Finite Group Databasebeta

← #11739110

Subgroups — #11739110

Order 896.

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

normal subgroups only

This group has 364 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
1#90419C211yesmaximal, characteristic, complemented
2#90371C211yesmaximal, characteristic
3C7×#489C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89832C211yesmaximal, characteristic, complemented
6#89588C212yesmaximal, complemented
7#89562C212yesmaximal, complemented
8#89552C212yesmaximal
9#89530C212yesmaximal, complemented
10#89479C211yesmaximal, characteristic, complemented
11#89475C211yesmaximal, characteristic, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C7×#124C2211yescharacteristic
14#6509—21no—
15#6509C2211yescharacteristic
16Q8×Dic7—21no—
17Q8×Dic7C2211yescharacteristic
18C2×#1144C2212yes—
19#6501C2211yescharacteristic
20D4×Dic7C2211yescharacteristic
21C2×#1142C2212yes—
22C2×#1131C2212yes—
23#6460C2212yes—
24#6455—21no—
25#6453—21no—
26#6450C2212yes—
27#6446—21no—
28(C22⋊D7)⋊C4—21no—
29#6442—21no—
30#6440—21no—
31C42×D7C2211yescharacteristic
32#6433C2211yescharacteristic
33#6432C2211yescharacteristic
34C7×#103C2212yes—
35C7×#102C2212yes—
36C7×#97C2211yescharacteristic
37#6390C2212yes—
38#6390C2212yes—
39C8×Dic7C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3172—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4C2311yescharacteristic
47C2×C22⋊D7—21no—
48C22×Dic7—21no—
49C2×C4×D7D411yescharacteristic
50C2×C4×D7D411yescharacteristic
51C2×C4×D7C2311yescharacteristic
52C2×Dic14C2312yes—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C7×C22⋊C4—21no—
57C7×C22⋊C4—21no—
58C42×C7C2311yescharacteristic
59C22⋊Dic7—21no—
60C22⋊Dic7—21no—
61#1144—24no—
62#1142—24no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—22no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7C2311yescharacteristic
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
76C4×Dic7—22no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81#1131—24no—
82#533—71no—
83#529—71no—
84#501—71no—
85#497—71no—
86#491—71no—
87#489D711yescharacteristic, complemented
88#470—72no—
89#468—72no—
90C4×Q16—72no—
91C4×SD16—72no—
92C4×M16—71no—
93C23×C7—21no—
94C22×D7C2×D411yescharacteristic
95C7×Q8—22no—
96C7×Q8—42no—
97C7×D4—22no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2411yescharacteristic
103C22⋊D7—42no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—42no—
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111C4×D7—22no—
112C4×D7—22no—
113C4×D7—41no—
114Dic14—22no—
115Dic14—22no—
116C7×C8—22no—
117C7⋊C8#4114yes—
118C2×Q16—72no—
119C2×SD16—72no—
120C2×M16—72no—
121C4⋊Q8—71no—
122C4⋊Q8C2×D711yescharacteristic
123#126—141no—
124#125—71no—
125#125—141no—
126#124C2×D711yescharacteristic
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#107—72no—
141#106—72no—
142#103—72no—
143#103—72no—
144#103C2×D712yes—
145#102C2×D712yes—
146#97—71no—
147#97C2×D711yescharacteristic
148C4×C8—72no—
149C22×C7—41no—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7C22×D411yescharacteristic
153C2×D7—21no—
154C2×D7—21no—
155C2×D7—21no—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—41no—
159C4×C7—42no—
160C4×C7C2×#4112yes—
161Dic7—21no—
162Dic7—21no—
163Dic7—42no—
164Dic7—41no—
165Dic7—41no—
166Dic7#13712yes—
167C2×Q8—21no—
168C2×Q8—72no—
169C2×Q8C22×D711yescharacteristic
170C2×D4—141no—
171C2×D4C22×D711yescharacteristic
172C22×C4—71no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—141no—
176Q16—144no—
177SD16—144no—
178M16—144no—
179C2×C8—72no—
180C2×C8C22×D712yes—
181C4⋊C4—21no—
182C4⋊C4—71no—
183C4⋊C4—71no—
184C4⋊C4—141no—
185C4⋊C4—142no—
186C4⋊C4—141no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4C22×D712yes—
193C22⋊C4—21no—
194C22⋊C4—21no—
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—81no—
206C2×C7#53211yescharacteristic
207C2×C7C2×#13711yescharacteristic
208C2×C7#57511yescharacteristic
209D7—41no—
210C23—21no—
211C23—71no—
212Q8—22no—
213Q8—42no—
214Q8—142no—
215Q8—142no—
216D4—22no—
217D4—282no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—71no—
221C2×C4—71no—
222C2×C4—71no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—141no—
226C2×C4—142no—
227C2×C4—141no—
228C2×C4—281no—
229C2×C4—282no—
230C2×C4D4×D711yescharacteristic
231C2×C4D4×D711yescharacteristic
232C2×C4C23×D711yescharacteristic
233C8—22no—
234C8—74no—
235C7#317211yescharacteristic, complemented, Sylow
236C22—41no—
237C22—41no—
238C22—41no—
239C22—141no—
240C22—141no—
241C22—141no—
242C22C2×D4×D711yescharacteristic
243C4—21no—
244C4—21no—
245C4—42no—
246C4—41no—
247C4—72no—
248C4—141no—
249C4—141no—
250C4—281no—
251C4—281no—
252C4—282no—
253C4D7×#4112yes—
254C2—81no—
255C2—281no—
256C2#9035911yescharacteristic
257C2#9046411yescharacteristic
258C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radical#6501C22
hypercenter#489D7
hypocenterC7#3172
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#489C2semidirect
4. #89852C2semidirect
5. #89832C2semidirect
6. #89588C2semidirect
7. #89562C2semidirect
9. #89530C2semidirect
10. #89479C2semidirect
11. #89475C2semidirect
87. #489D7semidirect
235. C7#3172semidirect

Maximal subgroups

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

subgroupindex
#904192
#903712
C7×#4892
#898522
#898322
#895882
#895622
#895522
#895302
#894792
#894752
#31727