Finite Group Databasebeta

← C89×#9766 #36387614

Subgroups — C89×#9766 #36387614

Order 22784.

396 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3167C211yesmaximal, characteristic
2C89×#1746C211yesmaximal, characteristic, complemented
3C89×#1568C211yesmaximal, characteristic, complemented
4C89×#1531C211yesmaximal, characteristic, complemented
5C89×#1530C211yesmaximal, characteristic
6C89×#1506C211yesmaximal, characteristic, complemented
7C89×#1505C211yesmaximal, characteristic
8#533×C89C2211yescharacteristic
9#532×C89C2211yescharacteristic
10#498×C89C411yescharacteristic, complemented
11#492×C89C411yescharacteristic, complemented
12#487×C89C411yescharacteristic, complemented
13#486×C89C411yescharacteristic, complemented
14#465×C89—21no—
15#465×C89—21no—
16C4×SD16×C89—21no—
17C4×SD16×C89—21no—
18#432×C89C2211yescharacteristic
19#432×C89C2211yescharacteristic
20C2×#97×C89C2211yescharacteristic
21C2×C4×C8×C89C2211yescharacteristic
22#376×C89C2211yescharacteristic
23#336×C89—21no—
24#336×C89—21no—
25#335×C89—21no—
26#334×C89—21no—
27#330×C89—22no—
28#329×C89—22no—
29#328×C89—22no—
30#327×C89—22no—
31C2×#41×C89—21no—
32C2×SD16×C89—21no—
33C2×SD16×C89—21no—
34C22×C8×C89—21no—
35C22×C8×C89D411yescharacteristic
36C22×C8×C89D411yescharacteristic
37C4⋊Q8×C89C2×C411yescharacteristic
38C4⋊Q8×C89C2×C411yescharacteristic
39C4⋊Q8×C89C2×C411yescharacteristic
40#127×C89C2×C411yescharacteristic
41#125×C89—21no—
42#124×C89—21no—
43C22⋊Q8×C89—21no—
44C22⋊Q8×C89—21no—
45#121×C89—21no—
46#121×C89—21no—
47C4×Q8×C89—21no—
48C4×Q8×C89—21no—
49C4×D4×C89—21no—
50C4×D4×C89—21no—
51#117×C89—21no—
52C2×C42×C89—21no—
53C2×C42×C89C2311yescharacteristic
54#106×C89—21no—
55#106×C89—21no—
56C4⋊C8×C89—21no—
57C4⋊C8×C89—21no—
58#103×C89—21no—
59#103×C89—21no—
60#102×C89—21no—
61#102×C89—21no—
62C22⋊C8×C89—21no—
63C22⋊C8×C89—21no—
64#97×C89—21no—
65#97×C89—21no—
66#97×C89—21no—
67#97×C89D412yes—
68C4×C8×C89C2×C411yescharacteristic
69C4×C8×C89C2×C411yescharacteristic
70C4×C8×C89D411yescharacteristic
71C4×C8×C89D411yescharacteristic
72#95×C89—21no—
73#95×C89D412yes—
74#41×C89—82no—
75C2×Q8×C89—21no—
76C2×Q8×C89—21no—
77C2×Q8×C89—21no—
78C2×D4×C89—21no—
79C2×D4×C89—21no—
80C2×D4×C89—21no—
81C2×D4×C89—41no—
82C22×C4×C89—22no—
83C22×C4×C89—41no—
84C22×C4×C89C2×D411yescharacteristic
85C22×C4×C89C2×D411yescharacteristic
86C22×C4×C89#4111yescharacteristic
87SD16×C89—42no—
88SD16×C89—42no—
89C2×C8×C89—22no—
90C2×C8×C89—41no—
91C2×C8×C89—41no—
92C2×C8×C89—41no—
93C2×C8×C89—41no—
94C2×C8×C89—41no—
95C2×C8×C89—41no—
96C2×C8×C89C22⋊C411yescharacteristic
97C2×C8×C89C22⋊C411yescharacteristic
98C2×C8×C89C22⋊C411yescharacteristic
99C2×C8×C89C22⋊C411yescharacteristic
100C4⋊C4×C89—21no—
101C4⋊C4×C89—21no—
102C4⋊C4×C89—21no—
103C4⋊C4×C89—21no—
104C4⋊C4×C89—21no—
105C4⋊C4×C89—21no—
106C4⋊C4×C89—41no—
107C4⋊C4×C89—41no—
108C22⋊C4×C89—41no—
109C22⋊C4×C89—41no—
110C22⋊C4×C89—41no—
111C42×C89—41no—
112C42×C89—41no—
113C42×C89—41no—
114C42×C89C22×C411yescharacteristic
115C42×C89C2×D411yescharacteristic
116C42×C89C2×D411yescharacteristic
117C42×C89#4111yescharacteristic
118C23×C89—41no—
119C23×C89#12111yescharacteristic
120Q8×C89—41no—
121Q8×C89—41no—
122Q8×C89—81no—
123Q8×C89—81no—
124D4×C89—41no—
125D4×C89—41no—
126D4×C89—81no—
127D4×C89—81no—
128D4×C89—82no—
129C2×C4×C89—22no—
130C2×C4×C89—21no—
131C2×C4×C89—21no—
132C2×C4×C89—41no—
133C2×C4×C89—42no—
134C2×C4×C89—42no—
135C2×C4×C89—41no—
136C2×C4×C89—41no—
137C2×C4×C89—41no—
138C2×C4×C89—42no—
139C2×C4×C89—82no—
140C2×C4×C89C2×C22⋊C411yescharacteristic
141C2×C4×C89C4×D411yescharacteristic
142C2×C4×C89C4×D411yescharacteristic
143C2×C4×C89#12111yescharacteristic
144C2×C4×C89#12411yescharacteristic
145C2×C4×C89#12711yescharacteristic
146C8×C89—21no—
147C8×C89—21no—
148C8×C89—21no—
149C8×C89—21no—
150C8×C89—42no—
151C8×C89—81no—
152C8×C89—81no—
153C22×C89—21no—
154C22×C89—21no—
155C22×C89—81no—
156C22×C89—81no—
157C22×C89—81no—
158C22×C89#39011yescharacteristic
159C22×C89#47112yes—
160C4×C89—21no—
161C4×C89—21no—
162C4×C89—21no—
163C4×C89—21no—
164C4×C89—21no—
