Finite Group Databasebeta

← #22595705

Subgroups — #22595705

Order 9088.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71⋊#401C211yesmaximal, characteristic, complemented
2#17985927C211yesmaximal, characteristic
3#17985927C211yesmaximal, characteristic
4#17985919C211yesmaximal, characteristic, complemented
5#17985911C211yesmaximal, characteristic
6#17985911C211yesmaximal, characteristic
7#17985904C211yesmaximal, characteristic, complemented
8#17985903C211yesmaximal, characteristic, complemented
9#17985903C211yesmaximal, characteristic, complemented
10#17985887C211yesmaximal, characteristic
11#17985887C211yesmaximal, characteristic
12#17985883C211yesmaximal, characteristic, complemented
13#17985883C211yesmaximal, characteristic, complemented
14#17985866C211yesmaximal, characteristic, complemented
15#388×C71C211yesmaximal, characteristic
16C22×Dic142C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×C22⋊Dic71C2211yescharacteristic
24C2×C22⋊Dic71C2211yescharacteristic
25C2×C22⋊Dic71C2211yescharacteristic
26C2×C4×Dic71C2211yescharacteristic
27C2×C4×Dic71C2211yescharacteristic
28C71⋊#95C2211yescharacteristic
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2211yescharacteristic
31C71⋊#95C2211yescharacteristic
32C71⋊#95C2211yescharacteristic
33C71⋊#95C2211yescharacteristic
34C71⋊#95C2211yescharacteristic
35C71⋊#95C2211yescharacteristic
36C71⋊#95C2211yescharacteristic
37C71⋊#95C2211yescharacteristic
38C71⋊#95C2211yescharacteristic
39C71⋊#95C2211yescharacteristic
40C71⋊#95C2211yescharacteristic
41C71⋊#95C2211yescharacteristic
42C71⋊#95C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C2×C4⋊C4×C71C2211yescharacteristic
45C2×C22⋊C4×C71C2211yescharacteristic
46C2×C22⋊C4×C71C2211yescharacteristic
47C2×C22⋊C4×C71C2211yescharacteristic
48C2×C42×C71C2211yescharacteristic
49#95×C71C2211yescharacteristic
50#95×C71C2211yescharacteristic
51C2×Dic142—22no—
52C2×Dic142—22no—
53C2×Dic142—22no—
54C2×Dic142—41no—
55C2×Dic142—41no—
56C2×Dic142—41no—
57C22×Dic71C2311yescharacteristic
58C22×Dic71C2311yescharacteristic
59C22×Dic71C2311yescharacteristic
60C22×Dic71C2311yescharacteristic
61C22×Dic71C2311yescharacteristic
62C22×Dic71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2311yescharacteristic
65Dic71⋊C4—22no—
66Dic71⋊C4—22no—
67Dic71⋊C4—41no—
68Dic71⋊C4—41no—
69C22⋊Dic71—22no—
70C22⋊Dic71—22no—
71C22⋊Dic71—41no—
72C22⋊Dic71—41no—
73C4×Dic71—22no—
74C4×Dic71—41no—
75C24×C71C2311yescharacteristic
76C22×C4×C71C2311yescharacteristic
77C22×C4×C71C2311yescharacteristic
78C22×C4×C71C2311yescharacteristic
79C22×C4×C71C2311yescharacteristic
80C22×C4×C71C2311yescharacteristic
81C22×C4×C71C2311yescharacteristic
82C4⋊C4×C71—41no—
83C22⋊C4×C71—22no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—41no—
86C42×C71—41no—
87C23×C71—21no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71C2411yescharacteristic
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
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#4112yes—
113C2×Dic71—21no—
114C2×Dic71—21no—
115C2×Dic71—21no—
116C2×Dic71—21no—
117C2×Dic71—21no—
118C2×Dic71—21no—
119C2×Dic71—21no—
120C2×Dic71—21no—
121C2×Dic71—21no—
122C2×Dic71—21no—
123C2×Dic71—21no—
124C2×Dic71—21no—
125C2×Dic71—21no—
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#4112yes—
136C2×Dic71#4112yes—
137Dic142—44no—
138Dic142—44no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71C2×#4111yescharacteristic
147C22×C71C2×#4111yescharacteristic
148C22×C71C2×#4111yescharacteristic
149C22×C71H(5,𝔽2)11yescharacteristic
150C22×C71H(5,𝔽2)11yescharacteristic
151C22×C712−1+411yescharacteristic
152C22×C712−1+411yescharacteristic
153C4×C71—22no—
154C4×C71—41no—
155C4×C71—41no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71—41no—
159Dic71—22no—
160Dic71—22no—
