Finite Group Databasebeta

← #36709148

Subgroups — #36709148

Order 22784.

534 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
1#26030791C211yesmaximal, characteristic, complemented
2#26022416C212yesmaximal, complemented
3#26022409C212yesmaximal, complemented
4#26022399C212yesmaximal, complemented
5#26022394C212yesmaximal, complemented
6#26022038C211yesmaximal, characteristic, complemented
7C2×#19151425C211yesmaximal, characteristic, complemented
8#26022011C211yesmaximal, characteristic, complemented
9#26021964C211yesmaximal, characteristic
10#26021959C211yesmaximal, characteristic
11C89×#2951C211yesmaximal, characteristic
12#19152226—21no—
13C2×#15667952C2211yescharacteristic
14C22×#13083549C2211yescharacteristic
15#19152007—22no—
16#19152003C2212yes—
17#19152000C2212yes—
18#19151640—21no—
19#19151636C2211yescharacteristic
20C2×#15667946C2211yescharacteristic
21#19151425C2212yes—
22#19151425C2212yes—
23#19151425C2212yes—
24#19151423C2212yes—
25C4⋊#13083549C2212yes—
26C4⋊#13083549C2212yes—
27#19151402C2211yescharacteristic
28#19151402C2212yes—
29#19151402C2211yescharacteristic
30#19151399C2211yescharacteristic
31#19151399C2212yes—
32#19151399C2211yescharacteristic
33C2×#135×C89C2211yescharacteristic
34#520×C89—21no—
35#440×C89—22no—
36C4×Q16×C89C2212yes—
37C4×D8×C89C2212yes—
38C2×#107×C89C2211yescharacteristic
39#417×C89—21no—
40#405×C89C2211yescharacteristic
41#15668012—22no—
42#15668011—41no—
43#15668004—22no—
44#15668003—41no—
45#15667989—42no—
46#15667971—21no—
47C2×C4⋊Dic89—21no—
48#15667952—22no—
49C2×#13083549—22no—
50C2×#13083549C2312yes—
51C2×#13083549C2311yescharacteristic
52C2×#13083549C2311yescharacteristic
53#15667946—21no—
54#15667946—22no—
55#15667946C2312yes—
56#15667946C2311yescharacteristic
57#15667946C2311yescharacteristic
58#15667944—21no—
59#15667944—22no—
60C2×#41×C89—21no—
61#135×C89—22no—
62#135×C89Q814yescomplemented
63C2×Q16×C89C2311yescharacteristic
64C2×SD16×C89—21no—
65C2×D8×C89C2311yescharacteristic
66C2×M16×C89D411yescharacteristic
67C22×C8×C89C2311yescharacteristic
68C4×Q8×C89—22no—
69C4×D4×C89—22no—
70C4×D4×C89—42no—
71#117×C89—41no—
72#117×C89D411yescharacteristic
73C2×C4⋊C4×C89—21no—
74C2×C4⋊C4×C89—41no—
75#107×C89—21no—
76#107×C89C2312yes—
77#103×C89—22no—
78#102×C89—22no—
79#97×C89D412yes—
80C4×C8×C89C2312yes—
81C2×Dic178—42no—
82C22×Dic89—41no—
83C4⋊Dic89—21no—
84C4⋊Dic89—21no—
85C4⋊Dic89—22no—
86C4⋊Dic89—42no—
87C4⋊Dic89—41no—
88#13083553—42no—
89#13083553—42no—
90C22⋊Dic89—41no—
91C4×Dic89—41no—
92#13083549—21no—
93#13083549—21no—
94#13083549—22no—
95#41×C89—24no—
96#41×C89—42no—
97C2×Q8×C89—21no—
98C2×D4×C89—21no—
99C2×D4×C89—41no—
100C22×C4×C89—41no—
101C22×C4×C89—41no—
102C22×C4×C89—42no—
103C22×C4×C89C2×D411yescharacteristic
104Q16×C89—21no—
105Q16×C89C2×Q812yes—
106SD16×C89—24no—
107D8×C89—21no—
108D8×C89C2×Q812yes—
109M16×C89—22no—
110C2×C8×C89C2×D412yes—
111C2×C8×C89C2×D411yescharacteristic
112C2×C8×C89C2×Q812yes—
113C2×C8×C89#4112yes—
114C2×C8×C89C2411yescharacteristic
115C4⋊C4×C89—22no—
116C4⋊C4×C89—41no—
117C4⋊C4×C89—42no—
118C4⋊C4×C89D811yescharacteristic
119C4⋊C4×C89D811yescharacteristic
120C22⋊C4×C89—42no—
121C22⋊C4×C89D812yes—
122C42×C89—42no—
123C42×C89C2×D412yes—
124C23×C89—41no—
125C23×C89C2×D811yescharacteristic
126Q8×C89—22no—
127Q8×C89—41no—
128D4×C89—22no—
129D4×C89—44no—
130D4×C89—41no—
131C2×C4×C89—41no—
132C2×C4×C89—44no—
133C2×C4×C89—42no—
134C2×C4×C89—162no—
135C2×C4×C89—161no—
136C2×C4×C89C22⋊Q812yes—
137C2×C4×C89C2×D812yes—
138C2×C4×C89C2×D812yes—
139C2×C4×C89C2×D811yescharacteristic
140C2×C4×C89#13712yes—
141C2×C4×C89C22×D411yescharacteristic
142C2×Dic89—41no—
143C2×Dic89—41no—
144C2×Dic89—42no—
145C2×Dic89—161no—
146Dic178—162no—
147C8×C89—22no—
148C8×C89C22⋊Q812yes—
149C8×C89C22×Q811yescharacteristic
150C8×C89C2×#4111yescharacteristic
151C22×C89—21no—
152C22×C89—21no—
153C22×C89—42no—
154C22×C89—81no—
155C22×C89—81no—
156C22×C89#47412yes—
157C22×C89C22×D811yescharacteristic
158C4×C89—22no—
159C4×C89—22no—
160C4×C89—42no—
161C4×C89—82no—
162C4×C89#47412yes—
163C4×C89C2×C22⋊Q811yescharacteristic
164C4×C89C2×#13711yescharacteristic
165Dic89—81no—
166Dic89—82no—
167Dic89—81no—
