Finite Group Databasebeta

← #22600511

Subgroups — #22600511

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986695C211yesmaximal, characteristic
2#17986367C211yesmaximal, characteristic, complemented
3#17986337C211yesmaximal, characteristic, complemented
4#17986332C211yesmaximal, characteristic, complemented
5#17986306C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986230C211yesmaximal, characteristic, complemented
8#17986201C211yesmaximal, characteristic, complemented
9#17986138C211yesmaximal, characteristic, complemented
10#17986138C211yesmaximal, characteristic, complemented
11#17986097C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986008C211yesmaximal, characteristic, complemented
14C2×#14556052C211yesmaximal, characteristic, complemented
15D4×Q8×C71C211yesmaximal, characteristic, complemented
16C2×#11765852C2211yescharacteristic
17C2×#11765852C2211yescharacteristic
18C2×#11765852C2211yescharacteristic
19C2×#11765849C2211yescharacteristic
20C22×C71⋊C8C2211yescharacteristic
21#14556123—21no—
22#14556120C2211yescharacteristic
23#14556110—21no—
24#14556102C2211yescharacteristic
25#14556102C2211yescharacteristic
26#14556100—21no—
27C4×D284C2211yescharacteristic
28(C22⋊D71)⋊C4—21no—
29#14556087C2211yescharacteristic
30C2×C4⋊Dic71C2211yescharacteristic
31#14556061C2211yescharacteristic
32#14556061C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556052C2211yescharacteristic
36#14556052C2211yescharacteristic
37#14556052C2211yescharacteristic
38#14556052C2211yescharacteristic
39#14556052C2211yescharacteristic
40#14556052C2211yescharacteristic
41#14556052C2211yescharacteristic
42#14556052C2211yescharacteristic
43#14556052C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
48C22×Q8×C71C2211yescharacteristic
49C22×Q8×C71C2211yescharacteristic
50C4⋊Q8×C71—21no—
51C4⋊Q8×C71C2211yescharacteristic
52C22⋊Q8×C71—21no—
53C22⋊Q8×C71—21no—
54C22⋊Q8×C71C2211yescharacteristic
55C22⋊Q8×C71C2211yescharacteristic
56C4×Q8×C71C2211yescharacteristic
57C4×D4×C71—21no—
58C4×D4×C71C2211yescharacteristic
59C2×D284C2311yescharacteristic
60C2×C22⋊D71—21no—
61C2×C22⋊D71—21no—
62C22×Dic71—21no—
63C2×C4×D71—21no—
64#11765852—22no—
65#11765852—41no—
66#11765852—41no—
67#11765849—41no—
68C2×C71⋊C8—41no—
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C2×C71⋊C8C2311yescharacteristic
73C4⋊Dic71—21no—
74C4⋊Dic71C2311yescharacteristic
75C4⋊Dic71C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82(C2×D71)⋊C4—21no—
83C4×Dic71—21no—
84C2×Q8×C71—21no—
85C2×Q8×C71—21no—
86C2×Q8×C71—22no—
87C2×Q8×C71—22no—
88C2×Q8×C71—41no—
89C2×Q8×C71—41no—
90C2×Q8×C71C2311yescharacteristic
91C2×Q8×C71C2311yescharacteristic
92C2×Q8×C71C2311yescharacteristic
93C2×D4×C71D411yescharacteristic
94C22×C4×C71—21no—
95C22×C4×C71—21no—
96C22×C4×C71C2311yescharacteristic
97C22×C4×C71C2311yescharacteristic
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71—21no—
102C4⋊C4×C71—21no—
103C4⋊C4×C71D411yescharacteristic
104C4⋊C4×C71C2311yescharacteristic
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71C2×D411yescharacteristic
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114Q8×C71—21no—
115Q8×C71—21no—
116Q8×C71—21no—
117Q8×C71—21no—
118Q8×C71—21no—
119Q8×C71—42no—
120Q8×C71—42no—
121Q8×C71#4112yes—
122D4×C71—22no—
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—42no—
132C2×C4×C71—42no—
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2411yescharacteristic
138C22⋊D71—42no—
139C22⋊D71—81no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—81no—
144D284—21no—
145D284—21no—
146C4×D71—81no—
147C71⋊C8—21no—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71#13712yes—
157C22×C71C22×D411yescharacteristic
158C2×D71—41no—
159C2×D71—41no—
160C2×D71—41no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—22no—
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71C2×#4111yescharacteristic
170C4×C71H(5,𝔽2)11yescharacteristic
