Finite Group Databasebeta

← #22595321

Subgroups — #22595321

Order 9088.

480 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556127C211yesmaximal, characteristic, complemented
2C2×#14556106C211yesmaximal, characteristic
3#17985941C211yesmaximal, characteristic, complemented
4#17985933C211yesmaximal, characteristic, complemented
5#17985921C211yesmaximal, characteristic
6#17985919C211yesmaximal, characteristic, complemented
7#17985912C211yesmaximal, characteristic
8#17985904C211yesmaximal, characteristic, complemented
9#17985894C211yesmaximal, characteristic, complemented
10#17985889C211yesmaximal, characteristic
11#17985888C211yesmaximal, characteristic
12#17985882C211yesmaximal, characteristic, complemented
13#17985873C211yesmaximal, characteristic
14#17985870C211yesmaximal, characteristic
15#398×C71C211yesmaximal, characteristic, complemented
16C22×Dic142C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C22×C4×D71C2211yescharacteristic
19#14556127—24no—
20#14556106—24no—
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2211yescharacteristic
23C2×Dic71⋊C4C2211yescharacteristic
24C2×Dic71⋊C4C2211yescharacteristic
25C2×Dic71⋊C4C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×(C2×D71)⋊C4C2211yescharacteristic
30C2×(C2×D71)⋊C4C2211yescharacteristic
31C2×(C2×D71)⋊C4C2211yescharacteristic
32C2×(C2×D71)⋊C4C2211yescharacteristic
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×C4×Dic71C2211yescharacteristic
36C2×C4×Dic71C2211yescharacteristic
37C71⋊#95C2211yescharacteristic
38C71⋊#95C2211yescharacteristic
39C71⋊#95C2211yescharacteristic
40C71⋊#95C2211yescharacteristic
41C71⋊#95C2211yescharacteristic
42C71⋊#95C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C71⋊#95C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C2×C4⋊C4×C71C2211yescharacteristic
47C2×C4⋊C4×C71C2211yescharacteristic
48C2×C4⋊C4×C71C2211yescharacteristic
49C2×C4⋊C4×C71C2211yescharacteristic
50#95×C71C2211yescharacteristic
51#95×C71C2211yescharacteristic
52#95×C71C2211yescharacteristic
53C23×D71C2311yescharacteristic
54C2×Dic142—22no—
55C2×Dic142—41no—
56C2×Dic142—41no—
57C2×Dic142—41no—
58C2×Dic142—41no—
59C2×Dic142D414yes—
60C22×Dic71C2311yescharacteristic
61C22×Dic71C2311yescharacteristic
62C22×Dic71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2311yescharacteristic
65C22×Dic71C2311yescharacteristic
66C22×Dic71C2311yescharacteristic
67C2×C4×D71—22no—
68C2×C4×D71—41no—
69C2×C4×D71—41no—
70C4⋊Dic71—22no—
71C4⋊Dic71—22no—
72Dic71⋊C4—22no—
73Dic71⋊C4—22no—
74Dic71⋊C4—22no—
75Dic71⋊C4—22no—
76Dic71⋊C4—22no—
77Dic71⋊C4—41no—
78(C2×D71)⋊C4—22no—
79(C2×D71)⋊C4—22no—
80(C2×D71)⋊C4—22no—
81(C2×D71)⋊C4—41no—
82(C2×D71)⋊C4—41no—
83(C2×D71)⋊C4Q814yescomplemented
84C4×Dic71—22no—
85C4×Dic71—41no—
86C22×C4×C71C2311yescharacteristic
87C22×C4×C71C2311yescharacteristic
88C22×C4×C71C2311yescharacteristic
89C22×C4×C71C2311yescharacteristic
90C22×C4×C71C2311yescharacteristic
91C22×C4×C71C2311yescharacteristic
92C22×C4×C71C2311yescharacteristic
93C4⋊C4×C71—22no—
94C4⋊C4×C71—22no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C23×C71C2411yescharacteristic
98C22×D71—21no—
99C22×D71—21no—
100C22×D71—21no—
101C22×D71—21no—
102C22×D71—21no—
103C22×D71—21no—
104C22×D71C2×Q812yes—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
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—21no—
123C2×C4×C71—21no—
124C2×C4×C71C2×D412yes—
125C2×C4×C71C2×Q812yes—
126C2×Dic71—21no—
127C2×Dic71—21no—
128C2×Dic71—21no—
129C2×Dic71—21no—
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71C2×D412yes—
144C2×Dic71C2×D412yes—
145C2×Dic71C2×Q812yes—
146C2×Dic71#4112yes—
147C4×D71—44no—
148Dic142—44no—
149Dic142—44no—
150C22×C71C22×D411yescharacteristic
151C22×C71C22×Q811yescharacteristic
152C22×C71C2×#4111yescharacteristic
153C22×C71H(5,𝔽2)11yescharacteristic
154C22×C71H(5,𝔽2)11yescharacteristic
155C22×C712−1+411yescharacteristic
156C22×C712−1+411yescharacteristic
157C2×D71—22no—
158C2×D71—22no—
159C2×D71—22no—
160C2×D71—41no—
161C2×D71—41no—
162C2×D71—41no—
163C2×D71—41no—
164C4×C71—22no—
