Finite Group Databasebeta

← #22600804

Subgroups — #22600804

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986653C211yesmaximal, characteristic, complemented
2#17986409C211yesmaximal, characteristic, complemented
3#17986323C211yesmaximal, characteristic, complemented
4#17986317C211yesmaximal, characteristic, complemented
5#17986315C211yesmaximal, characteristic, complemented
6#17986310C211yesmaximal, characteristic, complemented
7#17986296C211yesmaximal, characteristic, complemented
8#17986289C211yesmaximal, characteristic, complemented
9#17986250C211yesmaximal, characteristic, complemented
10#17986233C211yesmaximal, characteristic, complemented
11#17986109C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986042C211yesmaximal, characteristic, complemented
14C2×#14556059C211yesmaximal, characteristic, complemented
15#553×C71C211yesmaximal, characteristic, complemented
16C2×#11765864—21no—
17C2×#11765850C2211yescharacteristic
18C2×#11765849C2211yescharacteristic
19C22×C71⋊C8C2211yescharacteristic
20#14556128C2211yescharacteristic
21#14556112—21no—
22#14556112—21no—
23#14556108—21no—
24#14556102C2211yescharacteristic
25#14556102C2211yescharacteristic
26C4×D284C2211yescharacteristic
27C2×C4⋊Dic71C2211yescharacteristic
28C2×C4⋊Dic71C2211yescharacteristic
29#14556061C2211yescharacteristic
30#14556061C2211yescharacteristic
31#14556061C2211yescharacteristic
32#14556059C2211yescharacteristic
33#14556059C2211yescharacteristic
34#14556059C2211yescharacteristic
35#14556059C2211yescharacteristic
36#14556059C2211yescharacteristic
37#14556059C2211yescharacteristic
38C4⋊(C71⋊C8)C2211yescharacteristic
39#14556049C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
48#126×C71—21no—
49#125×C71C2211yescharacteristic
50#123×C71—21no—
51#121×C71—21no—
52#121×C71C2211yescharacteristic
53#121×C71C2211yescharacteristic
54C4×D4×C71—21no—
55C4×D4×C71C2211yescharacteristic
56C4×D4×C71C2211yescharacteristic
57#117×C71C2211yescharacteristic
58C2×C4⋊C4×C71C2211yescharacteristic
59#11765864—44no—
60C2×Dic142—21no—
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C22×Dic71—21no—
65C22×Dic71—21no—
66C2×C4×D71—21no—
67#11765850—22no—
68#11765849—22no—
69C2×C71⋊C8—22no—
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C2×C71⋊C8C2311yescharacteristic
73C2×C71⋊C8C2311yescharacteristic
74C4⋊Dic71—21no—
75C4⋊Dic71—21no—
76C4⋊Dic71—21no—
77C4⋊Dic71C2311yescharacteristic
78C4⋊Dic71C2311yescharacteristic
79C4⋊Dic71C2311yescharacteristic
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82(C2×D71)⋊C4—21no—
83C2×D4×C71—21no—
84C2×D4×C71—21no—
85C2×D4×C71D411yescharacteristic
86C2×D4×C71C2311yescharacteristic
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71C2311yescharacteristic
90C22×C4×C71C2311yescharacteristic
91C4⋊C4×C71—21no—
92C4⋊C4×C71—21no—
93C4⋊C4×C71—21no—
94C4⋊C4×C71D411yescharacteristic
95C4⋊C4×C71C2311yescharacteristic
96C4⋊C4×C71C2311yescharacteristic
97C4⋊C4×C71C2311yescharacteristic
98C22⋊C4×C71—21no—
99C22⋊C4×C71—21no—
100C22⋊C4×C71—21no—
101C22⋊C4×C71—21no—
102C22⋊C4×C71D411yescharacteristic
103C22⋊C4×C71D411yescharacteristic
104C42×C71—21no—
105C42×C71C2311yescharacteristic
106C23×C71—21no—
107C23×C71C2×D411yescharacteristic
108C23×C71C2×D411yescharacteristic
109C22×D71—21no—
110D4×C71—22no—
111D4×C71—21no—
112D4×C71—21no—
113D4×C71—42no—
114D4×C71—81no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—81no—
123C2×C4×C71—81no—
124C2×C4×C71C2×D411yescharacteristic
125C2×C4×C71C2×D411yescharacteristic
126C2×C4×C71C2×D411yescharacteristic
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2411yescharacteristic
129C22⋊D71—42no—
130C22⋊D71—42no—
131C2×Dic71—21no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—42no—
135C2×Dic71—42no—
136D284—21no—
137D284—21no—
138C4×D71—42no—
139Dic142—42no—
140C71⋊C8—21no—
141C71⋊C8—21no—
142C71⋊C8—21no—
143C71⋊C8#4112yescomplemented
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71AGL(1,ℤ/8)12yes—
153C22×C71C22×D411yescharacteristic
154C2×D71—41no—
155C2×D71—41no—
156C2×D71—41no—
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71—41no—
