Finite Group Databasebeta

← #11739314

Subgroups — #11739314

Order 896.

564 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#6482C212yesmaximal, complemented
2C2×#6479C212yesmaximal, complemented
3#90416C211yesmaximal, characteristic, complemented
4#90359C211yesmaximal, characteristic, complemented
5C7×#497C211yesmaximal, characteristic, complemented
6#89962C212yesmaximal, complemented
7#89948C212yesmaximal, complemented
8#89858C211yesmaximal, characteristic, complemented
9#89830C211yesmaximal, characteristic, complemented
10#89484C211yesmaximal, characteristic, complemented
11D7×#97C211yesmaximal, characteristic, complemented
12C2×#1161C2212yes—
13C2×#1161C2211yescharacteristic
14C2×#1159C2211yescharacteristic
15C2×C7×Q16C2212yes—
16C2×C7×SD16C2212yes—
17C7×C4⋊Q8C2211yescharacteristic
18C7×#124C2211yescharacteristic
19#6510C2212yes—
20#6507C2211yescharacteristic
21C2×#1144C2212yes—
22C2×#1143C2212yes—
23C2×#1143C2212yes—
24#6503C2211yescharacteristic
25C2×#1141C2212yes—
26#6482—28no—
27#6479—28no—
28C2×D56C2212yes—
29C2×#1132C2212yes—
30C2×C8×D7C2212yes—
31#6458—22no—
32#6456—22no—
33#6447—22no—
34(C22⋊D7)⋊C4—22no—
35#6438C2211yescharacteristic
36#6437C2211yescharacteristic
37C42×D7C2211yescharacteristic
38C7×#97C2211yescharacteristic
39#6389C2212yes—
40#6377C2211yescharacteristic
41#3150—71nomaximal, Sylow
42C2×C7×Q8C2311yescharacteristic
43C2×C7×Q8C2312yes—
44C2×C7×D4C2311yescharacteristic
45C2×C22⋊D7—21no—
46C2×C22⋊D7—22no—
47C22×Dic7—21no—
48#1161—24no—
49#1161—24no—
50#1161—24no—
51#1159—24no—
52C2×D28—21no—
53C2×D28—22no—
54C2×D28—22no—
55C2×D28C2312yes—
56C2×D28C2311yescharacteristic
57C2×C4×D7—21no—
58C2×C4×D7—22no—
59C2×C4×D7D412yes—
60C2×C4×D7C2311yescharacteristic
61C2×Dic14C2311yescharacteristic
62C7×Q16—24no—
63C7×SD16—24no—
64C2×C7×C8C2312yes—
65C7×C4⋊C4—22no—
66C7×C22⋊C4—22no—
67C42×C7C2311yescharacteristic
68#1144—24no—
69#1143—24no—
70#1143—24no—
71#1141—24no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—24no—
74Dic7⋊C4—22no—
75C4×Dic7D412yes—
76C4×Dic7C2311yescharacteristic
77C2×C7⋊C8C2312yes—
78D56—24no—
79#1132—24no—
80C8×D7D418yescomplemented
81C2×#135—72no—
82C2×#135—72no—
83#532—71no—
84#532—71no—
85#497—72no—
86#497—71no—
87#497D711yescharacteristic, complemented
88#496—72no—
89#496—71no—
90#496—71no—
91C2×#97—71no—
92C23×C7—21no—
93C22×D7—22no—
94C22×D7—21no—
95C22×D7C2×D411yescharacteristic
96C7×Q8—22no—
97C7×Q8—22no—
98C7×Q8—22no—
99C7×D4—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7C2×D412yes—
103C2×C4×C7C2411yescharacteristic
104C22⋊D7—44no—
105C22⋊D7—42no—
106C2×Dic7—21no—
107C2×Dic7—42no—
108C2×Dic7C2×D412yes—
109C2×Dic7C2×D411yescharacteristic
110D28—22no—
111D28—22no—
112D28—22no—
113D28—44no—
114D28—42no—
115D28—44no—
116C4×D7—42no—
117C4×D7—42no—
118C4×D7—44no—
119C4×D7C2×D414yes—
120Dic14—22no—
121C7×C8C2×D414yes—
122C7⋊C8C2×D414yes—
123C2×#41—72no—
124C2×#41—71no—
125C2×#41—71no—
126#135—148no—
127#135—148no—
128C2×Q16—72no—
129C2×Q16C2×D712yes—
