Finite Group Databasebeta

← #22598440

Subgroups — #22598440

Order 9088.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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×C4×D284C211yesmaximal, characteristic
3#17986335C212yesmaximal, complemented
4#17986280C212yesmaximal, complemented
5#17986205C212yesmaximal, complemented
6#17986152C212yesmaximal
7#17986101C214yesmaximal, complemented
8#432×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic
10C22×C4×D71—21no—
11C2×#11765854C2214yes—
12#14556128C2211yescharacteristic, complemented
13#14556128C2211yescharacteristic, complemented
14#14556120—21no—
15#14556107—21no—
16#14556107C2212yes—
17C4×Dic142—21no—
18C4×Dic142C2212yescomplemented
19C4×D284—21no—
20C4×D284—21no—
21C4×D284C2212yes—
22C4×D284C2212yes—
23#14556087—21no—
24C2×C4⋊Dic71C2211yescharacteristic
25C2×(C2×D71)⋊C4—21no—
26#14556062C2214yescomplemented
27#14556053C2214yescomplemented
28#14556050C2214yes—
29C2×C42×C71C2211yescharacteristic
30C4⋊C8×C71C2212yescomplemented
31C22⋊C8×C71C2212yes—
32#97×C71C2212yescomplemented
33C23×D71—21no—
34C2×Dic142—21no—
35C2×Dic142C2312yes—
36C2×D284—21no—
37C2×D284—24no—
38C2×D284C2312yes—
39C22×Dic71—21no—
40C2×C4×D71—22no—
41C2×C4×D71—22no—
42C2×C4×D71—44no—
43#11765854—28no—
44C4⋊Dic71—21no—
45C4⋊Dic71—21no—
46C4⋊Dic71—21no—
47C4⋊Dic71—21no—
48C4⋊Dic71C2312yes—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52C22⋊Dic71—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71—22no—
56C22×C4×C71D411yescharacteristic
57C22×C4×C71D411yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C2×C8×C71C2314yes—
60C42×C71D411yescharacteristic
61C42×C71D411yescharacteristic
62C42×C71C2311yescharacteristic
63C42×C71C2311yescharacteristic
64C23×C71C2×D411yescharacteristic
65C22×D71—24no—
66C22×D71—22no—
67C22×D71—42no—
68C22×D71—42no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2411yescharacteristic
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—42no—
82D284—24no—
83D284—22no—
84D284#4114yes—
85C4×D71—44no—
86C4×D71—44no—
87Dic142—24no—
88Dic142—42no—
89C8×C71—24no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71#13712yes—
93C22×C71C22×D411yescharacteristic
94C2×D71—24no—
95C2×D71—48no—
96C2×D71—42no—
97C2×D71—42no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71AGL(1,ℤ/8)12yes—
103C4×C71C2×#4112yes—
104Dic71—42no—
105Dic71—42no—
106Dic71—42no—
107C2×C71—22no—
108C2×C71#52511yescharacteristic
109C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C71C2×#13711yescharacteristic
111D71—44no—
112#3110—711nomaximal, Sylow
113C71#311011yescharacteristic, complemented, Sylow
114#533—711no—
115C2×C4×D4—711no—
116#484—712no—
117#474—712no—
118#461—712no—
119#450—712no—
120#440—714no—
121#432D7111yescharacteristic, complemented
122C22×D4—711no—
123C23×C4—1421no—
124C2×SD16—714no—
125C4⋊Q8—711no—
126C4⋊Q8—711no—
127#125—1421no—
128C22⋊Q8—712no—
129C22⋊Q8—1421no—
130C4×Q8—712no—
131C4×Q8—1421no—
132C4×D4—712no—
133C4×D4—712no—
134C4×D4—1421no—
135C4×D4—1421no—
136#117—1421no—
137C2×C4⋊C4—711no—
138C2×C22⋊C4—1421no—
139C2×C42C2×D7111yescharacteristic
140#107—714no—
141C4⋊C8C2×D7112yescomplemented
142#103—714no—
143#102—714no—
144C22⋊C8C2×D7112yes—
145#97C2×D7112yescomplemented
146C24—1421no—
147C2×Q8—712no—
148C2×Q8—1421no—
149C2×D4—712no—
150C2×D4—1424no—
151C2×D4—1421no—
152C22×C4—2844no—
153C22×C4—1421no—
154C22×C4—1422no—
155C22×C4—1422no—
156C22×C4D28411yescharacteristic
157C22×C4D28411yescharacteristic
158C22×C4C22×D7111yescharacteristic
159SD16—1428no—
160C2×C8C22×D7114yes—
161C4⋊C4—712no—
162C4⋊C4—712no—
163C4⋊C4—1422no—
164C4⋊C4—1422no—
165C4⋊C4—1421no—
166C4⋊C4—1421no—
167C4⋊C4—1421no—
168C4⋊C4—1421no—
169C22⋊C4—1422no—
170C22⋊C4—1422no—
171C22⋊C4—1422no—
172C42—1422no—
173C42D28411yescharacteristic
174C42D28411yescharacteristic
175C42C22×D7111yescharacteristic
176C42C22×D7111yescharacteristic
177C23—2842no—
178C23—2842no—
179C23—1424no—
180C23—1422no—
181C23C2×D28411yescharacteristic
182Q8—2842no—
183Q8—1424no—
184D4—714no—
185D4—1424no—
186D4—1422no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—2844no—
191C2×C4—2842no—
192C2×C4—2844no—
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—2848no—
206C22—2842no—
207C22—2842no—
208C22—1424no—
209C22#1455616112yes—
210C22C22×D28411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—2842no—
216C4—2842no—
217C4—2842no—
218C4#1455615612yes—
219C4#41×D7112yes—
220C2—22no—
221C2—2844no—
222C2#1798651111yescharacteristic
223C2C2×#1455615611yescharacteristic
224C2C2×#1455616111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3110
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. #17986335C2semidirect
4. #17986280C2semidirect
5. #17986205C2semidirect
7. #17986101C2semidirect
8. #432×C71C2semidirect
12. #14556128C22semidirect
13. #14556128C22semidirect
18. C4×Dic142C22semidirect
26. #14556062C22semidirect
27. #14556053C22semidirect
30. C4⋊C8×C71C22semidirect
32. #97×C71C22semidirect
113. C71#3110semidirect
121. #432D71semidirect
141. C4⋊C8C2×D71semidirect
145. #97C2×D71semidirect