Finite Group Databasebeta

← #22599061

Subgroups — #22599061

Order 9088.

402 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986554C211yesmaximal, characteristic, complemented
2#17986550C211yesmaximal, characteristic
3#17986402C211yesmaximal, characteristic, complemented
4#17986373C211yesmaximal, characteristic, complemented
5#17986346C211yesmaximal, characteristic, complemented
6#17986248C214yesmaximal, complemented
7#17986102C214yesmaximal, complemented
8C4×#11765848C211yesmaximal, characteristic, complemented
9#487×C71C211yesmaximal, characteristic, complemented
10C2×#11765866—21no—
11C2×#11765852C2214yes—
12C2×#11765848C2212yes—
13#14556128C2211yescharacteristic
14#14556125C2211yescharacteristic
15#14556119—21no—
16#14556117—21no—
17#14556108—21no—
18#14556104C2212yes—
19#14556103—21no—
20#14556103C2212yes—
21Q8×Dic71C2212yes—
22#14556093—21no—
23#14556090—21no—
24#14556090C2212yes—
25#14556086—21no—
26C42×D71C2211yescharacteristic
27#14556061C2214yes—
28#14556050C2214yes—
29#14556038C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#103×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35#11765866—44no—
36C2×Dic142—21no—
37C2×D284—22no—
38C2×D284—21no—
39C2×D284—21no—
40C2×D284C2312yes—
41C2×C4×D71—22no—
42C2×C4×D71D411yescharacteristic
43C2×C4×D71D411yescharacteristic
44C2×C4×D71C2311yescharacteristic
45#11765852—28no—
46#11765848—24no—
47C2×C71⋊C8C2312yes—
48C4⋊Dic71—21no—
49C4⋊Dic71—22no—
50C4⋊Dic71C2312yes—
51Dic71⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71—22no—
56C4×Dic71D411yescharacteristic
57C4×Dic71D411yescharacteristic
58C4×Dic71C2311yescharacteristic
59C2×Q8×C71—21no—
60C2×Q8×C71C2312yes—
61C2×C8×C71C2312yes—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71—21no—
65C4⋊C4×C71C2312yes—
66C42×C71C2311yescharacteristic
67C22×D71—22no—
68C22×D71C2×D411yescharacteristic
69Q8×C71—24no—
70Q8×C71—42no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2411yescharacteristic
76C2×Dic71—22no—
77C2×Dic71C2×D411yescharacteristic
78C2×Dic71C2×D411yescharacteristic
79C2×Dic71C2×D411yescharacteristic
80D284—24no—
81D284—44no—
82D284—42no—
83D284—42no—
84C4×D71—22no—
85C4×D71—22no—
86C4×D71—22no—
87C4×D71—44no—
88Dic142—42no—
89C8×C71—22no—
90C71⋊C8#4114yescomplemented
91C22×C71C22×D411yescharacteristic
92C2×D71—21no—
93C2×D71—21no—
94C2×D71—21no—
95C2×D71—42no—
96C2×D71—42no—
97C2×D71—42no—
98C4×C71—42no—
99C4×C71—42no—
100C4×C71AGL(1,ℤ/8)12yes—
101C4×C71#13712yes—
102C4×C71C2×#4112yes—
103Dic71—21no—
104Dic71—21no—
105Dic71—21no—
106Dic71—42no—
107C2×C71#53211yescharacteristic
108C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C71C2×#13711yescharacteristic
110D71—41no—
111D71—82no—
112#3182—711nomaximal, Sylow
113C71#318211yescharacteristic, complemented, Sylow
114#533—711no—
115#532—711no—
116#500—711no—
117#492—711no—
118#487D7111yescharacteristic, complemented
119#486—711no—
120#468—714no—
121#440—714no—
122C4×M16—711no—
123C2×#41—1421no—
124C2×SD16—714no—
125C2×M16—712no—
126C4⋊Q8—711no—
127C4⋊Q8C2×D7111yescharacteristic
128C4⋊Q8C2×D7111yescharacteristic
129#127—711no—
130#125—1421no—
131#124—1421no—
132C22⋊Q8—712no—
133C22⋊Q8—1421no—
134#121—712no—
135#121—1421no—
136C4×Q8—712no—
137C4×Q8—1421no—
138C4×D4—712no—
139C4×D4—1421no—
140#117—1421no—
141C2×C42—711no—
142#107—714no—
143#103C2×D7114yes—
144#102—714no—
145#97—712no—
146C4×C8—711no—
147C4×C8C2×D7111yescharacteristic
148#41—2844no—
149C2×Q8—21no—
150C2×Q8—1421no—
151C2×Q8C22×D7112yes—
152C2×D4—712no—
153C2×D4—1422no—
154C2×D4—1421no—
155C2×D4—1421no—
156C22×C4—711no—
157C22×C4—711no—
158C22×C4—711no—
159C22×C4—1422no—
160SD16—1428no—
161M16—1424no—
162C2×C8—712no—
163C2×C8C22×D7112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—712no—
168C4⋊C4—1422no—
169C4⋊C4—1422no—
170C4⋊C4—1421no—
171C4⋊C4C22×D7112yes—
172C22⋊C4—1422no—
173C22⋊C4—1422no—
174C22⋊C4—1422no—
175C42—711no—
176C42—711no—
177C42—711no—
178C42—1422no—
179C42C22×D7111yescharacteristic
180C23—711no—
181C23—1422no—
182Q8—24no—
183Q8—42no—
184Q8—2842no—
185D4—2844no—
186D4—2842no—
187D4—2842no—
188D4—1424no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—2844no—
192C2×C4—711no—
193C2×C4—711no—
194C2×C4—711no—
195C2×C4—1422no—
196C2×C4—1422no—
197C2×C4—1422no—
198C2×C4—1422no—
199C2×C4D4×D7111yescharacteristic
200C2×C4D4×D7111yescharacteristic
201C2×C4C23×D7111yescharacteristic
202C8—22no—
203C8—714no—
204C22—2842no—
205C22—2842no—
206C22—2842no—
207C22—1421no—
208C22—1421no—
209C22—1421no—
210C22C2×D4×D7111yescharacteristic
211C4—42no—
212C4—42no—
213C4—2842no—
214C4—1421no—
215C4—1421no—
216C4—1421no—
217C4#1455615512yes—
218C4#1455616012yes—
219C4#41×D7112yes—
220C2—2841no—
221C2—5682no—
222C2#1798654811yescharacteristic
223C2C2×#1455615511yescharacteristic
224C2C2×#1455616011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3182
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986402C2semidirect
4. #17986373C2semidirect
5. #17986346C2semidirect
6. #17986248C2semidirect
7. #17986102C2semidirect
8. C4×#11765848C2semidirect
9. #487×C71C2semidirect
90. C71⋊C8#41semidirect
113. C71#3182semidirect
118. #487D71semidirect