Finite Group Databasebeta

← #11733913

Subgroups — #11733913

Order 896.

666 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6500C211yesmaximal, characteristic
2C2×#6461C211yesmaximal, characteristic, complemented
3C7×#390C211yesmaximal, characteristic, complemented
4#89994C211yesmaximal, characteristic, complemented
5#89764C211yesmaximal, characteristic, complemented
6#89733C211yesmaximal, characteristic, complemented
7#89731C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89729C211yesmaximal, characteristic, complemented
10#89727C211yesmaximal, characteristic, complemented
11#89727C211yesmaximal, characteristic, complemented
12#89720C211yesmaximal, characteristic, complemented
13#89717C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15#89712C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic, complemented
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic, complemented
24C2×C7×C22⋊C4C2211yescharacteristic, complemented
25C2×C42×C7C2211yescharacteristic, complemented
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic, complemented
30C2×C22⋊Dic7C2211yescharacteristic, complemented
31#6500—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×Dic7⋊C4C2211yescharacteristic, complemented
44C2×Dic7⋊C4C2211yescharacteristic, complemented
45C2×Dic7⋊C4C2211yescharacteristic, complemented
46C2×Dic7⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic7C2211yescharacteristic, complemented
48#6461—24no—
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53#2685—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C23×D7—22no—
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—41no—
60C23×D7—41no—
61C23×D7—41no—
62C23×D7—41no—
63C23×D7C2311yescharacteristic
64C23×D7C2311yescharacteristic
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—41no—
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—22no—
76C2×C22⋊D7—41no—
77C2×C22⋊D7—41no—
78C2×C22⋊D7—41no—
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C7×C4⋊C4—22no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C42×C7—41no—
91C22⋊Dic7—22no—
92C22⋊Dic7—22no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C22⋊Dic7—41no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—41no—
101(C2×D7)⋊C4—41no—
102(C2×D7)⋊C4—41no—
103(C2×D7)⋊C4—41no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4D414yes—
106C4⋊Dic7—22no—
107Dic7⋊C4—22no—
108Dic7⋊C4—41no—
109Dic7⋊C4—41no—
110Dic7⋊C4—41no—
111C4×Dic7—22no—
112C2×#126—71no—
113C2×#120—71no—
114#399—71no—
115#396—71no—
116#394—71no—
117#394—71no—
118#394—71no—
119#394—71no—
120#390—71no—
121#390D711yescharacteristic, complemented
122#388—71no—
123#388—71no—
124#388—71no—
125#379—71no—
126#379—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7#4112yes—
141C23×C7C2411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
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—44no—
155C22×D7—44no—
156C22×D7—44no—
157C22×D7—44no—
158C22×D7—44no—
159C22×D7—44no—
160C22×D7—44no—
161C22×D7C2×D412yes—
162C22×D7#4112yes—
163C7×D4—44no—
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—21no—
177C2×C4×C7—21no—
178C2×C4×C7C2×D412yes—
179C22⋊D7—44no—
180C22⋊D7—44no—
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—21no—
198C2×Dic7C2×D412yes—
199C25—71no—
200C22×D4—71no—
201C22×D4—71no—
202C22×D4C2×D711yescharacteristic
203#126—144no—
204#120—144no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4—71no—
208C2×C4⋊C4—71no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4C2×D711yescharacteristic, complemented
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4—71no—
215C2×C22⋊C4—71no—
216C2×C22⋊C4—71no—
217C2×C22⋊C4—71no—
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⋊C4C2×D711yescharacteristic
227C2×C22⋊C4C2×D711yescharacteristic
228C2×C22⋊C4C2×D711yescharacteristic, complemented
229C2×C22⋊C4C2×D711yescharacteristic, complemented
230C2×C42—71no—
231C2×C42C2×D711yescharacteristic, complemented
232#95—71no—
233#95—71no—
234#95—71no—
235#95—71no—
236C22×C7—22no—
237C22×C7—22no—
238C22×C7—22no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7C22×D411yescharacteristic
251C22×C7C2×#4111yescharacteristic
252C22×C7C2×#4111yescharacteristic
253C22×C7H(5,𝔽2)11yescharacteristic
254C22×C7H(5,𝔽2)11yescharacteristic
255C22×C7H(5,𝔽2)11yescharacteristic
256C22×C7H(5,𝔽2)11yescharacteristic
