Finite Group Databasebeta

← #22594117

Subgroups — #22594117

Order 9088.

702 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556096C211yesmaximal, characteristic
2C22×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
3#17985928C212yesmaximal, complemented
4#17985926C212yesmaximal, complemented
5#17985915C212yesmaximal, complemented
6#17985914C212yesmaximal, complemented
7#17985879C211yesmaximal, characteristic, complemented
8#17985874C211yesmaximal, characteristic, complemented
9#17985872C211yesmaximal, characteristic, complemented
10#17985854C211yesmaximal, characteristic, complemented
11#386×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C23×Dic71C2211yescharacteristic
16#14556096—24no—
17C2×C4⋊Dic71C2212yescomplemented
18C2×Dic71⋊C4C2212yescomplemented
19C2×C22⋊Dic71C2211yescharacteristic
20C2×C22⋊Dic71C2211yescharacteristic
21C2×C22⋊Dic71C2212yescomplemented
22C2×(C2×D71)⋊C4—24no—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2211yescharacteristic
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C71⋊#95C2212yescomplemented
30C71⋊#95C2212yescomplemented
31C71⋊#95C2212yescomplemented
32C22×D4×C71C2211yescharacteristic
33C23×C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2212yescomplemented
36#95×C71C2212yescomplemented
37C23×D71—22no—
38C23×D71—22no—
39C23×D71—42no—
40C23×D71—42no—
41C23×D71D414yes—
42C23×D71C2311yescharacteristic
43C23×D71C2311yescharacteristic
44C2×C22⋊D71—22no—
45C2×C22⋊D71—22no—
46C2×C22⋊D71—42no—
47C2×C22⋊D71—42no—
48C22×Dic71—22no—
49C22×Dic71—42no—
50C22×Dic71C2312yes—
51C22×Dic71C2311yescharacteristic
52C22×Dic71C2311yescharacteristic
53C22×Dic71C2312yes—
54C4⋊Dic71—42no—
55Dic71⋊C4—42no—
56C22⋊Dic71—22no—
57C22⋊Dic71—22no—
58C22⋊Dic71—42no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—42no—
64(C2×D71)⋊C4—42no—
65C24×C71C2311yescharacteristic
66C24×C71C2311yescharacteristic
67C2×D4×C71—22no—
68C2×D4×C71—22no—
69C2×D4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—42no—
72C22×C4×C71C2311yescharacteristic
73C22×C4×C71C2312yes—
74C22×C4×C71C2312yes—
75C22⋊C4×C71—22no—
76C22⋊C4×C71—42no—
77C23×C71—22no—
78C23×C71—22no—
79C23×C71—21no—
80C23×C71—21no—
81C23×C71—21no—
82C23×C71—22no—
83C23×C71—22no—
84C23×C71C2×D412yes—
85C23×C71#4112yes—
86C23×C71#4112yes—
87C23×C71C2411yescharacteristic
88C22×D71—28no—
89C22×D71—21no—
90C22×D71—21no—
91C22×D71—21no—
92C22×D71—21no—
93C22×D71—22no—
94C22×D71—22no—
95C22×D71—22no—
96C22×D71—22no—
97C22×D71—44no—
98C22×D71—44no—
99C22×D71—48no—
100C22×D71—48no—
101C22×D71C2×D412yes—
102C22×D71C2×D412yes—
103D4×C71—44no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C2×C4×C71—22no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—44no—
114C22⋊D71—44no—
115C22⋊D71—44no—
116C2×Dic71—22no—
117C2×Dic71—21no—
118C2×Dic71—21no—
119C2×Dic71—22no—
120C2×Dic71—22no—
121C2×Dic71—22no—
122C2×Dic71—22no—
123C2×Dic71—22no—
124C2×Dic71—22no—
125C2×Dic71—22no—
126C2×Dic71—44no—
127C22×C71—22no—
128C22×C71—22no—
129C22×C71—24no—
130C22×C71—24no—
131C22×C71—41no—
132C22×C71—41no—
133C22×C71—41no—
134C22×C71—42no—
135C22×C71—42no—
136C22×C71#12314yes—
137C22×C71C22×D411yescharacteristic
138C22×C71C2×#4111yescharacteristic
139C22×C71C2×#4111yescharacteristic
140C22×C71H(5,𝔽2)12yes—
141C22×C71H(5,𝔽2)12yes—
142C2×D71—22no—
143C2×D71—22no—
144C2×D71—22no—
145C2×D71—22no—
146C2×D71—22no—
147C2×D71—22no—
148C2×D71—416no—
149C2×D71—42no—
150C2×D71—42no—
151C2×D71—42no—
152C2×D71—42no—
153C4×C71—41no—
154C4×C71—42no—
155C4×C71—42no—
156Dic71—42no—
157Dic71—41no—
158Dic71—41no—
159Dic71—42no—
160C2×C71—24no—
161C2×C71—81no—
162C2×C71C2×#12311yescharacteristic
163C2×C71#53411yescharacteristic
164C2×C71#53411yescharacteristic
165C2×C71#53812yes—
