Finite Group Databasebeta

← #22599138

Subgroups — #22599138

Order 9088.

514 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14556099C211yesmaximal, characteristic, complemented
2#121×D71C211yesmaximal, characteristic, complemented
3#17986318C211yesmaximal, characteristic, complemented
4#17986316C212yesmaximal, complemented
5#17986315C211yesmaximal, characteristic, complemented
6#17986251C212yesmaximal, complemented
7#17986249C212yesmaximal, complemented
8#17986133C211yesmaximal, characteristic, complemented
9#17986131C212yesmaximal, complemented
10#17985961C211yesmaximal, characteristic, complemented
11#447×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71—21no—
13C2×D4×D71—21no—
14C2×D4×D71C2211yescharacteristic, complemented
15C22×C4×D71C2211yescharacteristic
16C2×#11765850C2212yescomplemented
17C2×#11765848C2212yescomplemented
18#14556103C2211yescharacteristic, complemented
19#14556102C2211yescharacteristic
20#14556101—21no—
21#14556100—21no—
22#14556099—21no—
23#14556099—21no—
24#14556099C2212yes—
25#14556099C2211yescharacteristic, complemented
26#14556099C2212yescomplemented
27C2×C4⋊Dic71C2211yescharacteristic, complemented
28C4⋊C4×D71C2211yescharacteristic, complemented
29C2×C22⋊Dic71—21no—
30C22⋊C4×D71—21no—
31#14556060C2212yescomplemented
32#14556059C2212yescomplemented
33#14556050C2212yescomplemented
34#14556049C2212yescomplemented
35#14556049C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic, complemented
37(C2×D71)⋊C8C2212yescomplemented
38C22×D4×C71C2211yescharacteristic, complemented
39C2×D8×C71C2212yescomplemented
40#121×C71C2211yescharacteristic
41#102×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic, complemented
43C23×D71—21no—
44C23×D71D411yescharacteristic
45C2×D284—21no—
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C2×C22⋊D71—21no—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C2×C22⋊D71—21no—
52C2×C22⋊D71—44no—
53D4×D71—42no—
54D4×D71—44no—
55C22×Dic71—21no—
56C22×Dic71D411yescharacteristic
57C2×C4×D71—21no—
58C2×C4×D71—42no—
59C2×C4×D71D411yescharacteristic
60C2×C4×D71D411yescharacteristic
61C2×C4×D71C2311yescharacteristic
62C2×C4×D71C2311yescharacteristic
63#11765850—42no—
64#11765848—42no—
65C2×C71⋊C8C2312yes—
66C4⋊Dic71—21no—
67C4⋊Dic71C2311yescharacteristic
68C4⋊Dic71C2312yes—
69Dic71⋊C4—21no—
70C22⋊Dic71—21no—
71C22⋊Dic71—22no—
72C22⋊Dic71—22no—
73(C2×D71)⋊C4—21no—
74(C2×D71)⋊C4—21no—
75C24×C71—21no—
76C2×D4×C71—21no—
77C2×D4×C71—21no—
78C2×D4×C71—42no—
79C2×D4×C71C2311yescharacteristic
80C2×D4×C71C2312yes—
81C22×C4×C71C2311yescharacteristic
82D8×C71—42no—
83C2×C8×C71C2312yes—
84C4⋊C4×C71C2311yescharacteristic
85C22⋊C4×C71—21no—
86C23×C71—21no—
87C23×C71—22no—
88C23×C71—42no—
89C23×C71—42no—
90C23×C71—42no—
91C23×C71C2×D411yescharacteristic
92C22×D71—22no—
93C22×D71—22no—
94C22×D71—21no—
95C22×D71—41no—
96C22×D71—42no—
97C22×D71—42no—
98C22×D71—42no—
99C22×D71C2×D411yescharacteristic
100C22×D71C2×D411yescharacteristic
101D4×C71—22no—
102D4×C71—22no—
103D4×C71—22no—
104D4×C71—42no—
105D4×C71—42no—
106C2×C4×C71—22no—
107C2×C4×C71—21no—
108C2×C4×C71C2×D411yescharacteristic
109C2×C4×C71C2411yescharacteristic
110C22⋊D71—42no—
111C22⋊D71—42no—
112C22⋊D71—44no—
113C22⋊D71—42no—
114C22⋊D71—44no—
115C2×Dic71—21no—
116C2×Dic71—22no—
117C2×Dic71—41no—
118C2×Dic71—42no—
119C2×Dic71C2×D411yescharacteristic
120C2×Dic71C2×D411yescharacteristic
121D284—22no—
122D284—42no—
123C4×D71—22no—
124C4×D71—22no—
125C4×D71—41no—
126C4×D71—41no—
127C4×D71—42no—
128C8×C71—22no—
129C71⋊C8—22no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—41no—
133C22×C71—42no—
134C22×C71—42no—
135C22×C71—41no—
136C22×C71—41no—
137C22×C71—42no—
138C22×C71—84no—
139C22×C71AGL(1,ℤ/8)12yes—
140C22×C71C22×D411yescharacteristic
141C2×D71—21no—
142C2×D71—21no—
143C2×D71—21no—
144C2×D71—21no—
145C2×D71—21no—
146C2×D71—44no—
147C2×D71—41no—
148C2×D71—41no—
149C2×D71—41no—
150C2×D71—84no—
151C2×D71AGL(1,ℤ/8)12yes—
152C4×C71—21no—
153C4×C71—41no—
154C4×C71H(5,𝔽2)12yes—
155Dic71—21no—
156Dic71—21no—
157Dic71—41no—
158Dic71—42no—
159C2×C71—22no—
160C2×C71—81no—
161C2×C71—82no—
162C2×C71#53411yescharacteristic
163C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