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×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
272C4×C7—22no—
273C4×C7—41no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—41no—
277Dic7—22no—
278Dic7—41no—
279Dic7—41no—
280Dic7—41no—
281Dic7—41no—
282Dic7—41no—
283C24—71no—
284C24—71no—
285C24—142no—
286C24—142no—
287C24—142no—
288C24—281no—
289C24—281no—
290C24—281no—
291C24—281no—
292C24C22×D711yescharacteristic
293C24C22×D711yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—142no—
298C2×D4—142no—
299C2×D4—142no—
300C2×D4—281no—
301C2×D4—281no—
302C2×D4—281no—
303C22×C4—71no—
304C22×C4—71no—
305C22×C4—71no—
306C22×C4—71no—
307C22×C4—71no—
308C22×C4—71no—
309C22×C4C22×D711yescharacteristic
310C22×C4C22×D711yescharacteristic
311C22×C4C22×D711yescharacteristic
312C22×C4C22×D711yescharacteristic
313C22×C4C22×D711yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—142no—
316C4⋊C4—142no—
317C4⋊C4—281no—
318C4⋊C4—281no—
319C4⋊C4—281no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—74no—
325C22⋊C4—142no—
326C22⋊C4—142no—
327C22⋊C4—142no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C22⋊C4—142no—
331C22⋊C4—281no—
332C22⋊C4—281no—
333C22⋊C4—281no—
334C22⋊C4—281no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C42—41no—
340C42—142no—
341C2×C7—42no—
342C2×C7—81no—
343C2×C7#53511yescharacteristic
344C2×C7#53811yescharacteristic
345C2×C7#53811yescharacteristic
346C2×C7#54611yescharacteristic
347C2×C7#54611yescharacteristic
348C2×C7#55111yescharacteristic
349C2×C7#56011yescharacteristic
350D7—42no—
351D7—42no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—72no—
366C23—72no—
367C23—141no—
368C23—141no—
369C23—141no—
370C23—141no—
371C23—141no—
372C23—141no—
373C23—141no—
374C23—141no—
375C23—141no—
376C23—141no—
377C23—141no—
378C23—141no—
379C23—284no—
380C23—284no—
381C23—284no—
382C23—284no—
383C23—284no—
384C23—284no—
385C23—284no—
386C23#115712yes—
387C23C23×D711yescharacteristic
388D4—44no—
389D4—284no—
390D4—284no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—72no—
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—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4D4×D712yes—
424C7#268511yescharacteristic, complemented, Sylow
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—142no—
440C22—142no—
441C22—142no—
442C22—142no—
443C22—142no—
444C22—142no—
445C22—2816no—
446C22—281no—
447C22—281no—
448C22—281no—
449C22—281no—
450C22—281no—
451C22—281no—
452C22—281no—
453C22—281no—
454C22C2×#115711yescharacteristic
455C22C2×D4×D711yescharacteristic
456C22#654811yescharacteristic
457C22#654811yescharacteristic
458C22#654811yescharacteristic
459C22D7×#4111yescharacteristic
460C22#655311yescharacteristic
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—142no—
467C4—281no—
468C4—281no—
469C4—281no—
470C4—281no—
471C4—281no—
472C2—42no—
473C2—81no—
474C2—282no—
475C2—282no—
476C2#9018111yescharacteristic
477C2#9024711yescharacteristic
478C2#9025111yescharacteristic
479C2#9030411yescharacteristic
480C2#9031111yescharacteristic
481C2#9036711yescharacteristic
482C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC2×#6500C2
hypercenter#390D7
hypocenterC7#2685
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#6461C2semidirect
3. C7×#390C2semidirect
4. #89994C2semidirect
5. #89764C2semidirect
6. #89733C2semidirect
7. #89731C2semidirect
8. #89731C2semidirect
9. #89729C2semidirect
10. #89727C2semidirect
11. #89727C2semidirect
12. #89720C2semidirect
13. #89717C2semidirect
14. #89712C2semidirect
15. #89712C2semidirect
20. C2×C7×C4⋊C4C22semidirect
23. C2×C7×C22⋊C4C22semidirect
24. C2×C7×C22⋊C4C22semidirect
25. C2×C42×C7C22semidirect
29. C2×C22⋊Dic7C22semidirect
30. C2×C22⋊Dic7C22semidirect
42. C2×C4⋊Dic7C22semidirect
43. C2×Dic7⋊C4C22semidirect
44. C2×Dic7⋊C4C22semidirect
45. C2×Dic7⋊C4C22semidirect
46. C2×Dic7⋊C4C22semidirect
47. C2×C4×Dic7C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
121. #390D7semidirect
210. C2×C4⋊C4C2×D7semidirect
228. C2×C22⋊C4C2×D7semidirect
229. C2×C22⋊C4C2×D7semidirect
231. C2×C42C2×D7semidirect
424. C7#2685semidirect

Maximal subgroups

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

subgroupindex
C2×#65002
C2×#64612
C7×#3902
#899942
#897642
#897332
#897312
#897292
#897272
#897202
#897172
#897122
#26857