Finite Group Databasebeta

← #11737835

Subgroups — #11737835

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
1D7×#127C211yesmaximal, characteristic, complemented
2#90271C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic, complemented
4#89948C212yesmaximal, complemented
5#89941C212yesmaximal, complemented
6#89844C211yesmaximal, characteristic, complemented
7#89799C211yesmaximal, characteristic, complemented
8#89615C211yesmaximal, characteristic, complemented
9#89607C211yesmaximal, characteristic, complemented
10#89545C212yesmaximal, complemented
11#89536C212yesmaximal, complemented
12C2×D4×D7—21no—
13C2×D4×D7—21no—
14C2×D4×D7C2212yes—
15C2×C7×SD16C2212yescomplemented
16C7×#127C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic, complemented
18#6510C2211yescharacteristic, complemented
19#6509C2211yescharacteristic
20C2×#1143C2212yescomplemented
21#6503—21no—
22#6503—21no—
23#6503C2212yes—
24C2×#1141C2212yescomplemented
25C2×D56C2212yescomplemented
26#6461—21no—
27#6458C2211yescharacteristic
28#6457—21no—
29#6453C2211yescharacteristic, complemented
30#6437C2211yescharacteristic
31C4×D28—21no—
32C4×D28C2211yescharacteristic, complemented
33#6435—21no—
34C42×D7C2211yescharacteristic
35C7×C4⋊C8C2211yescharacteristic, complemented
36C7×#102C2212yescomplemented
37#6406C2212yescomplemented
38#6395C2212yescomplemented
39(C2×D7)⋊C8C2212yes—
40#6389C2212yescomplemented
41#6383C2212yescomplemented
42C4⋊(C7⋊C8)C2211yescharacteristic, complemented
43#3217—71nomaximal, Sylow
44C23×D7—22no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4—21no—
47C2×C7×D4—21no—
48C2×C7×D4C2312yes—
49C2×C22⋊D7—22no—
50C2×C22⋊D7—22no—
51C2×C22⋊D7—22no—
52D4×D7—42no—
53D4×D7—42no—
54D4×D7—44no—
55D4×D7—44no—
56C2×D28—21no—
57C2×D28—21no—
58C2×D28—21no—
59C2×D28C2311yescharacteristic
60C2×D28C2312yes—
61C2×C4×D7—21no—
62C2×C4×D7D411yescharacteristic
63C2×C4×D7D411yescharacteristic
64C2×C4×D7C2311yescharacteristic
65C7×SD16—42no—
66C2×C7×C8C2312yes—
67C7×C4⋊C4—21no—
68C7×C4⋊C4C2311yescharacteristic
69C42×C7—21no—
70C42×C7C2311yescharacteristic
71#1143—42no—
72#1141—42no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—21no—
76(C2×D7)⋊C4—21no—
77(C2×D7)⋊C4—21no—
78C4⋊Dic7—21no—
79C4⋊Dic7C2311yescharacteristic
80Dic7⋊C4—21no—
81Dic7⋊C4—21no—
82C4×Dic7D411yescharacteristic
83C4×Dic7D411yescharacteristic
84C4×Dic7C2311yescharacteristic
85C2×C7⋊C8C2312yes—
86D56—42no—
87C2×#127—71no—
88#529—71no—
89#496—72no—
90#490—72no—
91#460—71no—
92#460D711yescharacteristic, complemented
93#459—71no—
94#459—71no—
95#450—72no—
96#447—72no—
97#432—71no—
98C23×C7—22no—
99C22×D7—21no—
100C22×D7—22no—
101C22×D7—44no—
102C22×D7—44no—
103C22×D7—44no—
104C22×D7C2×D411yescharacteristic
105C7×Q8—21no—
106C7×Q8—21no—
107C7×D4—22no—
108C7×D4—22no—
109C7×D4—42no—
110C7×D4—42no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7C2×D411yescharacteristic
114C2×C4×C7C2×D411yescharacteristic
115C2×C4×C7C2411yescharacteristic
116C22⋊D7—44no—
117C22⋊D7—44no—
118C22⋊D7—44no—
119C2×Dic7—21no—
120C2×Dic7C2×D411yescharacteristic
121C2×Dic7C2×D411yescharacteristic
122C2×Dic7C2×D411yescharacteristic
123D28—21no—
124D28—22no—
125D28—21no—
126D28—22no—
127D28—42no—
128D28—42no—
129D28—42no—
130C4×D7—22no—
131C4×D7—22no—
132C4×D7—21no—
133C4×D7—21no—
134C4×D7—42no—
135C7×C8—22no—
136C7⋊C8—22no—
137C22×D4—72no—
138C22×D4—141no—
139C22×D4—141no—
140C2×SD16—72no—
141C2×SD16C2×D712yescomplemented
142C2×D8—72no—
143C2×D8—72no—
144#127—72no—
145#127—71no—
146#127—141no—
147#127—141no—
148#127C2×D711yescharacteristic
149#126—141no—
150#125—71no—
151#124—71no—
152#123—141no—
153C22⋊Q8—71no—
154#121—71no—
155C4×Q8C2×D711yescharacteristic, complemented
156C4×D4—71no—
157C4×D4—141no—
158#117—141no—
159C2×C42—71no—
160C4⋊C8—71no—
