Finite Group Databasebeta

← #22599569

Subgroups — #22599569

Order 9088.

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

normal subgroups only

This group has 362 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
1C4⋊Q8×D71C211yesmaximal, characteristic, complemented
2#17986531C211yesmaximal, characteristic
3#17986407C212yesmaximal, complemented
4#17986357C212yesmaximal, complemented
5#17986336C212yesmaximal, complemented
6#17986316C212yesmaximal
7#17986285C211yesmaximal, characteristic, complemented
8#17986281C211yesmaximal, characteristic, complemented
9#17986279C211yesmaximal, characteristic, complemented
10#17986055C211yesmaximal, characteristic, complemented
11#474×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13#14556128C2211yescharacteristic
14#14556127—21no—
15#14556127C2212yes—
16#14556126—21no—
17#14556121—21no—
18#14556119C2211yescharacteristic
19#14556115C2211yescharacteristic
20#14556110—21no—
21#14556108C2211yescharacteristic
22#14556099C2211yescharacteristic
23C4×Dic142C2211yescharacteristic
24C4×C22⋊D71—21no—
25#14556087—21no—
26C4⋊C4×D71—21no—
27C4⋊C4×D71C2212yes—
28C42×D71C2211yescharacteristic
29#14556061C2212yes—
30#14556060C2212yes—
31#14556059C2212yes—
32C4⋊(C71⋊C8)C2211yescharacteristic
33#14556053C2212yes—
34#14556051C2212yes—
35#14556049C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556038C2212yes—
38C4⋊Q8×C71C2211yescharacteristic
39C4×D4×C71C2211yescharacteristic
40#107×C71C2212yes—
41C4⋊C8×C71C2211yescharacteristic
42#102×C71C2212yes—
43C2×Dic142—21no—
44C2×Dic142—22no—
45C2×Dic142C2311yescharacteristic
46Q8×D71—44no—
47C2×C22⋊D71—21no—
48C2×C4×D71—22no—
49C2×C4×D71D411yescharacteristic
50C2×C4×D71D411yescharacteristic
51C2×C4×D71C2311yescharacteristic
52C2×C71⋊C8C2312yes—
53C4⋊Dic71—21no—
54C4⋊Dic71—21no—
55C4⋊Dic71—21no—
56C4⋊Dic71C2312yes—
57C4⋊Dic71C2311yescharacteristic
58Dic71⋊C4—21no—
59Dic71⋊C4—22no—
60Dic71⋊C4—22no—
61Dic71⋊C4—21no—
62C22⋊Dic71—21no—
63C22⋊Dic71—21no—
64C22⋊Dic71—21no—
65(C2×D71)⋊C4—21no—
66C4×Dic71—21no—
67C4×Dic71D411yescharacteristic
68C4×Dic71D411yescharacteristic
69C4×Dic71C2311yescharacteristic
70C2×Q8×C71—21no—
71C2×D4×C71C2311yescharacteristic
72C22×C4×C71—21no—
73C2×C8×C71C2312yes—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C4⋊C4×C71C2311yescharacteristic
77C22⋊C4×C71—21no—
78C42×C71C2311yescharacteristic
79C23×C71—21no—
80C22×D71C2×D411yescharacteristic
81Q8×C71—42no—
82D4×C71—21no—
83D4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C22⋊D71—42no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71C2×D411yescharacteristic
95C2×Dic71C2×D411yescharacteristic
96C2×Dic71C2×D411yescharacteristic
97C4×D71—22no—
98C4×D71—22no—
99C4×D71—21no—
100C4×D71—21no—
101C4×D71—44no—
102Dic142—21no—
103Dic142—21no—
104Dic142—44no—
105Dic142—42no—
106C8×C71—22no—
107C71⋊C8—22no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71—41no—
111C22×C71C22×D411yescharacteristic
112C2×D71—21no—
113C2×D71—21no—
114C2×D71#13712yes—
115C4×C71—21no—
116C4×C71—42no—
117C4×C71—41no—
118C4×C71AGL(1,ℤ/8)12yes—
119C4×C712−1+411yescharacteristic
120C4×C712−1+411yescharacteristic
121Dic71—21no—
122Dic71—21no—
