Finite Group Databasebeta

← #22601084

Subgroups — #22601084

Order 9088.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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
1Q8×Dic142C211yesmaximal, characteristic, complemented
2#17986723C211yesmaximal, characteristic
3#17986360C212yesmaximal
4#17986349C211yesmaximal, characteristic, complemented
5#17986290C211yesmaximal, characteristic
6#17986282C212yesmaximal, complemented
7#17986223C212yesmaximal
8#17986126C211yesmaximal, characteristic, complemented
9#17986106C212yesmaximal, complemented
10C4×#11765852C211yesmaximal, characteristic
11#462×C71C211yesmaximal, characteristic, complemented
12C2×#11765852C2211yescharacteristic
13#14556128—21no—
14#14556128C2211yescharacteristic
15#14556127—22no—
16#14556127C2212yes—
17#14556123—22no—
18#14556117C2212yes—
19#14556108—21no—
20Q8×Dic71C2212yes—
21Q8×Dic71C2212yes—
22C4×Dic142—21no—
23C4×Dic142C2211yescharacteristic
24C4×D284C2211yescharacteristic
25#14556086—21no—
26#14556060C2212yes—
27#14556059C2211yescharacteristic
28Dic71⋊C8C2212yes—
29#14556052C2211yescharacteristic
30#14556052C2212yes—
31#14556052C2212yes—
32#14556050C2212yes—
33#14556048C2211yescharacteristic
34#14556038C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C2×Q16×C71C2212yes—
37C4⋊Q8×C71C2211yescharacteristic
38C4×Q8×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142—21no—
42C2×Dic142—22no—
43C2×Dic142D411yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C4×D71—21no—
46#11765852—22no—
47C2×C71⋊C8C2311yescharacteristic
48C2×C71⋊C8C2311yescharacteristic
49C4⋊Dic71—21no—
50C4⋊Dic71—22no—
51C4⋊Dic71—22no—
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71C2312yes—
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—22no—
56Dic71⋊C4D412yes—
57(C2×D71)⋊C4—21no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60C4×Dic71—22no—
61C4×Dic71—22no—
62C4×Dic71C2312yes—
63C2×Q8×C71C2312yes—
64C2×Q8×C71C2311yescharacteristic
65Q16×C71—24no—
66C2×C8×C71C2312yes—
67C4⋊C4×C71—21no—
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71C2311yescharacteristic
71C42×C71—21no—
72C42×C71C2311yescharacteristic
73C22×D71—21no—
74Q8×C71—21no—
75Q8×C71—22no—
76Q8×C71—22no—
77Q8×C71#4112yes—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2411yescharacteristic
84C2×Dic71—21no—
85C2×Dic71—22no—
86C2×Dic71C2×D412yes—
87C2×Dic71C2×D412yes—
88D284—21no—
89D284—21no—
90C4×D71—42no—
91Dic142—22no—
92Dic142—82no—
93Dic142—81no—
94C8×C71—22no—
95C71⋊C8—21no—
96C71⋊C8—21no—
97C22×C71C22×D411yescharacteristic
98C2×D71—41no—
99C2×D71—41no—
100C2×D71—41no—
101C4×C71—22no—
102C4×C71—21no—
103C4×C71—42no—
104C4×C71—41no—
105C4×C71#13712yes—
106C4×C71C2×#4111yescharacteristic
107C4×C712−1+411yescharacteristic
108Dic71—22no—
109Dic71—22no—
110Dic71—41no—
111Dic71—42no—
112C2×C71#54811yescharacteristic
113C2×C71C2×#13711yescharacteristic
114C2×C71#57711yescharacteristic
115D71—81no—
116#3369—711nomaximal, Sylow
117C71#336911yescharacteristic, complemented, Sylow
118Q82—711no—
119#556—711no—
120#489—712no—
121#487—711no—
122#476—711no—
123#475—712no—
124#464—712no—
125#462D7111yescharacteristic, complemented
126#446—711no—
127#441—712no—
128C4×SD16—711no—
129C2×Q16C2×D7112yes—
130C2×SD16—711no—
131C4⋊Q8—712no—
132C4⋊Q8—711no—
133C4⋊Q8—1421no—
134C4⋊Q8—1422no—
135C4⋊Q8C2×D7111yescharacteristic
136#126—1422no—
137#124—712no—
138C22⋊Q8—1421no—
139C4×Q8—711no—
140C4×Q8—712no—
141C4×Q8—712no—
142C4×Q8—1421no—
143C4×Q8C2×D7111yescharacteristic
144C4×D4—711no—
145#117—1421no—
146#106—712no—
147#106—711no—
148C4⋊C8—712no—
149C4⋊C8C2×D7111yescharacteristic
150#103—711no—
151#103—712no—
152#103—712no—
153#103C2×D7112yes—
154#102—712no—
155#102—711no—
156#97—712no—
157C4×C8—711no—
158C2×Q8—711no—
159C2×Q8—1422no—
160C2×Q8—1421no—
161C2×Q8C22×D7112yes—
162C2×Q8C22×D7111yescharacteristic
163C2×D4—711no—
164C22×C4—1421no—
165Q16—24no—
166SD16—1422no—
167C2×C8—711no—
168C2×C8—711no—
169C2×C8C22×D7112yes—
170C4⋊C4—21no—
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—1421no—
181C4⋊C4C22×D7111yescharacteristic
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C22⋊C4—1421no—
185C42—21no—
186C42—712no—
187C42—1422no—
188C42—1422no—
189C42C22×D7111yescharacteristic
190C23—1421no—
191Q8—21no—
192Q8—22no—
193Q8—22no—
194Q8—5681no—
195Q8—5682no—
196Q8—1422no—
197Q8#1176586412yes—
198D4—1421no—
199D4—1421no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—2842no—
204C2×C4—712no—
205C2×C4—712no—
206C2×C4—1421no—
207C2×C4—1422no—
208C2×C4D4×D7111yescharacteristic
209C2×C4D4×D7111yescharacteristic
210C2×C4C23×D7111yescharacteristic
211C8—22no—
212C8—1421no—
213C8—1421no—
214C22—2841no—
215C22—2841no—
216C22—2841no—
217C22C2×D4×D7111yescharacteristic
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—2841no—
223C4—2842no—
224C4—1422no—
225C4—1422no—
226C4#1455616012yes—
227C4C2×#1176586411yescharacteristic
228C4#1455617511yescharacteristic
229C2—5681no—
230C2#1798665511yescharacteristic
231C2C2×#1455616011yescharacteristic
232C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#462×C71C2
socleC22×C71C22×D4
hypercenter#462D71
hypocenterC71#3369
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. Q8×Dic142C2semidirect
4. #17986349C2semidirect
6. #17986282C2semidirect
8. #17986126C2semidirect
9. #17986106C2semidirect
11. #462×C71C2semidirect
117. C71#3369semidirect
125. #462D71semidirect