Finite Group Databasebeta

← #22595360

Subgroups — #22595360

Order 9088.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556107C211yesmaximal, characteristic, complemented
2C2×#14556105C211yesmaximal, characteristic, complemented
3#17985935C211yesmaximal, characteristic, complemented
4#17985935C211yesmaximal, characteristic, complemented
5#17985933C211yesmaximal, characteristic, complemented
6#17985931C211yesmaximal, characteristic, complemented
7#17985922C211yesmaximal, characteristic, complemented
8#17985922C211yesmaximal, characteristic, complemented
9#17985903C211yesmaximal, characteristic, complemented
10#17985902C211yesmaximal, characteristic
11#17985896C211yesmaximal, characteristic, complemented
12#17985879C211yesmaximal, characteristic
13#17985872C211yesmaximal, characteristic, complemented
14#17985871C211yesmaximal, characteristic, complemented
15#386×C71C211yesmaximal, characteristic
16C22×Dic142C2211yescharacteristic
17C23×Dic71C2211yescharacteristic
18#14556107—24no—
19#14556105—24no—
20C2×C4⋊Dic71C2211yescharacteristic
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2211yescharacteristic
23C2×C4⋊Dic71C2211yescharacteristic
24C2×Dic71⋊C4C2211yescharacteristic
25C2×Dic71⋊C4C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×Dic71⋊C4C2211yescharacteristic
30C2×Dic71⋊C4C2211yescharacteristic
31C2×Dic71⋊C4C2211yescharacteristic
32C2×C22⋊Dic71C2211yescharacteristic
33C2×C22⋊Dic71C2211yescharacteristic
34C2×C22⋊Dic71C2211yescharacteristic
35C2×C22⋊Dic71C2211yescharacteristic
36C2×C22⋊Dic71C2211yescharacteristic
37C2×C22⋊Dic71C2211yescharacteristic
38C2×C22⋊Dic71C2211yescharacteristic
39C2×C4×Dic71C2211yescharacteristic
40C71⋊#95C2211yescharacteristic
41C71⋊#95C2211yescharacteristic
42C71⋊#95C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C71⋊#95C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C22×D4×C71C2211yescharacteristic
47C23×C4×C71C2211yescharacteristic
48C2×C22⋊C4×C71C2211yescharacteristic
49C2×C22⋊C4×C71C2211yescharacteristic
50C2×C22⋊C4×C71C2211yescharacteristic
51#95×C71C2211yescharacteristic
52#95×C71C2211yescharacteristic
53C2×Dic142—41no—
54C2×Dic142—41no—
55C2×Dic142D414yescomplemented
56C22×Dic71—22no—
57C22×Dic71—41no—
58C22×Dic71—41no—
59C22×Dic71C2311yescharacteristic
60C22×Dic71C2311yescharacteristic
61C22×Dic71C2311yescharacteristic
62C22×Dic71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2311yescharacteristic
65C22×Dic71C2311yescharacteristic
66C22×Dic71C2311yescharacteristic
67C4⋊Dic71—22no—
68C4⋊Dic71—22no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71Dic71⋊C4—22no—
72Dic71⋊C4—22no—
73Dic71⋊C4—22no—
74Dic71⋊C4—22no—
75Dic71⋊C4—22no—
76Dic71⋊C4—41no—
77Dic71⋊C4—41no—
78Dic71⋊C4—41no—
79C22⋊Dic71—22no—
80C22⋊Dic71—22no—
81C22⋊Dic71—22no—
82C22⋊Dic71—22no—
83C22⋊Dic71—22no—
84C22⋊Dic71—41no—
85C22⋊Dic71—41no—
86C4×Dic71—41no—
87C24×C71C2311yescharacteristic
88C24×C71C2311yescharacteristic
89C2×D4×C71—41no—
90C2×D4×C71—41no—
91C2×D4×C71Q814yescomplemented
92C22×C4×C71—22no—
93C22×C4×C71—41no—
94C22×C4×C71—41no—
95C22×C4×C71C2311yescharacteristic
96C22×C4×C71C2311yescharacteristic
97C22×C4×C71C2311yescharacteristic
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C22⋊C4×C71—22no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71C2×Q812yes—
116C23×C71C2×Q812yes—
117C23×C71C2411yescharacteristic
118D4×C71—44no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—44no—
133C2×C4×C71C2×D412yes—
134C2×C4×C71C2×Q812yes—
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×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—21no—
152C2×Dic71—21no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—21no—
156C2×Dic71—44no—
157C2×Dic71C2×D412yes—
158C2×Dic71C2×D412yes—
159C2×Dic71#4112yes—
160Dic142—44no—
161C22×C71—22no—
162C22×C71—22no—
163C22×C71—22no—
164C22×C71—22no—
165C22×C71—22no—
166C22×C71—22no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71—41no—
175C22×C71C22×D411yescharacteristic
176C22×C71C22×Q811yescharacteristic
177C22×C71C2×#4111yescharacteristic
178C22×C71H(5,𝔽2)11yescharacteristic
179C22×C71H(5,𝔽2)11yescharacteristic
180C22×C71H(5,𝔽2)11yescharacteristic
181C22×C712−1+411yescharacteristic
182C4×C71—22no—
183C4×C71—22no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187Dic71—22no—
188Dic71—22no—
189Dic71—22no—
190Dic71—41no—
191Dic71—41no—
