Finite Group Databasebeta

← #11732369

Subgroups — #11732369

Order 896.

680 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6500C211yesmaximal, characteristic
2C2×#6492C211yesmaximal, characteristic, complemented
3C7×#386C211yesmaximal, characteristic, complemented
4#89994C211yesmaximal, characteristic, complemented
5#89985C211yesmaximal, characteristic, complemented
6#89733C211yesmaximal, characteristic, complemented
7#89729C211yesmaximal, characteristic, complemented
8#89727C211yesmaximal, characteristic, complemented
9#89727C211yesmaximal, characteristic, complemented
10#89720C211yesmaximal, characteristic, complemented
11#89716C211yesmaximal, characteristic, complemented
12#89448C211yesmaximal, characteristic, complemented
13#89447C211yesmaximal, characteristic, complemented
14#89446C211yesmaximal, characteristic, complemented
15#89441C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic, complemented
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic, complemented
28C2×C22⋊Dic7C2211yescharacteristic, complemented
29#6500—24no—
30#6492—24no—
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
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×C4⋊Dic7C2211yescharacteristic, complemented
42C2×Dic7⋊C4C2211yescharacteristic, complemented
43C2×Dic7⋊C4C2211yescharacteristic, complemented
44C2×Dic7⋊C4C2211yescharacteristic, complemented
45C2×Dic7⋊C4C2211yescharacteristic, complemented
46C2×C4×Dic7C2211yescharacteristic, complemented
47C7×#95C2211yescharacteristic, complemented
48C7×#95C2211yescharacteristic, complemented
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53#2697—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×C7—22no—
69C22×C4×C7—41no—
70C22×C4×C7—41no—
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—41no—
79C2×C22⋊D7—41no—
80C2×C22⋊D7—41no—
81C2×C22⋊D7D414yes—
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C22⋊Dic7—22no—
92C22⋊Dic7—22no—
93C22⋊Dic7—22no—
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—22no—
101(C2×D7)⋊C4—41no—
102(C2×D7)⋊C4—41no—
103(C2×D7)⋊C4—41no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4—41no—
106C4⋊Dic7—41no—
107Dic7⋊C4—22no—
108Dic7⋊C4—22no—
109Dic7⋊C4—41no—
110Dic7⋊C4—41no—
111C4×Dic7—41no—
112C2×#123—71no—
113C2×#120—71no—
114#397—71no—
115#397—71no—
116#396—71no—
117#396—71no—
118#394—71no—
119#394—71no—
120#390—71no—
121#388—71no—
122#388—71no—
123#386—71no—
124#386D711yescharacteristic, complemented
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×C7C2×D412yes—
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×C7—21no—
179C2×C4×C7—44no—
180C22⋊D7—44no—
181C22⋊D7—44no—
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×Dic7—21no—
199C2×Dic7C2×D412yes—
200C25—71no—
201C22×D4—71no—
202C22×D4—71no—
203C22×D4C2×D711yescharacteristic
204C23×C4C2×D711yescharacteristic, complemented
205#123—144no—
206#120—144no—
207C2×C4⋊C4—71no—
208C2×C4⋊C4—71no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4—71no—
211C2×C4⋊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⋊C4—71no—
227C2×C22⋊C4C2×D711yescharacteristic
228C2×C22⋊C4C2×D711yescharacteristic
229C2×C22⋊C4C2×D711yescharacteristic, complemented
230C2×C42—71no—
231#95—71no—
232#95—71no—
233#95—71no—
234#95—71no—
235#95C2×D711yescharacteristic, complemented
236#95C2×D711yescharacteristic, complemented
237C22×C7—22no—
238C22×C7—22no—
239C22×C7—22no—
240C22×C7—22no—
241C22×C7—22no—
242C22×C7—22no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7C22×D411yescharacteristic
252C22×C7C2×#4111yescharacteristic
253C22×C7C2×#4111yescharacteristic
254C22×C7H(5,𝔽2)11yescharacteristic
255C22×C7H(5,𝔽2)11yescharacteristic
256C22×C7H(5,𝔽2)11yescharacteristic
257C22×C7H(5,𝔽2)11yescharacteristic
258C2×D7—22no—
259C2×D7—22no—
260C2×D7—22no—
261C2×D7—22no—
262C2×D7—22no—
263C2×D7—22no—
264C2×D7—416no—
265C2×D7—41no—
266C2×D7—41no—
267C2×D7—41no—
268C2×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—41no—
