Finite Group Databasebeta

← #22598508

Subgroups — #22598508

Order 9088.

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

normal subgroups only

This group has 386 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#17986530C211yesmaximal, characteristic
2C2×#14556093C211yesmaximal, characteristic, complemented
3#17986334C211yesmaximal, characteristic, complemented
4#17986309C212yesmaximal, complemented
5#17986220C212yesmaximal, complemented
6#17986157C211yesmaximal, characteristic, complemented
7#17986137C211yesmaximal, characteristic
8#17986105C212yesmaximal, complemented
9#17986096C212yesmaximal, complemented
10#17986053C211yesmaximal, characteristic, complemented
11#481×C71C211yesmaximal, characteristic
12C22×Dic142C2211yescharacteristic
13C2×Dic284C2212yes—
14C2×#11765853C2212yes—
15#14556121—21no—
16#14556119C2211yescharacteristic
17#14556115C2211yescharacteristic
18#14556109—21no—
19#14556107C2211yescharacteristic
20#14556093—21no—
21#14556093—21no—
22#14556093C2212yes—
23#14556093C2212yes—
24#14556093C2211yescharacteristic
25D4×Dic71—21no—
26D4×Dic71C2211yescharacteristic
27#14556084—21no—
28C2×Dic71⋊C4—21no—
29C2×C4×Dic71—21no—
30C2×C4×Dic71C2211yescharacteristic
31#14556061C2212yes—
32Dic71⋊C8C2212yes—
33#14556053C2212yes—
34#14556051C2212yes—
35#14556051C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37#14556038C2212yes—
38#121×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#106×C71C2212yes—
41#102×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142—24no—
45C2×Dic142C2311yescharacteristic
46C2×Dic142C2312yes—
47C22×Dic71—21no—
48C22×Dic71—21no—
49C22×Dic71D411yescharacteristic
50C22×Dic71D411yescharacteristic
51Dic284—24no—
52#11765853—24no—
53C2×C71⋊C8C2312yes—
54C4⋊Dic71—21no—
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—21no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61C22⋊Dic71—21no—
62C22⋊Dic71—21no—
63C22⋊Dic71—21no—
64C22⋊Dic71—21no—
65C4×Dic71—22no—
66C4×Dic71—22no—
67C4×Dic71—21no—
68C4×Dic71D411yescharacteristic
69C4×Dic71D411yescharacteristic
70C4×Dic71C2311yescharacteristic
71C4×Dic71C2311yescharacteristic
72C2×D4×C71—21no—
73C2×D4×C71C2311yescharacteristic
74C22×C4×C71—21no—
75C22×C4×C71C2311yescharacteristic
76C2×C8×C71C2312yes—
77C4⋊C4×C71—21no—
78C4⋊C4×C71C2312yes—
79C4⋊C4×C71C2311yescharacteristic
80C22⋊C4×C71—21no—
81C23×C71—21no—
82C23×C71C2×D411yescharacteristic
83D4×C71—22no—
84D4×C71—42no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—42no—
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2411yescharacteristic
91C2×Dic71—24no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—41no—
97C2×Dic71—42no—
98C2×Dic71C2×D411yescharacteristic
99C2×Dic71C2×D411yescharacteristic
100C2×Dic71C2×D411yescharacteristic
101C2×Dic71C2×D411yescharacteristic
102Dic142—24no—
103Dic142—22no—
104Dic142—22no—
105Dic142#4114yes—
106C8×C71—22no—
107C71⋊C8—22no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71—41no—
111C22×C71—41no—
112C22×C71—41no—
113C22×C71#13712yes—
114C22×C71C22×D411yescharacteristic
115C4×C71—21no—
116C4×C71—41no—
117C4×C71—42no—
118C4×C71C2×#4112yes—
119Dic71—24no—
120Dic71—21no—
121Dic71—21no—
122Dic71—21no—
123Dic71—21no—
124Dic71—41no—
125Dic71—41no—
126C2×C71—22no—
127C2×C71—81no—
128C2×C71#52511yescharacteristic
129C2×C71C2×#13711yescharacteristic
130C2×C71#57511yescharacteristic
131#3117—711nomaximal, Sylow
132C71#311711yescharacteristic, complemented, Sylow
133#529—711no—
134C2×C4×Q8—711no—
135#484—711no—
136#481D7111yescharacteristic
137#479—712no—
138#464—712no—
139#451—711no—
140#448—711no—
141#441—712no—
142#440—712no—
143#432—711no—
144C22×Q8—711no—
145C2×Q16—712no—
146C2×SD16—712no—
147#126—1421no—
148#125—711no—
149#124—711no—
150#123—1421no—
151C22⋊Q8—711no—
152#121C2×D7111yescharacteristic
153C4×Q8—712no—
154C4×Q8—712no—
155C4×Q8—711no—
156C4×Q8—1421no—
157C4×Q8—1421no—
158C4×D4—711no—
159C4×D4—1421no—
160#117—1421no—
161C2×C4⋊C4—1421no—
162C2×C4⋊C4C2×D7111yescharacteristic
163C2×C42—711no—
164C2×C42—1421no—
165#107—712no—
166#106C2×D7112yes—
167C4⋊C8—712no—
168#103—712no—
169#103—712no—
170#103—712no—
171#102C2×D7112yes—
172C22⋊C8—711no—
173C22⋊C8C2×D7111yescharacteristic
174#97—712no—
175C2×Q8—712no—
176C2×Q8—711no—
177C2×Q8—1424no—
178C2×Q8—1421no—
179C2×D4—21no—
180C2×D4C22×D7111yescharacteristic
181C22×C4—21no—
182C22×C4—711no—
183C22×C4—711no—
184C22×C4—1421no—
185C22×C4—1421no—
186C22×C4C22×D7111yescharacteristic
187Q16—1424no—
188SD16—1424no—
189C2×C8—712no—
190C2×C8C22×D7112yes—
191C4⋊C4—21no—
192C4⋊C4—711no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1421no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4—1421no—
199C4⋊C4C22×D7112yes—
200C4⋊C4C22×D7111yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1421no—
203C22⋊C4—1421no—
204C22⋊C4—1421no—
205C22⋊C4—1421no—
206C42—711no—
207C42—711no—
208C42—711no—
209C42—711no—
210C42—1422no—
211C42—1422no—
212C42—1421no—
213C23—21no—
214C23D4×D7111yescharacteristic
215Q8—714no—
216Q8—1424no—
217Q8—1422no—
218Q8—1422no—
219D4—22no—
220D4—42no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—2842no—
226C2×C4—2841no—
227C2×C4—711no—
228C2×C4—711no—
229C2×C4—711no—
230C2×C4—711no—
231C2×C4—1421no—
232C2×C4—1421no—
233C2×C4—1424no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4D4×D7111yescharacteristic
237C2×C4C23×D7111yescharacteristic
238C8—22no—
239C8—1422no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1455615712yes—
246C22C2×D4×D7111yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—2841no—
251C4—2841no—
252C4—1424no—
253C4—1421no—
254C4—1421no—
255C4—1421no—
256C4—1421no—
257C4#41×D7112yes—
258C2—22no—
259C2—81no—
260C2#1798650711yescharacteristic
261C2C2×#1455615711yescharacteristic
262C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#481×C71C2
socleC22×C71C22×D4
hypercenter#481D71
hypocenterC71#3117
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#14556093C2semidirect
3. #17986334C2semidirect
4. #17986309C2semidirect
5. #17986220C2semidirect
6. #17986157C2semidirect
8. #17986105C2semidirect
9. #17986096C2semidirect
10. #17986053C2semidirect
132. C71#3117semidirect