165D71—22no—
166D71—41no—
167D71—81no—
168#3192—711nomaximal, Sylow
169C71#319211yescharacteristic, complemented, Sylow
170C2×#121—711no—
171C2×#121—711no—
172#481—711no—
173#481—712no—
174#481—711no—
175#469—712no—
176#469—712no—
177#447—711no—
178#447—712no—
179#447D7111yescharacteristic, complemented
180#407—711no—
181C22×D4—711no—
182C22×D4—1421no—
183C22×D4—1421no—
184C22×D4C2×D7111yescharacteristic, complemented
185C23×C4—711no—
186C2×D8—712no—
187C2×D8C2×D7112yescomplemented
188C2×M16—712no—
189#121—712no—
190#121—711no—
191#121—712no—
192#121—711no—
193#121—711no—
194#121—1421no—
195#121—1421no—
196#121—1421no—
197#121—1421no—
198#121C2×D7111yescharacteristic
199C2×C4⋊C4—711no—
200C2×C4⋊C4—711no—
201C2×C22⋊C4—1421no—
202C2×C22⋊C4—1421no—
203#106—712no—
204#106—712no—
205#102—712no—
206#102—712no—
207#102—712no—
208#102C2×D7112yescomplemented
209C22⋊C8—711no—
210C22⋊C8—712no—
211C22⋊C8C2×D7111yescharacteristic, complemented
212C24—21no—
213C24—711no—
214C24—1421no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—42no—
218C2×D4—2844no—
219C2×D4—2844no—
220C2×D4—2842no—
221C2×D4—711no—
222C2×D4—1421no—
223C2×D4—1421no—
224C2×D4—1422no—
225C2×D4—1421no—
226C2×D4—1422no—
227C2×D4—1421no—
228C2×D4C22×D7111yescharacteristic
229C2×D4C22×D7112yes—
230C22×C4—2842no—
231C22×C4—711no—
232C22×C4—711no—
233C22×C4—711no—
234C22×C4—711no—
235C22×C4—711no—
236C22×C4—1421no—
237C22×C4—1421no—
238C22×C4C22×D7111yescharacteristic
239D8—42no—
240D8—2842no—
241M16—2842no—
242C2×C8—712no—
243C2×C8C22×D7112yes—
244C4⋊C4—711no—
245C4⋊C4—712no—
246C4⋊C4—1421no—
247C4⋊C4—1421no—
248C4⋊C4C22×D7111yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—1421no—
251C22⋊C4—1421no—
252C22⋊C4—1422no—
253C22⋊C4—1422no—
254C22⋊C4—1421no—
255C23—21no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—2841no—
261C23—2842no—
262C23—2842no—
263C23—2842no—
264C23—711no—
265C23—711no—
266C23—1422no—
267C23—1421no—
268C23—1422no—
269C23D4×D7111yescharacteristic
270D4—22no—
271D4—22no—
272D4—22no—
273D4—42no—
274D4—42no—
275D4—2842no—
276D4—2844no—
277D4—2844no—
278D4—2842no—
279D4—2842no—
280D4—2842no—
281D4—1422no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—2841no—
285C2×C4—2841no—
286C2×C4—2842no—
287C2×C4—2842no—
288C2×C4—2841no—
289C2×C4—711no—
290C2×C4—711no—
291C2×C4—1422no—
292C2×C4—1421no—
293C2×C4—1422no—
294C2×C4—1422no—
295C2×C4D4×D7111yescharacteristic
296C2×C4C23×D7111yescharacteristic
297C8—22no—
298C8—1422no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—84no—
308C22—2844no—
309C22—2841no—
310C22—2841no—
311C22—2841no—
312C22—5684no—
313C22—712no—
314C22—1421no—
315C22—1421no—
316C22—1421no—
317C22—1421no—
318C22—1421no—
319C22#1455615212yes—
320C22C2×D4×D7111yescharacteristic
321C4—21no—
322C4—41no—
323C4—2842no—
324C4—2841no—
325C4—1421no—
326C4—1421no—
327C4#1455617312yes—
328C2—22no—
329C2—82no—
330C2—81no—
331C2—2841no—
332C2—5681no—
333C2—1422no—
334C2#1798655911yescharacteristic
335C2AGL(1,ℤ/8)×D7111yescharacteristic
336C2C2×#1455615211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#447×C71C2
socleC22×C71C22×D4
hypercenter#447D71
hypocenterC71#3192
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#14556099C2semidirect
2. #121×D71C2semidirect
3. #17986318C2semidirect
4. #17986316C2semidirect
5. #17986315C2semidirect
6. #17986251C2semidirect
7. #17986249C2semidirect
8. #17986133C2semidirect
9. #17986131C2semidirect
10. #17985961C2semidirect
11. #447×C71C2semidirect
14. C2×D4×D71C22semidirect
16. C2×#11765850C22semidirect
17. C2×#11765848C22semidirect
18. #14556103C22semidirect
25. #14556099C22semidirect
26. #14556099C22semidirect
27. C2×C4⋊Dic71C22semidirect
28. C4⋊C4×D71C22semidirect
31. #14556060C22semidirect
32. #14556059C22semidirect
33. #14556050C22semidirect
34. #14556049C22semidirect
35. #14556049C22semidirect
36. C22⋊(C71⋊C8)C22semidirect
37. (C2×D71)⋊C8C22semidirect
38. C22×D4×C71C22semidirect
39. C2×D8×C71C22semidirect
41. #102×C71C22semidirect
42. C22⋊C8×C71C22semidirect
169. C71#3192semidirect
179. #447D71semidirect
184. C22×D4C2×D71semidirect
187. C2×D8C2×D71semidirect
208. #102C2×D71semidirect
211. C22⋊C8C2×D71semidirect