Finite Group Databasebeta

← #22599252

Subgroups — #22599252

Order 9088.

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

normal subgroups only

This group has 402 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
1#17986577C212yesmaximal, complemented
2#17986330C212yesmaximal, complemented
3#17986318C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic, complemented
5#17986252C211yesmaximal, characteristic, complemented
6#17986228C212yesmaximal, complemented
7#17986163C212yesmaximal, complemented
8#17986146C212yesmaximal, complemented
9#408×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2212yes—
11C2×Dic284C2211yescharacteristic
12C2×#11765854C2212yes—
13C2×D568C2211yescharacteristic
14#14556107C2212yes—
15#14556107C2212yes—
16#14556104—22no—
17#14556102C2212yes—
18#14556102C2212yes—
19#14556100—22no—
20#14556100—22no—
21#14556099—22no—
22C2×C4⋊Dic71C2212yes—
23#14556060C2211yescharacteristic
24#14556060C2212yes—
25#14556060C2211yescharacteristic
26#14556053C2212yes—
27#14556053C2212yes—
28#14556050C2212yes—
29#14556050C2212yes—
30C2×#41×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32C22×C8×C71C2211yescharacteristic
33C22⋊C8×C71C2212yes—
34C22⋊C8×C71C2212yes—
35#11765865—42no—
36#11765865—42no—
37C2×Dic142C2312yes—
38C2×D284C2312yes—
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C22×Dic71—22no—
43C2×C4×D71—22no—
44Dic284—41no—
45#11765854—42no—
46D568—41no—
47C4⋊Dic71—22no—
48C4⋊Dic71C2312yes—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52C22⋊Dic71—22no—
53C22⋊Dic71—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56#41×C71—41no—
57#41×C71—41no—
58C2×Q8×C71D411yescharacteristic
59C2×D4×C71D412yes—
60C2×D4×C71D411yescharacteristic
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63M16×C71—41no—
64C2×C8×C71—41no—
65C2×C8×C71C2312yes—
66C2×C8×C71C2311yescharacteristic
67C2×C8×C71C2311yescharacteristic
68C23×C71C2×D411yescharacteristic
69C23×C71C2×D412yes—
70C22×D71—22no—
71Q8×C71—41no—
72D4×C71—42no—
73D4×C71—41no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2411yescharacteristic
81C22⋊D71—44no—
82C22⋊D71—82no—
83C22⋊D71—82no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—82no—
88D284—22no—
89D284—22no—
90C4×D71—44no—
91Dic142—22no—
92Dic142—22no—
93C8×C71—22no—
94C8×C71—21no—
95C8×C71—21no—
96C22×C71—22no—
97C22×C71—21no—
98C22×C71—22no—
99C22×C71—22no—
100C22×C71—22no—
101C22×C71C22×D411yescharacteristic
102C2×D71—42no—
103C2×D71—42no—
104C2×D71—42no—
105C4×C71—22no—
106C4×C71—21no—
107C4×C71H(5,𝔽2)11yescharacteristic
108C4×C71H(5,𝔽2)11yescharacteristic
109Dic71—42no—
110Dic71—42no—
111Dic71—42no—
112C2×C71—42no—
113C2×C71—41no—
114C2×C71#53411yescharacteristic
115C2×C71#57712yes—
116D71—82no—
117#3204—711nomaximal, Sylow
118C71#320411yescharacteristic, complemented, Sylow
119#537—712no—
120#483—712no—
121#481—712no—
122#470—711no—
123#469—711no—
124#465—712no—
125#452—712no—
126#449—712no—
127#408D7111yescharacteristic, complemented
128C2×#41—712no—
129C2×#41C2×D7111yescharacteristic
130C2×Q16—711no—
131C2×SD16—712no—
132C2×D8—711no—
133C2×M16C2×D7111yescharacteristic
134C22×C8C2×D7111yescharacteristic
135C22⋊Q8—712no—
136C22⋊Q8—712no—
137C22⋊Q8—1422no—
138#121—712no—
139#121—712no—
140#121—1422no—
141#121—1422no—
142#121—1422no—
143C2×C4⋊C4—712no—
144#106—712no—
145#106—711no—
146#106—711no—
147#103—712no—
148#103—712no—
149#102—712no—
150#102—712no—
151C22⋊C8C2×D7112yes—
152C22⋊C8C2×D7112yes—
153#41—41no—
154#41—41no—
155#41—2842no—
156#41—2842no—
157C2×Q8—712no—
158C2×Q8D28411yescharacteristic
159C2×D4—712no—
160C2×D4—1422no—
161C2×D4—1422no—
162C2×D4—1422no—
163C2×D4D28411yescharacteristic
164C2×D4D28412yes—
165C22×C4—1422no—
166C22×C4—1422no—
167C22×C4C22×D7112yes—
168C22×C4C22×D7111yescharacteristic
169Q16—2841no—
170SD16—2842no—
171D8—2841no—
172M16—41no—
173C2×C8—41no—
174C2×C8C22×D7111yescharacteristic
175C2×C8C22×D7112yes—
176C2×C8C22×D7111yescharacteristic
177C4⋊C4—712no—
178C4⋊C4—712no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C23—1422no—
187C23C2×D28411yescharacteristic
188C23C2×D28412yes—
189Q8—41no—
190Q8—1422no—
191Q8—1422no—
192D4—42no—
193D4—41no—
194D4—2844no—
195D4—5682no—
196D4—5682no—
197D4—1422no—
198D4—1422no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—2844no—
204C2×C4—5682no—
205C2×C4—1422no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4C2×D28411yescharacteristic
209C2×C4C2×D28412yes—
210C2×C4C23×D7111yescharacteristic
211C8—22no—
212C8—21no—
213C8—21no—
214C22—22no—
215C22—21no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—2842no—
220C22—2842no—
221C22—2842no—
222C22C22×D28411yescharacteristic
223C4—22no—
224C4—21no—
225C4—2842no—
226C4—2842no—
227C4—2842no—
228C4#1455617311yescharacteristic
229C4#1455617311yescharacteristic
230C2—42no—
231C2—41no—
232C2—5682no—
233C2#1798656111yescharacteristic
234C2#1798681112yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#408×C71C2
socleC22×C71C22×D4
hypercenter#408D71
hypocenterC71#3204
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #17986577C2semidirect
2. #17986330C2semidirect
3. #17986318C2semidirect
4. #17986257C2semidirect
5. #17986252C2semidirect
6. #17986228C2semidirect
7. #17986163C2semidirect
8. #17986146C2semidirect
9. #408×C71C2semidirect
118. C71#3204semidirect
127. #408D71semidirect