Finite Group Databasebeta

← #22601370

Subgroups — #22601370

Order 9088.

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

normal subgroups only

This group has 350 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#17986727C211yesmaximal, characteristic, complemented
2#17986723C211yesmaximal, characteristic
3#17986392C212yesmaximal
4#17986378C211yesmaximal, characteristic, complemented
5#17986224C212yesmaximal
6#17986207C212yesmaximal, complemented
7#17986205C211yesmaximal, characteristic
8#17986126C211yesmaximal, characteristic, complemented
9#17986103C212yesmaximal, complemented
10C4×#11765852C211yesmaximal, characteristic
11#462×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2212yes—
13C2×#11765855C2212yes—
14C2×#11765854C2212yes—
15C2×#11765852C2211yescharacteristic
16#14556128C2211yescharacteristic
17#14556126C2212yes—
18#14556123—22no—
19#14556120—21no—
20#14556118—22no—
21#14556117C2212yes—
22#14556108—21no—
23Q8×Dic71C2212yes—
24C4×Dic142—21no—
25C4×Dic142C2211yescharacteristic
26#14556093C2212yes—
27C4×D284C2211yescharacteristic
28#14556086—21no—
29#14556059C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2212yes—
32#14556052C2211yescharacteristic
33#14556048C2211yescharacteristic
34#14556038C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C2×Q16×C71C2212yes—
37C4⋊Q8×C71C2211yescharacteristic
38C4×Q8×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142D411yescharacteristic
42C2×Dic142C2312yes—
43C2×D284C2311yescharacteristic
44C2×C4×D71—21no—
45#11765855—24no—
46#11765855—42no—
47#11765854—42no—
48#11765852—22no—
49C2×C71⋊C8C2311yescharacteristic
50C2×C71⋊C8C2311yescharacteristic
51C4⋊Dic71—21no—
52C4⋊Dic71—21no—
53C4⋊Dic71—22no—
54C4⋊Dic71D411yescharacteristic
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4D412yes—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63C4×Dic71—22no—
64C4×Dic71—22no—
65C4×Dic71C2312yes—
66C2×Q8×C71C2312yes—
67C2×Q8×C71C2311yescharacteristic
68Q16×C71—42no—
69C2×C8×C71C2312yes—
70C4⋊C4×C71—21no—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2311yescharacteristic
74C42×C71—21no—
75C42×C71C2311yescharacteristic
76C22×D71—21no—
77Q8×C71—21no—
78Q8×C71—22no—
79Q8×C71—21no—
80Q8×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—21no—
88C2×Dic71—22no—
89C2×Dic71C2×D412yes—
90C2×Dic71C2×D412yes—
91D284—21no—
92D284—21no—
93C4×D71—42no—
94Dic142—22no—
95Dic142—22no—
96Dic142—22no—
97C8×C71—22no—
98C71⋊C8—21no—
99C71⋊C8#4112yes—
100C22×C71C22×D411yescharacteristic
101C2×D71—41no—
102C2×D71—41no—
103C2×D71—41no—
104C4×C71—21no—
105C4×C71—41no—
106C4×C71—42no—
107C4×C71—41no—
108C4×C71#13712yes—
109C4×C71C2×#4111yescharacteristic
110C4×C71H(5,𝔽2)11yescharacteristic
111Dic71—22no—
112Dic71—22no—
113Dic71—42no—
114Dic71—41no—
115C2×C71#55011yescharacteristic
116C2×C71C2×#13711yescharacteristic
117C2×C71#57811yescharacteristic
118D71—81no—
119#3398—711nomaximal, Sylow
120C71#339811yescharacteristic, complemented, Sylow
121#557—711no—
122#556—711no—
123#497—712no—
124#494—711no—
125#464—712no—
126#462—712no—
127#462D7111yescharacteristic, complemented
128#461—711no—
129#446—711no—
130#441—712no—
131C4×SD16—711no—
132C2×Q16—712no—
133C2×Q16—712no—
134C2×Q16C2×D7112yes—
135C2×SD16—711no—
136C2×SD16—712no—
137C4⋊Q8—712no—
138C4⋊Q8—711no—
139C4⋊Q8C2×D7111yescharacteristic
140#126—1422no—
141#125—1421no—
142#125—1422no—
143#124—712no—
144C22⋊Q8—1421no—
145C4×Q8—711no—
146C4×Q8—712no—
147C4×Q8—712no—
148C4×Q8—1421no—
149C4×Q8C2×D7111yescharacteristic
150C4×D4—711no—
151#117—1421no—
152#106—711no—
153C4⋊C8—712no—
154C4⋊C8C2×D7111yescharacteristic
155#103—711no—
156#103—712no—
157#103C2×D7112yes—
158#102—711no—
159#97—712no—
160C4×C8—711no—
161C2×Q8—711no—
162C2×Q8—712no—
163C2×Q8C22×D7111yescharacteristic
164C2×Q8C22×D7112yes—
165C2×D4—711no—
166C22×C4—1421no—
167Q16—42no—
168Q16—2842no—
169Q16—1424no—
170SD16—2842no—
171SD16—1422no—
172C2×C8—711no—
173C2×C8—711no—
174C2×C8C22×D7112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—712no—
179C4⋊C4—711no—
180C4⋊C4—711no—
181C4⋊C4—1422no—
182C4⋊C4—1422no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C4⋊C4—1421no—
186C4⋊C4—1421no—
187C4⋊C4C22×D7111yescharacteristic
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C22⋊C4—1421no—
191C42—21no—
192C42—712no—
193C42—1422no—
194C42—1422no—
195C42C22×D7111yescharacteristic
196C23—1421no—
197Q8—21no—
198Q8—22no—
199Q8—21no—
200Q8—22no—
201Q8—1422no—
202Q8—1422no—
203Q8—1422no—
204D4—1421no—
205D4—1421no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—2842no—
210C2×C4—712no—
211C2×C4—712no—
212C2×C4—1421no—
213C2×C4—1422no—
214C2×C4D4×D7111yescharacteristic
215C2×C4D4×D7111yescharacteristic
216C2×C4C23×D7111yescharacteristic
217C8—22no—
218C8—712no—
219C8—1421no—
220C22—2841no—
221C22—2841no—
222C22—2841no—
223C22C2×D4×D7111yescharacteristic
224C4—21no—
225C4—41no—
226C4—42no—
227C4—41no—
228C4—2842no—
229C4—2841no—
230C4—1422no—
231C4—1422no—
232C4#1455616012yes—
233C4#41×D7111yescharacteristic
234C4#1455617311yescharacteristic
235C2—5681no—
236C2#1798666711yescharacteristic
237C2C2×#1455616011yescharacteristic
238C2#1798681511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#462×C71C2
socleC22×C71C22×D4
hypercenter#462D71
hypocenterC71#3398
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. #17986727C2semidirect
4. #17986378C2semidirect
6. #17986207C2semidirect
8. #17986126C2semidirect
9. #17986103C2semidirect
11. #462×C71C2semidirect
120. C71#3398semidirect
127. #462D71semidirect