192Dic71—41no—
193Dic71—41no—
194Dic71—41no—
195C2×C71—42no—
196C2×C71—42no—
197C2×C71#53411yescharacteristic
198C2×C71#53911yescharacteristic
199C2×C71#54311yescharacteristic
200C2×C71D4×Q811yescharacteristic
201C2×C71#55011yescharacteristic
202C2×C71#55411yescharacteristic
203C2×C71#56211yescharacteristic
204#2732—711nomaximal, Sylow
205C71#273211yescharacteristic, complemented, Sylow
206C2×C22⋊Q8—711no—
207C2×C22⋊Q8—711no—
208#399—711no—
209#399—711no—
210#398—711no—
211#398—711no—
212#396—711no—
213#396—711no—
214#391—711no—
215#390—711no—
216#389—711no—
217#386—711no—
218#386D7111yescharacteristic
219#385—711no—
220#385—711no—
221C22×Q8—711no—
222C22×D4C2×D7111yescharacteristic
223C23×C4—711no—
224C23×C4C2×D7111yescharacteristic
225C22⋊Q8—1424no—
226C22⋊Q8—1424no—
227C2×C4⋊C4—711no—
228C2×C4⋊C4—711no—
229C2×C4⋊C4—711no—
230C2×C4⋊C4—711no—
231C2×C4⋊C4—711no—
232C2×C4⋊C4—711no—
233C2×C4⋊C4—711no—
234C2×C4⋊C4—711no—
235C2×C4⋊C4—711no—
236C2×C4⋊C4—711no—
237C2×C4⋊C4—711no—
238C2×C4⋊C4—711no—
239C2×C22⋊C4—711no—
240C2×C22⋊C4—711no—
241C2×C22⋊C4—711no—
242C2×C22⋊C4—711no—
243C2×C22⋊C4—711no—
244C2×C22⋊C4—711no—
245C2×C22⋊C4—711no—
246C2×C22⋊C4C2×D7111yescharacteristic
247C2×C22⋊C4C2×D7111yescharacteristic
248C2×C22⋊C4C2×D7111yescharacteristic
249C2×C42—711no—
250#95—711no—
251#95—711no—
252#95—711no—
253#95—711no—
254#95—711no—
255#95—711no—
256#95C2×D7111yescharacteristic
257#95C2×D7111yescharacteristic
258C24C22×D7111yescharacteristic
259C24C22×D7111yescharacteristic
260C2×Q8—2841no—
261C2×Q8—2841no—
262C2×Q8—714no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4Dic14214yescomplemented
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—2841no—
270C22×C4—2841no—
271C22×C4—711no—
272C22×C4—711no—
273C22×C4—711no—
274C22×C4—711no—
275C22×C4—711no—
276C22×C4—711no—
277C22×C4—711no—
278C22×C4—711no—
279C22×C4—1422no—
280C22×C4C22×D7111yescharacteristic
281C22×C4C22×D7111yescharacteristic
282C22×C4C22×D7111yescharacteristic
283C22×C4C22×D7111yescharacteristic
284C22×C4C22×D7111yescharacteristic
285C4⋊C4—2841no—
286C4⋊C4—2841no—
287C4⋊C4—2841no—
288C4⋊C4—2841no—
289C4⋊C4—2841no—
290C4⋊C4—1422no—
291C4⋊C4—1422no—
292C4⋊C4—1422no—
293C4⋊C4—1422no—
294C4⋊C4—1422no—
295C4⋊C4—1422no—
296C4⋊C4—1422no—
297C22⋊C4—22no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—2841no—
301C22⋊C4—2841no—
302C22⋊C4—1422no—
303C22⋊C4—1422no—
304C22⋊C4—1422no—
305C22⋊C4—1422no—
306C22⋊C4—1422no—
307C42—2841no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×Dic14212yes—
321C23C2×Dic14212yes—
322C23C23×D7111yescharacteristic
323Q8—2844no—
324D4—44no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—44no—
339C2×C4—2844no—
340C2×C4—712no—
341C2×C4—712no—
342C2×C4—712no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4—1421no—
349C2×C4—1421no—
350C2×C4—1421no—
351C2×C4—1421no—
352C2×C4—1421no—
353C2×C4—1421no—
354C2×C4—1421no—
355C2×C4—1421no—
356C2×C4—1421no—
357C2×C4—1421no—
358C2×C4—1421no—
359C2×C4—1421no—
360C2×C4—1421no—
361C2×C4—1421no—
362C2×C4—1421no—
363C2×C4—1421no—
364C2×C4D4×D7112yes—
365C2×C4C2×Dic14212yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22C2×D4×D7111yescharacteristic
381C22C22×Dic14211yescharacteristic
382C22#41×D7111yescharacteristic
383C22#1455617311yescharacteristic
384C22#1455617311yescharacteristic
385C22#1455617311yescharacteristic
386C22#1455617511yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2841no—
393C4—2841no—
394C4—2841no—
395C4—2841no—
396C4—2841no—
397C4—1422no—
398C4—1422no—
399C4—1422no—
400C2—42no—
401C2—42no—
402C2#1798655911yescharacteristic
403C2#1798659211yescharacteristic
404C2#1798662511yescharacteristic
405C2D4×Dic14211yescharacteristic
406C2#1798666711yescharacteristic
407C2#1798670811yescharacteristic
408C2#1798674411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#386×C71C2
hypercenter#386D71
hypocenterC71#2732
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#14556107C2semidirect
2. C2×#14556105C2semidirect
3. #17985935C2semidirect
4. #17985935C2semidirect
5. #17985933C2semidirect
6. #17985931C2semidirect
7. #17985922C2semidirect
8. #17985922C2semidirect
9. #17985903C2semidirect
11. #17985896C2semidirect
13. #17985872C2semidirect
14. #17985871C2semidirect
55. C2×Dic142D4semidirect
91. C2×D4×C71Q8semidirect
205. C71#2732semidirect
265. C2×D4Dic142semidirect