Finite Group Databasebeta

← #22601260

Subgroups — #22601260

Order 9088.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986728C211yesmaximal, characteristic, complemented
2#17986664C211yesmaximal, characteristic
3#17986406C212yesmaximal, complemented
4#17986402C211yesmaximal, characteristic, complemented
5#17986298C212yesmaximal
6#17986292C212yesmaximal, complemented
7#17986280C211yesmaximal, characteristic
8#17986126C211yesmaximal, characteristic, complemented
9#17986102C212yesmaximal
10C4×#11765852C211yesmaximal, characteristic
11#475×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13C2×#11765852C2211yescharacteristic
14#14556128—21no—
15#14556128C2211yescharacteristic
16#14556127C2212yes—
17#14556119—22no—
18#14556119C2212yes—
19#14556106—22no—
20#14556104—21no—
21Q8×Dic71C2212yes—
22C4×Dic142—21no—
23C4×Dic142C2211yescharacteristic
24C4×D284C2211yescharacteristic
25#14556090C2212yes—
26#14556062C2212yes—
27#14556061C2212yes—
28#14556061C2212yes—
29#14556059C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556052C2211yescharacteristic
32#14556050C2212yes—
33#14556048C2211yescharacteristic
34#14556038C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C4⋊Q8×C71C2211yescharacteristic
37C4×Q8×C71C2211yescharacteristic
38#106×C71C2212yes—
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142—21no—
42C2×Dic142—22no—
43C2×Dic142D411yescharacteristic
44Q8×D71—44no—
45C2×D284C2311yescharacteristic
46C2×C4×D71—21no—
47C2×C4×D71—22no—
48#11765852Q814yescomplemented
49C2×C71⋊C8C2311yescharacteristic
50C2×C71⋊C8C2311yescharacteristic
51C4⋊Dic71—21no—
52C4⋊Dic71—22no—
53C4⋊Dic71—22no—
54C4⋊Dic71D411yescharacteristic
55C4⋊Dic71C2312yes—
56C4⋊Dic71C2311yescharacteristic
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4D412yes—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—22no—
62C4×Dic71—22no—
63C4×Dic71C2312yes—
64C2×Q8×C71—21no—
65C2×Q8×C71C2311yescharacteristic
66C2×C8×C71C2312yes—
67C4⋊C4×C71—21no—
68C4⋊C4×C71—21no—
69C4⋊C4×C71C2312yes—
70C4⋊C4×C71C2311yescharacteristic
71C42×C71—21no—
72C42×C71C2311yescharacteristic
73C22×D71—21no—
74Q8×C71—21no—
75Q8×C71—42no—
76Q8×C71C2×Q812yescomplemented
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C2×Dic71—22no—
84C2×Dic71—21no—
85C2×Dic71C2×D412yes—
86C2×Dic71C2×D412yes—
87D284—21no—
88D284C2×Q812yes—
89C4×D71—44no—
90C4×D71—42no—
91Dic142—22no—
92Dic142—44no—
93Dic142—42no—
94C8×C71—22no—
95C71⋊C8—21no—
96C71⋊C8C2×Q812yescomplemented
97C22×C71C22×D411yescharacteristic
98C2×D71—22no—
99C2×D71—41no—
100C2×D71—41no—
101C4×C71—22no—
102C4×C71—21no—
103C4×C71—42no—
104C4×C71—41no—
105C4×C71AGL(1,ℤ/8)12yes—
106C4×C71C22×Q811yescharacteristic
107C4×C712−1+411yescharacteristic
108Dic71—22no—
109Dic71—22no—
110Dic71—41no—
111Dic71—42no—
112C2×C71D4×Q811yescharacteristic
113C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C71#57811yescharacteristic
115D71—42no—
116#3386—711nomaximal, Sylow
117C71#338611yescharacteristic, complemented, Sylow
118#557—711no—
119D4×Q8—711no—
120#501—712no—
121#500—711no—
122#478—712no—
123#477—712no—
124#475D7111yescharacteristic, complemented
125#474—711no—
126#446—711no—
127#440—712no—
128C4×SD16—711no—
129C22×Q8—1421no—
130C2×SD16—711no—
131C4⋊Q8—712no—
132C4⋊Q8—711no—
133C4⋊Q8—1421no—
134C4⋊Q8C2×D7111yescharacteristic
135#125—712no—
136#125—1422no—
137C22⋊Q8—1422no—
138C22⋊Q8—1421no—
139C4×Q8—711no—
140C4×Q8—712no—
141C4×Q8—1421no—
142C4×Q8C2×D7111yescharacteristic
143C4×D4—712no—
144C4×D4—711no—
145#107—712no—
146#107—712no—
147#107—712no—
148#106—711no—
149#106C2×D7112yes—
150C4⋊C8—712no—
151C4⋊C8C2×D7111yescharacteristic
152#103—711no—
153#103C2×D7112yes—
154#102—712no—
155#102—711no—
156#97—712no—
157C4×C8—711no—
158C2×Q8—21no—
159C2×Q8—2844no—
160C2×Q8—711no—
161C2×Q8—1422no—
162C2×Q8—1421no—
163C2×Q8C22×D7111yescharacteristic
164C2×D4—711no—
165C22×C4—1422no—
166C22×C4—1421no—
167SD16—714no—
168C2×C8—711no—
169C2×C8—711no—
170C2×C8C22×D7112yes—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—712no—
174C4⋊C4—711no—
175C4⋊C4—712no—
176C4⋊C4—711no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1421no—
182C4⋊C4C22×D7111yescharacteristic
183C4⋊C4C22×D7112yes—
184C22⋊C4—1422no—
185C22⋊C4—1421no—
186C42—21no—
187C42—712no—
188C42—1422no—
189C42C22×D7111yescharacteristic
190C23—1421no—
191Q8—21no—
192Q8—42no—
193Q8—2844no—
194Q8—2842no—
195Q8—1422no—
196Q8Q8×D7112yescomplemented
197D4—712no—
198D4—1421no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—2844no—
203C2×C4—2842no—
204C2×C4—712no—
205C2×C4—712no—
206C2×C4—1422no—
207C2×C4—1421no—
208C2×C4D4×D7111yescharacteristic
209C2×C4D4×D7111yescharacteristic
210C2×C4C23×D7111yescharacteristic
211C8—22no—
212C8—712no—
213C8—1421no—
214C22—2841no—
215C22—2841no—
216C22—1422no—
217C22C2×D4×D7111yescharacteristic
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—2841no—
223C4—2842no—
224C4—1422no—
225C4—1422no—
226C4#1455615512yes—
227C4C2×Q8×D7111yescharacteristic
228C4#1455617511yescharacteristic
229C2—2842no—
230C2#1798666211yescharacteristic
231C2C2×#1455615511yescharacteristic
232C2#1798681511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#475×C71C2
socleC22×C71C22×D4
hypercenter#475D71
hypocenterC71#3386
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #17986728C2semidirect
3. #17986406C2semidirect
4. #17986402C2semidirect
6. #17986292C2semidirect
8. #17986126C2semidirect
11. #475×C71C2semidirect
48. #11765852Q8semidirect
76. Q8×C71C2×Q8semidirect
96. C71⋊C8C2×Q8semidirect
117. C71#3386semidirect
124. #475D71semidirect
196. Q8Q8×D71semidirect