123Dic71—21no—
124Dic71—41no—
125Dic71—42no—
126Dic71—41no—
127C2×C71—81no—
128C2×C71#53611yescharacteristic
129C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
130C2×C71C2×#13711yescharacteristic
131D71—22no—
132#3239—711nomaximal, Sylow
133C71#323911yescharacteristic, complemented, Sylow
134C2×C4⋊Q8—711no—
135#529—711no—
136#501—712no—
137#489—712no—
138#484—712no—
139#481—712no—
140#475—711no—
141#475—711no—
142#474—711no—
143#474D7111yescharacteristic, complemented
144#432—711no—
145C22×Q8—1421no—
146C4⋊Q8—712no—
147C4⋊Q8—711no—
148C4⋊Q8—1421no—
149C4⋊Q8—1421no—
150C4⋊Q8C2×D7111yescharacteristic
151#126—1421no—
152#125—711no—
153#124—711no—
154#123—1421no—
155C22⋊Q8—711no—
156#121—711no—
157C4×Q8—711no—
158C4×D4—1421no—
159C4×D4C2×D7111yescharacteristic
160#117—1421no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4—1421no—
163C2×C42—711no—
164#107—712no—
165#107C2×D7112yes—
166#106—712no—
167#106—712no—
168C4⋊C8—711no—
169C4⋊C8C2×D7111yescharacteristic
170#103—712no—
171#103—712no—
172#102—712no—
173#102C2×D7112yes—
174C22⋊C8—712no—
175#97—712no—
176C2×Q8—21no—
177C2×Q8—2844no—
178C2×Q8—711no—
179C2×Q8—1422no—
180C2×Q8—1421no—
181C2×D4—1421no—
182C2×D4C22×D7111yescharacteristic
183C22×C4—21no—
184C22×C4—711no—
185C22×C4—711no—
186C22×C4—711no—
187C22×C4—1422no—
188C2×C8—712no—
189C2×C8C22×D7112yes—
190C4⋊C4—21no—
191C4⋊C4—712no—
192C4⋊C4—711no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1421no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4—1421no—
199C4⋊C4—1421no—
200C4⋊C4C22×D7112yes—
201C4⋊C4C22×D7111yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—1421no—
204C22⋊C4—1421no—
205C22⋊C4—1421no—
206C22⋊C4—1421no—
207C42—711no—
208C42—711no—
209C42—711no—
210C42—1421no—
211C42C22×D7111yescharacteristic
212C23—21no—
213C23—711no—
214Q8—42no—
215Q8—2844no—
216Q8—2842no—
217Q8—1421no—
218Q8—1421no—
219D4—21no—
220D4—21no—
221D4—2842no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—2844no—
226C2×C4—711no—
227C2×C4—711no—
228C2×C4—711no—
229C2×C4—1422no—
230C2×C4—1421no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1421no—
234C2×C4—1421no—
235C2×C4—1422no—
236C2×C4D4×D7111yescharacteristic
237C2×C4D4×D7111yescharacteristic
238C2×C4C23×D7111yescharacteristic
239C8—22no—
240C8—1422no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—712no—
245C22—1421no—
246C22—1421no—
247C22C2×D4×D7111yescharacteristic
248C4—21no—
249C4—41no—
250C4—42no—
251C4—2841no—
252C4—2842no—
253C4—2841no—
254C4—1421no—
255C4—1421no—
256C4—1421no—
257C4#1455615212yes—
258C4#1455617511yescharacteristic
259C4#1455617611yescharacteristic
260C2—81no—
261C2—1422no—
262C2#1798657211yescharacteristic
263C2C2×#1455615211yescharacteristic
264C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#474×C71C2
socleC22×C71C22×D4
hypercenter#474D71
hypocenterC71#3239
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D71C2semidirect
3. #17986407C2semidirect
4. #17986357C2semidirect
5. #17986336C2semidirect
7. #17986285C2semidirect
8. #17986281C2semidirect
9. #17986279C2semidirect
10. #17986055C2semidirect
11. #474×C71C2semidirect
133. C71#3239semidirect
143. #474D71semidirect