Finite Group Databasebeta

← #22600707

Subgroups — #22600707

Order 9088.

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

normal subgroups only

This group has 342 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#17986687C212yesmaximal
2#17986343C212yesmaximal, complemented
3#17986331C212yesmaximal, complemented
4#17986303C212yesmaximal, complemented
5#17986241C211yesmaximal, characteristic, complemented
6#17986241C211yesmaximal, characteristic, complemented
7C4×Dic284C211yesmaximal, characteristic, complemented
8C8×D284C211yesmaximal, characteristic
9#17986036C211yesmaximal, characteristic, complemented
10#107×D71C211yesmaximal, characteristic, complemented
11#499×C71C211yesmaximal, characteristic, complemented
12C2×Dic284C2211yescharacteristic
13C2×C8×D71C2211yescharacteristic
14C2×C8×D71C2211yescharacteristic
15#14556123—22no—
16#14556119—22no—
17#14556113—22no—
18#14556108C2212yes—
19#14556108C2212yes—
20C4×Dic142C2212yes—
21C4×D284C2211yescharacteristic
22#14556086—22no—
23#14556086C2212yes—
24C4⋊C4×D71C2212yes—
25#14556062C2211yescharacteristic
26#14556061C2212yes—
27#14556059C2212yes—
28C4⋊(C71⋊C8)C2211yescharacteristic
29#14556053C2212yes—
30#14556051C2212yes—
31#14556051C2212yes—
32(C2×D71)⋊C8C2211yescharacteristic
33(C2×D71)⋊C8C2211yescharacteristic
34#125×C71C2212yes—
35#107×C71C2211yescharacteristic
36#107×C71C2211yescharacteristic
37#106×C71C2212yes—
38C4×C8×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40C2×D284D411yescharacteristic
41C2×C4×D71—22no—
42C2×C4×D71C2311yescharacteristic
43C2×C4×D71C2311yescharacteristic
44Dic284—22no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2311yescharacteristic
47C8×D71—22no—
48C8×D71—22no—
49C4⋊Dic71—22no—
50C4⋊Dic71—22no—
51C4⋊Dic71D411yescharacteristic
52C4⋊Dic71C2312yes—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4D411yescharacteristic
59(C2×D71)⋊C4D411yescharacteristic
60C4×Dic71—22no—
61C4×Dic71—22no—
62C2×C8×C71C2311yescharacteristic
63C2×C8×C71C2311yescharacteristic
64C4⋊C4×C71—22no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C4⋊C4×C71C2312yes—
68C42×C71C2311yescharacteristic
69C22×D71C2×D411yescharacteristic
70C22×D71C2×D411yescharacteristic
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2411yescharacteristic
76C2×Dic71—22no—
77C2×Dic71—22no—
78C2×Dic71C2×D411yescharacteristic
79C2×Dic71C2×D411yescharacteristic
80D284—22no—
81C4×D71—21no—
82C4×D71—21no—
83C4×D71—21no—
84C4×D71—21no—
85C4×D71—44no—
86Dic142—22no—
87Dic142—22no—
88C8×C71—21no—
89C8×C71#4112yes—
90C71⋊C8—21no—
91C71⋊C8—21no—
92C22×C71C22×D411yescharacteristic
93C2×D71—21no—
94C2×D71—21no—
95C2×D71—21no—
96C2×D71—21no—
97C2×D71—21no—
98C2×D71#13512yes—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—42no—
102C4×C71—42no—
103C4×C71C2×#4111yescharacteristic
104C4×C712−1+411yescharacteristic
105Dic71—21no—
106Dic71—21no—
107Dic71—42no—
108Dic71—42no—
109C2×C71#54711yescharacteristic
110C2×C71C2×#13511yescharacteristic
111C2×C71#57811yescharacteristic
112D71—22no—
113D71—41no—
114#3336—711nomaximal, Sylow
115C71#333611yescharacteristic, complemented, Sylow
116#552—712no—
117#499D7111yescharacteristic, complemented
118#485—712no—
119#483—712no—
120#479—712no—
121#467—711no—
122#467—711no—
123C4×Q16—711no—
124C8×D4—711no—
125#427—711no—
126C2×#107—711no—
127C2×Q16—711no—
128C22×C8—711no—
129C22×C8—711no—
130#126—1422no—
131#125—1422no—
132#125C2×D7112yes—
133#123—1422no—
134C22⋊Q8—712no—
135C22⋊Q8—712no—
136C4×Q8—712no—
137C4×D4—711no—
138#117—712no—
139#117—1422no—
140C2×C4⋊C4—712no—
141#107—711no—
142#107—712no—
143#107C2×D7111yescharacteristic
144#107C2×D7111yescharacteristic
145#106—712no—
146#106C2×D7112yes—
147C4⋊C8—711no—
148#103—712no—
149#103—712no—
150#103—712no—
151C22⋊C8—711no—
152C22⋊C8—711no—
153C4×C8C2×D7111yescharacteristic
154C2×Q8—712no—
155C2×D4—711no—
156C22×C4—711no—
157C22×C4—711no—
158C22×C4—1422no—
159Q16—1422no—
160C2×C8—711no—
161C2×C8—711no—
162C2×C8—1422no—
163C2×C8—1422no—
164C2×C8C22×D7111yescharacteristic
165C2×C8C22×D7111yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—711no—
169C4⋊C4—712no—
170C4⋊C4—1422no—
171C4⋊C4—1422no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C4⋊C4C22×D7112yes—
175C4⋊C4C22×D7112yes—
176C22⋊C4—711no—
177C22⋊C4—711no—
178C22⋊C4—1422no—
179C22⋊C4—1422no—
180C22⋊C4—1422no—
181C42—1422no—
182C42—1422no—
183C42C22×D7111yescharacteristic
184C23—711no—
185C23—711no—
186Q8—1422no—
187Q8—1422no—
188D4—1422no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—2844no—
192C2×C4—711no—
193C2×C4—711no—
194C2×C4—1422no—
195C2×C4—1422no—
196C2×C4—1421no—
197C2×C4—1421no—
198C2×C4—1421no—
199C2×C4—1421no—
200C2×C4D4×D7111yescharacteristic
201C2×C4D4×D7111yescharacteristic
202C2×C4C23×D7111yescharacteristic
203C8—21no—
204C8—1421no—
205C8—1421no—
206C8#1176586612yes—
207C22—712no—
208C22—1421no—
209C22—1421no—
210C22—1421no—
211C22—1421no—
212C22—1421no—
213C22C2×D4×D7111yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—42no—
218C4—2842no—
219C4—2842no—
220C4—1421no—
221C4—1421no—
222C4C2×#1176586611yescharacteristic
223C4#1455617611yescharacteristic
224C2—2841no—
225C2—1422no—
226C2#1798665211yescharacteristic
227C2#135×D7111yescharacteristic
228C2#1798681311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#499×C71C2
socleC22×C71C22×D4
hypercenter#499D71
hypocenterC71#3336
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
2. #17986343C2semidirect
3. #17986331C2semidirect
4. #17986303C2semidirect
5. #17986241C2semidirect
6. #17986241C2semidirect
7. C4×Dic284C2semidirect
9. #17986036C2semidirect
10. #107×D71C2semidirect
11. #499×C71C2semidirect
115. C71#3336semidirect
117. #499D71semidirect