Finite Group Databasebeta

← #22598997

Subgroups — #22598997

Order 9088.

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

normal subgroups only

This group has 338 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#17986607C212yesmaximal
2#17986399C211yesmaximal, characteristic, complemented
3#17986377C211yesmaximal, characteristic, complemented
4#17986366C212yesmaximal, complemented
5#17986258C211yesmaximal, characteristic
6#17986254C211yesmaximal, characteristic
7#17986239C212yesmaximal
8#17986107C211yesmaximal, characteristic, complemented
9C4×#11765855C212yesmaximal, complemented
10#17985956C211yesmaximal, characteristic
11#441×C71C211yesmaximal, characteristic, complemented
12C2×Dic284C2211yescharacteristic
13C2×#11765855C2212yes—
14C2×#11765848C2211yescharacteristic
15C2×C8×D71C2211yescharacteristic
16#14556126C2212yes—
17#14556123—22no—
18#14556122—22no—
19#14556120—22no—
20#14556119C2212yes—
21#14556118—22no—
22#14556108C2212yes—
23#14556104C2212yes—
24C4×Dic142C2212yes—
25#14556085C2211yescharacteristic
26#14556061C2212yes—
27#14556060C2211yescharacteristic
28#14556053C2212yes—
29#14556052C2212yes—
30#14556051C2212yes—
31#14556038C2211yescharacteristic
32C4×C71⋊C8C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×Q16×C71C2211yescharacteristic
35C4×Q8×C71C2212yes—
36#106×C71C2211yescharacteristic
37#103×C71C2212yes—
38#97×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40C2×C4×D71C2311yescharacteristic
41Dic284—22no—
42#11765855—24no—
43#11765848—22no—
44C2×C71⋊C8C2311yescharacteristic
45C2×C71⋊C8C2311yescharacteristic
46C8×D71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—22no—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4D412yes—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4D412yes—
59C4×Dic71—22no—
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×D71C2×D411yescharacteristic
70Q8×C71—22no—
71Q8×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71C2×D412yes—
75C2×C4×C71C2411yescharacteristic
76C2×Dic71—22no—
77C2×Dic71—22no—
78C2×Dic71C2×D412yes—
79C2×Dic71C2×D411yescharacteristic
80C4×D71—21no—
81C4×D71—21no—
82Dic142—22no—
83Dic142—22no—
84C8×C71—21no—
85C8×C71#4112yes—
86C71⋊C8—21no—
87C71⋊C8#4112yes—
88C22×C71C22×D411yescharacteristic
89C2×D71—21no—
90C2×D71—22no—
91C4×C71—22no—
92C4×C71—42no—
93C4×C71—42no—
94C4×C71C2×#4111yescharacteristic
95C4×C71C2×#4111yescharacteristic
96Dic71—21no—
97Dic71—22no—
98Dic71—42no—
99Dic71—42no—
100C2×C71#53211yescharacteristic
101C2×C71#57812yes—
102D71—41no—
103#3177—711nomaximal, Sylow
104C71#317711yescharacteristic, complemented, Sylow
105#541—712no—
106#499—711no—
107#494—711no—
108#491—712no—
109#470—711no—
110#470—711no—
111#467—712no—
112#441—711no—
113#441D7111yescharacteristic, complemented
114C4×Q16—712no—
115#405—711no—
116C2×Q16—712no—
117C2×Q16—711no—
118C2×Q16C2×D7111yescharacteristic
119C2×M16—711no—
120C22×C8—711no—
121C4⋊Q8—712no—
122#126—1422no—
123#126—1422no—
124#125—712no—
125#125—1422no—
126#125—1422no—
127C22⋊Q8—712no—
128C22⋊Q8—712no—
129C4×Q8—712no—
130C4×Q8C2×D7112yes—
131#117—711no—
132#107—712no—
133#106—711no—
134#106C2×D7111yescharacteristic
135#103—712no—
136#103—712no—
137#103—712no—
138#103C2×D7112yes—
139#97—711no—
140#97C2×D7111yescharacteristic
141C4×C8—711no—
142C4×C8—711no—
143C2×Q8—712no—
144C2×Q8C22×D7112yes—
145C22×C4—711no—
146Q16—22no—
147Q16—1424no—
148Q16—1422no—
149M16—1422no—
150C2×C8—711no—
151C2×C8—711no—
152C2×C8—1422no—
153C2×C8C22×D7111yescharacteristic
154C2×C8C22×D7111yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—712no—
157C4⋊C4—712no—
158C4⋊C4—1422no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4—1422no—
162C4⋊C4—1422no—
163C4⋊C4—1422no—
164C4⋊C4—1422no—
165C4⋊C4C22×D7112yes—
166C22⋊C4—712no—
167C22⋊C4—1422no—
168C22⋊C4—1422no—
169C42—22no—
170C42—711no—
171C42—1422no—
172C42C22×D7111yescharacteristic
173C23—711no—
174Q8—22no—
175Q8—22no—
176Q8—1422no—
177Q8—1422no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—712no—
181C2×C4—711no—
182C2×C4—1422no—
183C2×C4—1422no—
184C2×C4—1421no—
185C2×C4—1421no—
186C2×C4D4×D7112yes—
187C2×C4C23×D7111yescharacteristic
188C8—21no—
189C8—712no—
190C8—1421no—
191C8#1176586512yes—
192C22—1421no—
193C22—1422no—
194C22C2×D4×D7111yescharacteristic
195C4—22no—
196C4—42no—
197C4—42no—
198C4—2842no—
199C4—2842no—
200C4—1421no—
201C4—1422no—
202C4#41×D7111yescharacteristic
203C4C2×#1176586511yescharacteristic
204C2—2841no—
205C2#1798654611yescharacteristic
206C2#1798681512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#441×C71C2
socleC22×C71C22×D4
hypercenter#441D71
hypocenterC71#3177
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
2. #17986399C2semidirect
3. #17986377C2semidirect
4. #17986366C2semidirect
8. #17986107C2semidirect
9. C4×#11765855C2semidirect
11. #441×C71C2semidirect
104. C71#3177semidirect
113. #441D71semidirect