Finite Group Databasebeta

← #36471364

Subgroups — #36471364

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151834C211yesmaximal, characteristic
2C22×#15667946C211yesmaximal, characteristic, complemented
3C2×#19151400C212yesmaximal, complemented
4#26018344C212yesmaximal, complemented
5C2×C22⋊C16×C89C211yesmaximal, characteristic, complemented
6C22×D712C2211yescharacteristic
7C2×#15667986—21no—
8C22×C4⋊Dic89—21no—
9#19151834—22no—
10#19151834C414yescomplemented
11C2×#15667946—24no—
12C2×#15667946—21no—
13C2×#15667946C2211yescharacteristic
14C2×#15667946C2212yes—
15C2×#15667934—21no—
16#19151400—24no—
17C23×C8×C89C2211yescharacteristic
18C22×C16×C89C2212yes—
19C22⋊C16×C89—22no—
20C22×D356—21no—
21C22×C22⋊D89—41no—
22C23×Dic89—41no—
23C2×D712—22no—
24C2×D712—21no—
25C2×D712—22no—
26C2×D712C2×C412yescomplemented
27#15667986—24no—
28#15667986—42no—
29C2×C4⋊Dic89—24no—
30C2×C4⋊Dic89—22no—
31C2×C4⋊Dic89—21no—
32C2×C4⋊Dic89—42no—
33C2×C4⋊Dic89—41no—
34C2×#13083551—41no—
35#15667946—24no—
36#15667946—22no—
37#15667946—21no—
38#15667946C2×C412yescomplemented
39#15667946D414yescomplemented
40#15667934—24no—
41C23×C4×C89D411yescharacteristic
42C22×C8×C89—22no—
43C22×C8×C89—22no—
44C22×C8×C89C2×C412yes—
45C22×C8×C89D411yescharacteristic
46C22×C8×C89D412yes—
47C22×C8×C89C2311yescharacteristic
48C2×C16×C89—22no—
49C2×C16×C89—22no—
50C2×C16×C89—22no—
51C23×D89—41no—
52C2×D356—22no—
53C2×D356—41no—
54C2×D356—41no—
55C2×D356—41no—
56C2×D356—41no—
57C2×D356—41no—
58C2×C22⋊D89—44no—
59C2×C22⋊D89—82no—
60C2×C22⋊D89—82no—
61C22×Dic89—44no—
62C22×Dic89—42no—
63C22×Dic89—41no—
64C22×Dic89—82no—
65C22×Dic89—82no—
66D712—24no—
67D712—42no—
68C4⋊Dic89—22no—
69C4⋊Dic89—24no—
70C4⋊Dic89—44no—
71C4⋊Dic89—41no—
72C4⋊Dic89—42no—
73#13083551—44no—
74C24×C89D811yescharacteristic
75C22×C4×C89—21no—
76C22×C4×C89—21no—
77C22×C4×C89—21no—
78C22×C4×C89—21no—
79C22×C4×C89C22⋊C412yes—
80C22×C4×C89D811yescharacteristic
81C22×C4×C89SD1612yes—
82C22×C4×C89C2×D411yescharacteristic
83C2×C8×C89—21no—
84C2×C8×C89—21no—
85C2×C8×C89—24no—
86C2×C8×C89—24no—
87C2×C8×C89C22⋊C412yes—
88C2×C8×C89C22×C411yescharacteristic
89C2×C8×C89C2×D412yes—
90C2×C8×C89C2×D411yescharacteristic
91C2×C8×C89#4112yes—
92C16×C89—42no—
93C23×C89—21no—
94C23×C89—21no—
95C23×C89—21no—
96C23×C89—21no—
97C23×C89#10212yes—
98C23×C89D1612yes—
99C23×C89SD3212yes—
100C23×C89C2×D811yescharacteristic
101C22×D89—42no—
102C22×D89—81no—
103C22×D89—81no—
104C22×D89—81no—
105C22×D89—81no—
106C22×D89—81no—
107C22×D89—81no—
108C2×C4×C89—21no—
109C2×C4×C89—21no—
110C2×C4×C89—22no—
111C2×C4×C89—22no—
112C2×C4×C89—22no—
113C2×C4×C89—22no—
114C2×C4×C89#10212yes—
115C2×C4×C89C2×C22⋊C411yescharacteristic
116C2×C4×C89#12011yescharacteristic
117C2×C4×C89#12311yescharacteristic
118C2×C4×C89C2×SD1611yescharacteristic
