Finite Group Databasebeta

← #11737442

Subgroups — #11737442

Order 896.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6503C211yesmaximal, characteristic, complemented
2#90291C211yesmaximal, characteristic, complemented
3C7×#447C211yesmaximal, characteristic, complemented
4#89990C211yesmaximal, characteristic, complemented
5#89926C212yesmaximal, complemented
6#89922C212yesmaximal, complemented
7#89808C211yesmaximal, characteristic, complemented
8#89536C212yesmaximal, complemented
9#89534C212yesmaximal, complemented
10#89503C211yesmaximal, characteristic, complemented
11#89491C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7—21no—
14C22×C22⋊D7—21no—
15C22×D28C2211yescharacteristic
16C2×C7×D8C2212yescomplemented
17C7×#121C2211yescharacteristic
18#6503—21no—
19#6503—21no—
20#6503C2212yes—
21#6503C2212yes—
22#6499C2211yescharacteristic, complemented
23#6498—21no—
24D4×Dic7—21no—
25D4×Dic7C2211yescharacteristic, complemented
26C2×#1142C2212yescomplemented
27C2×#1141C2212yescomplemented
28#6487C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×#1132C2212yescomplemented
31#6453C2211yescharacteristic, complemented
32#6450C2211yescharacteristic, complemented
33#6442—21no—
34#6440—21no—
35C7×#102C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37#6406C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6395C2212yescomplemented
40#6389C2212yescomplemented
41Dic7⋊C8C2212yescomplemented
42#6383C2212yescomplemented
43#3217—71nomaximal, Sylow
44C24×C7—21no—
45C23×D7—21no—
46C2×C7×D4—21no—
47C2×C7×D4—21no—
48C2×C7×D4—42no—
49C2×C7×D4C2312yes—
50C2×C7×D4C2311yescharacteristic
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—22no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7—22no—
55C2×C22⋊D7—22no—
56C2×C22⋊D7—44no—
57C2×C22⋊D7—44no—
58C22×Dic7—21no—
59C22×Dic7D411yescharacteristic
60C22×Dic7D411yescharacteristic
61C2×D28—21no—
62C2×D28—42no—
63C2×D28C2312yes—
64C2×Dic14C2311yescharacteristic
65C7×D8—42no—
66C2×C7×C8C2312yes—
67C7×C4⋊C4C2311yescharacteristic
68C7×C22⋊C4—21no—
69C22⋊Dic7—21no—
70C22⋊Dic7—21no—
71C22⋊Dic7—21no—
72C22⋊Dic7—21no—
73#1142—42no—
74#1141—42no—
75C4⋊Dic7—21no—
76C4⋊Dic7C2311yescharacteristic
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81C4×Dic7D411yescharacteristic
82C4×Dic7D411yescharacteristic
83C4×Dic7C2311yescharacteristic
84C4×Dic7C2311yescharacteristic
85C2×C7⋊C8C2312yes—
86#1132—42no—
87C2×#127—71no—
88#529—71no—
89#496—72no—
90#490—72no—
91#460—72no—
92#459—72no—
93#450—71no—
94#450—71no—
95#447—71no—
96#447D711yescharacteristic, complemented
97#432—71no—
98C23×C7—21no—
99C23×C7—22no—
100C23×C7—42no—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7C2×D411yescharacteristic
104C22×D7—22no—
105C22×D7—42no—
106C22×D7—42no—
107C22×D7—42no—
108C7×D4—22no—
109C7×D4—22no—
110C7×D4—22no—
111C7×D4—42no—
112C7×D4—42no—
113C2×C4×C7—22no—
114C2×C4×C7—21no—
115C2×C4×C7C2×D411yescharacteristic
116C2×C4×C7C2411yescharacteristic
117C22⋊D7—44no—
118C22⋊D7—44no—
119C22⋊D7—44no—
120C22⋊D7—44no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—21no—
124C2×Dic7—21no—
125C2×Dic7—42no—
126C2×Dic7C2×D411yescharacteristic
127C2×Dic7C2×D411yescharacteristic
128C2×Dic7C2×D411yescharacteristic
129C2×Dic7C2×D411yescharacteristic
130D28—22no—
131D28—22no—
132D28—42no—
133Dic14—22no—
134C7×C8—22no—
135C7⋊C8—22no—
136C22×D4—71no—
137C22×D4—141no—
138C22×D4—141no—
139C22×D4C2×D711yescharacteristic
140C2×SD16—72no—
141C2×SD16—72no—
142C2×D8—72no—
143C2×D8C2×D712yescomplemented
144#127—72no—
145#127—72no—
146#127—141no—
147#127—141no—
148#126—141no—
149#125—71no—
150#124—71no—
151#123—141no—
152C22⋊Q8—71no—
