharacteristic
360C2×C7#53811yescharacteristic
361C2×C7D4211yescharacteristic
362C2×C7#54611yescharacteristic
363C2×C7#54611yescharacteristic
364C2×C7#55111yescharacteristic
365D7—24no—
366D7—42no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—72no—
380C23—72no—
381C23—72no—
382C23—72no—
383C23—148no—
384C23—141no—
385C23—141no—
386C23—141no—
387C23—141no—
388C23—141no—
389C23—141no—
390C23—141no—
391C23—141no—
392C23—141no—
393C23—141no—
394C23—284no—
395C23—284no—
396C23—284no—
397C23—284no—
398C23—284no—
399C23—284no—
400C23D4×D712yes—
401C23D4×D712yes—
402C23C23×D711yescharacteristic
403D4—44no—
404D4—148no—
405D4—284no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—72no—
422C2×C4—72no—
423C2×C4—72no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—141no—
427C2×C4—141no—
428C2×C4—141no—
429C2×C4—141no—
430C2×C4—141no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—141no—
438C2×C4—141no—
439C2×C4—284no—
440C2×C4—284no—
441C2×C4—284no—
442C7#241811yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—74no—
458C22—142no—
459C22—142no—
460C22—142no—
461C22—142no—
462C22—142no—
463C22—142no—
464C22—142no—
465C22—142no—
466C22—142no—
467C22—2816no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22—281no—
472C22C2×D4×D711yescharacteristic
473C22C2×D4×D711yescharacteristic
474C22#654811yescharacteristic
475C22#654811yescharacteristic
476C22D7×#4111yescharacteristic
477C22D7×#4111yescharacteristic
478C22D7×#4111yescharacteristic
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—142no—
485C4—142no—
486C4—142no—
487C4—281no—
488C4—281no—
489C4—281no—
490C2—42no—
491C2—42no—
492C2—144no—
493C2—282no—
494C2#9028111yescharacteristic
495C2#9030111yescharacteristic
496C2#9030111yescharacteristic
497C2#9030411yescharacteristic
498C2D7×#12311yescharacteristic
499C2#9034711yescharacteristic
500C2#9037011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6500C2
hypercenter#394D7
hypocenterC7#2418
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6498C2semidirect
3. C2×(C22⋊D7)⋊C4C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#394C2semidirect
6. #89764C2semidirect
7. #89761C2semidirect
8. #89731C2semidirect
9. #89730C2semidirect
10. #89728C2semidirect
11. #89727C2semidirect
12. #89724C2semidirect
13. #89717C2semidirect
14. #89441C2semidirect
15. #89439C2semidirect
20. C23×Dic7C22semidirect
23. C2×C7×C4⋊C4C22semidirect
26. C2×C7×C22⋊C4C22semidirect
27. C2×C7×C22⋊C4C22semidirect
31. C2×C22⋊Dic7C22semidirect
42. C2×Dic7⋊C4C22semidirect
43. C2×Dic7⋊C4C22semidirect
44. C2×Dic7⋊C4C22semidirect
45. C2×Dic7⋊C4C22semidirect
46. C2×C4×Dic7C22semidirect
49. C7×#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
104. C22⋊Dic7D4semidirect
125. #394D7semidirect
222. C2×C4⋊C4C2×D7semidirect
238. C2×C22⋊C4C2×D7semidirect
239. C2×C22⋊C4C2×D7semidirect
246. #95C2×D7semidirect
282. C2×D7#123semidirect
442. C7#2418semidirect

Maximal subgroups

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

subgroupindex
C2×#65002
C2×#64982
C2×(C22⋊D7)⋊C42
C2×D7×C22⋊C42
C7×#3942
#897642
#897612
#897312
#897302
#897282
#897272
#897242
#897172
#894412
#894392
#24187