Finite Group Databasebeta

← #22599043

Subgroups — #22599043

Order 9088.

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

normal subgroups only

This group has 366 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#17986405C211yesmaximal, characteristic, complemented
3#17986391C211yesmaximal, characteristic, complemented
4#17986367C211yesmaximal, characteristic, complemented
5#17986361C211yesmaximal, characteristic, complemented
6#17986238C212yesmaximal, complemented
7#17986230C212yesmaximal
8#17986104C212yesmaximal, complemented
9#17986097C212yesmaximal, complemented
10C2×#14556039C211yesmaximal, 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—
27#14556052C2212yes—
28#14556052C2212yes—
29#14556051C2212yes—
30#14556048C2212yes—
31#14556039C2212yes—
32#14556039C2211yescharacteristic
33#14556039C2211yescharacteristic
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—24no—
48#11765852—24no—
49C2×C71⋊C8—24no—
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—22no—
89C2×C4×C71—22no—
90C2×C4×C71—41no—
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2411yescharacteristic
97C22⋊D71—42no—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101D284—22no—
102C4×D71—42no—
103Dic142—22no—
104C71⋊C8—22no—
105C71⋊C8#4114yes—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71#13712yes—
109C22×C71C22×D411yescharacteristic
110C2×D71—41no—
111C2×D71—41no—
112C2×D71—41no—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—21no—
116C4×C71—21no—
117C4×C71—21no—
118C4×C71—42no—
119C4×C71—42no—
120C4×C71C2×#4112yes—
121Dic71—41no—
122Dic71—41no—
123Dic71—41no—
124C2×C71—22no—
125C2×C71#53211yescharacteristic
126C2×C71C2×#13711yescharacteristic
127C2×C71#57511yescharacteristic
128D71—81no—
129#3181—711nomaximal, Sylow
130C71#318111yescharacteristic, complemented, Sylow
131#533D7111yescharacteristic, complemented
132#529—711no—
133#501—711no—
134#497—711no—
135#491—711no—
136#489—711no—
137#467—712no—
138#465—712no—
139#441—712no—
140#440—712no—
141C2×#97—711no—
142C2×Q16—712no—
143C2×SD16—712no—
144C22×C8—712no—
145C4⋊Q8C2×D7111yescharacteristic
146C4⋊Q8C2×D7111yescharacteristic
147#126—1421no—
148#125—21no—
149#125—711no—
150#124—711no—
151#123—1421no—
152C22⋊Q8—21no—
153C22⋊Q8—711no—
154C22⋊Q8C2×D7112yes—
155#121—711no—
156C4×Q8—21no—
157C4×Q8—711no—
158C4×Q8C2×D7112yes—
159C4×D4—711no—
160C4×D4—1421no—
161#117—21no—
162#117—1421no—
163C2×C42C2×D7111yescharacteristic
164#107—712no—
165#106—712no—
166#103—712no—
167#103—712no—
168#103—712no—
169#102—712no—
170#97—712no—
171#97—711no—
172#97—711no—
173C2×Q8—21no—
174C2×Q8—711no—
175C2×Q8C22×D7112yes—
176C2×D4—711no—
177C2×D4—1421no—
178C22×C4—1421no—
179C22×C4C22⋊D7111yescharacteristic
180C22×C4C22⋊D7111yescharacteristic
181C22×C4C22×D7111yescharacteristic
182Q16—1424no—
183SD16—1424no—
184C2×C8—712no—
185C2×C8—712no—
186C2×C8—1424no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—711no—
193C4⋊C4—711no—
194C4⋊C4—1421no—
195C4⋊C4—1421no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4C22×D7112yes—
199C22⋊C4—22no—
200C22⋊C4—1421no—
201C22⋊C4—1421no—
202C22⋊C4—1421no—
203C22⋊C4—1421no—
204C22⋊C4—1421no—
205C42—22no—
206C42—1421no—
207C42C22⋊D7111yescharacteristic
208C42C22⋊D7111yescharacteristic
209C42C22×D7111yescharacteristic
210C42C22×D7111yescharacteristic
211C23—1421no—
212C23C2×C22⋊D7111yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—1422no—
217D4—2842no—
218D4—1422no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—2842no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4—1421no—
228C2×C4C2×C22⋊D7111yescharacteristic
229C2×C4C2×C22⋊D7111yescharacteristic
230C2×C4C2×C22⋊D7111yescharacteristic
231C2×C4C2×C22⋊D7111yescharacteristic
232C2×C4C2×C22⋊D7111yescharacteristic
233C2×C4C23×D7111yescharacteristic
234C8—714no—
235C8—1422no—
236C22—21no—
237C22—21no—
238C22—2841no—
239C22—2841no—
240C22—2841no—
241C22#1455615912yes—
242C22C22×C22⋊D7111yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—2841no—
251C4—2841no—
252C4—2841no—
253C4#41×D7112yes—
254C2—22no—
255C2—5681no—
256C2#1798654511yescharacteristic
257C2C2×#1455615911yescharacteristic
258C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3181
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. #17986405C2semidirect
3. #17986391C2semidirect
4. #17986367C2semidirect
5. #17986361C2semidirect
6. #17986238C2semidirect
8. #17986104C2semidirect
9. #17986097C2semidirect
10. C2×#14556039C2semidirect
11. #533×C71C2semidirect
130. C71#3181semidirect
131. #533D71semidirect