Finite Group Databasebeta

← #11738272

Subgroups — #11738272

Order 896.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90333C211yesmaximal, characteristic, complemented
2#90303C211yesmaximal, characteristic
3C7×#468C211yesmaximal, characteristic, complemented
4#89941C211yesmaximal, characteristic, complemented
5#89938C211yesmaximal, characteristic, complemented
6#89905C211yesmaximal, characteristic, complemented
7#89900C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89806C211yesmaximal, characteristic, complemented
10#89663C211yesmaximal, characteristic, complemented
11#89660C211yesmaximal, characteristic, complemented
12#89588C211yesmaximal, characteristic, complemented
13#89560C211yesmaximal, characteristic, complemented
14#89542C211yesmaximal, characteristic, complemented
15#89540C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×M16C2211yescharacteristic, complemented
20C7×C22⋊Q8C2211yescharacteristic, complemented
21C7×#121C2211yescharacteristic
22#6509C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic, complemented
24C2×#1144C2211yescharacteristic, complemented
25C2×#1143C2211yescharacteristic
26#6501—21no—
27#6500—21no—
28#6499C2211yescharacteristic
29C4×C22⋊D7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic, complemented
31C2×#1131C2211yescharacteristic
32#6461—21no—
33#6460C2211yescharacteristic
34#6456C2211yescharacteristic
35#6455C2211yescharacteristic
36#6453—21no—
37#6453C2211yescharacteristic, complemented
38#6449—21no—
39#6446—21no—
40D7×C22⋊C4—21no—
41#6440—21no—
42C7×#107C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic, complemented
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6390C2211yescharacteristic, complemented
52#6390C2211yescharacteristic, complemented
53Dic7⋊C8C2211yescharacteristic, complemented
54C8×Dic7C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic, complemented
58#6382C2211yescharacteristic, complemented
59#3323—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7D411yescharacteristic
68C22×Dic7—21no—
69D4×D7—22no—
70D4×D7—41no—
71#1157—22no—
72#1157—41no—
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7C2311yescharacteristic
77C2×Dic14C2311yescharacteristic
78C7×SD16—41no—
79C7×M16—41no—
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7D411yescharacteristic
90#1144—41no—
91#1143—22no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4D411yescharacteristic
97C4⋊Dic7—21no—
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4D411yescharacteristic
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1131—41no—
112#552—71no—
113#546—71no—
114#488—71no—
115#485—71no—
116#481—71no—
117#478—71no—
118#470—71no—
119#468D711yescharacteristic, complemented
120#464—71no—
121#452—71no—
122#450—71no—
123C4×SD16—71no—
124#436—71no—
125#418—71no—
126#417—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—42no—
131C22×D7—42no—
132C22×D7—81no—
133C22×D7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—81no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—41no—
147C22⋊D7—42no—
148C22⋊D7—42no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—42no—
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156D28—21no—
157D28#4112yes—
158C4×D7—21no—
159C4×D7—21no—
160C4×D7—42no—
161C4×D7—81no—
162Dic14—21no—
163Dic14—21no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×D4—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×SD16C2×D711yescharacteristic
173C2×M16—71no—
174C2×M16C2×D711yescharacteristic, complemented
175#126—141no—
176#125—71no—
177#125—141no—
178#124—71no—
179#123—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8C2×D711yescharacteristic, complemented
184#121—141no—
185#121C2×D711yescharacteristic
186#120—141no—
187C4×Q8—71no—
188C4×D4—71no—
189C4×D4—71no—
190#117—71no—
191#117—141no—
192C2×C4⋊C4—71no—
193C2×C22⋊C4—141no—
194#107C2×D711yescharacteristic, complemented
195#106—71no—
196#106—71no—
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103C2×D711yescharacteristic, complemented
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102C2×D711yescharacteristic
208C22⋊C8—71no—
209C22⋊C8—71no—
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C2×D7—21no—
219C2×D7—22no—
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—84no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7C2×#4111yescharacteristic
230C4×C7H(5,𝔽2)11yescharacteristic
231Dic7—21no—
232Dic7—21no—
233Dic7—21no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237C24—141no—
238#41—142no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8C22×D711yescharacteristic
242C2×D4—21no—
243C2×D4—71no—
244C2×D4—71no—
245C2×D4—141no—
246C2×D4—142no—
247C2×D4—141no—
248C2×D4—281no—
249C2×D4C22×D711yescharacteristic
250C22×C4—71no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255Q16—281no—
256SD16—41no—
257SD16—142no—
258M16—41no—
259M16—281no—
260C2×C8—71no—
261C2×C8—71no—
262C2×C8C22×D711yescharacteristic
263C2×C8C22×D711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4C22×D711yescharacteristic
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—71no—
279C22⋊C4—71no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C42—71no—
287C42—141no—
288C42—141no—
289C2×C7—41no—
290C2×C7—81no—
291C2×C7#54611yescharacteristic
292C2×C7#57511yescharacteristic
293C2×C7#57711yescharacteristic
294D7—41no—
295D7—42no—
296C23—21no—
297C23—71no—
298C23—141no—
299C23—282no—
300C23—282no—
301C23—561no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—141no—
306Q8—141no—
307D4—21no—
308D4—21no—
309D4—72no—
310D4—81no—
311D4—141no—
312D4—281no—
313D4—282no—
314D4—282no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—71no—
321C2×C4—71no—
322C2×C4—71no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4—561no—
331C2×C4D4×D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—21no—
334C8—21no—
335C8—141no—
336C8—141no—
337C7#332311yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—141no—
345C22—141no—
346C22—141no—
347C22—142no—
348C22—281no—
349C22—281no—
350C22—564no—
351C22C2×D4×D711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—141no—
357C4—141no—
358C4—141no—
359C4—281no—
360C4—281no—
361C4—281no—
362C4#654811yescharacteristic
363C4D7×#4111yescharacteristic
364C2—41no—
365C2—81no—
366C2—281no—
367C2—282no—
368C2#9030111yescharacteristic
369C2#9046611yescharacteristic
370C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#468C2
socleC22×C7C22×D4
prime radical#90303C2
hypercenter#468D7
hypocenterC7#3323
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #90333C2semidirect
3. C7×#468C2semidirect
4. #89941C2semidirect
5. #89938C2semidirect
6. #89905C2semidirect
7. #89900C2semidirect
8. #89819C2semidirect
9. #89806C2semidirect
10. #89663C2semidirect
11. #89660C2semidirect
12. #89588C2semidirect
13. #89560C2semidirect
14. #89542C2semidirect
15. #89540C2semidirect
19. C2×C7×M16C22semidirect
20. C7×C22⋊Q8C22semidirect
23. Q8×Dic7C22semidirect
24. C2×#1144C22semidirect
30. C2×C4⋊Dic7C22semidirect
37. #6453C22semidirect
42. C7×#107C22semidirect
43. C7×#103C22semidirect
47. C22⋊(C7⋊C8)C22semidirect
51. #6390C22semidirect
52. #6390C22semidirect
53. Dic7⋊C8C22semidirect
54. C8×Dic7C22semidirect
55. #6384C22semidirect
57. #6382C22semidirect
58. #6382C22semidirect
119. #468D7semidirect
174. C2×M16C2×D7semidirect
183. C22⋊Q8C2×D7semidirect
194. #107C2×D7semidirect
201. #103C2×D7semidirect
337. C7#3323semidirect

Maximal subgroups

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

subgroupindex
#903332
#903032
C7×#4682
#899412
#899382
#899052
#899002
#898192
#898062
#896632
#896602
#895882
#895602
#895422
#895402
#33237