273C4×C7—41no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—41no—
278Dic7—22no—
279Dic7—41no—
280Dic7—41no—
281Dic7—41no—
282Dic7—41no—
283Dic7—41no—
284C24—71no—
285C24—71no—
286C24—142no—
287C24—142no—
288C24—142no—
289C24—281no—
290C24—281no—
291C24—281no—
292C24—281no—
293C24C22×D711yescharacteristic
294C24C22×D711yescharacteristic
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—74no—
299C2×D4—142no—
300C2×D4—142no—
301C2×D4—281no—
302C2×D4—281no—
303C2×D4—281no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—71no—
308C22×C4—71no—
309C22×C4—71no—
310C22×C4—71no—
311C22×C4—71no—
312C22×C4—71no—
313C22×C4C22×D711yescharacteristic
314C22×C4C22×D711yescharacteristic
315C22×C4C22×D711yescharacteristic
316C22×C4C22×D711yescharacteristic
317C22×C4C22×D711yescharacteristic
318C4⋊C4—142no—
319C4⋊C4—142no—
320C4⋊C4—281no—
321C4⋊C4—281no—
322C4⋊C4—281no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—142no—
327C22⋊C4—142no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C22⋊C4—142no—
331C22⋊C4—142no—
332C22⋊C4—142no—
333C22⋊C4—142no—
334C22⋊C4—281no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C22⋊C4—281no—
341C42—281no—
342C2×C7—42no—
343C2×C7—42no—
344C2×C7#53411yescharacteristic
345C2×C7#53811yescharacteristic
346C2×C7#53811yescharacteristic
347C2×C7#54611yescharacteristic
348C2×C7#54611yescharacteristic
349C2×C7#55111yescharacteristic
350C2×C7#56011yescharacteristic
351D7—42no—
352D7—42no—
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—
387C23D4×D712yes—
388C23C23×D711yescharacteristic
389D4—44no—
390D4—284no—
391D4—284no—
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—21no—
406C2×C4—21no—
407C2×C4—44no—
408C2×C4—72no—
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×C4—141no—
424C2×C4—141no—
425C2×C4—141no—
426C7#269711yescharacteristic, complemented, Sylow
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—142no—
442C22—142no—
443C22—142no—
444C22—142no—
445C22—142no—
446C22—142no—
447C22—2816no—
448C22—281no—
449C22—281no—
450C22—281no—
451C22—281no—
452C22—281no—
453C22—281no—
454C22—281no—
455C22—281no—
456C22C2×#115711yescharacteristic
457C22C2×D4×D711yescharacteristic
458C22#654811yescharacteristic
459C22#654811yescharacteristic
460C22#654811yescharacteristic
461C22D7×#4111yescharacteristic
462C22#655311yescharacteristic
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—142no—
469C4—281no—
470C4—281no—
471C4—281no—
472C4—281no—
473C4—281no—
474C2—42no—
475C2—42no—
476C2—282no—
477C2—282no—
478C2#9018111yescharacteristic
479C2#9024711yescharacteristic
480C2#9025111yescharacteristic
481C2#9028011yescharacteristic
482C2#9030411yescharacteristic
483C2#9031111yescharacteristic
484C2#9034511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC2×#6500C2
hypercenter#386D7
hypocenterC7#2697
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#6492C2semidirect
3. C7×#386C2semidirect
4. #89994C2semidirect
5. #89985C2semidirect
6. #89733C2semidirect
7. #89729C2semidirect
8. #89727C2semidirect
9. #89727C2semidirect
10. #89720C2semidirect
11. #89716C2semidirect
12. #89448C2semidirect
13. #89447C2semidirect
14. #89446C2semidirect
15. #89441C2semidirect
18. C23×C4×C7C22semidirect
23. C2×C7×C22⋊C4C22semidirect
27. C2×C22⋊Dic7C22semidirect
28. C2×C22⋊Dic7C22semidirect
41. C2×C4⋊Dic7C22semidirect
42. C2×Dic7⋊C4C22semidirect
43. C2×Dic7⋊C4C22semidirect
44. C2×Dic7⋊C4C22semidirect
45. C2×Dic7⋊C4C22semidirect
46. C2×C4×Dic7C22semidirect
47. C7×#95C22semidirect
48. C7×#95C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
124. #386D7semidirect
204. C23×C4C2×D7semidirect
229. C2×C22⋊C4C2×D7semidirect
235. #95C2×D7semidirect
236. #95C2×D7semidirect
426. C7#2697semidirect

Maximal subgroups

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

subgroupindex
C2×#65002
C2×#64922
C7×#3862
#899942
#899852
#897332
#897292
#897272
#897202
#897162
#894482
#894472
#894462
#894412
#26977