165C4×C71—22no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171Dic71—22no—
172Dic71—22no—
173Dic71—22no—
174Dic71—22no—
175Dic71—41no—
176Dic71—41no—
177Dic71—41no—
178C2×C71#53411yescharacteristic
179C2×C71#54211yescharacteristic
180C2×C71#54411yescharacteristic
181C2×C71#54811yescharacteristic
182C2×C71D4×Q811yescharacteristic
183C2×C71#55411yescharacteristic
184C2×C71#56311yescharacteristic
185D71—42no—
186#2729—711nomaximal, Sylow
187C71#272911yescharacteristic, complemented, Sylow
188C2×C4⋊Q8—711no—
189C2×C22⋊Q8—711no—
190#400—711no—
191#398—711no—
192#398D7111yescharacteristic, complemented
193#396—711no—
194#395—711no—
195#393—711no—
196#391—711no—
197#389—711no—
198#388—711no—
199#388—711no—
200#387—711no—
201#385—711no—
202#385—711no—
203C22×Q8—711no—
204C22×Q8—711no—
205C23×C4—711no—
206C4⋊Q8—1424no—
207C22⋊Q8—1424no—
208C2×C4⋊C4—711no—
209C2×C4⋊C4—711no—
210C2×C4⋊C4—711no—
211C2×C4⋊C4—711no—
212C2×C4⋊C4—711no—
213C2×C4⋊C4—711no—
214C2×C4⋊C4—711no—
215C2×C4⋊C4—711no—
216C2×C4⋊C4C2×D7111yescharacteristic
217C2×C4⋊C4C2×D7111yescharacteristic
218C2×C4⋊C4C2×D7111yescharacteristic
219C2×C4⋊C4C2×D7111yescharacteristic
220C2×C22⋊C4—711no—
221C2×C22⋊C4—711no—
222C2×C22⋊C4—711no—
223C2×C22⋊C4—711no—
224C2×C22⋊C4—711no—
225C2×C22⋊C4—711no—
226C2×C42—711no—
227C2×C42—711no—
228#95—711no—
229#95—711no—
230#95—711no—
231#95—711no—
232#95—711no—
233#95—711no—
234#95—711no—
235#95—711no—
236#95—711no—
237#95C2×D7111yescharacteristic
238#95C2×D7111yescharacteristic
239#95C2×D7111yescharacteristic
240C24—711no—
241C2×Q8—2841no—
242C2×Q8—2841no—
243C2×Q8—2841no—
244C2×Q8—2841no—
245C2×Q8—714no—
246C2×Q8—1422no—
247C22×C4—2841no—
248C22×C4—2841no—
249C22×C4—711no—
250C22×C4—711no—
251C22×C4—711no—
252C22×C4—711no—
253C22×C4—711no—
254C22×C4—711no—
255C22×C4—711no—
256C22×C4—1422no—
257C22×C4C22×D7111yescharacteristic
258C22×C4C22×D7111yescharacteristic
259C22×C4C22×D7111yescharacteristic
260C22×C4C22×D7111yescharacteristic
261C22×C4C22×D7111yescharacteristic
262C22×C4C22×D7111yescharacteristic
263C22×C4C22×D7111yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—2841no—
269C4⋊C4—1422no—
270C4⋊C4—1422no—
271C4⋊C4—1422no—
272C4⋊C4—1422no—
273C4⋊C4—1422no—
274C4⋊C4—1422no—
275C4⋊C4—1422no—
276C22⋊C4—2841no—
277C22⋊C4—2841no—
278C22⋊C4—714no—
279C22⋊C4—1422no—
280C22⋊C4—1422no—
281C22⋊C4—1422no—
282C42—2841no—
283C42—1422no—
284C23—712no—
285C23—1421no—
286C23—1421no—
287C23—1421no—
288C23—1421no—
289C23—1421no—
290C23—1421no—
291C23C23×D7111yescharacteristic
292Q8—2844no—
293Q8—2844no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—2844no—
314C2×C4—712no—
315C2×C4—712no—
316C2×C4—712no—
317C2×C4—712no—
318C2×C4—1421no—
319C2×C4—1421no—
320C2×C4—1421no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4—1421no—
324C2×C4—1421no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4—1421no—
329C2×C4—1421no—
330C2×C4—1421no—
331C2×C4—1421no—
332C2×C4—1421no—
333C2×C4—1421no—
334C2×C4—1421no—
335C2×C4D4×D7112yes—
336C2×C4Q8×D7112yes—
337C22—2841no—
338C22—2841no—
339C22—2841no—
340C22—2841no—
341C22—1422no—
342C22—1422no—
343C22—1422no—
344C22C2×D4×D7111yescharacteristic
345C22C2×Q8×D7111yescharacteristic
346C22#41×D7111yescharacteristic
347C22#1455617311yescharacteristic
348C22#1455617411yescharacteristic
349C22#1455617511yescharacteristic
350C22#1455617611yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—2841no—
359C4—2841no—
360C4—2841no—
361C4—1422no—
362C4—1422no—
363C4—1422no—
364C4—1422no—
365C2—2842no—
366C2#1798655711yescharacteristic
367C2#1798661711yescharacteristic
368C2#1798662611yescharacteristic
369C2#1798665911yescharacteristic
370C2#1798666211yescharacteristic
371C2#1798670511yescharacteristic
372C2#1798675311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2729
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14556127C2semidirect
3. #17985941C2semidirect
4. #17985933C2semidirect
6. #17985919C2semidirect
8. #17985904C2semidirect
9. #17985894C2semidirect
12. #17985882C2semidirect
15. #398×C71C2semidirect
83. (C2×D71)⋊C4Q8semidirect
187. C71#2729semidirect
192. #398D71semidirect