119C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C89#13711yescharacteristic
121C22⋊D89—88no—
122C2×Dic89—42no—
123C2×Dic89—44no—
124C2×Dic89—88no—
125C2×Dic89—82no—
126C2×Dic89—81no—
127C2×Dic89—81no—
128C2×Dic89—82no—
129D356—42no—
130D356—42no—
131D356—81no—
132D356—81no—
133C8×C89—22no—
134C8×C89C4×D412yes—
135C8×C89#12112yes—
136C22×C89—22no—
137C22×C89—22no—
138C22×C89—22no—
139C22×C89—22no—
140C22×C89—22no—
141C22×C89—22no—
142C22×C89#35714yescomplemented
143C22×C89C2×#10211yescharacteristic
144C22×C89#44711yescharacteristic
145C22×C89#48011yescharacteristic
146C22×C89C2×D1611yescharacteristic
147C22×C89C2×SD3211yescharacteristic
148C22×C89#50911yescharacteristic
149C22×C89#51011yescharacteristic
150C2×D89—82no—
151C2×D89—82no—
152C2×D89—82no—
153C2×D89—161no—
154C2×D89—161no—
155C2×D89—161no—
156C2×D89—161no—
157C4×C89—22no—
158C4×C89#38611yescharacteristic
159C4×C89#41711yescharacteristic
160C4×C89#44811yescharacteristic
161C4×C89#48111yescharacteristic
162Dic89—82no—
163Dic89—84no—
164#12991—891nomaximal, Sylow
165C2×C89—24no—
166C2×C89#187411yescharacteristic
167C2×C89C2×#35711yescharacteristic
168C2×C89#213911yescharacteristic
169C2×C89#218411yescharacteristic
170C2×C89#218511yescharacteristic
171C2×C89#223211yescharacteristic
172C2×C89#223311yescharacteristic
173D89—162no—
174C2×#466—891no—
175C22×#107—891no—
176C2×#357—892no—
177C2×C22⋊C16D8911yescharacteristic, complemented
178#1380—892no—
179C89#1299111yescharacteristic, complemented, Sylow
180C22×D8—891no—
181C23×C8C2×D8911yescharacteristic
182C2×#121—1781no—
183C22×C4⋊C4—1781no—
184C22×C16C2×D8912yes—
185#466—894no—
186#466—1782no—
187C2×#107—892no—
188C2×#107—891no—
189C2×#107—1784no—
190C2×#107—1781no—
191C2×#102—1781no—
192#357—1784no—
193C22⋊C16—22no—
194C22×D4—3561no—
195C22×D4—1781no—
196C23×C4—3561no—
197C23×C4D35611yescharacteristic
198C2×D8—892no—
199C2×D8—1782no—
200C2×D8—1781no—
201C2×D8—1782no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C4×D8912yes—
205C22×C8D35611yescharacteristic
206C22×C8C22⋊D8912yes—
207C22×C8C22×D8911yescharacteristic
208#121—3562no—
209#121—1784no—
210C2×C4⋊C4—3562no—
211C2×C4⋊C4—3561no—
212C2×C4⋊C4—1784no—
213C2×C4⋊C4—1782no—
214C2×C4⋊C4—1781no—
215C2×C22⋊C4—3561no—
216C2×C16—22no—
217C2×C16—22no—
218C2×C16—22no—
219#107—892no—
220#107—894no—
221#107—1784no—
222#107—1782no—
223#107—1781no—
224#102—1784no—
225C24—3561no—
226C24D71211yescharacteristic
227C2×D4—3564no—
228C2×D4—3561no—
229C2×D4—3561no—
230C2×D4—3561no—
231C2×D4—3561no—
232C2×D4—3561no—
233C2×D4—1782no—
234C2×D4—7122no—
235C2×D4—7122no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—3564no—
241C22×C4—3561no—
242C22×C4—3562no—
243C22×C4—7122no—
244C22×C4—7122no—
245C22×C4#1308355112yes—
246C22×C4D71211yescharacteristic
247C22×C4#1308356312yes—
