Finite Group Databasebeta

← #22594267

Subgroups — #22594267

Order 9088.

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

normal subgroups only

This group has 576 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×#14556108C211yesmaximal, characteristic, complemented
2C2×#14556101C211yesmaximal, characteristic
3#17985941C212yesmaximal, complemented
4#17985925C211yesmaximal, characteristic, complemented
5#17985915C212yesmaximal, complemented
6#17985912C211yesmaximal, characteristic, complemented
7#17985911C211yesmaximal, characteristic, complemented
8#17985877C212yesmaximal, complemented
9#17985875C212yesmaximal, complemented
10#95×D71C211yesmaximal, characteristic, complemented
11#397×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C22×C22⋊D71C2211yescharacteristic
16C22×C4×D71C2212yes—
17C22×C4×D71C2211yescharacteristic
18#14556108—24no—
19#14556101—24no—
20C2×C4⋊Dic71C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×C22⋊Dic71C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C23×D71C2311yescharacteristic
39C2×Dic142—22no—
40C2×Dic142—42no—
41C2×D284—22no—
42C2×D284—24no—
43C2×C22⋊D71—22no—
44C2×C22⋊D71—24no—
45C2×C22⋊D71—22no—
46C2×C22⋊D71—42no—
47C22×Dic71C2312yes—
48C22×Dic71C2311yescharacteristic
49C22×Dic71C2312yes—
50C22×Dic71C2311yescharacteristic
51C2×C4×D71—24no—
52C2×C4×D71—42no—
53C2×C4×D71—42no—
54C2×C4×D71—42no—
55C2×C4×D71D414yes—
56C4⋊Dic71—24no—
57Dic71⋊C4—22no—
58C22⋊Dic71—42no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—42no—
62(C2×D71)⋊C4—42no—
63C24×C71C2311yescharacteristic
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2312yes—
67C22×C4×C71C2311yescharacteristic
68C4⋊C4×C71—22no—
69C22⋊C4×C71—22no—
70C22⋊C4×C71—42no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—21no—
74C23×C71—22no—
75C23×C71—21no—
76C23×C71C2411yescharacteristic
77C22×D71—21no—
78C22×D71—22no—
79C22×D71—21no—
80C22×D71—21no—
81C22×D71—22no—
82C22×D71—21no—
83C22×D71—22no—
84C22×D71—21no—
85C22×D71C2×D412yes—
86C22×D71#4114yes—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71C2×D412yes—
97C22⋊D71—44no—
98C22⋊D71—44no—
99C2×Dic71—22no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71C2×D412yes—
109D284—44no—
110C4×D71—44no—
111C4×D71—48no—
112Dic142—44no—
113C22×C71—41no—
114C22×C71—41no—
115C22×C71—41no—
116C22×C71—42no—
117C22×C71—42no—
118C22×C71C22×D411yescharacteristic
119C22×C71C2×#4112yes—
120C22×C71H(5,𝔽2)11yescharacteristic
121C22×C71H(5,𝔽2)12yes—
122C22×C712−1+411yescharacteristic
123C2×D71—22no—
124C2×D71—22no—
125C2×D71—24no—
126C2×D71—24no—
127C2×D71—41no—
128C2×D71—41no—
129C2×D71—41no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71#12414yes—
133C4×C71—22no—
134C4×C71—41no—
135C4×C71—42no—
136C4×C71—42no—
137Dic71—22no—
138Dic71—41no—
139Dic71—42no—
140Dic71—42no—
141C2×C71—81no—
142C2×C71C2×#12411yescharacteristic
143C2×C71#53411yescharacteristic
144C2×C71#53711yescharacteristic
145C2×C71#53812yes—
146C2×C71#53912yes—
147D71—24no—
148D71—81no—
149#2639—711nomaximal, Sylow
150C71#263911yescharacteristic, complemented, Sylow
151C2×C22⋊Q8—711no—
152C2×#121—711no—
153#400—712no—
154#397—711no—
155#397D7111yescharacteristic, complemented
156#394—712no—
157#393—711no—
158#393—711no—
159#386—712no—
160#386—712no—
161C2×#95—711no—
162C22×Q8—711no—
163C22×D4—711no—
164C22×D4—711no—
165C22×D4—711no—
166C23×C4—711no—
167C23×C4—712no—
168C22⋊Q8—1424no—
169#121—1424no—
170C2×C4⋊C4—711no—
171C2×C4⋊C4—712no—
172C2×C4⋊C4C2×D7111yescharacteristic
173C2×C22⋊C4—712no—
174C2×C22⋊C4—712no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—711no—
177C2×C22⋊C4—712no—
178C2×C22⋊C4C2×D7111yescharacteristic
179C2×C22⋊C4C2×D7112yes—
180#95—711no—
181#95—712no—
182#95—712no—
183#95—712no—
184#95—712no—
185#95C2×D7111yescharacteristic
186#95C2×D7112yes—
187C24—711no—
188C24—711no—
189C24C22×D7111yescharacteristic
190C2×Q8—2842no—
191C2×Q8—1422no—
192C2×D4—2842no—
193C2×D4—1422no—
194C2×D4—1424no—
195C2×D4—1422no—
196C2×D4—1422no—
197C2×D4—1424no—
198C22×C4—2842no—
199C22×C4—2842no—
200C22×C4—2842no—
201C22×C4—714no—
202C22×C4—711no—
203C22×C4—712no—
204C22×C4—711no—
205C22×C4—712no—
206C22×C4—1424no—
207C22×C4C22×D7112yes—
208C22×C4C22×D7111yescharacteristic
209C22×C4C22×D7112yes—
210C22×C4C22×D7111yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—1422no—
213C4⋊C4—1424no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—2842no—
217C22⋊C4—2842no—
218C22⋊C4—2842no—
219C22⋊C4—1424no—
220C22⋊C4—1422no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—712no—
227C23—714no—
228C23—1421no—
229C23—1422no—
230C23—1421no—
231C23—1422no—
232C23—1421no—
233C23—1421no—
234C23—1422no—
235C23—1421no—
236C23C23×D7111yescharacteristic
237Q8—2844no—
238D4—2844no—
239D4—2844no—
240D4—2844no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—2844no—
251C2×C4—2848no—
252C2×C4—712no—
253C2×C4—1421no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4—1422no—
262C2×C4D4×D7112yes—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—2841no—
269C22—2841no—
270C22—2842no—
271C22—2842no—
272C22—2841no—
273C22—714no—
274C22—1422no—
275C22—1422no—
276C22—1424no—
277C22—1424no—
278C22C2×D4×D7111yescharacteristic
279C22#41×D7112yes—
280C22#1455617311yescharacteristic
281C22#1455617412yes—
282C22#1455617511yescharacteristic
283C4—22no—
284C4—42no—
285C4—42no—
286C4—41no—
287C4—2842no—
288C4—2842no—
289C4—2841no—
290C4—1422no—
291C2—81no—
292C2—5681no—
293C2—1424no—
294C2#124×D7111yescharacteristic
295C2#1798656011yescharacteristic
296C2#1798657811yescharacteristic
297C2#1798658512yes—
298C2#1798659012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#397×C71C2
hypercenter#397D71
hypocenterC71#2639
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14556108C2semidirect
3. #17985941C2semidirect
4. #17985925C2semidirect
5. #17985915C2semidirect
6. #17985912C2semidirect
7. #17985911C2semidirect
8. #17985877C2semidirect
9. #17985875C2semidirect
10. #95×D71C2semidirect
11. #397×C71C2semidirect
150. C71#2639semidirect
155. #397D71semidirect