Finite Group Databasebeta

← #22599519

Subgroups — #22599519

Order 9088.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986398C212yesmaximal, complemented
3#17986356C212yesmaximal, complemented
4#17986340C212yesmaximal, complemented
5#17986323C212yesmaximal
6#17986291C211yesmaximal, characteristic, complemented
7#17986286C211yesmaximal, characteristic, complemented
8#17986281C211yesmaximal, characteristic, complemented
9#17986276C211yesmaximal, characteristic, complemented
10#17986052C211yesmaximal, characteristic, complemented
11#533×C71C211yesmaximal, characteristic, complemented
12#14556122—21no—
13#14556120C2211yescharacteristic
14#14556116C2211yescharacteristic
15#14556111—21no—
16#14556107C2211yescharacteristic
17#14556102C2211yescharacteristic
18C4×Dic142C2211yescharacteristic
19C4×D284C2211yescharacteristic
20C4×C22⋊D71—21no—
21#14556085—21no—
22#14556061C2212yes—
23#14556061C2212yes—
24#14556059C2212yes—
25#14556059C2212yes—
26C4⋊(C71⋊C8)C2211yescharacteristic
27C4⋊(C71⋊C8)C2211yescharacteristic
28#14556051C2212yes—
29#14556051C2212yes—
30#14556048C2212yes—
31#14556048C2212yes—
32C22⋊(C71⋊C8)C2212yes—
33C4×C71⋊C8C2212yes—
34C4⋊Q8×C71C2211yescharacteristic
35C4⋊Q8×C71C2211yescharacteristic
36#125×C71C2212yes—
37C22⋊Q8×C71—21no—
38C22⋊Q8×C71—21no—
39C4×Q8×C71—21no—
40C4×Q8×C71—21no—
41#117×C71C2212yes—
42C2×C42×C71C2211yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C4×D71—21no—
47C2×C71⋊C8C2312yes—
48C2×C71⋊C8C2312yes—
49C4⋊Dic71C2311yescharacteristic
50C4⋊Dic71C2311yescharacteristic
51Dic71⋊C4—21no—
52Dic71⋊C4—21no—
53Dic71⋊C4—21no—
54Dic71⋊C4—21no—
55C22⋊Dic71—21no—
56(C2×D71)⋊C4—21no—
57(C2×D71)⋊C4—21no—
58(C2×D71)⋊C4—21no—
59(C2×D71)⋊C4—21no—
60C4×Dic71—21no—
61C2×Q8×C71—21no—
62C2×Q8×C71—21no—
63C22×C4×C71D411yescharacteristic
64C22×C4×C71D411yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71C2312yes—
71C4⋊C4×C71C2312yes—
72C22⋊C4×C71—22no—
73C42×C71—22no—
74C42×C71D411yescharacteristic
75C42×C71D411yescharacteristic
76C42×C71C2311yescharacteristic
77C42×C71C2311yescharacteristic
78C23×C71C2×D411yescharacteristic
79C22×D71—21no—
80Q8×C71—42no—
81Q8×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—41no—
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2411yescharacteristic
94C22⋊D71—42no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98D284—21no—
99D284—21no—
100C4×D71—42no—
101Dic142—21no—
102Dic142—21no—
103C71⋊C8—22no—
104C71⋊C8—22no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71C22×D411yescharacteristic
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—41no—
112C4×C71—21no—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—21no—
116C4×C71—42no—
117C4×C71—42no—
118C4×C71#13512yes—
119C4×C712−1+411yescharacteristic
120C4×C712−1+411yescharacteristic
121Dic71—41no—
122Dic71—41no—
123Dic71—41no—
124C2×C71—41no—
125C2×C71#53611yescharacteristic
126C2×C71C2×#13511yescharacteristic
127C2×C71#57511yescharacteristic
128D71—81no—
129#3233—711nomaximal, Sylow
130C71#323311yescharacteristic, complemented, Sylow
131#533D7111yescharacteristic, complemented
132#529—711no—
133#499—712no—
134#488—712no—
135#485—712no—
136#482—712no—
137#477—711no—
138#476—711no—
139#475—711no—
140#474—711no—
141#431—711no—
142C4⋊Q8C2×D7111yescharacteristic
143C4⋊Q8C2×D7111yescharacteristic
144#126—1421no—
145#125—711no—
146#125C2×D7112yes—
147#124—711no—
148#123—1421no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8—711no—
152#121—711no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—711no—
156C4×D4—711no—
157C4×D4—1421no—
158#117—1421no—
159#117C2×D7112yes—
160C2×C42C2×D7111yescharacteristic
161#107—712no—
162#107—712no—
163#106—712no—
164#106—712no—
165C4⋊C8—711no—
166C4⋊C8—711no—
167#103—712no—
168#103—712no—
169#102—712no—
170#102—712no—
171C22⋊C8—712no—
172C4×C8—712no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—711no—
176C2×D4—711no—
177C2×D4—1421no—
178C22×C4—1421no—
179C22×C4C22⋊D7111yescharacteristic
180C22×C4C22⋊D7111yescharacteristic
181C22×C4C22×D7111yescharacteristic
182C2×C8—712no—
183C2×C8—712no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—711no—
189C4⋊C4—711no—
190C4⋊C4—1421no—
191C4⋊C4—1421no—
192C4⋊C4—1421no—
193C4⋊C4—1421no—
194C4⋊C4C22×D7112yes—
195C4⋊C4C22×D7112yes—
196C22⋊C4—22no—
197C22⋊C4—1421no—
198C22⋊C4—1421no—
199C22⋊C4—1421no—
200C22⋊C4—1421no—
201C22⋊C4—1421no—
202C42—22no—
203C42—1421no—
204C42C22⋊D7111yescharacteristic
205C42C22⋊D7111yescharacteristic
206C42C22×D7111yescharacteristic
207C42C22×D7111yescharacteristic
208C23—1421no—
209C23C2×C22⋊D7111yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—1421no—
213Q8—1421no—
214D4—2842no—
215D4—1421no—
216D4—1421no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—2842no—
224C2×C4—1421no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4C2×C22⋊D7111yescharacteristic
228C2×C4C2×C22⋊D7111yescharacteristic
229C2×C4C2×C22⋊D7111yescharacteristic
230C2×C4C2×C22⋊D7111yescharacteristic
231C2×C4C2×C22⋊D7111yescharacteristic
232C2×C4C23×D7111yescharacteristic
233C8—1422no—
234C8—1422no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—2841no—
239C22—2841no—
240C22—2841no—
241C22C22×C22⋊D7111yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—2841no—
249C4—2841no—
250C4—2841no—
251C4#1455614812yes—
252C4#1455617611yescharacteristic
253C4#1455617611yescharacteristic
254C2—41no—
255C2—5681no—
256C2#1798657311yescharacteristic
257C2C2×#1455614811yescharacteristic
258C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3233
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. #17986398C2semidirect
3. #17986356C2semidirect
4. #17986340C2semidirect
6. #17986291C2semidirect
7. #17986286C2semidirect
8. #17986281C2semidirect
9. #17986276C2semidirect
10. #17986052C2semidirect
11. #533×C71C2semidirect
130. C71#3233semidirect
131. #533D71semidirect