165C4×C89—82no—
166C4×C89—81no—
167C4×C89—81no—
168C4×C89—81no—
169C4×C89#42111yescharacteristic
170C4×C89#42111yescharacteristic
171#9766C8911yesmaximal, characteristic, direct product factor, Sylow
172C2×C89—22no—
173C2×C89—161no—
174C2×C89#196311yescharacteristic
175C2×C89#197611yescharacteristic
176C2×C89#197711yescharacteristic
177#3167C2×C8911yescharacteristic
178#1746C2×C8911yescharacteristic, complemented
179#1568C2×C8911yescharacteristic, complemented
180#1531C2×C8911yescharacteristic, complemented
181#1530C2×C8911yescharacteristic
182#1506C2×C8911yescharacteristic, complemented
183#1505C2×C8911yescharacteristic
184C89#976611yescharacteristic, direct product factor, Sylow
185#533C22×C8911yescharacteristic
186#532C22×C8911yescharacteristic
187#498C4×C8911yescharacteristic, complemented
188#492C4×C8911yescharacteristic, complemented
189#487C4×C8911yescharacteristic, complemented
190#486C4×C8911yescharacteristic, complemented
191#465—21no—
192#465—21no—
193C4×SD16—21no—
194C4×SD16—21no—
195#432C22×C8911yescharacteristic
196#432C22×C8911yescharacteristic
197C2×#97C22×C8911yescharacteristic
198C2×C4×C8C22×C8911yescharacteristic
199#376C22×C8911yescharacteristic
200#336—21no—
201#336—21no—
202#335—21no—
203#334—21no—
204#330—22no—
205#329—22no—
206#328—22no—
207#327—22no—
208C2×#41—21no—
209C2×SD16—21no—
210C2×SD16—21no—
211C22×C8—21no—
212C22×C8D4×C8911yescharacteristic
213C22×C8D4×C8911yescharacteristic
214C4⋊Q8C2×C4×C8911yescharacteristic
215C4⋊Q8C2×C4×C8911yescharacteristic
216C4⋊Q8C2×C4×C8911yescharacteristic
217#127C2×C4×C8911yescharacteristic
218#125—21no—
219#124—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×D4—21no—
227C4×D4—21no—
228#117—21no—
229C2×C42—21no—
230C2×C42C23×C8911yescharacteristic
231#106—21no—
232#106—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241#97—21no—
242#97—21no—
243#97—21no—
244#97D4×C8912yes—
245C4×C8C2×C4×C8911yescharacteristic
246C4×C8C2×C4×C8911yescharacteristic
247C4×C8D4×C8911yescharacteristic
248C4×C8D4×C8911yescharacteristic
249#95—21no—
250#95D4×C8912yes—
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4C2×D4×C8911yescharacteristic
262C22×C4C2×D4×C8911yescharacteristic
263C22×C4#41×C8911yescharacteristic
264SD16—42no—
265SD16—42no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8C22⋊C4×C8911yescharacteristic
274C2×C8C22⋊C4×C8911yescharacteristic
275C2×C8C22⋊C4×C8911yescharacteristic
276C2×C8C22⋊C4×C8911yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42C22×C4×C8911yescharacteristic
292C42C2×D4×C8911yescharacteristic
293C42C2×D4×C8911yescharacteristic
294C42#41×C8911yescharacteristic
295C23—41no—
296C23#121×C8911yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—81no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4C2×C22⋊C4×C8911yescharacteristic
318C2×C4C4×D4×C8911yescharacteristic
319C2×C4C4×D4×C8911yescharacteristic
320C2×C4#121×C8911yescharacteristic
321C2×C4#124×C8911yescharacteristic
322C2×C4#127×C8911yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C8—42no—
328C8—81no—
329C8—81no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#390×C8911yescharacteristic
336C22#471×C8912yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4#421×C8911yescharacteristic
347C4#421×C8911yescharacteristic
348C2—22no—
349C2—161no—
350C2C89×#196311yescharacteristic
351C2C89×#197611yescharacteristic
352C2C89×#197711yescharacteristic
subgroupG/H
center, socleC22×C89#390
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×#1746C2semidirect
3. C89×#1568C2semidirect
4. C89×#1531C2semidirect
6. C89×#1506C2semidirect
10. #498×C89C4semidirect
11. #492×C89C4semidirect
12. #487×C89C4semidirect
13. #486×C89C4semidirect
171. #9766C89direct (N × H)
178. #1746C2×C89semidirect
179. #1568C2×C89semidirect
180. #1531C2×C89semidirect
182. #1506C2×C89semidirect
184. C89#9766direct (N × H)
187. #498C4×C89semidirect
188. #492C4×C89semidirect
189. #487C4×C89semidirect
190. #486C4×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#31672
C89×#17462
C89×#15682
C89×#15312
C89×#15302
C89×#15062
C89×#15052
#976689