166C2×C71#53812yes—
167D71—42no—
168D71—42no—
169#2622—711nomaximal, Sylow
170C71#262211yescharacteristic, complemented, Sylow
171C2×#120—711no—
172C22×C22⋊C4—711no—
173#397—712no—
174#397—712no—
175#394—712no—
176#394—712no—
177#386—711no—
178#386D7111yescharacteristic, complemented
179#385—711no—
180#385—711no—
181#379—711no—
182C25—711no—
183C22×D4—711no—
184C22×D4—711no—
185C22×D4C2×D7111yescharacteristic
186C23×C4—711no—
187C23×C4C2×D7111yescharacteristic
188#120—1424no—
189C2×C4⋊C4—712no—
190C2×C4⋊C4—712no—
191C2×C22⋊C4—712no—
192C2×C22⋊C4—712no—
193C2×C22⋊C4—712no—
194C2×C22⋊C4—711no—
195C2×C22⋊C4—711no—
196C2×C22⋊C4—712no—
197C2×C22⋊C4—712no—
198C2×C22⋊C4—711no—
199C2×C22⋊C4—711no—
200C2×C22⋊C4—1424no—
201C2×C22⋊C4C2×D7111yescharacteristic
202C2×C22⋊C4C2×D7112yescomplemented
203#95—712no—
204#95—712no—
205#95—712no—
206#95C2×D7112yescomplemented
207C24—2842no—
208C24—2842no—
209C24—714no—
210C24—711no—
211C24—711no—
212C24—1422no—
213C24—1422no—
214C24C22×D7111yescharacteristic
215C24C22×D7111yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—2842no—
220C2×D4—2842no—
221C2×D4—1422no—
222C2×D4—1422no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—2842no—
226C22×C4—712no—
227C22×C4—712no—
228C22×C4—711no—
229C22×C4—711no—
230C22×C4—1422no—
231C22×C4C22×D7111yescharacteristic
232C22×C4C22×D7112yes—
233C22×C4C22×D7112yes—
234C4⋊C4—2842no—
235C4⋊C4—2842no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—2842no—
239C22⋊C4—2842no—
240C22⋊C4—2842no—
241C22⋊C4—1424no—
242C22⋊C4—1424no—
243C22⋊C4—1422no—
244C22⋊C4—1422no—
245C22⋊C4—1422no—
246C22⋊C4—1422no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—2848no—
255C23—2848no—
256C23—2844no—
257C23—2844no—
258C23—712no—
259C23—712no—
260C23—1421no—
261C23—1421no—
262C23—1421no—
263C23—1421no—
264C23—1422no—
265C23—1422no—
266C23—1422no—
267C23—1422no—
268C23—1428no—
269C23D4×D7112yes—
270C23#1176586412yes—
271C23#1176586512yes—
272C23C23×D7111yescharacteristic
273D4—44no—
274D4—2844no—
275D4—2844no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—2844no—
287C2×C4—1422no—
288C2×C4—1421no—
289C2×C4—1421no—
290C2×C4—1422no—
291C2×C4—1422no—
292C2×C4—1422no—
293C2×C4—1422no—
294C2×C4—1422no—
295C2×C4—1422no—
296C2×C4—1422no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—22no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—28416no—
307C22—2842no—
308C22—2842no—
309C22—2842no—
310C22—2842no—
311C22—1422no—
312C22—1422no—
313C22—1422no—
314C22—1422no—
315C22—1422no—
316C22—1422no—
317C22#1455611014yes—
318C22C2×D4×D7111yescharacteristic
319C22C2×#1176586411yescharacteristic
320C22C2×#1176586511yescharacteristic
321C22#1455617312yes—
322C22#1455617412yes—
323C4—42no—
324C4—41no—
325C4—42no—
326C4—2842no—
327C4—2841no—
328C4—2842no—
329C4—2841no—
330C2—24no—
331C2—81no—
332C2—2842no—
333C2—2842no—
334C2C2×#1455611011yescharacteristic
335C2#1798655911yescharacteristic
336C2#1798656011yescharacteristic
337C2#1798658212yes—
338C2#1798658612yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#386×C71C2
hypercenter#386D71
hypocenterC71#2622
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×(C2×D71)⋊C4C2semidirect
3. #17985928C2semidirect
4. #17985926C2semidirect
5. #17985915C2semidirect
6. #17985914C2semidirect
7. #17985879C2semidirect
8. #17985874C2semidirect
9. #17985872C2semidirect
10. #17985854C2semidirect
11. #386×C71C2semidirect
17. C2×C4⋊Dic71C22semidirect
18. C2×Dic71⋊C4C22semidirect
21. C2×C22⋊Dic71C22semidirect
29. C71⋊#95C22semidirect
30. C71⋊#95C22semidirect
31. C71⋊#95C22semidirect
35. C2×C22⋊C4×C71C22semidirect
36. #95×C71C22semidirect
170. C71#2622semidirect
178. #386D71semidirect
202. C2×C22⋊C4C2×D71semidirect
206. #95C2×D71semidirect