Finite Group Databasebeta

← #11739265

Subgroups — #11739265

Order 896.

572 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 141 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×#6482C211yesmaximal, characteristic, complemented
2C2×#6479C212yesmaximal, complemented
3C2×#6475C211yesmaximal, characteristic, complemented
4#90359C212yesmaximal, complemented
5C7×#495C211yesmaximal, characteristic, complemented
6#89953C211yesmaximal, characteristic, complemented
7#89947C212yesmaximal, complemented
8#89923C211yesmaximal, characteristic, complemented
9#89831C212yesmaximal, complemented
10#89468C211yesmaximal, characteristic, complemented
11C4×C8×D7C211yesmaximal, characteristic, complemented
12C2×#1161C2212yes—
13C2×#1159C2212yes—
14C2×C7×Q16C2211yescharacteristic
15C2×C7×SD16C2212yes—
16C2×C7×D8C2211yescharacteristic
17C7×#124C2212yes—
18#6507C2212yes—
19C2×#1144C2212yes—
20C2×#1143C2212yes—
21#6503C2212yes—
22C2×#1142C2212yes—
23C2×#1141C2212yes—
24#6482—24no—
25#6479—28no—
26#6475—24no—
27C2×Dic28C2211yescharacteristic
28C2×D56C2211yescharacteristic
29C2×#1132C2212yes—
30C2×C8×D7C2211yescharacteristic
31C2×C8×D7C2211yescharacteristic
32#6447—24no—
33(C22⋊D7)⋊C4—24no—
34#6438C2212yes—
35C42×D7C2211yescharacteristic
36C4×C7×C8C2211yescharacteristic
37C8×Dic7C2211yescharacteristic
38C8×Dic7C2211yescharacteristic
39C4×C7⋊C8C2211yescharacteristic
40#3147—71nomaximal, Sylow
41C2×C7×Q8C2312yes—
42C2×C7×D4C2312yes—
43C2×C22⋊D7—22no—
44C2×C22⋊D7—24no—
45C22×Dic7—22no—
46#1161—24no—
47#1161—24no—
48#1159—24no—
49#1159—24no—
50C2×D28—22no—
51C2×D28C2312yes—
52C2×C4×D7—22no—
53C2×C4×D7D412yes—
54C2×C4×D7C2311yescharacteristic
55C2×Dic14C2312yes—
56C7×Q16—22no—
57C7×SD16—24no—
58C7×D8—22no—
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2311yescharacteristic
61C7×C22⋊C4—24no—
62C42×C7C2311yescharacteristic
63#1144—24no—
64#1143—24no—
65#1142—24no—
66#1141—24no—
67(C2×D7)⋊C4—24no—
68Dic7⋊C4—24no—
69C4×Dic7D412yes—
70C4×Dic7C2311yescharacteristic
71C2×C7⋊C8C2311yescharacteristic
72C2×C7⋊C8C2311yescharacteristic
73Dic28—22no—
74D56—22no—
75#1132—24no—
76C8×D7D414yescomplemented
77C8×D7D414yescomplemented
78C2×#135—71no—
79C2×#135—72no—
80C2×#135—71no—
81#532—72no—
82#495—72no—
83#495—71no—
84#495D711yescharacteristic, complemented
85#494—71no—
86#493—71no—
87#492—72no—
88C2×C4×C8—71no—
89C23×C7—22no—
90C22×D7—22no—
91C22×D7C2×D411yescharacteristic
92C7×Q8—22no—
93C7×Q8—22no—
94C7×D4—22no—
95C7×D4—22no—
96C2×C4×C7—22no—
97C2×C4×C7C2×D412yes—
98C2×C4×C7C2411yescharacteristic
99C22⋊D7—44no—
100C22⋊D7—48no—
101C2×Dic7—22no—
102C2×Dic7—44no—
103C2×Dic7C2×D411yescharacteristic
104C2×Dic7C2×D412yes—
105D28—22no—
106D28—22no—
107D28—44no—
108C4×D7—24no—
109C4×D7—44no—
110C4×D7C2×D412yes—
111C4×D7C2×D412yes—
112Dic14—22no—
113Dic14—22no—
114C7×C8C2×D412yes—
115C7×C8C2×D412yes—
116C7⋊C8C2×D412yes—
117C7⋊C8C2×D412yes—
118C2×#41—72no—
119C2×#41—72no—
120#135—144no—
121#135—148no—
122#135—144no—