248C22×C4C2×D35611yescharacteristic
249D8—3562no—
250D8—1784no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8#1308355112yes—
256C2×C8C2×C4×D8911yescharacteristic
257C2×C8D4×D8912yes—
258C2×C8C2×C22⋊D8911yescharacteristic
259C2×C8#1308357312yes—
260C4⋊C4—3564no—
261C4⋊C4—3561no—
262C4⋊C4—3562no—
263C4⋊C4—1784no—
264C4⋊C4—1782no—
265C22⋊C4—3564no—
266C16—42no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—3562no—
272C23—7121no—
273C23—7121no—
274C23—7121no—
275C23—7121no—
276C23—7121no—
277C23—7121no—
278C23#1566793412yes—
279C23D142412yes—
280C23#1566795712yes—
281C23C2×D71211yescharacteristic
282D4—3562no—
283D4—3562no—
284D4—7128no—
285D4—7121no—
286D4—7121no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—3564no—
294C2×C4—3562no—
295C2×C4—7128no—
296C2×C4—7122no—
297C2×C4—7121no—
298C2×C4—7121no—
299C2×C4—7122no—
300C2×C4#1566793412yes—
301C2×C4C2×#1308355111yescharacteristic
302C2×C4#1566798211yescharacteristic
303C2×C4#1566799611yescharacteristic
304C2×C4C2×#1308356311yescharacteristic
305C2×C4#1566804011yescharacteristic
306C2×C4#1566804511yescharacteristic
307C8—22no—
308C8#1566797212yes—
309C8#1566798412yes—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—14241no—
317C22—14241no—
318C22—14241no—
319C22—14241no—
320C22—7122no—
321C22—7122no—
322C22—7122no—
323C22#1915140014yescomplemented
324C22C2×#1566793411yescharacteristic
325C22#1915173411yescharacteristic
326C22#1915191211yescharacteristic
327C22C2×D142411yescharacteristic
328C22C2×#1566795711yescharacteristic
329C22#1915203611yescharacteristic
330C22#1915204111yescharacteristic
331C4—22no—
332C4—7124no—
333C4—7122no—
334C4#1915147511yescharacteristic
335C4#1915160011yescharacteristic
336C4#1915173911yescharacteristic
337C4#1915191711yescharacteristic
338C2—24no—
339C2—14242no—
340C2#2602056711yescharacteristic
341C2C2×#1915140011yescharacteristic
342C2#2602196511yescharacteristic
343C2#2602216611yescharacteristic
344C2#2602217111yescharacteristic
345C2#2602243311yescharacteristic
346C2#2602243811yescharacteristic
subgroupG/H
centerC23C2×D712
commutatorC2×C8×C89C22×C4
FrattiniC22×C8C22×D89
FittingC2×C22⋊C16×C89C2
socleC23×C89C2×D8
prime radicalC2×#19151834C2
cosocleC22×C8×C89C23
hypercenterC2×C22⋊C16D89
hypocenterC89#12991
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C22×#15667946C2semidirect
3. C2×#19151400C2semidirect
4. #26018344C2semidirect
5. C2×C22⋊C16×C89C2semidirect
10. #19151834C4semidirect
26. C2×D712C2×C4semidirect
38. #15667946C2×C4semidirect
39. #15667946D4semidirect
142. C22×C89#357semidirect
177. C2×C22⋊C16D89semidirect
179. C89#12991semidirect
323. C22#19151400semidirect

Maximal subgroups

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

subgroupindex
C2×#191518342
C22×#156679462
C2×#191514002
#260183442
C2×C22⋊C16×C892
#1299189