Finite Group Databasebeta

← #22591581

Subgroups — #22591581

Order 9088.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#14556090C211yesmaximal, characteristic, complemented
2C2×#14556090C212yesmaximal, complemented
3C2×#14556086C211yesmaximal, characteristic, complemented
4#17985925C211yesmaximal, characteristic, complemented
5#17985921C212yesmaximal, complemented
6#17985917C212yesmaximal, complemented
7#17985916C211yesmaximal, characteristic, complemented
8#17985909C211yesmaximal, characteristic
9#17985850C211yesmaximal, characteristic, complemented
10#17985846C212yesmaximal, complemented
11#398×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×D284C2212yes—
14C22×C4×D71C2212yes—
15C22×C4×D71C2211yescharacteristic
16#14556090—28no—
17#14556090—24no—
18#14556086—24no—
19C2×C4⋊Dic71C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2211yescharacteristic
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×C4×Dic71C2211yescharacteristic
29C2×C4×Dic71C2212yes—
30C2×C4×Dic71C2211yescharacteristic
31C71⋊#95C2212yes—
32C71⋊#95C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C4⋊C4×C71C2212yes—
35C2×C4⋊C4×C71C2211yescharacteristic
36#95×C71C2212yes—
37#95×C71C2211yescharacteristic
38C23×D71C2311yescharacteristic
39C23×D71C2312yes—
40C2×D284—24no—
41C2×D284—24no—
42C2×D284—24no—
43C2×D284—22no—
44C2×D284—24no—
45C22×Dic71C2312yes—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—24no—
52C2×C4×D71—24no—
53C2×C4×D71—24no—
54C2×C4×D71D414yescomplemented
55C4⋊Dic71—22no—
56Dic71⋊C4—24no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—24no—
64C4×Dic71—22no—
65C4×Dic71—24no—
66C4×Dic71D414yescomplemented
67C22×C4×C71C2312yes—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2311yescharacteristic
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2312yes—
72C4⋊C4×C71—22no—
73C4⋊C4×C71—24no—
74C4⋊C4×C71—22no—
75C23×C71C2411yescharacteristic
76C22×D71—21no—
77C22×D71—22no—
78C22×D71—22no—
79C22×D71—22no—
80C22×D71—21no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71C2×D412yes—
87C22×D71#4114yes—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71C2×D412yes—
99C2×C4×C71C2×D412yes—
100C2×C4×C71#4114yes—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—21no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71C2×D412yes—
108C2×Dic71C2×D412yes—
109C2×Dic71C2×D412yes—
110C2×Dic71#4112yes—
111D284—28no—
112D284—216no—
113C4×D71—28no—
114C4×D71—48no—
115C22×C71C22×D411yescharacteristic
116C22×C71C22×D411yescharacteristic
117C22×C71C2×#4111yescharacteristic
118C22×C71C2×#4112yes—
119C22×C71C2×#4112yes—
120C2×D71—22no—
121C2×D71—24no—
122C2×D71—24no—
123C2×D71—24no—
124C2×D71—24no—
125C2×D71—42no—
126C2×D71—41no—
127C2×D71—42no—
128C2×D71—42no—
129C2×D71—42no—
130C2×D71—41no—
131C2×D71—42no—
132C4×C71—22no—
133C4×C71—22no—
134C4×C71—24no—
135C4×C71—42no—
136C4×C71—41no—
137Dic71—22no—
138Dic71—42no—
139Dic71—41no—
140Dic71#12114yes—
141C2×C71C2×#12111yescharacteristic
142C2×C71#52511yescharacteristic
143C2×C71#52512yes—
144C2×C71#52912yes—
145C2×C71#53211yescharacteristic
146D71—42no—
147D71—44no—
148#2392—711nomaximal, Sylow
