Finite Group Databasebeta

← #22600202

Subgroups — #22600202

Order 9088.

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

normal subgroups only

This group has 420 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
1Q8×D284C211yesmaximal, characteristic, complemented
2#17986652C211yesmaximal, characteristic
3#17986351C211yesmaximal, characteristic, complemented
4#17986331C212yesmaximal, complemented
5#17986229C212yesmaximal, complemented
6#17986205C211yesmaximal, characteristic, complemented
7#17986139C212yesmaximal, complemented
8C4×#11765852C211yesmaximal, characteristic, complemented
9#17986064C211yesmaximal, characteristic, complemented
10#103×D71C212yesmaximal, complemented
11#475×C71C211yesmaximal, characteristic, complemented
12C2×#11765866—21no—
13C2×Q8×D71C2212yes—
14C2×#11765854C2212yes—
15C2×#11765852C2211yescharacteristic
16C2×C8×D71C2212yes—
17#14556128—21no—
18#14556128C2211yescharacteristic
19#14556113—22no—
20#14556108—22no—
21#14556108C2212yes—
22#14556104—21no—
23#14556103C2212yes—
24C4×D284—21no—
25C4×D284C2211yescharacteristic
26C4×D284C2211yescharacteristic
27C4⋊C4×D71C2212yes—
28#14556059C2211yescharacteristic
29#14556053C2212yes—
30#14556052C2211yescharacteristic
31#14556051C2212yes—
32#14556051C2212yes—
33#14556048C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C4⋊Q8×C71C2211yescharacteristic
37C4×Q8×C71C2211yescharacteristic
38#106×C71C2212yes—
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41#11765866—44no—
42C2×Dic142—22no—
43C2×Dic142C2312yes—
44Q8×D71—24no—
45Q8×D71—42no—
46C2×D284—22no—
47C2×D284D411yescharacteristic
48C2×D284C2311yescharacteristic
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—21no—
52C2×C4×D71C2312yes—
53#11765854—42no—
54#11765852—22no—
55C2×C71⋊C8C2311yescharacteristic
56C2×C71⋊C8C2311yescharacteristic
57C8×D71—42no—
58C4⋊Dic71—21no—
59C4⋊Dic71—21no—
60C4⋊Dic71—22no—
61C4⋊Dic71D411yescharacteristic
62C4⋊Dic71C2311yescharacteristic
63Dic71⋊C4—22no—
64(C2×D71)⋊C4—22no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4D412yes—
68C2×Q8×C71—21no—
69C2×Q8×C71C2311yescharacteristic
70C2×C8×C71C2312yes—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2312yes—
74C4⋊C4×C71C2311yescharacteristic
75C42×C71—21no—
76C42×C71C2311yescharacteristic
77C22×D71—21no—
78C22×D71C2×D412yes—
79Q8×C71—21no—
80Q8×C71—42no—
81Q8×C71#4112yescomplemented
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2411yescharacteristic
88C2×Dic71—21no—
89C2×Dic71—22no—
90C2×Dic71C2×D412yes—
91D284—21no—
92D284—21no—
93D284—44no—
94D284SD1614yescomplemented
95C4×D71—22no—
96C4×D71—22no—
97C4×D71—44no—
98C4×D71—44no—
99C4×D71—42no—
100Dic142—22no—
101Dic142—22no—
102Dic142—44no—
103C8×C71—22no—
104C71⋊C8—21no—
105C71⋊C8—21no—
106C22×C71C22×D411yescharacteristic
107C2×D71—22no—
108C2×D71—22no—
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—41no—
112C2×D71C2×SD1614yes—
113C4×C71—22no—
114C4×C71—21no—
115C4×C71—42no—
116C4×C71—41no—
117C4×C71C2×SD1612yes—
118C4×C71C2×#4111yescharacteristic
119C4×C71H(5,𝔽2)11yescharacteristic
120Dic71—22no—
121Dic71—42no—
122Dic71—41no—
