Finite Group Databasebeta

← #22599232

Subgroups — #22599232

Order 9088.

378 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986572C212yesmaximal
2#17986343C212yesmaximal, complemented
3#17986338C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic, complemented
5#17986238C212yesmaximal, complemented
6#17986156C212yesmaximal, complemented
7#17986153C212yesmaximal, complemented
8#17985967C211yesmaximal, characteristic
9#470×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C2×Q8×D71C2212yes—
12C2×Dic284C2211yescharacteristic
13C2×#11765855C2212yes—
14C22×C71⋊C8C2211yescharacteristic
15#14556127—22no—
16#14556114—22no—
17#14556113—22no—
18#14556110—22no—
19#14556108C2212yes—
20#14556107C2212yes—
21#14556104C2212yes—
22#14556086C2212yes—
23#14556061C2212yes—
24#14556060C2211yescharacteristic
25#14556053C2212yes—
26#14556052C2212yes—
27#14556051C2212yes—
28(C2×D71)⋊C8C2212yes—
29(C2×D71)⋊C8C2212yes—
30C2×Q16×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#106×C71C2211yescharacteristic
34#103×C71C2212yes—
35#11765865—41no—
36#11765865—41no—
37C2×Dic142—22no—
38C2×Dic142D411yescharacteristic
39C2×Dic142C2312yes—
40Q8×D71—42no—
41Q8×D71—42no—
42C2×D284D411yescharacteristic
43C2×C22⋊D71D412yes—
44C2×C4×D71—22no—
45C2×C4×D71C2312yes—
46Dic284—41no—
47#11765855—42no—
48C2×C71⋊C8—41no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—22no—
51C4⋊Dic71C2312yes—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58C4×Dic71—22no—
59C2×Q8×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61Q16×C71—41no—
62M16×C71—41no—
63C2×C8×C71C2311yescharacteristic
64C2×C8×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C23×C71C2×D411yescharacteristic
69C22×D71C2×D412yes—
70Q8×C71—22no—
71Q8×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—42no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71C2×D412yes—
82D284—41no—
83C4×D71—22no—
84C4×D71—22no—
85C4×D71—44no—
86Dic142—22no—
87Dic142—22no—
88Dic142—41no—
89Dic142—44no—
90C8×C71—21no—
91C8×C71—21no—
92C71⋊C8—22no—
93C22×C71—21no—
94C22×C71—22no—
95C22×C71C22×D411yescharacteristic
96C2×D71—22no—
97C2×D71—24no—
98C4×C71—21no—
99C4×C71—42no—
100C4×C71—42no—
101C4×C71H(5,𝔽2)11yescharacteristic
102C4×C71H(5,𝔽2)11yescharacteristic
103Dic71—22no—
104Dic71—42no—
105Dic71—42no—
106C2×C71—41no—
107C2×C71#53411yescharacteristic
108C2×C71#57812yes—
109D71—42no—
110#3202—711nomaximal, Sylow
111C71#320211yescharacteristic, complemented, Sylow
112#536—712no—
113#485—712no—
114#485—712no—
115#470—711no—
116#470D7111yescharacteristic, complemented
117#467—712no—
118#451—712no—
119#451—712no—
120#408—711no—
121C2×#41—711no—
122C22×Q8—712no—
123C2×Q16—712no—
124C2×Q16—711no—
125C2×Q16C2×D7111yescharacteristic
126C2×M16C2×D7111yescharacteristic
127C22×C8—711no—
128C4⋊Q8—1422no—
129#124—1422no—
130#123—1422no—
131#123—1422no—
132C22⋊Q8—712no—
133C22⋊Q8—712no—
134C22⋊Q8—712no—
135C22⋊Q8C2×D7112yes—
136#117—712no—
137#107—712no—
138#106—711no—
139#106C2×D7111yescharacteristic
140#103—712no—
141#103—712no—
142#103—712no—
143#103C2×D7112yes—
144C22⋊C8—712no—
145C22⋊C8—712no—
146#41—2841no—
147#41—2841no—
148C2×Q8—2842no—
149C2×Q8—2842no—
150C2×Q8—711no—
151C2×Q8—712no—
152C2×Q8—1422no—
153C2×Q8C22×D7112yes—
154C2×D4—712no—
155C2×D4—711no—
156C22×C4—712no—
157C22×C4—1422no—
158C22×C4C22×D7111yescharacteristic
159Q16—41no—
160Q16—2841no—
161Q16—2842no—
162M16—41no—
163C2×C8—2841no—
164C2×C8—712no—
165C2×C8C22×D7111yescharacteristic
166C2×C8C22×D7111yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—712no—
169C4⋊C4—1422no—
170C4⋊C4—1422no—
171C4⋊C4—1422no—
172C4⋊C4C22×D7112yes—
173C22⋊C4—22no—
174C22⋊C4—1422no—
175C22⋊C4—1422no—
176C22⋊C4—1422no—
177C22⋊C4—1422no—
178C42—1422no—
179C23—712no—
180C23D4×D7111yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—2841no—
184Q8—2844no—
185Q8—1422no—
186Q8—1422no—
187D4—2841no—
188D4—2842no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—2844no—
194C2×C4—712no—
195C2×C4—1422no—
196C2×C4—1422no—
197C2×C4—1422no—
198C2×C4—1422no—
199C2×C4D4×D7111yescharacteristic
200C2×C4C23×D7111yescharacteristic
201C8—21no—
202C8—21no—
203C8—1422no—
204C22—21no—
205C22—22no—
206C22—1422no—
207C22—1424no—
208C22C2×D4×D7111yescharacteristic
209C4—21no—
210C4—42no—
211C4—42no—
212C4—2842no—
213C4—2842no—
214C4—1422no—
215C4#1455617311yescharacteristic
216C4#1455617411yescharacteristic
217C2—41no—
218C2—2842no—
219C2#1798655711yescharacteristic
220C2#1798681512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#470×C71C2
socleC22×C71C22×D4
hypercenter#470D71
hypocenterC71#3202
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #17986343C2semidirect
3. #17986338C2semidirect
4. #17986257C2semidirect
5. #17986238C2semidirect
6. #17986156C2semidirect
7. #17986153C2semidirect
9. #470×C71C2semidirect
111. C71#3202semidirect
116. #470D71semidirect