Finite Group Databasebeta

← C89×#9306 #36387154

Subgroups — C89×#9306 #36387154

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3037C211yesmaximal, characteristic
2C89×#2982C211yesmaximal, characteristic, complemented
3C89×#1805C211yesmaximal, characteristic, complemented
4C89×#1517C211yesmaximal, characteristic
5C89×#1480C211yesmaximal, characteristic
6C89×#1301C211yesmaximal, characteristic, complemented
7C89×#1286C211yesmaximal, characteristic, complemented
8C2×#137×C89—21no—
9#532×C89C2211yescharacteristic
10C2×C4×Q8×C89C2211yescharacteristic
11#464×C89—22no—
12#462×C89C412yescomplemented
13#460×C89C412yescomplemented
14C8×Q8×C89—22no—
15C8×Q8×C89—22no—
16#431×C89—22no—
17#431×C89C2211yescharacteristic
18#431×C89C2211yescharacteristic
19#423×C89C2211yescharacteristic
20C2×C4⋊C8×C89C2211yescharacteristic
21#417×C89—21no—
22C2×#103×C89—21no—
23#382×C89C2211yescharacteristic
24#340×C89—21no—
25#327×C89—22no—
26D4⋊C8×C89—22no—
27C2×#41×C89—21no—
28C22×Q8×C89D411yescharacteristic
29#137×C89—42no—
30#137×C89—42no—
31C2×Q16×C89—22no—
32C2×SD16×C89—22no—
33C2×M16×C89—21no—
34C22×C8×C89—21no—
35C4⋊Q8×C89C2×C411yescharacteristic
36#127×C89C2×C411yescharacteristic
37#124×C89—21no—
38#121×C89—21no—
39#121×C89—21no—
40C4×Q8×C89—21no—
41C4×Q8×C89—21no—
42C4×Q8×C89C2×C412yes—
43C4×Q8×C89D412yes—
44C4×D4×C89—21no—
45C4×D4×C89—21no—
46C2×C4⋊C4×C89—21no—
47C2×C4⋊C4×C89—21no—
48C2×C4⋊C4×C89D411yescharacteristic
49C2×C42×C89—21no—
50C2×C42×C89C2311yescharacteristic
51C4⋊C8×C89—24no—
52C4⋊C8×C89—21no—
53C4⋊C8×C89—21no—
54C4⋊C8×C89—21no—
55C4⋊C8×C89—21no—
56C4⋊C8×C89—21no—
57C4⋊C8×C89C2×C412yes—
58#103×C89—22no—
59#103×C89—42no—
60#102×C89—22no—
61C22⋊C8×C89—21no—
62C22⋊C8×C89D412yes—
63C4×C8×C89—24no—
64C4×C8×C89—21no—
65C4×C8×C89D412yes—
66#95×C89—21no—
67#95×C89—21no—
68#41×C89—41no—
69#41×C89—41no—
70#41×C89—41no—
71#41×C89—41no—
72C2×Q8×C89—21no—
73C2×Q8×C89—21no—
74C2×Q8×C89—21no—
75C2×Q8×C89—21no—
76C2×Q8×C89C22⋊C412yes—
77C2×Q8×C89D812yes—
78C2×Q8×C89SD1612yes—
79C2×D4×C89—21no—
80C2×D4×C89—21no—
81C2×D4×C89—21no—
82C2×D4×C89—41no—
83C22×C4×C89—21no—
84C22×C4×C89—21no—
85C22×C4×C89—41no—
86C22×C4×C89—41no—
87C22×C4×C89C2×D411yescharacteristic
88C22×C4×C89C2×D411yescharacteristic
89C22×C4×C89#4111yescharacteristic
90Q16×C89—44no—
91SD16×C89—44no—
92M16×C89—41no—
93M16×C89—41no—
94C2×C8×C89—22no—
95C2×C8×C89—22no—
96C2×C8×C89—22no—
97C2×C8×C89—41no—
98C2×C8×C89—41no—
99C2×C8×C89—41no—
100C4⋊C4×C89—22no—
101C4⋊C4×C89—21no—
102C4⋊C4×C89—22no—
103C4⋊C4×C89—21no—
104C4⋊C4×C89—21no—
105C4⋊C4×C89—41no—
106C4⋊C4×C89—41no—
107C4⋊C4×C89C22⋊C412yes—
108C22⋊C4×C89—41no—
109C22⋊C4×C89—41no—
110C42×C89—22no—
111C42×C89—22no—
112C42×C89C22×C411yescharacteristic
113C42×C89C2×D411yescharacteristic
114C42×C89C2×D411yescharacteristic
115C42×C89#4111yescharacteristic
116C23×C89—41no—
