Finite Group Databasebeta

← #22600204

Subgroups — #22600204

Order 9088.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986664C211yesmaximal, characteristic, complemented
2#17986652C211yesmaximal, characteristic
3#17986328C212yesmaximal, complemented
4#17986281C211yesmaximal, characteristic, complemented
5#17986231C212yesmaximal, complemented
6#17986201C211yesmaximal, characteristic, complemented
7#17986139C212yesmaximal, complemented
8C4×#11765854C211yesmaximal, characteristic, complemented
9C8×D284C211yesmaximal, characteristic, complemented
10#103×D71C212yesmaximal, complemented
11#487×C71C211yesmaximal, characteristic, complemented
12C2×#11765866—21no—
13C2×Q8×D71C2212yes—
14C2×#11765854C2211yescharacteristic
15C2×#11765852C2212yes—
16C2×C8×D71C2212yes—
17#14556127—21no—
18#14556113—22no—
19#14556106—22no—
20#14556104—21no—
21#14556104C2212yes—
22#14556103C2212yes—
23C4×Dic142C2211yescharacteristic
24C4×D284C2211yescharacteristic
25C4×D284C2211yescharacteristic
26#14556090—21no—
27C4⋊C4×D71C2212yes—
28#14556060C2211yescharacteristic
29#14556059C2212yes—
30C4⋊(C71⋊C8)C2211yescharacteristic
31#14556053C2211yescharacteristic
32#14556052C2212yes—
33#14556051C2212yes—
34#14556050C2211yescharacteristic
35(C2×D71)⋊C8C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#103×C71C2212yes—
39#103×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41#11765866—44no—
42C2×Dic142—22no—
43C2×Dic142C2311yescharacteristic
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—22no—
54#11765852—42no—
55C2×C71⋊C8C2312yes—
56C8×D71—42no—
57C4⋊Dic71D411yescharacteristic
58C4⋊Dic71C2311yescharacteristic
59C4⋊Dic71C2311yescharacteristic
60Dic71⋊C4—21no—
61Dic71⋊C4—22no—
62Dic71⋊C4—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—22no—
66(C2×D71)⋊C4D412yes—
67C4×Dic71—21no—
68C2×Q8×C71—21no—
69C2×Q8×C71C2312yes—
70C2×C8×C71C2311yescharacteristic
71C2×C8×C71C2311yescharacteristic
72C4⋊C4×C71—21no—
73C4⋊C4×C71—21no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C42×C71C2311yescharacteristic
77C22×D71—21no—
78C22×D71C2×D412yes—
79Q8×C71—22no—
80Q8×C71—22no—
81Q8×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—21no—
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—21no—
101Dic142—44no—
102Dic142#4112yescomplemented
103C8×C71—21no—
104C8×C71—21no—
105C71⋊C8—22no—
106C22×C71C22×D411yescharacteristic
107C2×D71—22no—
108C2×D71—22no—
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—41no—
112C2×D71C2×SD1614yes—
113C4×C71—21no—
114C4×C71—42no—
115C4×C71—42no—
116C4×C71C2×SD1612yes—
117C4×C71C2×#4111yescharacteristic
118C4×C71H(5,𝔽2)11yescharacteristic
119Dic71—22no—
120Dic71—22no—
121Dic71—41no—
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#487D7111yescharacteristic, complemented
133#483—712no—
134#475—711no—
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—712no—
144C2×SD16—711no—
145C22×C8—712no—
146C4⋊Q8—1421no—
147C4⋊Q8C2×D7111yescharacteristic
148C4⋊Q8C2×D7111yescharacteristic
149#123—1422no—
150C22⋊Q8—712no—
151C22⋊Q8—1422no—
152C22⋊Q8—1421no—
153#121—712no—
154C4×Q8—711no—
155C4×D4—711no—
156C4×D4—711no—
157C4×D4—1421no—
158C2×C4⋊C4—712no—
159#106—711no—
160#106—712no—
161C4⋊C8—711no—
162#103—712no—
163#103—711no—
164#103—712no—
165#103C2×D7112yes—
166#103C2×D7112yes—
167#102—711no—
168C22⋊C8—712no—
169C4×C8C2×D7111yescharacteristic
170#41—2844no—
171C2×Q8—21no—
172C2×Q8—2842no—
173C2×Q8—711no—
174C2×Q8—1422no—
175C2×Q8—1424no—
176C2×Q8C22×D7112yes—
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—712no—
188C2×C8C22×D7111yescharacteristic
189C2×C8C22×D7111yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—711no—
194C4⋊C4—711no—
195C4⋊C4—711no—
196C4⋊C4—1422no—
197C4⋊C4—1422no—
198C4⋊C4—1421no—
199C4⋊C4C22×D7112yes—
200C22⋊C4—712no—
201C22⋊C4—1422no—
202C22⋊C4—1422no—
203C22⋊C4—1421no—
204C42—1421no—
205C42C22×D7111yescharacteristic
206C23—712no—
207C23—1421no—
208Q8—22no—
209Q8—22no—
210Q8—42no—
211Q8—2844no—
212Q8—712no—
213Q8—1421no—
214D4—2844no—
215D4—714no—
216D4—1421no—
217D4—1421no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—2844no—
221C2×C4—2844no—
222C2×C4—2842no—
223C2×C4—712no—
224C2×C4—1421no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4D4×D7111yescharacteristic
230C2×C4D4×D7111yescharacteristic
231C2×C4C23×D7111yescharacteristic
232C8—21no—
233C8—21no—
234C8—1422no—
235C22—2841no—
236C22—2841no—
237C22—2841no—
238C22—714no—
239C22—1422no—
240C22—1422no—
241C22C2×D4×D7111yescharacteristic
242C4—21no—
243C4—42no—
244C4—42no—
245C4—2841no—
246C4—2841no—
247C4—1422no—
248C4—1422no—
249C4SD16×D7112yes—
250C4#41×D7111yescharacteristic
251C4#1455617411yescharacteristic
252C2—5681no—
253C2—1424no—
254C2#1798664211yescharacteristic
255C2C2×SD16×D7111yescharacteristic
256C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3298
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #17986664C2semidirect
3. #17986328C2semidirect
4. #17986281C2semidirect
5. #17986231C2semidirect
6. #17986201C2semidirect
7. #17986139C2semidirect
8. C4×#11765854C2semidirect
9. C8×D284C2semidirect
10. #103×D71C2semidirect
11. #487×C71C2semidirect
94. D284SD16semidirect
102. Dic142#41semidirect
129. C71#3298semidirect
132. #487D71semidirect