Finite Group Databasebeta

← #37543950

Subgroups — #37543950

Order 23168.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041900C211yesmaximal, characteristic
2#26041620C211yesmaximal, characteristic, complemented
3#26041603C211yesmaximal, characteristic
4#26041574C211yesmaximal, characteristic, complemented
5#26041564C211yesmaximal, characteristic, complemented
6#26041501C211yesmaximal, characteristic, complemented
7#26041501C211yesmaximal, characteristic, complemented
8#26041435C211yesmaximal, characteristic, complemented
9#26041418C211yesmaximal, characteristic
10#26041384C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12C4×#15668439C211yesmaximal, characteristic, complemented
13C4×#15668436C211yesmaximal, characteristic, complemented
14#26041340C211yesmaximal, characteristic, complemented
15#519×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20#19153816C2211yescharacteristic
21#19153811—21no—
22#19153804C2211yescharacteristic
23#19153802—21no—
24#19153798—21no—
25#19153795C2211yescharacteristic
26#19153794—21no—
27#19153790C2211yescharacteristic
28C4×Dic362C2211yescharacteristic
29C4×D724C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35C4⋊#13089678C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153739C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48C22×Q8×C181C2211yescharacteristic
49C4×Q8×C181—21no—
50C4×Q8×C181—21no—
51C4×Q8×C181C2211yescharacteristic
52C4×Q8×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55#117×C181—21no—
56#117×C181—21no—
57#117×C181C2211yescharacteristic
58#117×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C4×D181—21no—
64#15668439—22no—
65#15668439—22no—
66#15668436—22no—
67#15668436—22no—
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C4⋊Dic181—21no—
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75#15668428—21no—
76#15668428—21no—
77C22⋊Dic181—21no—
78#15668426—21no—
79#15668426—21no—
80#15668426—21no—
81#15668426—21no—
82C4×Dic181—21no—
83C2×Q8×C181—22no—
84C2×Q8×C181—22no—
85C2×Q8×C181—21no—
86C2×Q8×C181C2311yescharacteristic
87C2×Q8×C181C2311yescharacteristic
88C22×C4×C181—21no—
89C22×C4×C181C2311yescharacteristic
90C4⋊C4×C181—21no—
91C4⋊C4×C181—21no—
92C4⋊C4×C181—21no—
93C4⋊C4×C181—21no—
94C4⋊C4×C181D411yescharacteristic
95C4⋊C4×C181D411yescharacteristic
96C4⋊C4×C181C2311yescharacteristic
97C4⋊C4×C181C2311yescharacteristic
98C22⋊C4×C181—21no—
99C22⋊C4×C181D411yescharacteristic
100C22⋊C4×C181D411yescharacteristic
101C42×C181—21no—
102C42×C181—21no—
103C42×C181—21no—
104C42×C181—21no—
105C42×C181—21no—
106C42×C181C2311yescharacteristic
107C42×C181C2311yescharacteristic
108C22×D181—21no—
109Dic362—21no—
110Dic362—21no—
111Dic362—81no—
112D724—21no—
113D724—21no—
114C22⋊D181—81no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C4×D181—81no—
119#13089678—21no—
120#13089678—21no—
121#13089678—21no—
122#13089678—21no—
123C23×C181C2×D411yescharacteristic
124Q8×C181—24no—
125Q8×C181—21no—
126Q8×C181—21no—
127Q8×C181#4112yes—
128Q8×C181#4112yes—
129C2×C4×C181—24no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2411yescharacteristic
142C22×C181—21no—
143C22×C181—21no—
144C22×C181—21no—
145C22×C181C22×D411yescharacteristic
146C2×D181—41no—
147C2×D181—41no—
148C2×D181—41no—
149C4×C181—22no—
150C4×C181—22no—
151C4×C181—21no—
152C4×C181—21no—
153C4×C181—21no—
154C4×C181—21no—
155C4×C181—21no—
156C4×C181—41no—
157C4×C181—41no—
158C4×C181C2×#4111yescharacteristic
159C4×C181C2×#4111yescharacteristic
160Dic181—41no—
161Dic181—41no—
162Dic181—41no—
163C2×C181—41no—
164C2×C181#52511yescharacteristic
165C2×C181#57711yescharacteristic
166C2×C181#57811yescharacteristic
167D181—81no—
168C181#312111yescharacteristic, complemented, Sylow