117C23×C89#12111yescharacteristic
118Q8×C89—22no—
119Q8×C89—22no—
120Q8×C89—22no—
121Q8×C89—41no—
122Q8×C89—41no—
123Q8×C89#10214yes—
124D4×C89—42no—
125D4×C89—41no—
126D4×C89—41no—
127D4×C89—81no—
128D4×C89—81no—
129D4×C89—81no—
130C2×C4×C89—22no—
131C2×C4×C89—22no—
132C2×C4×C89—22no—
133C2×C4×C89—22no—
134C2×C4×C89—41no—
135C2×C4×C89—41no—
136C2×C4×C89—41no—
137C2×C4×C89—81no—
138C2×C4×C89—81no—
139C2×C4×C89—81no—
140C2×C4×C89—81no—
141C2×C4×C89—81no—
142C2×C4×C89—81no—
143C2×C4×C89C2×C22⋊C411yescharacteristic
144C2×C4×C89C4×D411yescharacteristic
145C2×C4×C89C4×D411yescharacteristic
146C2×C4×C89#12011yescharacteristic
147C2×C4×C89#12111yescharacteristic
148C2×C4×C89#12311yescharacteristic
149C2×C4×C89C2×D811yescharacteristic
150C2×C4×C89C2×SD1611yescharacteristic
151C2×C4×C89#13511yescharacteristic
152C2×C4×C89#13511yescharacteristic
153C8×C89—42no—
154C8×C89—42no—
155C8×C89—42no—
156C8×C89—81no—
157C22×C89—21no—
158C22×C89—21no—
159C22×C89—81no—
160C22×C89—81no—
161C22×C89—81no—
162C22×C89#38611yescharacteristic
163C22×C89#46511yescharacteristic
164C22×C89#46611yescharacteristic
165C4×C89—21no—
166C4×C89—21no—
167C4×C89—24no—
168C4×C89—21no—
169C4×C89—21no—
170C4×C89—42no—
171C4×C89—81no—
172C4×C89—161no—
173C4×C89C2×#10211yescharacteristic
174C4×C89#41611yescharacteristic
175C4×C89#46511yescharacteristic
176C4×C89#46611yescharacteristic
177#9306C8911yesmaximal, characteristic, direct product factor, Sylow
178C2×C89—21no—
179C2×C89—21no—
180C2×C89—161no—
181C2×C89#188111yescharacteristic
182C2×C89#189311yescharacteristic
183C2×C89#190511yescharacteristic
184#3037C2×C8911yescharacteristic
185#2982C2×C8911yescharacteristic, complemented
186#1805C2×C8911yescharacteristic, complemented
187#1517C2×C8911yescharacteristic
188#1480C2×C8911yescharacteristic
189#1301C2×C8911yescharacteristic, complemented
190#1286C2×C8911yescharacteristic, complemented
191C89#930611yescharacteristic, direct product factor, Sylow
192C2×#137—21no—
193#532C22×C8911yescharacteristic
194C2×C4×Q8C22×C8911yescharacteristic
195#464—22no—
196#462C4×C8912yescomplemented
197#460C4×C8912yescomplemented
198C8×Q8—22no—
199C8×Q8—22no—
200#431—22no—
201#431C22×C8911yescharacteristic
202#431C22×C8911yescharacteristic
203#423C22×C8911yescharacteristic
204C2×C4⋊C8C22×C8911yescharacteristic
205#417—21no—
206C2×#103—21no—
207#382C22×C8911yescharacteristic
208#340—21no—
209#327—22no—
210D4⋊C8—22no—
211C2×#41—21no—
212C22×Q8D4×C8911yescharacteristic
213#137—42no—
214#137—42no—
215C2×Q16—22no—
216C2×SD16—22no—
217C2×M16—21no—
218C22×C8—21no—
219C4⋊Q8C2×C4×C8911yescharacteristic
220#127C2×C4×C8911yescharacteristic
221#124—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C2×C4×C8912yes—
227C4×Q8D4×C8912yes—
228C4×D4—21no—
229C4×D4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4D4×C8911yescharacteristic