164C4×C71C2×#4111yescharacteristic
165C4×C712−1+411yescharacteristic
166Dic71—41no—
167Dic71—41no—
168Dic71—41no—
169C2×C71—22no—
170C2×C71—41no—
171C2×C71—81no—
172C2×C71#54711yescharacteristic
173C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C71#57711yescharacteristic
175D71—81no—
176#3345—711nomaximal, Sylow
177C71#334511yescharacteristic, complemented, Sylow
178#553D7111yescharacteristic, complemented
179#547—711no—
180#501—711no—
181#482—711no—
182#481—711no—
183#481—711no—
184#480—711no—
185#478—711no—
186#476—711no—
187#469—711no—
188#466—711no—
189#442—711no—
190#435—711no—
191#428—711no—
192C2×#106—711no—
193C2×#41—1421no—
194C2×D8—711no—
195C2×M16—711no—
196C22×C8—711no—
197C4⋊Q8—711no—
198#126—21no—
199#125C2×D7111yescharacteristic
200#123—21no—
201#123—1421no—
202#123—1421no—
203C22⋊Q8—1421no—
204#121—21no—
205#121—711no—
206#121—711no—
207#121C2×D7111yescharacteristic
208#121C2×D7111yescharacteristic
209C4×D4—21no—
210C4×D4—711no—
211C4×D4C2×D7111yescharacteristic
212C4×D4C2×D7111yescharacteristic
213#117C2×D7111yescharacteristic
214C2×C4⋊C4—711no—
215C2×C4⋊C4—711no—
216C2×C4⋊C4C2×D7111yescharacteristic
217#107—711no—
218#107—711no—
219#107—711no—
220#106—711no—
221#106—711no—
222#106—711no—
223#106—711no—
224#106—711no—
225#106—711no—
226C4⋊C8—711no—
227#102—711no—
228#102—711no—
229#102—711no—
230#102—711no—
231#102—711no—
232#102—711no—
233C22⋊C8—711no—
234C22⋊C8—711no—
235#97—711no—
236#41—2844no—
237C2×Q8—1421no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—711no—
241C2×D4—1421no—
242C2×D4—1421no—
243C2×D4C22⋊D7111yescharacteristic
244C2×D4C22×D7111yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1421no—
248C22×C4—1421no—
249C22×C4—1421no—
250C22×C4C22×D7111yescharacteristic
251C22×C4C22×D7111yescharacteristic
252D8—1422no—
253M16—1422no—
254C2×C8—711no—
255C2×C8—711no—
256C2×C8—711no—
257C2×C8—711no—
258C2×C8—1422no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—711no—
263C4⋊C4—711no—
264C4⋊C4—711no—
265C4⋊C4—1421no—
266C4⋊C4—1421no—
267C4⋊C4—1421no—
268C4⋊C4C22⋊D7111yescharacteristic
269C4⋊C4C22×D7111yescharacteristic
270C4⋊C4C22×D7111yescharacteristic
271C4⋊C4C22×D7111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—1421no—
277C22⋊C4—1421no—
278C22⋊C4—1421no—
279C22⋊C4C22⋊D7111yescharacteristic
280C22⋊C4C22⋊D7111yescharacteristic
281C42—21no—
282C42C22×D7111yescharacteristic
283C23—21no—
284C23—1421no—
285C23C2×C22⋊D7111yescharacteristic
286C23C2×C22⋊D7111yescharacteristic
287Q8—2842no—
288D4—22no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—81no—
293D4—2842no—
294D4—2842no—
295D4—1421no—
296D4—1421no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—2842no—
307C2×C4—2842no—
308C2×C4—2842no—
309C2×C4—1421no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4C2×C22⋊D7111yescharacteristic
313C2×C4C2×C22⋊D7111yescharacteristic
314C2×C4C2×C22⋊D7111yescharacteristic
315C2×C4C2×C22⋊D7111yescharacteristic
316C2×C4C23×D7111yescharacteristic
317C8—712no—
318C8—1421no—
319C8—1421no—
320C8—1421no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—2841no—
330C22—2841no—
331C22—2841no—
332C22#1455615312yes—
333C22C22×C22⋊D7111yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—2841no—
342C4—2841no—
343C4—2841no—
344C4#41×D7111yescharacteristic
345C4#1455617511yescharacteristic
346C2—22no—
347C2—41no—
348C2—81no—
349C2—5681no—
350C2#1798664811yescharacteristic
351C2C2×#1455615311yescharacteristic
352C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#553×C71C2
socleC22×C71C22×D4
hypercenter#553D71
hypocenterC71#3345
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #17986653C2semidirect
2. #17986409C2semidirect
3. #17986323C2semidirect
4. #17986317C2semidirect
5. #17986315C2semidirect
6. #17986310C2semidirect
7. #17986296C2semidirect
8. #17986289C2semidirect
9. #17986250C2semidirect
10. #17986233C2semidirect
11. #17986109C2semidirect
12. #17986071C2semidirect
13. #17986042C2semidirect
14. C2×#14556059C2semidirect
15. #553×C71C2semidirect
143. C71⋊C8#41semidirect
177. C71#3345semidirect
178. #553D71semidirect