Finite Group Databasebeta

← #11735949

Subgroups — #11735949

Order 896.

366 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 129 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#90170C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3C2×C7×#97C211yesmaximal, characteristic, complemented
4#89888C212yesmaximal, complemented
5#89886C212yesmaximal
6#89488C212yesmaximal, complemented
7#89487C211yesmaximal, characteristic, complemented
8#89486C211yesmaximal, characteristic, complemented
9#89482C212yesmaximal, complemented
10#89475C211yesmaximal, characteristic, complemented
11#89474C211yesmaximal, characteristic, complemented
12C22×C7×C8C2212yes—
13C2×C42×C7C2211yescharacteristic
14#6493C2211yescharacteristic
15#6492—21no—
16C4×C22⋊D7—21no—
17#6489—21no—
18#6487—21no—
19#6487C2212yes—
20#6487C2211yescharacteristic
21C2×Dic28C2212yes—
22C2×#1132C2212yes—
23#6439—21no—
24#6438C2211yescharacteristic
25C4×D28C2211yescharacteristic
26#6435—21no—
27#6433—21no—
28#6433C2211yescharacteristic
29#6432C2211yescharacteristic
30#6432C2211yescharacteristic
31C4×Dic14—21no—
32C4×Dic14C2212yes—
33C4×Dic14C2211yescharacteristic
34C7×#97C2211yescharacteristic
35C7×#97C2211yescharacteristic
36C7×#97C2212yes—
37#6395C2212yes—
38#6392C2212yes—
39#6391C2212yes—
40#6390C2212yes—
41#6390C2212yes—
42#6390C2212yes—
43#3181—71nomaximal, Sylow
44C22×C4×C7D411yescharacteristic
45C22×C4×C7D411yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×Dic14—21no—
51C2×Dic14C2312yes—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8—24no—
54C2×C7×C8C2312yes—
55C2×C7×C8C2312yes—
56C42×C7D411yescharacteristic
57C42×C7D411yescharacteristic
58C42×C7C2311yescharacteristic
59C42×C7C2311yescharacteristic
60C22⋊Dic7—22no—
61C22⋊Dic7—21no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71C4⋊Dic7C2311yescharacteristic
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78C4×Dic7—22no—
79C4×Dic7—21no—
80Dic28—24no—
81#1132—24no—
82#533—71no—
83#529—71no—
84#501—71no—
85#497—71no—
86#491—71no—
87#489—71no—
88#467—72no—
89#465—72no—
90#441—72no—
91#440—72no—
92C2×#97D711yescharacteristic, complemented
93C23×C7C2×D411yescharacteristic
94C22×D7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—41no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2411yescharacteristic
104C22⋊D7—42no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110D28—22no—
111C4×D7—42no—
112Dic14—22no—
113Dic14—22no—
114Dic14—22no—
115Dic14—42no—
116C7×C8—22no—
117C7×C8#4114yes—
118C2×Q16—72no—
119C2×SD16—72no—
120C22×C8C2×D712yes—
121C4⋊Q8—71no—
122C4⋊Q8—71no—
123#126—141no—
124#125—71no—
125#125—141no—
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×C42C2×D711yescharacteristic
140#107—72no—
141#106—72no—
142#103—72no—
143#103—72no—
144#103—72no—
145#102—72no—
146#97C2×D712yes—
147#97C2×D711yescharacteristic
148#97C2×D711yescharacteristic
149C22×C7—21no—
150C22×C7—21no—
151C22×C7#13712yes—
152C22×C7C22×D411yescharacteristic
153C2×D7—41no—
154C2×D7—41no—
155C2×D7—41no—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—21no—
161C4×C7C2×#4112yes—
162Dic7—42no—
163Dic7—41no—
164Dic7—41no—
165Dic7—42no—
166Dic7—41no—
167C2×Q8—71no—
168C2×Q8—72no—
169C2×Q8—141no—
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—141no—
173C22×C4D2811yescharacteristic
174C22×C4D2811yescharacteristic
175C22×C4C22×D711yescharacteristic
176Q16—144no—
177SD16—144no—
178C2×C8—24no—
179C2×C8C22×D712yes—
180C2×C8C22×D712yes—
181C4⋊C4—72no—
182C4⋊C4—71no—
183C4⋊C4—71no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—142no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4—141no—
193C22⋊C4—142no—
194C22⋊C4—141no—
195C22⋊C4—141no—
196C22⋊C4—141no—
197C22⋊C4—141no—
198C22⋊C4—141no—
199C42—141no—
200C42—142no—
201C42D2811yescharacteristic
202C42D2811yescharacteristic
203C42C22×D711yescharacteristic
204C42C22×D711yescharacteristic
205C2×C7—22no—
206C2×C7#53211yescharacteristic
207C2×C7C2×#13711yescharacteristic
208C2×C7#57511yescharacteristic
209D7—81no—
210C23—141no—
211C23C2×D2811yescharacteristic
212Q8—142no—
213Q8—142no—
214Q8—142no—
215Q8—282no—
216D4—142no—
217D4—282no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—141no—
222C2×C4—141no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—141no—
226C2×C4—282no—
227C2×C4C2×D2811yescharacteristic
228C2×C4C2×D2811yescharacteristic
229C2×C4C2×D2811yescharacteristic
230C2×C4C2×D2811yescharacteristic
231C2×C4C2×D2811yescharacteristic
232C2×C4C23×D711yescharacteristic
233C8—22no—
234C8#115714yes—
235C7#318111yescharacteristic, complemented, Sylow
236C22—21no—
237C22—21no—
238C22—281no—
239C22—281no—
240C22—281no—
241C22#647212yes—
242C22C22×D2811yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—281no—
249C4—282no—
250C4—281no—
251C4—281no—
252C4—282no—
253C4C2×#115712yes—
254C2—22no—
255C2—561no—
256C2#9016811yescharacteristic
257C2C2×#647211yescharacteristic
258C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#97C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenterC2×#97D7
hypocenterC7#3181
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. C2×C7×#97C2semidirect
4. #89888C2semidirect
6. #89488C2semidirect
7. #89487C2semidirect
8. #89486C2semidirect
9. #89482C2semidirect
10. #89475C2semidirect
11. #89474C2semidirect
92. C2×#97D7semidirect
235. C7#3181semidirect

Maximal subgroups

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

subgroupindex
#901702
#901612
C2×C7×#972
#898882
#898862
#894882
#894872
#894862
#894822
#894752
#894742
#31817