Finite Group Databasebeta

← #11734859

Subgroups — #11734859

Order 896.

786 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
1C2×#6457C211yesmaximal, characteristic, complemented
2C2×#6455C211yesmaximal, characteristic, complemented
3C2×#6445C211yesmaximal, characteristic
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#396C211yesmaximal, characteristic, complemented
6#89766C211yesmaximal, characteristic, complemented
7#89763C211yesmaximal, characteristic, complemented
8#89761C211yesmaximal, characteristic, complemented
9#89760C211yesmaximal, characteristic, complemented
10#89733C211yesmaximal, characteristic, complemented
11#89729C211yesmaximal, characteristic, complemented
12#89727C211yesmaximal, characteristic, complemented
13#89716C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89437C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×D28C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic, complemented
23C2×C7×C4⋊C4C2211yescharacteristic, complemented
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
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, complemented
39C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
42C2×C4⋊Dic7C2211yescharacteristic, complemented
43C2×Dic7⋊C4C2211yescharacteristic, complemented
44C2×Dic7⋊C4C2211yescharacteristic, complemented
45C2×C4×Dic7C2211yescharacteristic, complemented
46C2×C4×Dic7C2211yescharacteristic, complemented
47#6457—24no—
48#6455—24no—
49#6445—24no—
50D7×C22⋊C4—24no—
51C7×#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2411—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—41no—
63C23×D7—41no—
64C23×D7C2311yescharacteristic
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—41no—
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C2×D28—22no—
82C2×D28—22no—
83C2×D28—22no—
84C2×D28—22no—
85C2×D28—22no—
86C2×D28—41no—
87C2×C4×D7—22no—
88C2×C4×D7—22no—
89C2×C4×D7—22no—
90C2×C4×D7—41no—
91C2×C4×D7—41no—
92C2×C4×D7D414yescomplemented
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—41no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—41no—
99C22⋊Dic7—22no—
100C22⋊Dic7—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—22no—
108(C2×D7)⋊C4—22no—
109(C2×D7)⋊C4—41no—
110(C2×D7)⋊C4—41no—
111(C2×D7)⋊C4—41no—
112(C2×D7)⋊C4D414yes—
113C4⋊Dic7—22no—
114Dic7⋊C4—22no—
115Dic7⋊C4—41no—
116C4×Dic7—22no—
117C4×Dic7—22no—
118C2×#123—71no—
119C2×#120—71no—
120C2×#117—71no—
121C22×C22⋊C4—71no—
122#396—71no—
123#396D711yescharacteristic, complemented
124#394—71no—
125#394—71no—
126#390—71no—
127#390—71no—
128#388—71no—
129#388—71no—
130#386—71no—
131#385—71no—
132#379—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7#4112yes—
140C23×C7C2411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—28no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
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×D7C2×D412yes—
167C22×D7#4112yes—
168C22×D7#4112yes—
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—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—
185C2×C4×C7C2×D412yes—
186C2×C4×C7C2×D412yes—
187C22⋊D7—44no—
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×Dic7C2×D412yes—
202C2×Dic7C2×D412yes—
203D28—44no—
204D28—44no—
205C4×D7—28no—
206C4×D7—44no—
207C25—71no—
208C22×D4—71no—
209C22×D4—71no—
210C22×D4—71no—
211C23×C4—71no—
212C23×C4—71no—
213#123—144no—
214#120—144no—
215#117—144no—
216C2×C4⋊C4—71no—
217C2×C4⋊C4—71no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4C2×D711yescharacteristic, complemented
220C2×C4⋊C4C2×D711yescharacteristic, complemented
221C2×C4⋊C4C2×D711yescharacteristic, complemented
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—71no—
231C2×C22⋊C4—71no—
232C2×C22⋊C4—71no—
233C2×C22⋊C4—71no—
234C2×C22⋊C4—71no—
235C2×C22⋊C4—71no—
236C2×C22⋊C4—144no—
237C2×C22⋊C4C2×D711yescharacteristic
238C2×C22⋊C4C2×D711yescharacteristic
239C2×C22⋊C4C2×D711yescharacteristic
240C2×C42—71no—
241C2×C42—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95C2×D711yescharacteristic, complemented
246C22×C7—22no—
247C22×C7—22no—
248C22×C7—22no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7—41no—
253C22×C7C22×D411yescharacteristic
254C22×C7C22×D411yescharacteristic
255C22×C7C2×#4111yescharacteristic
256C22×C7C2×#4111yescharacteristic
257C22×C7C2×#4111yescharacteristic
258C22×C7H(5,𝔽2)11yescharacteristic
259C22×C7H(5,𝔽2)11yescharacteristic
260C2×D7—22no—
261C2×D7—22no—
262C2×D7—22no—
263C2×D7—22no—
264C2×D7—22no—
265C2×D7—22no—
266C2×D7—22no—