123C2×C71#54611yescharacteristic
124C2×C71C22×SD1611yescharacteristic
125C2×C71#57811yescharacteristic
126D71—24no—
127D71—81no—
128#3298—711nomaximal, Sylow
129C71#329811yescharacteristic, complemented, Sylow
130D4×Q8—711no—
131#547—711no—
132#487—711no—
133#483—712no—
134#475D7111yescharacteristic, complemented
135#465—712no—
136#461—711no—
137#448—712no—
138C4×SD16—711no—
139C8×D4—711no—
140C2×#103—712no—
141C2×#41—1421no—
142C22×Q8—712no—
143C2×SD16—711no—
144C2×SD16—712no—
145C22×C8—712no—
146C4⋊Q8—711no—
147C4⋊Q8—1421no—
148C4⋊Q8C2×D7111yescharacteristic
149#123—1422no—
150C22⋊Q8—712no—
151C22⋊Q8—1422no—
152C22⋊Q8—1421no—
153#121—712no—
154C4×Q8C2×D7111yescharacteristic
155C4×D4—711no—
156C4×D4—711no—
157C4×D4—1421no—
158C2×C4⋊C4—712no—
159#106—711no—
160#106C2×D7112yes—
161C4⋊C8C2×D7111yescharacteristic
162#103—711no—
163#103—712no—
164#103—712no—
165#103—712no—
166#103C2×D7112yes—
167#102—711no—
168C22⋊C8—712no—
169C4×C8—711no—
170#41—2844no—
171C2×Q8—21no—
172C2×Q8—2842no—
173C2×Q8—712no—
174C2×Q8—1422no—
175C2×Q8—1424no—
176C2×Q8C22×D7111yescharacteristic
177C2×D4—711no—
178C2×D4—711no—
179C2×D4—1422no—
180C22×C4—712no—
181C22×C4—1422no—
182C22×C4—1422no—
183C22×C4—1421no—
184SD16—2842no—
185SD16—1422no—
186C2×C8—2842no—
187C2×C8—711no—
188C2×C8—711no—
189C2×C8C22×D7112yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—711no—
193C4⋊C4—711no—
194C4⋊C4—1422no—
195C4⋊C4—1422no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4C22×D7111yescharacteristic
199C4⋊C4C22×D7112yes—
200C22⋊C4—712no—
201C22⋊C4—1422no—
202C22⋊C4—1422no—
203C22⋊C4—1421no—
204C42—21no—
205C42C22×D7111yescharacteristic
206C23—712no—
207C23—1421no—
208Q8—21no—
209Q8—42no—
210Q8—2844no—
211Q8—1422no—
212Q8—1422no—
213Q8#1176586612yescomplemented
214D4—2844no—
215D4—714no—
216D4—1421no—
217D4—1421no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—2844no—
222C2×C4—2844no—
223C2×C4—2842no—
224C2×C4—712no—
225C2×C4—1421no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4D4×D7111yescharacteristic
230C2×C4D4×D7111yescharacteristic
231C2×C4C23×D7111yescharacteristic
232C8—22no—
233C8—1421no—
234C8—1421no—
235C22—2841no—
236C22—2841no—
237C22—2841no—
238C22—714no—
239C22—1422no—
240C22—1422no—
241C22C2×D4×D7111yescharacteristic
242C4—22no—
243C4—21no—
244C4—42no—
245C4—41no—
246C4—2842no—
247C4—2841no—
248C4—1422no—
249C4SD16×D7112yes—
250C4C2×#1176586611yescharacteristic
251C4#1455617411yescharacteristic
252C2—5681no—
253C2—1424no—
254C2#1798664011yescharacteristic
255C2C2×SD16×D7111yescharacteristic
256C2#1798681511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#475×C71C2
socleC22×C71C22×D4
hypercenter#475D71
hypocenterC71#3298
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×D284C2semidirect
3. #17986351C2semidirect
4. #17986331C2semidirect
5. #17986229C2semidirect
6. #17986205C2semidirect
7. #17986139C2semidirect
8. C4×#11765852C2semidirect
9. #17986064C2semidirect
10. #103×D71C2semidirect
11. #475×C71C2semidirect
81. Q8×C71#41semidirect
94. D284SD16semidirect
129. C71#3298semidirect
134. #475D71semidirect
213. Q8#11765866semidirect