Finite Group Databasebeta

← #22594479

Subgroups — #22594479

Order 9088.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556122C211yesmaximal, characteristic, complemented
2#17985922C212yesmaximal, complemented
3#17985918C211yesmaximal, characteristic, complemented
4#17985915C212yesmaximal, complemented
5#17985913C211yesmaximal, characteristic, complemented
6#17985910C211yesmaximal, characteristic, complemented
7#17985907C211yesmaximal, characteristic
8#17985899C212yesmaximal, complemented
9#17985893C212yesmaximal, complemented
10C4×C4⋊Dic71C211yesmaximal, characteristic, complemented
11C2×#124×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2212yes—
13#14556122—24no—
14C2×C4⋊Dic71C2212yes—
15C2×Dic71⋊C4C2211yescharacteristic
16C2×Dic71⋊C4C2211yescharacteristic
17C2×Dic71⋊C4C2211yescharacteristic
18C2×Dic71⋊C4C2212yes—
19C2×C22⋊Dic71C2212yes—
20C2×C22⋊Dic71C2212yes—
21C2×C22⋊Dic71C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2211yescharacteristic
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2212yes—
26C2×C4×Dic71C2212yes—
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C22×Q8×C71C2211yescharacteristic
32C22×D4×C71C2211yescharacteristic
33#124×C71—24no—
34C2×C22⋊C4×C71C2212yes—
35C2×C22⋊C4×C71C2212yes—
36C2×C42×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—24no—
39C2×C22⋊D71—42no—
40C2×C22⋊D71—42no—
41C22×Dic71C2312yes—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C4⋊Dic71—24no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—42no—
51C22⋊Dic71—24no—
52C22⋊Dic71—42no—
53C22⋊Dic71—42no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—42no—
58C4×Dic71—42no—
59C24×C71C2312yes—
60C2×Q8×C71—22no—
61C2×Q8×C71—42no—
62C2×D4×C71—22no—
63C2×D4×C71—42no—
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68C22⋊C4×C71—24no—
69C22⋊C4×C71—24no—
70C42×C71D414yescomplemented
71C23×C71—22no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71—22no—
78C23×C71C2411yescharacteristic
79C22×D71—21no—
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—21no—
83C22×D71—21no—
84Q8×C71—44no—
85D4×C71—44no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71C2×D412yes—
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2×D412yes—
96C22⋊D71—48no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—21no—
107C2×Dic71—22no—
108C2×Dic71—21no—
109C2×Dic71#4114yes—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71—42no—
113C22×C71—42no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71—42no—
117C22×C71C22×D411yescharacteristic
118C22×C71C2×#4112yes—
119C22×C71H(5,𝔽2)12yes—
120C22×C71H(5,𝔽2)11yescharacteristic
121C22×C712−1+411yescharacteristic
122C2×D71—41no—
123C2×D71—41no—
124C2×D71—41no—
125C2×D71—42no—
126C2×D71—42no—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—22no—
130C4×C71—42no—
131Dic71—24no—
132Dic71—41no—
133Dic71—41no—
134Dic71—41no—
135Dic71—42no—
136C2×C71—82no—
137C2×C71#53211yescharacteristic
138C2×C71#53511yescharacteristic
139C2×C71#53711yescharacteristic
140C2×C71#53812yes—
141C2×C71#54212yes—
142D71—81no—
143#2662—711nomaximal, Sylow
144C71#266211yescharacteristic, complemented, Sylow
145C2×#126—711no—
146C2×#124D7111yescharacteristic, complemented
147#396—712no—
148#395—711no—
149#394—711no—
150#394—712no—
151#393—711no—
152#392—711no—
153#390—712no—
154#388—712no—
155C4×C4⋊C4—711no—
156C22×Q8C2×D7111yescharacteristic
157C22×D4—712no—
158C22×D4C2×D7111yescharacteristic
159#126—1424no—
160#124—24no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4—711no—
163C2×C4⋊C4—711no—
164C2×C4⋊C4—711no—
165C2×C4⋊C4—712no—
166C2×C22⋊C4—712no—
167C2×C22⋊C4—712no—
168C2×C22⋊C4—712no—
169C2×C22⋊C4—712no—
170C2×C22⋊C4—711no—
171C2×C22⋊C4—711no—
172C2×C22⋊C4—711no—
173C2×C22⋊C4C2×D7112yes—
174C2×C22⋊C4C2×D7112yes—
175C2×C42—712no—
176C2×C42C2×D7111yescharacteristic
177#95—711no—
178#95—711no—
179#95—712no—
180#95—712no—
181C24—711no—
182C24C22×D7112yes—
183C2×Q8—22no—
184C2×Q8—42no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—2842no—
188C2×D4—2842no—
189C2×D4—1424no—
190C22×C4—712no—
191C22×C4—712no—
192C22×C4—711no—
193C22×C4—711no—
194C22×C4—711no—
195C22×C4C22×D7112yes—
196C22×C4C22×D7111yescharacteristic
197C22×C4C22×D7111yescharacteristic
198C22×C4C22×D7111yescharacteristic
199C4⋊C4—2842no—
200C4⋊C4—1424no—
201C4⋊C4—1422no—
202C4⋊C4—1422no—
203C4⋊C4—1422no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—2842no—
207C22⋊C4—2842no—
208C22⋊C4—2842no—
209C22⋊C4—1424no—
210C22⋊C4—1422no—
211C22⋊C4—1422no—
212C22⋊C4—1422no—
213C42—2842no—
214C42C22⋊D7114yescomplemented
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—1421no—
223C23—1422no—
224C23—1422no—
225C23—1421no—
226C23—1421no—
227C23C23×D7111yescharacteristic
228Q8—44no—
229D4—44no—
230D4—2848no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—714no—
239C2×C4—1421no—
240C2×C4—1421no—
241C2×C4—1421no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1422no—
247C2×C4—1421no—
248C2×C4—1421no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4C2×C22⋊D7112yes—
252C2×C4C2×C22⋊D7112yes—
253C2×C4C2×C22⋊D7112yes—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—2841no—
262C22—2841no—
263C22—2842no—
264C22—2842no—
265C22—2841no—
266C22C22×C22⋊D7111yescharacteristic
267C22#41×D7112yes—
268C22#1455617312yes—
269C22#1455617411yescharacteristic
270C22#1455617511yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—2841no—
276C4—2841no—
277C4—2842no—
278C4—2841no—
279C4—1424no—
280C2—82no—
281C2—5681no—
282C2#1798654511yescharacteristic
283C2#1798656611yescharacteristic
284C2#1798657711yescharacteristic
285C2#1798658312yes—
286C2#1798661312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#124×C71C2
hypercenterC2×#124D71
hypocenterC71#2662
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14556122C2semidirect
2. #17985922C2semidirect
3. #17985918C2semidirect
4. #17985915C2semidirect
5. #17985913C2semidirect
6. #17985910C2semidirect
8. #17985899C2semidirect
9. #17985893C2semidirect
10. C4×C4⋊Dic71C2semidirect
11. C2×#124×C71C2semidirect
70. C42×C71D4semidirect
144. C71#2662semidirect
146. C2×#124D71semidirect
214. C42C22⋊D71semidirect