Finite Group Databasebeta

← #22599414

Subgroups — #22599414

Order 9088.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986611C211yesmaximal, characteristic, complemented
2#17986568C211yesmaximal, characteristic
3#17986393C211yesmaximal, characteristic, complemented
4#17986372C211yesmaximal, characteristic, complemented
5#17986364C211yesmaximal, characteristic, complemented
6#17986354C211yesmaximal, characteristic, complemented
7#17986222C212yesmaximal, complemented
8#17986200C212yesmaximal, complemented
9#17986163C212yesmaximal, complemented
10#17986152C212yesmaximal, complemented
11#433×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C2×Dic284C2212yescomplemented
15C2×#11765854C2212yes—
16C2×#11765854C2212yes—
17C2×#11765854C2212yescomplemented
18#14556128C2211yescharacteristic, complemented
19#14556125C2211yescharacteristic
20#14556125C2211yescharacteristic
21#14556121—22no—
22#14556120C2211yescharacteristic, complemented
23#14556118—22no—
24#14556116C2211yescharacteristic
25#14556116C2211yescharacteristic
26#14556107C2212yes—
27#14556103—22no—
28#14556098—22no—
29C4×Dic142C2212yescomplemented
30#14556053C2212yescomplemented
31#14556050C2212yes—
32#14556050C2212yes—
33#14556050C2212yescomplemented
34#117×C71C2211yescharacteristic
35C4⋊C8×C71C2212yescomplemented
36C22⋊C8×C71C2212yes—
37#97×C71C2211yescharacteristic, complemented
38C4×C8×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40#11765865—42no—
41C2×Dic142C2311yescharacteristic
42C2×Dic142C2312yes—
43C2×D284—21no—
44C2×D284—22no—
45C2×D284—22no—
46C2×D284—42no—
47C2×D284C2311yescharacteristic
48C2×D284C2311yescharacteristic
49C2×D284C2311yescharacteristic
50C2×C22⋊D71—21no—
51C2×C4×D71—21no—
52Dic284—42no—
53#11765854—42no—
54#11765854—42no—
55#11765854—42no—
56C4⋊Dic71—22no—
57C4⋊Dic71C2312yes—
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60Dic71⋊C4—22no—
61C22⋊Dic71—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—22no—
64C4×Dic71—22no—
65C22×C4×C71C2311yescharacteristic
66C2×C8×C71C2312yes—
67C2×C8×C71C2312yes—
68C4⋊C4×C71D412yes—
69C22⋊C4×C71D412yes—
70C42×C71C2311yescharacteristic
71C42×C71C2311yescharacteristic
72C23×C71C2×D411yescharacteristic
73C22×D71—21no—
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—42no—
77C22×D71—82no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2×D412yes—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C22⋊D71—42no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—21no—
87D284—22no—
88D284—22no—
89D284—22no—
90D284—42no—
91D284—82no—
92D284—82no—
93C4×D71—42no—
94Dic142—22no—
95Dic142—22no—
96Dic142—22no—
97C8×C71—22no—
98C8×C71—22no—
99C22×C71—21no—
100C22×C71—22no—
101C22×C71C22×D411yescharacteristic
102C2×D71—41no—
103C2×D71—41no—
104C2×D71—41no—
105C2×D71—42no—
106C2×D71—42no—
107C2×D71—42no—
108C2×D71—84no—
109C4×C71—21no—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71H(5,𝔽2)12yes—
113Dic71—41no—
114Dic71—42no—
115Dic71—42no—
116C2×C71—41no—
117C2×C71#53511yescharacteristic
118C2×C71#57712yes—
119D71—81no—
120D71—81no—
121D71—81no—
122#3222—711nomaximal, Sylow
