Finite Group Databasebeta

← #22599345

Subgroups — #22599345

Order 9088.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986556C211yesmaximal, characteristic, complemented
2C2×#14556116C211yesmaximal, characteristic
3#17986393C212yesmaximal, complemented
4#17986359C212yesmaximal, complemented
5#17986222C214yesmaximal, complemented
6#17986157C212yesmaximal, complemented
7#17986152C212yesmaximal
8#432×C71C211yesmaximal, characteristic, complemented
9C22×Dic142C2211yescharacteristic
10C22×D284C2211yescharacteristic
11C2×Dic284C2214yescomplemented
12C2×#11765854C2214yes—
13#14556128C2211yescharacteristic, complemented
14#14556128C2211yescharacteristic, complemented
15#14556120—21no—
16#14556116—21no—
17#14556116—21no—
18#14556116C2212yes—
19#14556116C2212yes—
20#14556107—21no—
21#14556107C2212yes—
22C4×Dic142—21no—
23C4×Dic142C2212yescomplemented
24#14556087—21no—
25C2×(C2×D71)⋊C4—21no—
26C2×(C2×D71)⋊C4—21no—
27#14556053C2214yescomplemented
28#14556050C2214yes—
29C2×C42×C71C2211yescharacteristic
30C4⋊C8×C71C2212yescomplemented
31C22⋊C8×C71C2212yes—
32#97×C71C2212yescomplemented
33C23×D71—21no—
34C2×Dic142—21no—
35C2×Dic142—21no—
36C2×Dic142—42no—
37C2×Dic142C2312yes—
38C2×Dic142C2312yes—
39C2×D284—21no—
40C2×D284—42no—
41C2×D284C2312yes—
42C22×Dic71—21no—
43Dic284—44no—
44#11765854—44no—
45C4⋊Dic71—21no—
46C4⋊Dic71—21no—
47C4⋊Dic71—21no—
48C4⋊Dic71C2312yes—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51C22⋊Dic71—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56C4×Dic71—22no—
57C22×C4×C71D411yescharacteristic
58C22×C4×C71D411yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C2×C8×C71C2314yes—
61C42×C71D411yescharacteristic
62C42×C71D411yescharacteristic
63C42×C71C2311yescharacteristic
64C42×C71C2311yescharacteristic
65C23×C71C2×D411yescharacteristic
66C22×D71—22no—
67C22×D71—42no—
68C22×D71—42no—
69C22×D71—42no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—41no—
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2411yescharacteristic
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—42no—
83D284—24no—
84D284—42no—
85Dic142—24no—
86Dic142—24no—
87Dic142—42no—
88Dic142—42no—
89C8×C71—24no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71#13712yes—
93C22×C71C22×D411yescharacteristic
94C2×D71—42no—
95C2×D71—42no—
96C2×D71—42no—
97C2×D71—84no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—21no—
103C4×C71H(5,𝔽2)12yes—
104Dic71—42no—
105Dic71—42no—
106Dic71—42no—
107C2×C71—22no—
108C2×C71#53511yescharacteristic
109C2×C71C2×#13711yescharacteristic
110C2×C71#57511yescharacteristic
111D71—82no—
112#3215—711nomaximal, Sylow
113C71#321511yescharacteristic, complemented, Sylow
114#533—711no—
115C2×#124—711no—
116#497—712no—
117#489—712no—
118#464—714no—
119#451—712no—
120#450—712no—
121#432D7111yescharacteristic, complemented
122C22×Q8—711no—
123C22×D4—711no—
124C2×Q16—714no—
125C2×SD16—714no—
126C4⋊Q8—711no—
127C4⋊Q8—711no—
128#125—1421no—
129#124—712no—
130#124—712no—
131#124—1421no—
132#124—1421no—
133C22⋊Q8—712no—
134C22⋊Q8—1421no—
135C4×Q8—712no—
136C4×Q8—1421no—
137#117—1421no—
138C2×C22⋊C4—1421no—
139C2×C22⋊C4—1421no—
140C2×C42C2×D7111yescharacteristic
141C4⋊C8C2×D7112yescomplemented
142#103—714no—
143#102—714no—
144C22⋊C8C2×D7112yes—
145#97C2×D7112yescomplemented
146C24—1421no—
147C2×Q8—2842no—
148C2×Q8—712no—
149C2×Q8—712no—
150C2×Q8—1421no—
151C2×Q8—1421no—
152C2×D4—2842no—
153C2×D4—712no—
154C2×D4—1421no—
155C22×C4—1421no—
156C22×C4D28411yescharacteristic
157C22×C4D28411yescharacteristic
158C22×C4C22×D7111yescharacteristic
159Q16—2844no—
160SD16—2844no—
161C2×C8C22×D7114yes—
162C4⋊C4—712no—
163C4⋊C4—1422no—
164C4⋊C4—1422no—
165C4⋊C4—1421no—
166C4⋊C4—1421no—
167C4⋊C4—1421no—
168C22⋊C4—1422no—
169C22⋊C4—1422no—
170C22⋊C4—1422no—
171C22⋊C4—1422no—
172C22⋊C4—1422no—
173C42—1422no—
174C42D28411yescharacteristic
175C42D28411yescharacteristic
176C42C22×D7111yescharacteristic
177C42C22×D7111yescharacteristic
178C23—2842no—
179C23—2842no—
180C23—2842no—
181C23—1422no—
182C23C2×D28411yescharacteristic
183Q8—2842no—
184Q8—2842no—
185Q8—1424no—
186Q8—1424no—
187D4—2842no—
188D4—1424no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—2842no—
193C2×C4—1422no—
194C2×C4—1422no—
195C2×C4—1422no—
196C2×C4C2×D28411yescharacteristic
197C2×C4C2×D28411yescharacteristic
198C2×C4C2×D28411yescharacteristic
199C2×C4C2×D28411yescharacteristic
200C2×C4C2×D28411yescharacteristic
201C2×C4C23×D7111yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—2842no—
206C22—2842no—
207C22—2842no—
208C22—5684no—
209C22#1455616112yes—
210C22C22×D28411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—2842no—
217C4—2842no—
218C4—2842no—
219C4#1455617412yes—
220C2—22no—
221C2—5682no—
222C2#1798656811yescharacteristic
223C2C2×#1455616111yescharacteristic
224C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3215
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #17986556C2semidirect
3. #17986393C2semidirect
4. #17986359C2semidirect
5. #17986222C2semidirect
6. #17986157C2semidirect
8. #432×C71C2semidirect
11. C2×Dic284C22semidirect
13. #14556128C22semidirect
14. #14556128C22semidirect
23. C4×Dic142C22semidirect
27. #14556053C22semidirect
30. C4⋊C8×C71C22semidirect
32. #97×C71C22semidirect
113. C71#3215semidirect
121. #432D71semidirect
141. C4⋊C8C2×D71semidirect
145. #97C2×D71semidirect