153#121C2×D711yescharacteristic
154C4×Q8—71no—
155C4×D4—71no—
156C4×D4—141no—
157#117—141no—
158C2×C42—71no—
159C4⋊C8—72no—
160#102—72no—
161#102—72no—
162#102—72no—
163#102C2×D712yescomplemented
164C22⋊C8—71no—
165C22⋊C8C2×D711yescharacteristic
166#97—72no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7—42no—
170C22×C7—41no—
171C22×C7—41no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—41no—
175C22×C7—84no—
176C22×C7AGL(1,ℤ/8)12yes—
177C22×C7C22×D411yescharacteristic
178C2×D7—42no—
179C2×D7—42no—
180C2×D7—42no—
181C2×D7—84no—
182C4×C7—21no—
183C4×C7—41no—
184C4×C7H(5,𝔽2)12yes—
185Dic7—21no—
186Dic7—21no—
187Dic7—21no—
188Dic7—41no—
189Dic7—41no—
190Dic7AGL(1,ℤ/8)12yes—
191C24—21no—
192C24—141no—
193C2×Q8—71no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—42no—
197C2×D4—72no—
198C2×D4—142no—
199C2×D4—142no—
200C2×D4—142no—
201C2×D4—142no—
202C2×D4—141no—
203C2×D4—282no—
204C2×D4—284no—
205C2×D4—284no—
206C2×D4C22×D712yes—
207C2×D4C22×D711yescharacteristic
208C22×C4—71no—
209C22×C4—71no—
210C22×C4—141no—
211C22×C4C22×D711yescharacteristic
212SD16—282no—
213SD16—282no—
214D8—42no—
215D8—282no—
216C2×C8—72no—
217C2×C8C22×D712yes—
218C4⋊C4—71no—
219C4⋊C4—141no—
220C4⋊C4—141no—
221C4⋊C4—141no—
222C4⋊C4—141no—
223C4⋊C4C22×D711yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—141no—
226C22⋊C4—141no—
227C22⋊C4—141no—
228C22⋊C4—141no—
229C42—71no—
230C42—71no—
231C42—71no—
232C42—71no—
233C42—141no—
234C2×C7—22no—
235C2×C7—81no—
236C2×C7—82no—
237C2×C7#53511yescharacteristic
238C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
239C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
240D7—82no—
241C23—21no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—142no—
247C23—282no—
248C23—282no—
249C23—282no—
250C23D4×D711yescharacteristic
251Q8—142no—
252D4—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—42no—
257D4—142no—
258D4—142no—
259D4—282no—
260D4—284no—
261D4—284no—
262D4—284no—
263D4—284no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—71no—
267C2×C4—71no—
268C2×C4—71no—
269C2×C4—71no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—282no—
275C2×C4D4×D711yescharacteristic
276C2×C4C23×D711yescharacteristic
277C8—22no—
278C8—142no—
279C7#321711yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—84no—
289C22—282no—
290C22—282no—
291C22—282no—
292C22—564no—
293C22#647412yes—
294C22C2×D4×D711yescharacteristic
295C4—21no—
296C4—41no—
297C4—72no—
298C4—141no—
299C4—141no—
300C4—141no—
301C4—281no—
302C4—281no—
303C4#654812yes—
304C2—22no—
305C2—82no—
306C2—81no—
307C2—562no—
308C2#9028311yescharacteristic
309C2C2×#647411yescharacteristic
310C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#447C2
socleC22×C7C22×D4
hypercenter#447D7
hypocenterC7#3217
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#6503C2semidirect
2. #90291C2semidirect
3. C7×#447C2semidirect
4. #89990C2semidirect
5. #89926C2semidirect
6. #89922C2semidirect
7. #89808C2semidirect
8. #89536C2semidirect
9. #89534C2semidirect
10. #89503C2semidirect
11. #89491C2semidirect
16. C2×C7×D8C22semidirect
22. #6499C22semidirect
25. D4×Dic7C22semidirect
26. C2×#1142C22semidirect
27. C2×#1141C22semidirect
30. C2×#1132C22semidirect
31. #6453C22semidirect
32. #6450C22semidirect
35. C7×#102C22semidirect
37. #6406C22semidirect
39. #6395C22semidirect
40. #6389C22semidirect
41. Dic7⋊C8C22semidirect
42. #6383C22semidirect
96. #447D7semidirect
143. C2×D8C2×D7semidirect
163. #102C2×D7semidirect
279. C7#3217semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
#902912
C7×#4472
#899902
#899262
#899222
#898082
#895362
#895342
#895032
#894912
#32177