Finite Group Databasebeta

← #22594141

Subgroups — #22594141

Order 9088.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14556106C211yesmaximal, characteristic
2C2×#14556086C211yesmaximal, characteristic
3#17985937C211yesmaximal, characteristic, complemented
4#17985931C211yesmaximal, characteristic, complemented
5#17985929C211yesmaximal, characteristic, complemented
6#17985928C212yesmaximal
7#17985926C212yesmaximal
8#17985903C212yesmaximal, complemented
9#17985896C212yesmaximal, complemented
10#17985858C211yesmaximal, characteristic
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14#14556106—24no—
15#14556086—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2212yes—
26C2×C4×Dic71C2212yes—
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—24no—
40C22×Dic71C2311yescharacteristic
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C2×C4×D71—24no—
46C2×C4×D71D414yes—
47C4⋊Dic71—22no—
48C4⋊Dic71—42no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—24no—
52Dic71⋊C4—42no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—42no—
57C4×Dic71—24no—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2312yes—
60C22×C4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C23×C71C2411yescharacteristic
66C22×D71—22no—
67C22×D71—21no—
68C22×D71—22no—
69C22×D71—21no—
70C22×D71C2×D412yes—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—24no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71C2×D412yes—
83C2×Dic71—22no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—21no—
94C2×Dic71C2×D412yes—
95C2×Dic71#4114yes—
96C4×D71—28no—
97Dic142—44no—
98C22×C71C22×D411yescharacteristic
99C22×C71C2×#4112yes—
100C22×C71H(5,𝔽2)11yescharacteristic
101C22×C712−1+412yes—
102C22×C712−1+411yescharacteristic
103C2×D71—22no—
104C2×D71—22no—
105C2×D71—22no—
106C2×D71—42no—
107C2×D71—42no—
108C4×C71—22no—
109C4×C71—41no—
110C4×C71—42no—
111C4×C71—41no—
112C4×C71—42no—
113Dic71—41no—
114Dic71—42no—
115Dic71—41no—
116Dic71—42no—
117Dic71#12314yes—
118C2×C71C2×#12311yescharacteristic
119C2×C71#53611yescharacteristic
120C2×C71#53711yescharacteristic
121C2×C71#54112yes—
122C2×C71#54212yes—
123D71—42no—
124#2625—711nomaximal, Sylow
125C71#262511yescharacteristic, complemented, Sylow
126C2×C22⋊Q8—711no—
127C2×#117—711no—
128#400—711no—
129#400D7111yescharacteristic, complemented
130#398—711no—
131#398—711no—
132#397—712no—
133#397—712no—
134#391—712no—
135#389—712no—
136#380—711no—
137C22×Q8—711no—
138C23×C4—711no—
139C22⋊Q8—1424no—
140#117—1424no—
141C2×C4⋊C4—712no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—711no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4C2×D7111yescharacteristic
148C2×C4⋊C4C2×D7111yescharacteristic
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—711no—
152C2×C22⋊C4—711no—
153C2×C42—712no—
154#95—711no—
155#95—712no—
156#95—712no—
157#95—712no—
158#95—712no—
159#95C2×D7111yescharacteristic
160#95C2×D7112yes—
161#95C2×D7112yes—
162C24—711no—
163C2×Q8—1422no—
164C2×Q8—1424no—
165C22×C4—711no—
166C22×C4—714no—
167C22×C4—712no—
168C22×C4—711no—
169C22×C4—712no—
170C22×C4—711no—
171C22×C4—1424no—
172C22×C4C22×D7112yes—
173C22×C4C22×D7111yescharacteristic
174C22×C4C22×D7112yes—
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7111yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—2842no—
180C4⋊C4—2842no—
181C4⋊C4—1424no—
182C4⋊C4—1422no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C22⋊C4—2842no—
186C22⋊C4—1424no—
187C22⋊C4—1422no—
188C22⋊C4—1422no—
189C42—1424no—
190C23—712no—
191C23—1422no—
192C23—1421no—
193C23—1422no—
194C23—1421no—
195C23C23×D7111yescharacteristic
196Q8—2844no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—712no—
209C2×C4—714no—
210C2×C4—1428no—
211C2×C4—1422no—
212C2×C4—1421no—
213C2×C4—1421no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1421no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4D4×D7112yes—
223C22—2842no—
224C22—2842no—
225C22—1422no—
226C22—1422no—
227C22—1422no—
228C22C2×D4×D7111yescharacteristic
229C22#41×D7112yes—
230C22#1455617411yescharacteristic
231C22#1455617512yes—
232C22#1455617611yescharacteristic
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—2842no—
239C4—2841no—
240C4—2842no—
241C4—2841no—
242C4—714no—
243C2—2842no—
244C2#123×D7111yescharacteristic
245C2#1798656911yescharacteristic
246C2#1798657911yescharacteristic
247C2#1798660812yes—
248C2#1798661512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2625
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #17985937C2semidirect
4. #17985931C2semidirect
5. #17985929C2semidirect
8. #17985903C2semidirect
9. #17985896C2semidirect
11. #400×C71C2semidirect
125. C71#2625semidirect
129. #400D71semidirect