171Dic71—41no—
172Dic71—41no—
173Dic71—41no—
174C2×C71—22no—
175C2×C71—41no—
176C2×C71#54611yescharacteristic
177C2×C71C2×#13711yescharacteristic
178C2×C71#57711yescharacteristic
179D71—81no—
180#3320—711nomaximal, Sylow
181C71#332011yescharacteristic, complemented, Sylow
182#553—711no—
183D4×Q8D7111yescharacteristic, complemented
184#491—711no—
185#485—711no—
186#484—711no—
187#479—711no—
188#468—711no—
189#465—711no—
190#461—711no—
191#448—711no—
192#448—711no—
193#440—711no—
194#435—711no—
195#419—711no—
196C2×#103—711no—
197C22×Q8C2×D7111yescharacteristic
198C22×Q8C2×D7111yescharacteristic
199C2×SD16—711no—
200C2×SD16—711no—
201C2×SD16—711no—
202C2×M16—711no—
203C22×C8—711no—
204C4⋊Q8—21no—
205C4⋊Q8C2×D7111yescharacteristic
206#126—1421no—
207#125—711no—
208#123—1421no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D7111yescharacteristic
212C22⋊Q8C2×D7111yescharacteristic
213#121—711no—
214#121—711no—
215#121—1421no—
216C4×Q8C2×D7111yescharacteristic
217C4×D4—21no—
218C4×D4—711no—
219C4×D4—1421no—
220C4×D4C2×D7111yescharacteristic
221#117—711no—
222C2×C4⋊C4—711no—
223#107—711no—
224#107—711no—
225#106—711no—
226C4⋊C8—711no—
227#103—711no—
228#103—711no—
229#103—711no—
230#103—711no—
231#103—711no—
232#103—711no—
233#103—711no—
234#103—711no—
235#103—711no—
236#102—711no—
237C22⋊C8—711no—
238C22⋊C8—711no—
239#97—711no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C22×D7111yescharacteristic
247C2×Q8C22×D7111yescharacteristic
248C2×Q8C22×D7111yescharacteristic
249C2×D4—711no—
250C2×D4—1421no—
251C2×D4—1421no—
252C2×D4C22⋊D7111yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1421no—
256C22×C4—1421no—
257C22×C4C22×D7111yescharacteristic
258C22×C4C22×D7111yescharacteristic
259SD16—2841no—
260SD16—2841no—
261SD16—1422no—
262M16—2841no—
263C2×C8—2841no—
264C2×C8—711no—
265C2×C8—711no—
266C2×C8—711no—
267C2×C8—711no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—711no—
274C4⋊C4—711no—
275C4⋊C4—711no—
276C4⋊C4—1421no—
277C4⋊C4—1421no—
278C4⋊C4—1421no—
279C4⋊C4C22⋊D7111yescharacteristic
280C4⋊C4C22×D7111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—1421no—
284C22⋊C4—1421no—
285C22⋊C4—1421no—
286C22⋊C4—1421no—
287C22⋊C4C22⋊D7111yescharacteristic
288C22⋊C4C22⋊D7111yescharacteristic
289C42—21no—
290C42—1421no—
291C42C22×D7111yescharacteristic
292C23—1421no—
293C23C2×C22⋊D7111yescharacteristic
294C23C2×C22⋊D7111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—42no—
302Q8#1176586412yes—
303D4—22no—
304D4—2842no—
305D4—5681no—
306D4—1421no—
307D4—1421no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—5681no—
319C2×C4—5681no—
320C2×C4—1421no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4C2×C22⋊D7111yescharacteristic
324C2×C4C2×C22⋊D7111yescharacteristic
325C2×C4C2×C22⋊D7111yescharacteristic
326C2×C4C2×C22⋊D7111yescharacteristic
327C2×C4C23×D7111yescharacteristic
328C8—1421no—
329C8—1421no—
330C8—1421no—
331C8—1421no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—2841no—
338C22—2841no—
339C22—2841no—
340C22#1455615912yes—
341C22C22×C22⋊D7111yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—2841no—
351C4—2841no—
352C4—2841no—
353C4C2×#1176586411yescharacteristic
354C4#1455617311yescharacteristic
355C2—22no—
356C2—41no—
357C2—5681no—
358C2#1798664311yescharacteristic
359C2C2×#1455615911yescharacteristic
360C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingD4×Q8×C71C2
socleC22×C71C22×D4
hypercenterD4×Q8D71
hypocenterC71#3320
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #17986367C2semidirect
3. #17986337C2semidirect
4. #17986332C2semidirect
5. #17986306C2semidirect
6. #17986247C2semidirect
7. #17986230C2semidirect
8. #17986201C2semidirect
9. #17986138C2semidirect
10. #17986138C2semidirect
11. #17986097C2semidirect
12. #17986071C2semidirect
13. #17986008C2semidirect
14. C2×#14556052C2semidirect
15. D4×Q8×C71C2semidirect
181. C71#3320semidirect
183. D4×Q8D71semidirect