130C2×SD16—72no—
131C2×SD16—72no—
132C2×SD16—72no—
133C2×SD16C2×D712yes—
134C2×D8—72no—
135C2×D8—72no—
136C22×C8—72no—
137C4⋊Q8—71no—
138C4⋊Q8C2×D711yescharacteristic
139#127—71no—
140#127—71no—
141#124—72no—
142#124—71no—
143#124C2×D711yescharacteristic
144#121—142no—
145#121—142no—
146C4×D4—142no—
147C4×D4—142no—
148C2×C42—71no—
149#97—72no—
150#97—71no—
151#97C2×D711yescharacteristic
152C22×C7—41no—
153C22×C7—42no—
154C22×C7C22×D411yescharacteristic
155C2×D7—22no—
156C2×D7—42no—
157C2×D7—41no—
158C2×D7—44no—
159C2×D7—42no—
160C2×D7#12712yes—
161C4×C7—22no—
162C4×C7—41no—
163C4×C7—42no—
164C4×C7C22×D412yes—
165Dic7—22no—
166Dic7—41no—
167Dic7#12712yes—
168#41—144no—
169#41—144no—
170#41—144no—
171#41—144no—
172C2×Q8—71no—
173C2×Q8C22×D712yes—
174C2×Q8C22×D711yescharacteristic
175C2×D4—72no—
176C2×D4—71no—
177C2×D4—141no—
178C2×D4—142no—
179C2×D4—142no—
180C2×D4—141no—
181C2×D4—142no—
182C2×D4C22×D711yescharacteristic
183C22×C4—71no—
184C22×C4—72no—
185C22×C4—141no—
186C22×C4—142no—
187C22×C4—141no—
188Q16—24no—
189Q16—144no—
190SD16—24no—
191SD16—144no—
192SD16—144no—
193SD16—144no—
194D8—144no—
195D8—144no—
196C2×C8—72no—
197C2×C8—78no—
198C2×C8C22×D712yes—
199C4⋊C4—22no—
200C4⋊C4—142no—
201C22⋊C4—22no—
202C22⋊C4—144no—
203C22⋊C4—142no—
204C42—72no—
205C42—71no—
206C42C22×D711yescharacteristic
207C2×C7—81no—
208C2×C7C2×#12711yescharacteristic
209C2×C7#57512yes—
210D7—22no—
211D7—82no—
212D7—81no—
213C23—21no—
214C23—71no—
215C23—142no—
216C23—141no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—142no—
221D4—22no—
222D4—142no—
223D4—142no—
224D4—142no—
225D4—284no—
226D4—282no—
227D4—284no—
228D4—284no—
229D4—282no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—71no—
233C2×C4—74no—
234C2×C4—72no—
235C2×C4—141no—
236C2×C4—282no—
237C2×C4—282no—
238C2×C4—284no—
239C2×C4—282no—
240C2×C4D4×D712yes—
241C2×C4C23×D711yescharacteristic
242C8—74no—
243C8D4×D714yes—
244C7#315011yescharacteristic, complemented, Sylow
245C22—41no—
246C22—42no—
247C22—72no—
248C22—142no—
249C22—282no—
250C22—281no—
251C22—284no—
252C22—282no—
253C22C2×D4×D711yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—72no—
258C4—142no—
259C4—281no—
260C4C2×D4×D712yes—
261C2—81no—
262C2—142no—
263C2—562no—
264C2—561no—
265C2D7×#12711yescharacteristic
266C2#9046812yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#497C2
socleC22×C7C22×D4
hypercenter#497D7
hypocenterC7#3150
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#6482C2semidirect
2. C2×#6479C2semidirect
3. #90416C2semidirect
4. #90359C2semidirect
5. C7×#497C2semidirect
6. #89962C2semidirect
7. #89948C2semidirect
8. #89858C2semidirect
9. #89830C2semidirect
10. #89484C2semidirect
11. D7×#97C2semidirect
80. C8×D7D4semidirect
87. #497D7semidirect
244. C7#3150semidirect

Maximal subgroups

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

subgroupindex
C2×#64822
C2×#64792
#904162
#903592
C7×#4972
#899622
#899482
#898582
#898302
#894842
D7×#972
#31507