161Dic71—41no—
162Dic71—41no—
163Dic71—41no—
164Dic71—41no—
165Dic71—41no—
166Dic71—41no—
167C2×C71—81no—
168C2×C71#53811yescharacteristic
169C2×C71#54211yescharacteristic
170C2×C71#54211yescharacteristic
171C2×C71#55111yescharacteristic
172C2×C71#55611yescharacteristic
173C2×C71#55611yescharacteristic
174C2×C71#56311yescharacteristic
175#2759—711nomaximal, Sylow
176C71#275911yescharacteristic, complemented, Sylow
177#401—711no—
178#397—711no—
179#397—711no—
180#395—711no—
181#393—711no—
182#393—711no—
183#391—711no—
184#391—711no—
185#391—711no—
186#388—711no—
187#388—711no—
188#388D7111yescharacteristic
189#387—711no—
190#387—711no—
191#383—711no—
192C22×Q8—711no—
193C22×Q8—711no—
194C2×C4⋊C4—711no—
195C2×C4⋊C4—711no—
196C2×C4⋊C4—711no—
197C2×C4⋊C4—711no—
198C2×C4⋊C4C2×D7111yescharacteristic
199C2×C22⋊C4—711no—
200C2×C22⋊C4—711no—
201C2×C22⋊C4—711no—
202C2×C22⋊C4—711no—
203C2×C22⋊C4C2×D7111yescharacteristic
204C2×C22⋊C4C2×D7111yescharacteristic
205C2×C22⋊C4C2×D7111yescharacteristic
206C2×C42—711no—
207C2×C42—711no—
208C2×C42C2×D7111yescharacteristic
209#95—711no—
210#95—711no—
211#95—711no—
212#95—711no—
213#95—711no—
214#95—711no—
215#95—711no—
216#95—711no—
217#95—711no—
218#95—711no—
219#95—711no—
220#95—711no—
221#95—711no—
222#95—711no—
223#95—711no—
224#95—711no—
225#95C2×D7111yescharacteristic
226#95C2×D7111yescharacteristic
227C24C22×D7111yescharacteristic
228C2×Q8—2841no—
229C2×Q8—2841no—
230C2×Q8—2841no—
231C2×Q8—1422no—
232C2×Q8—1422no—
233C2×Q8—1422no—
234C22×C4—711no—
235C22×C4—711no—
236C22×C4—711no—
237C22×C4—711no—
238C22×C4—711no—
239C22×C4—711no—
240C22×C4—711no—
241C22×C4—711no—
242C22×C4C22×D7111yescharacteristic
243C22×C4C22×D7111yescharacteristic
244C22×C4C22×D7111yescharacteristic
245C22×C4C22×D7111yescharacteristic
246C22×C4C22×D7111yescharacteristic
247C22×C4C22×D7111yescharacteristic
248C4⋊C4—41no—
249C4⋊C4—2841no—
250C4⋊C4—2841no—
251C4⋊C4—1422no—
252C4⋊C4—1422no—
253C22⋊C4—22no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—2841no—
257C22⋊C4—2841no—
258C22⋊C4—1422no—
259C22⋊C4—1422no—
260C42—41no—
261C42—2841no—
262C42—1422no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23C23×D7111yescharacteristic
271Q8—2844no—
272Q8—2844no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—712no—
291C2×C4—712no—
292C2×C4—1421no—
293C2×C4—1421no—
294C2×C4—1421no—
295C2×C4—1421no—
296C2×C4—1421no—
297C2×C4—1421no—
298C2×C4—1421no—
299C2×C4—1421no—
300C2×C4—1421no—
301C2×C4—1421no—
302C2×C4—1421no—
303C2×C4—1421no—
304C2×C4—1421no—
305C2×C4—1421no—
306C2×C4—1421no—
307C2×C4—1421no—
308C2×C4—1421no—
309C2×C4—1421no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4—1421no—
313C2×C4—1421no—
314C2×C4#1176586512yes—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22#41×D7111yescharacteristic
323C22#41×D7111yescharacteristic
324C22C2×#1176586511yescharacteristic
325C22#1455617311yescharacteristic
326C22#1455617311yescharacteristic
327C22#1455617511yescharacteristic
328C22#1455617511yescharacteristic
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—2841no—
336C4—2841no—
337C4—2841no—
338C4—2841no—
339C4—2841no—
340C4—2841no—
341C4—1422no—
342C4—1422no—
343C2—81no—
344C2#1798658111yescharacteristic
345C2#1798661411yescharacteristic
346C2#1798661711yescharacteristic
347C2#1798667811yescharacteristic
348C2#1798671611yescharacteristic
349C2#1798671711yescharacteristic
350C2#1798674911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#388×C71C2
hypercenter#388D71
hypocenterC71#2759
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C71⋊#401C2semidirect
4. #17985919C2semidirect
7. #17985904C2semidirect
8. #17985903C2semidirect
9. #17985903C2semidirect
12. #17985883C2semidirect
13. #17985883C2semidirect
14. #17985866C2semidirect
176. C71#2759semidirect