Finite Group Databasebeta

← #22599072

Subgroups — #22599072

Order 9088.

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

normal subgroups only

This group has 370 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#17986536C212yesmaximal, complemented
2#17986399C211yesmaximal, characteristic, complemented
3#17986384C211yesmaximal, characteristic, complemented
4#17986355C212yesmaximal, complemented
5#17986254C211yesmaximal, characteristic, complemented
6#17986253C211yesmaximal, characteristic
7#17986248C212yesmaximal, complemented
8#17986112C211yesmaximal, characteristic, complemented
9#17986097C212yesmaximal, complemented
10#17985951C211yesmaximal, characteristic, complemented
11#441×C71C211yesmaximal, characteristic, complemented
12C2×#11765852C2212yes—
13C2×D568C2211yescharacteristic
14C2×#11765848C2211yescharacteristic
15C2×#11765848C2211yescharacteristic
16#14556123—22no—
17#14556119C2212yes—
18#14556117C2212yes—
19#14556113—22no—
20#14556104C2212yes—
21#14556103C2212yes—
22C4×D284—22no—
23C4×D284C2212yes—
24#14556085—22no—
25C42×D71C2211yescharacteristic
26#14556061C2212yes—
27#14556060C2211yescharacteristic
28#14556052C2212yes—
29#14556050C2212yes—
30#14556048C2212yes—
31#14556039C2211yescharacteristic
32C8×Dic71C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×Q16×C71C2211yescharacteristic
35C4×Q8×C71C2212yes—
36#106×C71C2211yescharacteristic
37#103×C71C2212yes—
38#97×C71C2211yescharacteristic
39C2×D284—22no—
40C2×D284C2312yes—
41C2×C4×D71—22no—
42C2×C4×D71D412yes—
43C2×C4×D71C2311yescharacteristic
44#11765852—24no—
45D568—22no—
46#11765848—22no—
47#11765848—22no—
48C2×C71⋊C8C2311yescharacteristic
49C2×C71⋊C8C2311yescharacteristic
50C4⋊Dic71—22no—
51C4⋊Dic71C2312yes—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59C4×Dic71D412yes—
60C4×Dic71C2311yescharacteristic
61C2×Q8×C71C2312yes—
62Q16×C71—22no—
63C2×C8×C71C2311yescharacteristic
64C2×C8×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C42×C71—22no—
68C42×C71C2311yescharacteristic
69C22×D71—22no—
70C22×D71C2×D411yescharacteristic
71Q8×C71—22no—
72Q8×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71C2×D412yes—
76C2×C4×C71C2411yescharacteristic
77C2×Dic71—22no—
78C2×Dic71C2×D411yescharacteristic
79C2×Dic71C2×D412yes—
80D284—22no—
81D284—22no—
82D284—44no—
83C4×D71—21no—
84C4×D71—21no—
85C4×D71—42no—
86C4×D71—44no—
87C8×C71—21no—
88C8×C71#4112yes—
89C71⋊C8—21no—
90C71⋊C8#4112yes—
91C22×C71C22×D411yescharacteristic
92C2×D71—21no—
93C2×D71—22no—
94C2×D71—42no—
95C2×D71—42no—
96C2×D71—42no—
97C4×C71—22no—
98C4×C71—42no—
99C4×C71—42no—
100C4×C71C2×#4111yescharacteristic
101C4×C71C2×#4111yescharacteristic
102Dic71—21no—
103Dic71—22no—
104Dic71—42no—
105C2×C71#53211yescharacteristic
106C2×C71#57512yes—
107D71—41no—
108D71—82no—
109#3183—711nomaximal, Sylow
110C71#318311yescharacteristic, complemented, Sylow
111#529—712no—
112#499—711no—
113#495—711no—
114#488—712no—
115#470—711no—
116#469—711no—
117#468—712no—
118#442—711no—
119#441D7111yescharacteristic, complemented
120#440—712no—
121C4×M16—711no—
122C2×Q16C2×D7111yescharacteristic
123C2×SD16—712no—
124C2×D8—711no—
125C2×M16—711no—
126C2×M16—711no—
127#126—1422no—
128#125—712no—
129#124—712no—
130#123—1422no—
131C22⋊Q8—712no—
132#121—712no—
133C4×Q8C2×D7112yes—
134C4×D4—712no—
135C4×D4—1422no—
136#117—1422no—
137C2×C42—711no—
138#107—712no—
139#106—711no—
140#106C2×D7111yescharacteristic
141#103—712no—
142#103C2×D7112yes—
143#102—712no—
144#102—712no—
145#97—711no—
146#97C2×D7111yescharacteristic
147C4×C8—711no—
148C4×C8—711no—
149C2×Q8C22×D7112yes—
150C2×D4—712no—
151C2×D4—1422no—
152C22×C4—712no—
153C22×C4—711no—
154C22×C4—1422no—
155Q16—22no—
156SD16—1424no—
157D8—1422no—
158M16—1422no—
159M16—1422no—
160C2×C8—711no—
161C2×C8—711no—
162C2×C8C22×D7111yescharacteristic
163C2×C8C22×D7111yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—712no—
166C4⋊C4—1422no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4C22×D7112yes—
170C22⋊C4—1422no—
171C22⋊C4—1422no—
172C22⋊C4—1422no—
173C22⋊C4—1422no—
174C22⋊C4—1422no—
175C42—22no—
176C42—712no—
177C42—711no—
178C42C22×D7111yescharacteristic
179C23—711no—
180C23—1422no—
181Q8—22no—
182Q8—22no—
183D4—2844no—
184D4—1422no—
185D4—1422no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—2844no—
189C2×C4—2842no—
190C2×C4—712no—
191C2×C4—711no—
192C2×C4—1422no—
193C2×C4—1421no—
194C2×C4—1421no—
195C2×C4D4×D7112yes—
196C2×C4C23×D7111yescharacteristic
197C8—21no—
198C8—712no—
199C8—1421no—
200C8#1176586512yes—
201C22—2842no—
202C22—2842no—
203C22—2842no—
204C22—1422no—
205C22—1421no—
206C22C2×D4×D7111yescharacteristic
207C4—22no—
208C4—42no—
209C4—42no—
210C4—2842no—
211C4—1422no—
212C4—1421no—
213C4#41×D7111yescharacteristic
214C4C2×#1176586511yescharacteristic
215C2—2841no—
216C2—5682no—
217C2#1798654611yescharacteristic
218C2#1798679812yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#441×C71C2
socleC22×C71C22×D4
hypercenter#441D71
hypocenterC71#3183
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #17986536C2semidirect
2. #17986399C2semidirect
3. #17986384C2semidirect
4. #17986355C2semidirect
5. #17986254C2semidirect
7. #17986248C2semidirect
8. #17986112C2semidirect
9. #17986097C2semidirect
10. #17985951C2semidirect
11. #441×C71C2semidirect
110. C71#3183semidirect
119. #441D71semidirect