Finite Group Databasebeta

← #22599671

Subgroups — #22599671

Order 9088.

354 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986533C211yesmaximal, characteristic
2#17986306C212yesmaximal, complemented
3#17986281C211yesmaximal, characteristic, complemented
4#17986276C211yesmaximal, characteristic, complemented
5#17986238C212yesmaximal, complemented
6#17986230C212yesmaximal
7#17986221C212yesmaximal, complemented
8#17986206C211yesmaximal, characteristic, complemented
9#17986201C211yesmaximal, characteristic, complemented
10C2×C4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
11#533×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2212yes—
13C2×#11765852C2212yes—
14C22×C71⋊C8C2212yes—
15#14556122—21no—
16#14556120C2211yescharacteristic
17#14556116C2211yescharacteristic
18#14556111—21no—
19#14556107C2211yescharacteristic
20#14556102C2211yescharacteristic
21C4×Dic142C2211yescharacteristic
22C4×D284C2211yescharacteristic
23C4×C22⋊D71—21no—
24#14556085—21no—
25#14556061C2212yes—
26#14556059C2212yes—
27C4⋊(C71⋊C8)C2212yes—
28C4⋊(C71⋊C8)C2211yescharacteristic
29C4⋊(C71⋊C8)C2211yescharacteristic
30#14556052C2212yes—
31#14556052C2212yes—
32#14556051C2212yes—
33#14556048C2212yes—
34C4⋊Q8×C71C2211yescharacteristic
35C4⋊Q8×C71C2211yescharacteristic
36#125×C71—21no—
37C22⋊Q8×C71—21no—
38C22⋊Q8×C71C2212yes—
39C4×Q8×C71—21no—
40C4×Q8×C71C2212yes—
41#117×C71—21no—
42C2×C42×C71C2211yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C4×D71—21no—
47#11765855—42no—
48#11765852—42no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8C2312yes—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54Dic71⋊C4—21no—
55Dic71⋊C4—21no—
56Dic71⋊C4—21no—
57Dic71⋊C4—21no—
58C22⋊Dic71—21no—
59(C2×D71)⋊C4—21no—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—21no—
62(C2×D71)⋊C4—21no—
63C4×Dic71—21no—
64C2×Q8×C71—21no—
65C2×Q8×C71C2312yes—
66C22×C4×C71D411yescharacteristic
67C22×C4×C71D411yescharacteristic
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—21no—
70C4⋊C4×C71—21no—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—22no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71C2312yes—
75C22⋊C4×C71—22no—
76C42×C71—22no—
77C42×C71D411yescharacteristic
78C42×C71D411yescharacteristic
79C42×C71C2311yescharacteristic
80C42×C71C2311yescharacteristic
81C23×C71C2×D411yescharacteristic
82C22×D71—21no—
83Q8×C71—22no—
84Q8×C71—22no—
85Q8×C71—42no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2411yescharacteristic
98C22⋊D71—42no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—21no—
102D284—21no—
103D284—21no—
104C4×D71—42no—
105Dic142—21no—
106Dic142—21no—
107C71⋊C8—22no—
108C71⋊C8—22no—
109C22×C71—21no—
110C22×C71—21no—
111C22×C71#13712yes—
112C22×C71C22×D411yescharacteristic
113C2×D71—41no—
114C2×D71—41no—
115C2×D71—41no—
116C4×C71—21no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—21no—
120C4×C71—42no—
121C4×C71—42no—
122C4×C71#13512yes—
123C4×C71H(5,𝔽2)11yescharacteristic
124C4×C712−1+411yescharacteristic
125Dic71—41no—
126Dic71—41no—