123C71#322211yescharacteristic, complemented, Sylow
124#541—711no—
125#535—711no—
126#497—711no—
127#492—711no—
128#490—711no—
129#488—711no—
130#464—712no—
131#460—712no—
132#452—712no—
133#450—712no—
134#433D7111yescharacteristic, complemented
135C2×#41—711no—
136C22×D4—711no—
137C2×Q16—712no—
138C2×SD16—712no—
139C2×SD16—712no—
140C2×SD16—712no—
141C4⋊Q8—711no—
142#127—711no—
143#127—711no—
144#126—1422no—
145#125—711no—
146#125—1422no—
147#124—711no—
148#124—711no—
149C22⋊Q8—712no—
150#121—1422no—
151#120—1422no—
152C4×Q8—712no—
153#117C2×D7111yescharacteristic
154C4⋊C8C2×D7112yescomplemented
155#103—712no—
156#102—712no—
157#102—712no—
158#102—712no—
159C22⋊C8C2×D7112yes—
160#97C2×D7111yescharacteristic, complemented
161C4×C8C2×D7111yescharacteristic, complemented
162C24—1421no—
163#41—2842no—
164C2×Q8—711no—
165C2×Q8—712no—
166C2×D4—2842no—
167C2×D4—711no—
168C2×D4—711no—
169C2×D4—711no—
170C2×D4—1421no—
171C2×D4—1421no—
172C2×D4—1422no—
173C2×D4—1422no—
174C22×C4—1421no—
175C22×C4C22×D7111yescharacteristic
176Q16—2842no—
177SD16—2842no—
178SD16—2842no—
179SD16—2842no—
180C2×C8C22×D7112yes—
181C2×C8C22×D7112yes—
182C4⋊C4—712no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4D28412yes—
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C22⋊C4—1422no—
191C22⋊C4D28412yes—
192C42—1422no—
193C42C22×D7111yescharacteristic
194C42C22×D7111yescharacteristic
195C23—2842no—
196C23—5682no—
197C23—1421no—
198C23—1421no—
199C23—1421no—
200C23C2×D28411yescharacteristic
201Q8—1422no—
202Q8—1422no—
203Q8—1422no—
204D4—2842no—
205D4—2842no—
206D4—5682no—
207D4—5682no—
208D4—1422no—
209D4—1422no—
210D4—1422no—
211C2×C4—22no—
212C2×C4—2842no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1421no—
216C2×C4C2×D28412yes—
217C2×C4C2×D28412yes—
218C2×C4C2×D28411yescharacteristic
219C2×C4C23×D7111yescharacteristic
220C8—22no—
221C8—22no—
222C22—21no—
223C22—22no—
224C22—2841no—
225C22—2841no—
226C22—2841no—
227C22—2842no—
228C22—2842no—
229C22—2842no—
230C22—5684no—
231C22C22×D28411yescharacteristic
232C4—21no—
233C4—22no—
234C4—22no—
235C4—2841no—
236C4—2842no—
237C4—2842no—
238C4#1455617412yes—
239C2—41no—
240C2—5681no—
241C2—5681no—
242C2—5681no—
243C2#1798656811yescharacteristic
244C2#1798681112yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#433×C71C2
socleC22×C71C22×D4
hypercenter#433D71
hypocenterC71#3222
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #17986611C2semidirect
3. #17986393C2semidirect
4. #17986372C2semidirect
5. #17986364C2semidirect
6. #17986354C2semidirect
7. #17986222C2semidirect
8. #17986200C2semidirect
9. #17986163C2semidirect
10. #17986152C2semidirect
11. #433×C71C2semidirect
14. C2×Dic284C22semidirect
17. C2×#11765854C22semidirect
18. #14556128C22semidirect
22. #14556120C22semidirect
29. C4×Dic142C22semidirect
30. #14556053C22semidirect
33. #14556050C22semidirect
35. C4⋊C8×C71C22semidirect
37. #97×C71C22semidirect
38. C4×C8×C71C22semidirect
123. C71#3222semidirect
134. #433D71semidirect
154. C4⋊C8C2×D71semidirect
160. #97C2×D71semidirect
161. C4×C8C2×D71semidirect