Finite Group Databasebeta

← #11739408

Subgroups — #11739408

Order 896.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#90419C211yesmaximal, characteristic, complemented
2#90382C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89841C211yesmaximal, characteristic, complemented
6#89661C212yesmaximal, complemented
7#89659C212yesmaximal, complemented
8#89636C212yesmaximal
9#89625C212yesmaximal, complemented
10#89487C211yesmaximal, characteristic, complemented
11D7×#97C211yesmaximal, characteristic, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C7×#125C2211yescharacteristic
14#6509—21no—
15Q8×Dic7—21no—
16C2×Dic28C2212yes—
17C2×#1132C2212yes—
18C2×C8×D7C2212yes—
19#6461—21no—
20#6460C2211yescharacteristic
21#6460C2212yes—
22#6458C2211yescharacteristic
23#6457—21no—
24#6456—21no—
25#6456C2211yescharacteristic
26#6455—21no—
27#6455—21no—
28#6453—21no—
29#6450C2211yescharacteristic
30#6450C2212yes—
31#6438C2211yescharacteristic
32C42×D7C2211yescharacteristic
33#6432C2211yescharacteristic
34C7×#107C2212yes—
35C7×#106C2212yes—
36C7×#97C2211yescharacteristic
37#6389C2212yes—
38#6384C2212yes—
39#6384C2212yes—
40#6384C2212yes—
41#6383C2212yes—
42#6377C2211yescharacteristic
43#3181—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×D28—21no—
46C2×D28C2311yescharacteristic
47C2×C4×D7—21no—
48C2×C4×D7D411yescharacteristic
49C2×C4×D7D411yescharacteristic
50C2×C4×D7C2311yescharacteristic
51C2×Dic14C2312yes—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2311yescharacteristic
58C7×C4⋊C4C2312yes—
59C7×C4⋊C4C2311yescharacteristic
60C42×C7C2311yescharacteristic
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7—22no—
70C4⋊Dic7—21no—
71Dic7⋊C4—22no—
72Dic7⋊C4—21no—
73C4×Dic7—21no—
74C4×Dic7—22no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C2×C7⋊C8C2312yes—
79Dic28—24no—
80#1132—24no—
81C8×D7—24no—
82#533—71no—
83#529—71no—
84#501D711yescharacteristic, complemented
85#497—71no—
86#491—71no—
87#489—71no—
88#467—72no—
89#465—72no—
90#441—72no—
91#440—72no—
92C2×#97—71no—
93C22×D7—21no—
94C22×D7C2×D411yescharacteristic
95C7×Q8—42no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7C2×D411yescharacteristic
105C2×Dic7C2×D411yescharacteristic
106C2×Dic7C2×D411yescharacteristic
107D28—22no—
108D28—42no—
109C4×D7—22no—
110C4×D7—22no—
111C4×D7—41no—
112C4×D7—42no—
113Dic14—22no—
114Dic14—22no—
115Dic14—22no—
116C7×C8#4114yes—
117C7⋊C8—22no—
118C2×Q16—72no—
119C2×SD16—72no—
120C22×C8—72no—
121C4⋊Q8—71no—
122C4⋊Q8C2×D711yescharacteristic
123#126—141no—
124#125—141no—
125#125C2×D711yescharacteristic
126#124—71no—
127#123—141no—
128C22⋊Q8—71no—
129C22⋊Q8—72no—
130C22⋊Q8—141no—
131#121—71no—
132C4×Q8—71no—
133C4×Q8—72no—
134C4×Q8—141no—
135C4×D4—71no—
136C4×D4—141no—
137#117—141no—
138#117—141no—
139C2×C42—71no—
140#107C2×D712yes—
141#106C2×D712yes—
142#103—72no—
143#103—72no—
144#103—72no—
145#102—72no—
146#97—72no—
147#97—71no—
148#97C2×D711yescharacteristic
149C22×C7C22×D411yescharacteristic
150C2×D7—21no—
151C2×D7—21no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—41no—
155C2×D7#13712yes—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—41no—
159C4×C7—41no—
160C4×C7—42no—
161C4×C7C2×#4112yes—
162Dic7—21no—
163Dic7—21no—
164Dic7—21no—
165Dic7—42no—
166Dic7—41no—
167C2×Q8—21no—
168C2×Q8—71no—
169C2×Q8—72no—
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—71no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—141no—
176Q16—144no—
177SD16—144no—
178C2×C8—72no—
179C2×C8—144no—
180C2×C8C22×D712yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—142no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4C22×D712yes—
191C4⋊C4C22×D711yescharacteristic
192C4⋊C4C22×D711yescharacteristic
193C22⋊C4—141no—
194C22⋊C4—141no—
195C22⋊C4—142no—
196C22⋊C4—141no—
197C22⋊C4—141no—
198C22⋊C4—141no—
199C42—71no—
200C42—71no—
201C42—71no—
202C42—141no—
203C42—142no—
204C42C22×D711yescharacteristic
205C2×C7#53211yescharacteristic
206C2×C7C2×#13711yescharacteristic
207C2×C7#57511yescharacteristic
208D7—22no—
209D7—81no—
210C23—71no—
211C23—141no—
212Q8—42no—
213Q8—142no—
214Q8—142no—
215Q8—142no—
216D4—142no—
217D4—282no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—71no—
222C2×C4—71no—
223C2×C4—71no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—141no—
227C2×C4—142no—
228C2×C4—281no—
229C2×C4—282no—
230C2×C4D4×D711yescharacteristic
231C2×C4D4×D711yescharacteristic
232C2×C4C23×D711yescharacteristic
233C8—142no—
234C8#116114yes—
235C7#318111yescharacteristic, complemented, Sylow
236C22—72no—
237C22—141no—
238C22—141no—
239C22—281no—
240C22—281no—
241C22—281no—
242C22C2×D4×D711yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—42no—
247C4—41no—
248C4—141no—
249C4—141no—
250C4—141no—
251C4—281no—
252C4—282no—
253C4C2×#116112yes—
254C2—142no—
255C2—561no—
256C2#9041611yescharacteristic
257C2#9046411yescharacteristic
258C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#501D7
hypocenterC7#3181
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. #90419C2semidirect
3. C7×#501C2semidirect
4. #89861C2semidirect
5. #89841C2semidirect
6. #89661C2semidirect
7. #89659C2semidirect
9. #89625C2semidirect
10. #89487C2semidirect
11. D7×#97C2semidirect
84. #501D7semidirect
235. C7#3181semidirect

Maximal subgroups

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

subgroupindex
#904192
#903822
C7×#5012
#898612
#898412
#896612
#896592
#896362
#896252
#894872
D7×#972
#31817