127Dic71—41no—
128C2×C71—22no—
129C2×C71#53711yescharacteristic
130C2×C71C2×#13511yescharacteristic
131C2×C71C2×#13711yescharacteristic
132D71—81no—
133#3248—711nomaximal, Sylow
134C71#324811yescharacteristic, complemented, Sylow
135#533D7111yescharacteristic, complemented
136#529—711no—
137#479—712no—
138#475—711no—
139#474—711no—
140#467—712no—
141#465—712no—
142#464—712no—
143#462—711no—
144#461—711no—
145C2×C4⋊C8—711no—
146C2×Q16—712no—
147C2×SD16—712no—
148C22×C8—712no—
149C4⋊Q8C2×D7111yescharacteristic
150C4⋊Q8C2×D7111yescharacteristic
151#126—1421no—
152#125—21no—
153#125—711no—
154#124—711no—
155#123—1421no—
156C22⋊Q8—21no—
157C22⋊Q8—711no—
158C22⋊Q8C2×D7112yes—
159#121—711no—
160C4×Q8—21no—
161C4×Q8—711no—
162C4×Q8C2×D7112yes—
163C4×D4—711no—
164C4×D4—1421no—
165#117—21no—
166#117—1421no—
167C2×C42C2×D7111yescharacteristic
168#107—712no—
169#106—712no—
170C4⋊C8—711no—
171C4⋊C8—711no—
172C4⋊C8—712no—
173#103—712no—
174#103—712no—
175#103—712no—
176#102—712no—
177C2×Q8—21no—
178C2×Q8—711no—
179C2×Q8C22×D7112yes—
180C2×D4—711no—
181C2×D4—1421no—
182C22×C4—1421no—
183C22×C4C22⋊D7111yescharacteristic
184C22×C4C22⋊D7111yescharacteristic
185C22×C4C22×D7111yescharacteristic
186Q16—2842no—
187SD16—2842no—
188C2×C8—2842no—
189C2×C8—712no—
190C2×C8—712no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—711no—
197C4⋊C4—711no—
198C4⋊C4—1421no—
199C4⋊C4—1421no—
200C4⋊C4—1421no—
201C4⋊C4—1421no—
202C4⋊C4C22×D7112yes—
203C22⋊C4—22no—
204C22⋊C4—1421no—
205C22⋊C4—1421no—
206C22⋊C4—1421no—
207C22⋊C4—1421no—
208C22⋊C4—1421no—
209C42—22no—
210C42—1421no—
211C42C22⋊D7111yescharacteristic
212C42C22⋊D7111yescharacteristic
213C42C22×D7111yescharacteristic
214C42C22×D7111yescharacteristic
215C23—1421no—
216C23C2×C22⋊D7111yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—1421no—
221Q8—1421no—
222D4—2842no—
223D4—1421no—
224D4—1421no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—2842no—
232C2×C4—1421no—
233C2×C4—1421no—
234C2×C4—1421no—
235C2×C4C2×C22⋊D7111yescharacteristic
236C2×C4C2×C22⋊D7111yescharacteristic
237C2×C4C2×C22⋊D7111yescharacteristic
238C2×C4C2×C22⋊D7111yescharacteristic
239C2×C4C2×C22⋊D7111yescharacteristic
240C2×C4C23×D7111yescharacteristic
241C8—1422no—
242C8—1422no—
243C22—21no—
244C22—21no—
245C22—2841no—
246C22—2841no—
247C22—2841no—
248C22#1455615912yes—
249C22C22×C22⋊D7111yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—2841no—
257C4—2841no—
258C4—2841no—
259C4#1455614812yes—
260C4#1455617411yescharacteristic
261C4#1455617511yescharacteristic
262C2—22no—
263C2—5681no—
264C2#1798657711yescharacteristic
265C2C2×#1455614811yescharacteristic
266C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3248
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #17986306C2semidirect
3. #17986281C2semidirect
4. #17986276C2semidirect
5. #17986238C2semidirect
7. #17986221C2semidirect
8. #17986206C2semidirect
9. #17986201C2semidirect
10. C2×C4⋊(C71⋊C8)C2semidirect
11. #533×C71C2semidirect
134. C71#3248semidirect
135. #533D71semidirect