Finite Group Databasebeta

← #11739427

Subgroups — #11739427

Order 896.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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×C4⋊Q8C211yesmaximal, characteristic, complemented
2#90382C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89837C211yesmaximal, characteristic, complemented
6#89654C212yesmaximal
7#89647C212yesmaximal, complemented
8#89642C212yesmaximal, complemented
9#89634C212yesmaximal, complemented
10#89486C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×Q8×D7—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#125C2211yescharacteristic
15#6507—21no—
16#6461—21no—
17#6460C2211yescharacteristic
18#6458C2211yescharacteristic
19#6457—21no—
20#6456—21no—
21#6456C2211yescharacteristic
22#6455—21no—
23D7×C4⋊C4—21no—
24D7×C4⋊C4C2212yes—
25#6451—21no—
26#6451C2212yes—
27#6450C2211yescharacteristic
28#6438C2211yescharacteristic
29C42×D7C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#107C2212yes—
32C7×#106C2212yes—
33C7×#97C2211yescharacteristic
34#6395C2212yes—
35(C2×D7)⋊C8C2212yes—
36#6390C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3238—71nomaximal, Sylow
44C2×C7×Q8—21no—
45Q8×D7—44no—
46C2×D28—21no—
47C2×D28C2311yescharacteristic
48C2×C4×D7—22no—
49C2×C4×D7—21no—
50C2×C4×D7D411yescharacteristic
51C2×C4×D7D411yescharacteristic
52C2×C4×D7C2311yescharacteristic
53C2×Dic14—21no—
54C2×Dic14—22no—
55C2×Dic14C2311yescharacteristic
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2311yescharacteristic
61C7×C4⋊C4C2312yes—
62C7×C4⋊C4C2311yescharacteristic
63C42×C7C2311yescharacteristic
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7C2312yes—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80C2×C4⋊Q8—71no—
81#529—71no—
82#501—71no—
83#501D711yescharacteristic, complemented
84#489—71no—
85#489—71no—
86#483—72no—
87#480—72no—
88#477—72no—
89#476—72no—
90#432—71no—
91C22×D7—21no—
92C22×D7C2×D411yescharacteristic
93C7×Q8—42no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2411yescharacteristic
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7C2×D411yescharacteristic
103C2×Dic7C2×D411yescharacteristic
104C2×Dic7C2×D411yescharacteristic
105D28—22no—
106D28—42no—
107C4×D7—22no—
108C4×D7—22no—
109C4×D7—22no—
110C4×D7—44no—
111C4×D7—42no—
112Dic14—22no—
113Dic14—44no—
114Dic14—42no—
115C7×C8—22no—
116C7⋊C8—22no—
117C22×Q8—141no—
118C4⋊Q8—72no—
119C4⋊Q8—71no—
120C4⋊Q8—141no—
121C4⋊Q8—141no—
122C4⋊Q8C2×D711yescharacteristic
123#126—141no—
124#125C2×D711yescharacteristic
125#124—71no—
126#123—141no—
127C22⋊Q8—71no—
128#121—71no—
129C4×Q8—71no—
130C4×D4—71no—
131C4×D4—141no—
132#117—141no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4—141no—
135C2×C42—71no—
136#107—72no—
137#107C2×D712yes—
138#106—72no—
139#106C2×D712yes—
140C4⋊C8—72no—
141#103—72no—
142#103—72no—
143#102—72no—
144#102—72no—
145C22⋊C8—72no—
146#97—71no—
147#97C2×D711yescharacteristic
148C22×C7C22×D411yescharacteristic
149C2×D7—21no—
150C2×D7—21no—
151C2×D7—41no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7AGL(1,ℤ/8)12yes—
155C4×C7—21no—
156C4×C7—21no—
157C4×C7—42no—
158C4×C7—41no—
159C4×C7—41no—
160C4×C72−1+412yes—
161Dic7—21no—
162Dic7—21no—
163Dic7—41no—
164Dic7—42no—
165Dic7#13712yes—
166C2×Q8—21no—
167C2×Q8—71no—
168C2×Q8—142no—
169C2×Q8—141no—
170C2×Q8—284no—
171C2×D4—71no—
172C2×D4—141no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—71no—
176C22×C4—141no—
177C22×C4—142no—
178C2×C8—72no—
179C2×C8C22×D712yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—72no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4C22×D712yes—
190C4⋊C4C22×D711yescharacteristic
191C4⋊C4C22×D711yescharacteristic
192C22⋊C4—141no—
193C22⋊C4—141no—
194C22⋊C4—141no—
195C22⋊C4—141no—
196C22⋊C4—141no—
197C42—71no—
198C42—71no—
199C42—71no—
200C42—141no—
201C42C22×D711yescharacteristic
202C2×C7#53611yescharacteristic
203C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
204C2×C7C2×#13711yescharacteristic
205D7—22no—
206D7—81no—
207C23—71no—
208C23—141no—
209Q8—42no—
210Q8—142no—
211Q8—282no—
212Q8—284no—
213D4—142no—
214D4—282no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—71no—
219C2×C4—71no—
220C2×C4—71no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—141no—
225C2×C4—142no—
226C2×C4—282no—
227C2×C4—284no—
228C2×C4D4×D711yescharacteristic
229C2×C4D4×D711yescharacteristic
230C2×C4C23×D711yescharacteristic
231C8—22no—
232C8—142no—
233C7#323811yescharacteristic, complemented, Sylow
234C22—72no—
235C22—141no—
236C22—141no—
237C22—281no—
238C22—281no—
239C22—281no—
240C22C2×D4×D711yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—72no—
247C4—141no—
248C4—141no—
249C4—281no—
250C4—282no—
251C4#655112yes—
252C2—142no—
253C2—561no—
254C2#9041511yescharacteristic
255C2D7×AGL(1,ℤ/8)11yescharacteristic
256C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#501D7
hypocenterC7#3238
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊Q8C2semidirect
3. C7×#501C2semidirect
4. #89861C2semidirect
5. #89837C2semidirect
7. #89647C2semidirect
8. #89642C2semidirect
9. #89634C2semidirect
10. #89486C2semidirect
11. #89477C2semidirect
83. #501D7semidirect
233. C7#3238semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#903822
C7×#5012
#898612
#898372
#896542
#896472
#896422
#896342
#894862
#894772
#32387