149C71#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—711no—
151C2×C4×D4—712no—
152C2×#117—711no—
153#398D7111yescharacteristic, complemented
154#397—711no—
155#396—712no—
156#394—711no—
157#394—712no—
158#392—711no—
159C4×C4⋊C4—711no—
160C4×C22⋊C4—712no—
161C22×D4—711no—
162C22×D4—712no—
163C23×C4—711no—
164C23×C4—712no—
165C4×D4—1424no—
166C4×D4—1428no—
167#117—1424no—
168C2×C4⋊C4—711no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4C2×D7111yescharacteristic
171C2×C4⋊C4C2×D7112yes—
172C2×C4⋊C4C2×D7111yescharacteristic
173C2×C22⋊C4—711no—
174C2×C22⋊C4—711no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C22⋊C4—712no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4—712no—
180C2×C42—711no—
181C2×C42—711no—
182C2×C42—712no—
183#95—712no—
184#95—711no—
185#95C2×D7112yes—
186#95C2×D7111yescharacteristic
187C24—711no—
188C24—712no—
189C2×D4—1424no—
190C2×D4—1424no—
191C2×D4—1424no—
192C2×D4—1422no—
193C2×D4—1424no—
194C22×C4—711no—
195C22×C4—711no—
196C22×C4—714no—
197C22×C4—712no—
198C22×C4—711no—
199C22×C4—1424no—
200C22×C4—1424no—
201C22×C4—1424no—
202C22×C4—1422no—
203C22×C4—1422no—
204C22×C4C22×D7112yes—
205C22×C4C22×D7111yescharacteristic
206C22×C4C22×D7111yescharacteristic
207C22×C4C22×D7111yescharacteristic
208C22×C4C22×D7112yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—1422no—
213C4⋊C4—1424no—
214C22⋊C4—1422no—
215C22⋊C4—1424no—
216C22⋊C4—1424no—
217C22⋊C4—1424no—
218C22⋊C4—1422no—
219C22⋊C4—1424no—
220C22⋊C4—1424no—
221C42—714no—
222C42—1422no—
223C42—1424no—
224C23—712no—
225C23—714no—
226C23—1421no—
227C23—1422no—
228C23—1422no—
229C23—1422no—
230C23—1421no—
231C23—1422no—
232C23—1422no—
233C23—1422no—
234C23—1422no—
235C23—1422no—
236C23C23×D7111yescharacteristic
237D4—1428no—
238D4—14216no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—2848no—
250C2×C4—712no—
251C2×C4—712no—
252C2×C4—712no—
253C2×C4—712no—
254C2×C4—1428no—
255C2×C4—1422no—
256C2×C4—1421no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4D4×D7112yes—
262C2×C4D4×D7112yes—
263C2×C4#1176586614yes—
264C22—2842no—
265C22—2841no—
266C22—2842no—
267C22—2842no—
268C22—2842no—
269C22—2841no—
270C22—2842no—
271C22—1422no—
272C22—1424no—
273C22—1424no—
274C22—1424no—
275C22—1424no—
276C22C2×D4×D7111yescharacteristic
277C22C2×D4×D7111yescharacteristic
278C22#41×D7111yescharacteristic
279C22#41×D7112yes—
280C22C2×#1176586612yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—2842no—
287C4—2841no—
288C4—714no—
289C4—1422no—
290C2—2844no—
291C2—2842no—
292C2#121×D7111yescharacteristic
293C2#1798650711yescharacteristic
294C2#1798651212yes—
295C2#1798653412yes—
296C2#1798655011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2392
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14556090C2semidirect
2. C2×#14556090C2semidirect
3. C2×#14556086C2semidirect
4. #17985925C2semidirect
5. #17985921C2semidirect
6. #17985917C2semidirect
7. #17985916C2semidirect
9. #17985850C2semidirect
10. #17985846C2semidirect
11. #398×C71C2semidirect
54. C2×C4×D71D4semidirect
66. C4×Dic71D4semidirect
149. C71#2392semidirect
153. #398D71semidirect