Finite Group Databasebeta

← #36472345

Subgroups — #36472345

Order 22784.

394 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151400C211yesmaximal, characteristic
2C2×#19151398C211yesmaximal, characteristic
3C2×C16×Dic89C211yesmaximal, characteristic, complemented
4#26021116C211yesmaximal, characteristic
5#26020986C211yesmaximal, characteristic, complemented
6#26018343C211yesmaximal, characteristic, complemented
7C89×#1380C211yesmaximal, characteristic, complemented
8C22×D712C2211yescharacteristic
9C22×#13083549C2211yescharacteristic
10C2×#15667946C2211yescharacteristic
11C2×#15667932—21no—
12C2×C8×Dic89C2211yescharacteristic
13#19151499—21no—
14#19151497—21no—
15#19151400—22no—
16#19151398C414yescomplemented
17C16×Dic89—22no—
18#19151360—21no—
19C22×C16×C89C2211yescharacteristic
20C2×#107×C89C2211yescharacteristic
21C2×#107×C89C2211yescharacteristic
22C22×D356—21no—
23C2×D712—21no—
24C2×D712—21no—
25C2×D712—21no—
26C2×D712—21no—
27C2×D712—21no—
28C2×D712C2×C412yes—
29C22×#10623494D411yescharacteristic
30C2×C4⋊Dic89—21no—
31C2×C4⋊Dic89—41no—
32C2×#13083551—41no—
33C2×#13083551—41no—
34C2×C4×Dic89D411yescharacteristic
35C2×#13083549—21no—
36C2×#13083549—21no—
37C2×#13083549C2×C412yes—
38#15667946—21no—
39#15667946—21no—
40#15667932—42no—
41C8×Dic89—21no—
42C8×Dic89—21no—
43#15667919—41no—
44C22×C8×C89C2311yescharacteristic
45C2×C4⋊C4×C89—21no—
46C2×C4⋊C4×C89—21no—
47C2×C16×C89—21no—
48C2×C16×C89—21no—
49C2×C16×C89D412yes—
50#107×C89—21no—
51#107×C89—21no—
52#107×C89—21no—
53#107×C89C2×C412yes—
54C23×D89—41no—
55C2×D356—22no—
56C2×D356—41no—
57C2×D356—41no—
58C2×D356—41no—
59C2×D356—41no—
60C2×D356—41no—
61C22×Dic89—41no—
62C22×Dic89D811yescharacteristic
63C22×Dic89D811yescharacteristic
64D712—22no—
65D712—22no—
66D712—41no—
67D712—41no—
68C2×#10623494—21no—
69C2×#10623494—21no—
70C2×#10623494SD1612yes—
71C4⋊Dic89—41no—
72C4⋊Dic89—41no—
73C4⋊Dic89—82no—
74#13083551—82no—
75#13083551—82no—
76C4×Dic89—21no—
77C4×Dic89—21no—
78#13083549—22no—
79C22×C4×C89—41no—
80C22×C4×C89—41no—
81C22×C4×C89C2×D411yescharacteristic
82C2×C8×C89C22×C411yescharacteristic
83C2×C8×C89C2×D411yescharacteristic
84C2×C8×C89#4111yescharacteristic
85C2×C8×C89#4111yescharacteristic
86C2×C8×C89#4111yescharacteristic
87C2×C8×C89#4111yescharacteristic
88C4⋊C4×C89—22no—
89C4⋊C4×C89—41no—
90C4⋊C4×C89—41no—
91C4⋊C4×C89—41no—
92C16×C89—22no—
93C23×C89C2×D811yescharacteristic
94C22×D89—42no—
95C22×D89—81no—
96C22×D89—81no—
97C22×D89—81no—
98C22×D89—81no—
99C22×D89—81no—
100C22×D89—81no—
101C2×C4×C89—42no—
102C2×C4×C89—81no—
103C2×C4×C89—81no—
104C2×C4×C89—81no—
105C2×C4×C89—81no—
106C2×C4×C89—81no—
107C2×C4×C89C4×D411yescharacteristic
108C2×C4×C89#12111yescharacteristic
109C2×C4×C89#12411yescharacteristic
110C2×C4×C89C2×SD1611yescharacteristic
111C2×C4×C89#13511yescharacteristic
112C2×C4×C89#13511yescharacteristic
113C2×Dic89—21no—
114C2×Dic89—21no—
115C2×Dic89—21no—
116C2×Dic89—21no—
117C2×Dic89—81no—
118C2×Dic89—81no—
119C2×Dic89—81no—
120C2×Dic89D1612yes—
121C2×Dic89SD3212yes—
122D356—42no—
123D356—42no—
124D356—81no—
125D356—81no—
126C8×C89#11711yescharacteristic
127C8×C89C4×D411yescharacteristic
128C8×C89#12311yescharacteristic
129C8×C89#12611yescharacteristic
130#10623494—22no—
131C22×C89C4×D811yescharacteristic
132C22×C89#46611yescharacteristic
133C22×C89#48611yescharacteristic
134C22×C89C2×D1611yescharacteristic
135C22×C89C2×SD3211yescharacteristic
136C22×C89#50811yescharacteristic
137C22×C89#50811yescharacteristic
138C2×D89—82no—
139C2×D89—82no—
140C2×D89—82no—
141C2×D89—161no—
142C2×D89—161no—
143C2×D89—161no—
144C2×D89—161no—
145C4×C89—82no—
146C4×C89—161no—
147C4×C89#38811yescharacteristic
148C4×C89C4×SD1611yescharacteristic
149C4×C89#46511yescharacteristic
150C4×C89#48811yescharacteristic
151Dic89—22no—
152Dic89—22no—
153Dic89—161no—
154#13147—891nomaximal, Sylow
155C2×C89#193411yescharacteristic
156C2×C89C4×D1611yescharacteristic
157C2×C89C4×SD3211yescharacteristic
158C2×C89#221511yescharacteristic
159C2×C89#221711yescharacteristic
160C2×C89#224011yescharacteristic
