Finite Group Databasebeta

← #22599251

Subgroups — #22599251

Order 9088.

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

normal subgroups only

This group has 402 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#17986578C212yesmaximal, complemented
2#17986331C212yesmaximal, complemented
3#17986316C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic, complemented
5#17986226C212yesmaximal, complemented
6#17986162C212yesmaximal, complemented
7#17986141C212yesmaximal, complemented
8#17985967C211yesmaximal, characteristic, complemented
9#469×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C2×#11765864C2212yes—
12C2×Dic284C2211yescharacteristic
13C2×#11765853C2212yes—
14C22×C71⋊C8C2211yescharacteristic
15#14556108C2212yes—
16#14556107C2212yes—
17#14556105—22no—
18#14556101—22no—
19#14556100—22no—
20#14556099—22no—
21#14556099C2212yes—
22C4⋊C4×D71C2212yes—
23#14556060C2211yescharacteristic
24#14556059C2212yes—
25#14556053C2212yes—
26#14556051C2212yes—
27#14556049C2212yes—
28(C2×D71)⋊C8C2212yes—
29(C2×D71)⋊C8C2212yes—
30C2×D8×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32#121×C71C2212yes—
33#106×C71C2211yescharacteristic
34#102×C71C2212yes—
35#11765865—41no—
36#11765865—41no—
37#11765864—42no—
38#11765864—42no—
39C2×Dic142D411yescharacteristic
40C2×Dic142C2312yes—
41C2×D284D411yescharacteristic
42C2×C22⋊D71—22no—
43C2×C22⋊D71—22no—
44C2×C22⋊D71D412yes—
45C22×Dic71—22no—
46C2×C4×D71—22no—
47C2×C4×D71C2312yes—
48Dic284—41no—
49#11765853—42no—
50C2×C71⋊C8—41no—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71—22no—
53C4⋊Dic71C2312yes—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56C22⋊Dic71—22no—
57C22⋊Dic71—22no—
58(C2×D71)⋊C4—22no—
59C2×D4×C71—22no—
60C2×D4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62D8×C71—41no—
63M16×C71—41no—
64C2×C8×C71C2311yescharacteristic
65C2×C8×C71C2311yescharacteristic
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C23×C71—22no—
69C23×C71C2×D411yescharacteristic
70C22×D71C2×D412yes—
71D4×C71—22no—
72D4×C71—22no—
73D4×C71—82no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2411yescharacteristic
79C22⋊D71—42no—
80C22⋊D71—44no—
81C22⋊D71—82no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—44no—
85C2×Dic71C2×D412yes—
86D284—41no—
87C4×D71—22no—
88C4×D71—22no—
89C4×D71—82no—
90Dic142—22no—
91Dic142—22no—
92Dic142—41no—
93C8×C71—21no—
94C8×C71—21no—
95C71⋊C8—22no—
96C22×C71—21no—
97C22×C71—22no—
98C22×C71—42no—
99C22×C71—42no—
100C22×C71—42no—
101C22×C71C22×D411yescharacteristic
102C2×D71—22no—
103C2×D71—22no—
104C2×D71—22no—
105C4×C71—21no—
106C4×C71—42no—
107C4×C71H(5,𝔽2)11yescharacteristic
108C4×C71H(5,𝔽2)11yescharacteristic
109Dic71—22no—
110Dic71—42no—
111Dic71—42no—
112C2×C71—41no—
113C2×C71—82no—
114C2×C71#53411yescharacteristic
115C2×C71#57712yes—
116D71—42no—
117#3204—711nomaximal, Sylow
118C71#320411yescharacteristic, complemented, Sylow
119#537—712no—
120#483—712no—
121#481—712no—
122#470—711no—
123#469D7111yescharacteristic, complemented
124#465—712no—
125#452—712no—
126#449—712no—
127#408—711no—
128C2×#41—711no—
129C2×#41—712no—
130C2×Q16—711no—
131C2×SD16—712no—
132C2×D8C2×D7111yescharacteristic
133C2×M16C2×D7111yescharacteristic
134C22×C8—711no—
135C22⋊Q8—712no—
136C22⋊Q8—712no—
137C22⋊Q8—1422no—
138#121—712no—
139#121—1422no—
140#121—1422no—
141#121—1422no—
142#121C2×D7112yes—
143C2×C4⋊C4—712no—
144#106—711no—
145#106—712no—
146#106C2×D7111yescharacteristic
147#103—712no—
148#103—712no—
149#102—712no—
150#102C2×D7112yes—
151C22⋊C8—712no—
152C22⋊C8—712no—
153#41—2841no—
154#41—2841no—
155#41—2842no—
156#41—2842no—
157C2×Q8—711no—
158C2×Q8—712no—
159C2×D4—22no—
160C2×D4—712no—
161C2×D4—711no—
162C2×D4—1422no—
163C2×D4—1422no—
164C2×D4C22×D7112yes—
165C22×C4—712no—
166C22×C4—1422no—
167C22×C4—1422no—
168C22×C4C22×D7111yescharacteristic
169Q16—2841no—
170SD16—2842no—
171D8—41no—
172M16—41no—
173C2×C8—2841no—
174C2×C8—712no—
175C2×C8C22×D7111yescharacteristic
176C2×C8C22×D7111yescharacteristic
177C4⋊C4—712no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4C22×D7112yes—
182C22⋊C4—22no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C23—22no—
187C23—712no—
188C23D4×D7111yescharacteristic
189Q8—2841no—
190Q8—1422no—
191Q8—1422no—
192D4—22no—
193D4—22no—
194D4—82no—
195D4—2841no—
196D4—2844no—
197D4—2842no—
198D4—5682no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—2844no—
203C2×C4—5682no—
204C2×C4—712no—
205C2×C4—1422no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4D4×D7111yescharacteristic
210C2×C4C23×D7111yescharacteristic
211C8—21no—
212C8—21no—
213C8—1422no—
214C22—22no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—1422no—
220C22—1422no—
221C22—1422no—
222C22C2×D4×D7111yescharacteristic
223C4—21no—
224C4—42no—
225C4—2842no—
226C4—2842no—
227C4—1422no—
228C4#1455617311yescharacteristic
229C4#1455617411yescharacteristic
230C2—41no—
231C2—82no—
232C2—2842no—
233C2#1798655711yescharacteristic
234C2#1798680912yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3204
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #17986578C2semidirect
2. #17986331C2semidirect
3. #17986316C2semidirect
4. #17986257C2semidirect
5. #17986226C2semidirect
6. #17986162C2semidirect
7. #17986141C2semidirect
8. #17985967C2semidirect
9. #469×C71C2semidirect
118. C71#3204semidirect
123. #469D71semidirect