161C4⋊C8C2×D711yescharacteristic, complemented
162#102—72no—
163#102—72no—
164#102—72no—
165#102C2×D712yescomplemented
166C22⋊C8—72no—
167#97—72no—
168C22×C7—42no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7C22×D411yescharacteristic
172C2×D7—21no—
173C2×D7—21no—
174C2×D7—42no—
175C2×D7—41no—
176C2×D7—41no—
177C2×D7—42no—
178C2×D7—42no—
179C2×D7—41no—
180C2×D7—88no—
181C2×D7AGL(1,ℤ/8)12yes—
182C4×C7—21no—
183C4×C7—41no—
184C4×C7—41no—
185C4×C7AGL(1,ℤ/8)12yes—
186C4×C7H(5,𝔽2)11yescharacteristic
187C4×C7H(5,𝔽2)11yescharacteristic
188Dic7—21no—
189Dic7—21no—
190Dic7—21no—
191Dic7—41no—
192C24—142no—
193C2×Q8C22×D711yescharacteristic
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—72no—
197C2×D4—71no—
198C2×D4—141no—
199C2×D4—142no—
200C2×D4—142no—
201C2×D4—142no—
202C2×D4—141no—
203C2×D4—141no—
204C2×D4—282no—
205C2×D4—282no—
206C2×D4—284no—
207C2×D4—284no—
208C2×D4C22×D712yes—
209C22×C4—71no—
210C22×C4—71no—
211C22×C4—71no—
212C22×C4—141no—
213SD16—42no—
214SD16—282no—
215D8—282no—
216D8—282no—
217C2×C8—72no—
218C2×C8C22×D712yes—
219C4⋊C4—21no—
220C4⋊C4—71no—
221C4⋊C4—141no—
222C4⋊C4—141no—
223C4⋊C4—141no—
224C4⋊C4C22×D711yescharacteristic
225C22⋊C4—141no—
226C22⋊C4—141no—
227C22⋊C4—141no—
228C22⋊C4—141no—
229C22⋊C4—141no—
230C42—21no—
231C42—71no—
232C42—71no—
233C42—71no—
234C42C22×D711yescharacteristic
235C2×C7—82no—
236C2×C7#53511yescharacteristic
237C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
238C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
239D7—22no—
240D7—81no—
241D7—82no—
242C23—22no—
243C23—71no—
244C23—141no—
245C23—142no—
246C23—284no—
247C23—284no—
248C23—284no—
249Q8—21no—
250Q8—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—42no—
255D4—141no—
256D4—142no—
257D4—141no—
258D4—142no—
259D4—282no—
260D4—282no—
261D4—284no—
262D4—284no—
263D4—284no—
264D4—282no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—71no—
268C2×C4—71no—
269C2×C4—71no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—282no—
276C2×C4D4×D711yescharacteristic
277C2×C4D4×D711yescharacteristic
278C2×C4C23×D711yescharacteristic
279C8—22no—
280C8—142no—
281C7#321711yescharacteristic, complemented, Sylow
282C22—42no—
283C22—42no—
284C22—42no—
285C22—72no—
286C22—141no—
287C22—141no—
288C22—281no—
289C22—282no—
290C22—282no—
291C22—281no—
292C22—282no—
293C22—281no—
294C22—568no—
295C22C2×D4×D711yescharacteristic
296C4—21no—
297C4—41no—
298C4—41no—
299C4—141no—
300C4—141no—
301C4—141no—
302C4—281no—
303C4#647712yes—
304C4#654811yescharacteristic
305C4#655311yescharacteristic
306C2—82no—
307C2—142no—
308C2—562no—
309C2—561no—
310C2#9029811yescharacteristic
311C2C2×#647711yescharacteristic
312C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
hypercenter#460D7
hypocenterC7#3217
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. D7×#127C2semidirect
2. #90271C2semidirect
3. C7×#460C2semidirect
4. #89948C2semidirect
5. #89941C2semidirect
6. #89844C2semidirect
7. #89799C2semidirect
8. #89615C2semidirect
9. #89607C2semidirect
10. #89545C2semidirect
11. #89536C2semidirect
15. C2×C7×SD16C22semidirect
17. C4×C7×Q8C22semidirect
18. #6510C22semidirect
20. C2×#1143C22semidirect
24. C2×#1141C22semidirect
25. C2×D56C22semidirect
29. #6453C22semidirect
32. C4×D28C22semidirect
35. C7×C4⋊C8C22semidirect
36. C7×#102C22semidirect
37. #6406C22semidirect
38. #6395C22semidirect
40. #6389C22semidirect
41. #6383C22semidirect
42. C4⋊(C7⋊C8)C22semidirect
92. #460D7semidirect
141. C2×SD16C2×D7semidirect
155. C4×Q8C2×D7semidirect
161. C4⋊C8C2×D7semidirect
165. #102C2×D7semidirect
281. C7#3217semidirect

Maximal subgroups

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

subgroupindex
D7×#1272
#902712
C7×#4602
#899482
#899412
#898442
#897992
#896152
#896072
#895452
#895362
#32177