267C2×D7—22no—
268C2×D7—22no—
269C2×D7—416no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—41no—
273C2×D7—41no—
274C2×D7—41no—
275C2×D7—41no—
276C2×D7—41no—
277C2×D7—41no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7#12114yes—
282C4×C7—22no—
283C4×C7—22no—
284C4×C7—41no—
285C4×C7—41no—
286C4×C7—41no—
287C4×C7—41no—
288Dic7—22no—
289Dic7—22no—
290Dic7—41no—
291Dic7—41no—
292Dic7—41no—
293C24—71no—
294C24—71no—
295C24—71no—
296C24—142no—
297C24—142no—
298C24—142no—
299C24—142no—
300C24—142no—
301C24—281no—
302C24—281no—
303C24C22×D711yescharacteristic
304C2×D4—142no—
305C2×D4—142no—
306C2×D4—142no—
307C2×D4—142no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—281no—
312C2×D4—281no—
313C22×C4—71no—
314C22×C4—71no—
315C22×C4—74no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—142no—
320C22×C4—142no—
321C22×C4—142no—
322C22×C4—281no—
323C22×C4—281no—
324C22×C4C22×D711yescharacteristic
325C22×C4C22×D711yescharacteristic
326C22×C4C22×D711yescharacteristic
327C22×C4C22×D711yescharacteristic
328C22×C4C22×D711yescharacteristic
329C22×C4C22×D711yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—142no—
334C4⋊C4—142no—
335C4⋊C4—281no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—74no—
340C22⋊C4—142no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4—142no—
349C22⋊C4—142no—
350C22⋊C4—281no—
351C22⋊C4—281no—
352C22⋊C4—281no—
353C42—142no—
354C42—142no—
355C2×C7—42no—
356C2×C7C2×#12111yescharacteristic
357C2×C7#52511yescharacteristic
358C2×C7#53511yescharacteristic
359C2×C7#54611yescharacteristic
360C2×C7#54611yescharacteristic
361C2×C7#55111yescharacteristic
362C2×C7#55111yescharacteristic
363D7—24no—
364D7—42no—
365D7—81no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—72no—
373C23—72no—
374C23—72no—
375C23—72no—
376C23—148no—
377C23—141no—
378C23—141no—
379C23—141no—
380C23—141no—
381C23—141no—
382C23—141no—
383C23—141no—
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—
400C23#116112yes—
401C23C23×D711yescharacteristic
402D4—284no—
403D4—284no—
404D4—284no—
405C2×C4—21no—
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—148no—
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—284no—
438C2×C4D4×D712yes—
439C2×C4D4×D712yes—
440C7#241111yescharacteristic, complemented, Sylow
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—74no—
449C22—142no—
450C22—142no—
451C22—142no—
452C22—142no—
453C22—142no—
454C22—142no—
455C22—142no—
456C22—142no—
457C22—142no—
458C22—2816no—
459C22—281no—
460C22—281no—
461C22—281no—
462C22—281no—
463C22—281no—
464C22—281no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22—281no—
470C22C2×D4×D711yescharacteristic
471C22C2×D4×D711yescharacteristic
472C22#654811yescharacteristic
473C22C2×#116111yescharacteristic
474C22D7×#4111yescharacteristic
475C22D7×#4111yescharacteristic
476C22#655311yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—142no—
484C4—142no—
485C4—281no—
486C4—281no—
487C4—281no—
488C2—42no—
489C2—144no—
490C2—282no—
491C2—561no—
492C2D7×#12111yescharacteristic
493C2#9029811yescharacteristic
494C2#9030111yescharacteristic
495C2#9031811yescharacteristic
496C2#9032111yescharacteristic
497C2#9032811yescharacteristic
498C2#9039611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#396C2
prime radicalC2×#6445C2
hypercenter#396D7
hypocenterC7#2411
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#6457C2semidirect
2. C2×#6455C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#396C2semidirect
6. #89766C2semidirect
7. #89763C2semidirect
8. #89761C2semidirect
9. #89760C2semidirect
10. #89733C2semidirect
11. #89729C2semidirect
12. #89727C2semidirect
13. #89716C2semidirect
14. #89441C2semidirect
15. #89437C2semidirect
22. C2×C7×C4⋊C4C22semidirect
23. C2×C7×C4⋊C4C22semidirect
24. C2×C7×C4⋊C4C22semidirect
38. C2×(C2×D7)⋊C4C22semidirect
39. C2×(C2×D7)⋊C4C22semidirect
40. C2×(C2×D7)⋊C4C22semidirect
41. C2×(C2×D7)⋊C4C22semidirect
42. C2×C4⋊Dic7C22semidirect
43. C2×Dic7⋊C4C22semidirect
44. C2×Dic7⋊C4C22semidirect
45. C2×C4×Dic7C22semidirect
46. C2×C4×Dic7C22semidirect
51. C7×#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
92. C2×C4×D7D4semidirect
123. #396D7semidirect
219. C2×C4⋊C4C2×D7semidirect
220. C2×C4⋊C4C2×D7semidirect
221. C2×C4⋊C4C2×D7semidirect
245. #95C2×D7semidirect
440. C7#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#64572
C2×#64552
C2×#64452
C2×D7×C22⋊C42
C7×#3962
#897662
#897632
#897612
#897602
#897332
#897292
#897272
#897162
#894412
#894372
#24117