233C2×C42—21no—
234C2×C42C23×C8911yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8C2×C4×C8912yes—
242#103—22no—
243#103—42no—
244#102—22no—
245C22⋊C8—21no—
246C22⋊C8D4×C8912yes—
247C4×C8—24no—
248C4×C8—21no—
249C4×C8D4×C8912yes—
250#95—21no—
251#95—21no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41—41no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8C22⋊C4×C8912yes—
261C2×Q8D8×C8912yes—
262C2×Q8SD16×C8912yes—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C2×D4×C8911yescharacteristic
272C22×C4C2×D4×C8911yescharacteristic
273C22×C4#41×C8911yescharacteristic
274Q16—44no—
275SD16—44no—
276M16—41no—
277M16—41no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4C22⋊C4×C8912yes—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—22no—
295C42—22no—
296C42C22×C4×C8911yescharacteristic
297C42C2×D4×C8911yescharacteristic
298C42C2×D4×C8911yescharacteristic
299C42#41×C8911yescharacteristic
300C23—41no—
301C23#121×C8911yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—22no—
305Q8—41no—
306Q8—41no—
307Q8#102×C8914yes—
308D4—42no—
309D4—41no—
310D4—41no—
311D4—81no—
312D4—81no—
313D4—81no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4C2×C22⋊C4×C8911yescharacteristic
328C2×C4C4×D4×C8911yescharacteristic
329C2×C4C4×D4×C8911yescharacteristic
330C2×C4#120×C8911yescharacteristic
331C2×C4#121×C8911yescharacteristic
332C2×C4#123×C8911yescharacteristic
333C2×C4C2×D8×C8911yescharacteristic
334C2×C4C2×SD16×C8911yescharacteristic
335C2×C4#135×C8911yescharacteristic
336C2×C4#135×C8911yescharacteristic
337C8—42no—
338C8—42no—
339C8—42no—
340C8—81no—
341C22—21no—
342C22—21no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#386×C8911yescharacteristic
347C22#465×C8911yescharacteristic
348C22#466×C8911yescharacteristic
349C4—21no—
350C4—21no—
351C4—24no—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—81no—
356C4—161no—
357C4C2×#102×C8911yescharacteristic
358C4#416×C8911yescharacteristic
359C4#465×C8911yescharacteristic
360C4#466×C8911yescharacteristic
361C2—21no—
362C2—21no—
363C2—161no—
364C2C89×#188111yescharacteristic
365C2C89×#189311yescharacteristic
366C2C89×#190511yescharacteristic
subgroupG/H
center, socleC22×C89#386
commutatorC42C22×C4×C89
Frattini, cosocleC2×C42C23×C89
prime radical#532×C89C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2982C2semidirect
3. C89×#1805C2semidirect
6. C89×#1301C2semidirect
7. C89×#1286C2semidirect
12. #462×C89C4semidirect
13. #460×C89C4semidirect
177. #9306C89direct (N × H)
185. #2982C2×C89semidirect
186. #1805C2×C89semidirect
189. #1301C2×C89semidirect
190. #1286C2×C89semidirect
191. C89#9306direct (N × H)
196. #462C4×C89semidirect
197. #460C4×C89semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C89×#30372
C89×#29822
C89×#18052
C89×#15172
C89×#14802
C89×#13012
C89×#12862
#930689