168#33634—891nomaximal, Sylow
169C2×C89—22no—
170C2×C89—82no—
171C2×C89C2×#47411yescharacteristic
172C2×C89#341611yescharacteristic
173C2×C89#341611yescharacteristic
174#3164—891no—
175#2951D8911yescharacteristic
176#2229—892no—
177#2228—892no—
178#2226—892no—
179#2225—892no—
180#2159—891no—
181C2×#367—891no—
182#2151—891no—
183#2139—891no—
184#2138—891no—
185C89#3363411yescharacteristic, complemented, Sylow
186C2×#135C2×D8911yescharacteristic
187#544—1781no—
188#520—21no—
189C2×M32—891no—
190C22×C16—891no—
191#501—1782no—
192#500—892no—
193#499—892no—
194#440—22no—
195C4×Q16C2×D8912yes—
196C4×D8C2×D8912yes—
197#428—1781no—
198#427—891no—
199C2×#107—891no—
200C2×#107C2×D8911yescharacteristic
201#417—21no—
202#405C2×D8911yescharacteristic
203#367—892no—
204#367—892no—
205#367—892no—
206#366—892no—
207C4⋊C16—892no—
208C4⋊C16—892no—
209#358—891no—
210#358—891no—
211#358—892no—
212#357—891no—
213#357—891no—
214#357—892no—
215C2×#41—21no—
216#135—22no—
217#135Dic17814yescomplemented
218C2×Q16C22×D8911yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D8911yescharacteristic
221C2×M16C22⋊D8911yescharacteristic
222C22×C8C22×D8911yescharacteristic
223C4⋊Q8—3561no—
224C4⋊Q8—1782no—
225#125—3561no—
226#125—1782no—
227C22⋊Q8—3562no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—42no—
231#117—41no—
232#117—1781no—
233#117C22⋊D8911yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—1781no—
237M32—1782no—
238C2×C16—892no—
239C2×C16—891no—
240C2×C16—891no—
241C2×C16—1782no—
242#107—21no—
243#107—891no—
244#107—892no—
245#107—891no—
246#107—1781no—
247#107—1782no—
248#107C22×D8912yes—
249#106—1781no—
250#106—1782no—
251#103—22no—
252#102—22no—
253#97C22⋊D8912yes—
254C4×C8C22×D8912yes—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—3562no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—3561no—
265C22×C4C2×C22⋊D8911yescharacteristic
266Q16—21no—
267Q16C2×Dic17812yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic17812yes—
271M16—22no—
272C2×C8C2×C22⋊D8912yes—
273C2×C8C2×C22⋊D8911yescharacteristic
274C2×C8C2×Dic17812yes—
275C2×C8#1308357412yes—
276C2×C8C23×D8911yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—3561no—
281C4⋊C4—3562no—
282C4⋊C4—3562no—
283C4⋊C4—3562no—
284C4⋊C4—1781no—
285C4⋊C4—1782no—
286C4⋊C4—1781no—
287C4⋊C4#1308355911yescharacteristic
288C4⋊C4#1308355911yescharacteristic
289C22⋊C4—42no—
290C22⋊C4—3561no—
291C22⋊C4#1308355912yes—
292C42—42no—
293C42—3561no—
294C42C2×C22⋊D8912yes—
295C16—1781no—
296C16—1781no—
297C16—1782no—
298C23—41no—
299C23C2×#1308355911yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—14242no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—162no—
310C2×C4—161no—
311C2×C4—3561no—
312C2×C4—3561no—
313C2×C4—3562no—
314C2×C4—14241no—
315C2×C4#1566799112yes—
316C2×C4C2×#1308355912yes—
317C2×C4C2×#1308355912yes—
318C2×C4C2×#1308355911yescharacteristic
319C2×C4#1566804212yes—
320C2×C4C22×C22⋊D8911yescharacteristic
321C8—22no—
322C8#1566799112yes—
323C8C22×Dic17811yescharacteristic
324C8C2×#1308357411yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1915187812yes—
331C22C22×#1308355911yescharacteristic
332C4—22no—
333C4—22no—
334C4—42no—
335C4—82no—
336C4—7122no—
337C4—7121no—
338C4—7121no—
339C4#1915187812yes—
340C4C2×#1566799111yescharacteristic
341C4C2×#1566804211yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1915187811yescharacteristic
345C2#2603339611yescharacteristic
346C2#2603339911yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#2951C2
socleC22×C89C22×D8
prime radicalC2×#135×C89C22
hypercenter#2951D89
hypocenterC89#33634
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #26030791C2semidirect
2. #26022416C2semidirect
3. #26022409C2semidirect
4. #26022399C2semidirect
5. #26022394C2semidirect
6. #26022038C2semidirect
7. C2×#19151425C2semidirect
8. #26022011C2semidirect
62. #135×C89Q8semidirect
185. C89#33634semidirect
217. #135Dic178semidirect

Maximal subgroups

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

subgroupindex
#260307912
#260224162
#260224092
#260223992
#260223942
#260220382
C2×#191514252
#260220112
#260219642
#260219592
C89×#29512
#3363489