123C2×Q16—72no—
124C2×Q16—71no—
125C2×Q16C2×D711yescharacteristic
126C2×SD16—72no—
127C2×SD16—72no—
128C2×SD16—72no—
129C2×SD16C2×D712yes—
130C2×D8—72no—
131C2×D8—71no—
132C2×D8C2×D711yescharacteristic
133C22×C8—71no—
134C22×C8—71no—
135C4⋊Q8—72no—
136#127—72no—
137#124—72no—
138#124C2×D712yes—
139#121—144no—
140C4×D4—144no—
141C2×C42—71no—
142C4×C8—71no—
143C4×C8—71no—
144C4×C8—71no—
145C4×C8C2×D711yescharacteristic
146C22×C7—42no—
147C22×C7—44no—
148C22×C7C22×D411yescharacteristic
149C2×D7—22no—
150C2×D7—42no—
151C2×D7—44no—
152C2×D7#12712yes—
153C4×C7—22no—
154C4×C7—42no—
155C4×C7C22×D411yescharacteristic
156C4×C7C22×D411yescharacteristic
157Dic7—42no—
158Dic7#12712yes—
159Dic7#13514yescomplemented
160#41—144no—
161#41—144no—
162#41—144no—
163#41—144no—
164C2×Q8—72no—
165C2×Q8C22×D712yes—
166C2×D4—72no—
167C2×D4—142no—
168C2×D4—144no—
169C2×D4—142no—
170C2×D4C22×D712yes—
171C22×C4—71no—
172C22×C4—72no—
173C22×C4—142no—
174C22×C4—142no—
175Q16—22no—
176Q16—144no—
177Q16—142no—
178SD16—24no—
179SD16—144no—
180SD16—144no—
181SD16—144no—
182D8—22no—
183D8—144no—
184D8—142no—
185C2×C8—71no—
186C2×C8—71no—
187C2×C8—74no—
188C2×C8—74no—
189C2×C8C22×D711yescharacteristic
190C2×C8C22×D711yescharacteristic
191C4⋊C4—144no—
192C22⋊C4—24no—
193C22⋊C4—144no—
194C42—72no—
195C42—71no—
196C42C22×D711yescharacteristic
197C2×C7—82no—
198C2×C7C2×#12711yescharacteristic
199C2×C7C2×#13512yes—
200D7—22no—
201D7—82no—
202C23—22no—
203C23—71no—
204C23—142no—
205Q8—22no—
206Q8—22no—
207Q8—142no—
208Q8—142no—
209D4—22no—
210D4—22no—
211D4—142no—
212D4—142no—
213D4—284no—
214D4—288no—
215D4—284no—
216C2×C4—22no—
217C2×C4—72no—
218C2×C4—71no—
219C2×C4—72no—
220C2×C4—72no—
221C2×C4—144no—
222C2×C4—142no—
223C2×C4—284no—
224C2×C4—284no—
225C2×C4D4×D712yes—
226C2×C4C23×D711yescharacteristic
227C8—72no—
228C8—72no—
229C8D4×D712yes—
230C8D4×D712yes—
231C7#314711yescharacteristic, complemented, Sylow
232C22—42no—
233C22—44no—
234C22—72no—
235C22—142no—
236C22—282no—
237C22—284no—
238C22C2×D4×D711yescharacteristic
239C4—22no—
240C4—42no—
241C4—72no—
242C4—74no—
243C4—282no—
244C4C2×D4×D711yescharacteristic
245C4C2×D4×D711yescharacteristic
246C2—82no—
247C2—142no—
248C2—562no—
249C2D7×#12711yescharacteristic
250C2D7×#13512yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#495C2
socleC22×C7C22×D4
hypercenter#495D7
hypocenterC7#3147
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#6482C2semidirect
2. C2×#6479C2semidirect
3. C2×#6475C2semidirect
4. #90359C2semidirect
5. C7×#495C2semidirect
6. #89953C2semidirect
7. #89947C2semidirect
8. #89923C2semidirect
9. #89831C2semidirect
10. #89468C2semidirect
11. C4×C8×D7C2semidirect
76. C8×D7D4semidirect
77. C8×D7D4semidirect
84. #495D7semidirect
159. Dic7#135semidirect
231. C7#3147semidirect

Maximal subgroups

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

subgroupindex
C2×#64822
C2×#64792
C2×#64752
#903592
C7×#4952
#899532
#899472
#899232
#898312
#894682
C4×C8×D72
#31477