161C2×C89#224211yescharacteristic
162D89—162no—
163C2×#357—891no—
164C2×#357—891no—
165C2×C4×C16—891no—
166#1970—891no—
167#1942—891no—
168#1380—891no—
169#1380D8911yescharacteristic, complemented
170C89#1314711yescharacteristic, complemented, Sylow
171C22×D8—891no—
172C22×C16—891no—
173C22×C16C2×D8911yescharacteristic
174C2×#107—891no—
175C2×#107C2×D8911yescharacteristic
176C2×#107C2×D8911yescharacteristic
177C2×#102—1781no—
178C2×C4×C8—891no—
179#390—1781no—
180#389—1781no—
181#357—894no—
182#357—1782no—
183C4×C16—1782no—
184#340—1781no—
185C22×D4—1781no—
186C2×D8—892no—
187C2×D8—1781no—
188C2×D8—1781no—
189C2×D8—1781no—
190C2×D8—1781no—
191C2×D8—1781no—
192C22×C8—891no—
193C22×C8C22×D8911yescharacteristic
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—3561no—
197C2×C4⋊C4—1781no—
198C2×C22⋊C4—3561no—
199C2×C22⋊C4—3561no—
200C2×C42—891no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—892no—
204C2×C16—1781no—
205C2×C16—1781no—
206C2×C16D35612yes—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—1781no—
211#107—1781no—
212#107C4×D8912yes—
213#102—3562no—
214C4×C8—1781no—
215C4×C8—1781no—
216#95—3561no—
217C24—3561no—
218C2×D4—3561no—
219C2×D4—3561no—
220C2×D4—3561no—
221C2×D4—3561no—
222C2×D4—3561no—
223C2×D4—1782no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—891no—
227C22×C4—891no—
228C22×C4—3561no—
229C22×C4D4×D8911yescharacteristic
230D8—3561no—
231D8—3561no—
232D8—1782no—
233D8—1782no—
234C2×C8—892no—
235C2×C8—1781no—
236C2×C8—1781no—
237C2×C8C2×C4×D8911yescharacteristic
238C2×C8C2×D35611yescharacteristic
239C2×C8#1308357311yescharacteristic
240C2×C8#1308357311yescharacteristic
241C2×C8#1308357411yescharacteristic
242C2×C8#1308357511yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—3561no—
248C4⋊C4—3561no—
249C4⋊C4—7122no—
250C22⋊C4—7122no—
251C22⋊C4—7122no—
252C42—1781no—
253C42—1781no—
254C16—22no—
255C16—1782no—
256C23—3562no—
257C23—7121no—
258C23—7121no—
259C23—7121no—
260C23—7121no—
261C23—7121no—
262C23—7121no—
263C23D8×D8911yescharacteristic
264D4—3562no—
265D4—3562no—
266D4—7121no—
267D4—7121no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—892no—
275C2×C4—892no—
276C2×C4—1781no—
277C2×C4—1781no—
278C2×C4—1781no—
279C2×C4—1781no—
280C2×C4—7121no—
281C2×C4—7121no—
282C2×C4—7121no—
283C2×C4#1566797211yescharacteristic
284C2×C4#1566798711yescharacteristic
285C2×C4#1566799811yescharacteristic
286C2×C4SD16×D8911yescharacteristic
287C2×C4#1566803311yescharacteristic
288C2×C4#1566803511yescharacteristic
289C8—1782no—
290C8#1566797011yescharacteristic
291C8C4×D35611yescharacteristic
292C8#1566799611yescharacteristic
293C8#1566800711yescharacteristic
294C22—14241no—
295C22—14241no—
296C22—14241no—
297C22—14241no—
298C22—7122no—
299C22—7122no—
300C22—7122no—
301C22#1915167811yescharacteristic
302C22#1915183511yescharacteristic
303C22#1915194311yescharacteristic
304C22D16×D8911yescharacteristic
305C22SD32×D8911yescharacteristic
306C22#1915202911yescharacteristic
307C22#1915203111yescharacteristic
308C4—82no—
309C4—161no—
310C4—14241no—
311C4—1782no—
312C4—1782no—
313C4#1915149011yescharacteristic
314C4#1915168311yescharacteristic
315C4#1915182811yescharacteristic
316C4#1915195411yescharacteristic
317C2—14242no—
318C2#2602093611yescharacteristic
319C2#2602209611yescharacteristic
320C2#2602210111yescharacteristic
321C2#2602233911yescharacteristic
322C2#2602234911yescharacteristic
323C2#2602247411yescharacteristic
324C2#2602248511yescharacteristic
subgroupG/H
centerC23D8×D89
commutatorC2×C8×C89C22×C4
FrattiniC22×C8C22×D89
FittingC89×#1380C2
socleC23×C89C2×D8
prime radicalC22×D712C22
cosocleC22×C8×C89C23
hypercenter#1380D89
hypocenterC89#13147
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×C16×Dic89C2semidirect
5. #26020986C2semidirect
6. #26018343C2semidirect
7. C89×#1380C2semidirect
16. #19151398C4semidirect
169. #1380D89semidirect
170. C89#13147semidirect

Maximal subgroups

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

subgroupindex
C2×#191514002
C2×#191513982
C2×C16×Dic892
#260211162
#260209862
#260183432
C89×#13802
#1314789