169#3121—1811nomaximal, Sylow
170#542—1811no—
171#519D18111yescharacteristic, complemented
172#485—1811no—
173#482—1811no—
174#477—1811no—
175#475—1811no—
176#464—1811no—
177#464—1811no—
178#451—1811no—
179#448—1811no—
180#441—1811no—
181#440—1811no—
182C4×Q16—1811no—
183C4×SD16—1811no—
184#433—1811no—
185C22×Q8C2×D18111yescharacteristic
186C2×Q16—1811no—
187C2×Q16—1811no—
188C2×SD16—1811no—
189C2×SD16—1811no—
190C4⋊Q8—1811no—
191#126—3621no—
192#124—3621no—
193#124—1811no—
194#123—3621no—
195C22⋊Q8—3621no—
196C22⋊Q8—1811no—
197#121—1811no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—1811no—
201C4×Q8C2×D18111yescharacteristic
202C4×Q8C2×D18111yescharacteristic
203C4×Q8C2×D18111yescharacteristic
204C4×Q8C2×D18111yescharacteristic
205C4×D4—1811no—
206#117—21no—
207#117—21no—
208#117C2×D18111yescharacteristic
209#117C2×D18111yescharacteristic
210#107—1811no—
211#107—1811no—
212#106—1811no—
213#106—1811no—
214C4⋊C8—1811no—
215C4⋊C8—1811no—
216#103—1811no—
217#103—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#103—1811no—
222#102—1811no—
223#102—1811no—
224C22⋊C8—1811no—
225C22⋊C8—1811no—
226#97—1811no—
227C4×C8—1811no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—3621no—
232C2×Q8—1811no—
233C2×Q8C22×D18111yescharacteristic
234C2×Q8C22×D18111yescharacteristic
235C2×D4—3621no—
236C2×D4—1811no—
237C22×C4—21no—
238C22×C4—3621no—
239C22×C4C22×D18111yescharacteristic
240Q16—3622no—
241Q16—3622no—
242SD16—3622no—
243SD16—3622no—
244C2×C8—1811no—
245C2×C8—1811no—
246C2×C8—1811no—
247C2×C8—1811no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4C22⋊D18111yescharacteristic
258C4⋊C4C22⋊D18111yescharacteristic
259C4⋊C4C22×D18111yescharacteristic
260C4⋊C4C22×D18111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4C22⋊D18111yescharacteristic
268C22⋊C4C22⋊D18111yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—3621no—
275C42C22×D18111yescharacteristic
276C42C22×D18111yescharacteristic
277C23—3621no—
278C23C2×C22⋊D18111yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—3621no—
283Q8—3621no—
284Q8—14481no—
285Q8#1566844912yes—
286Q8#1566844912yes—
287D4—3621no—
288D4—3621no—
289D4—14481no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3621no—
300C2×C4—14481no—
301C2×C4C2×C22⋊D18111yescharacteristic
302C2×C4C2×C22⋊D18111yescharacteristic
303C2×C4C2×C22⋊D18111yescharacteristic
304C2×C4C2×C22⋊D18111yescharacteristic
305C2×C4C2×C22⋊D18111yescharacteristic
306C2×C4C23×D18111yescharacteristic
307C8—3621no—
308C8—3621no—
309C8—3621no—
310C8—3621no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—7241no—
315C22—7241no—
316C22—7241no—
317C22C22×C22⋊D18111yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—7241no—
328C4—7241no—
329C4—7241no—
330C4C2×#1566844911yescharacteristic
331C4C2×#1566844911yescharacteristic
332C2—41no—
333C2—14481no—
334C2#2604179011yescharacteristic
335C2#2604208711yescharacteristic
336C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#519×C181C2
socleC22×C181C22×D4
prime radical#19153790C22
hypercenter#519D181
hypocenterC181#3121
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #26041620C2semidirect
4. #26041574C2semidirect
5. #26041564C2semidirect
6. #26041501C2semidirect
7. #26041501C2semidirect
8. #26041435C2semidirect
10. #26041384C2semidirect
11. #26041377C2semidirect
12. C4×#15668439C2semidirect
13. C4×#15668436C2semidirect
14. #26041340C2semidirect
15. #519×C181C2semidirect
168. C181#3121semidirect
171. #519D181semidirect

Maximal subgroups

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

subgroupindex
#260419002
#260416202
#260416032
#260415742
#260415642
#260415012
#260414352
#260414182
#260413842
#260413772
C4×#156684392
C4×#156684362
#260413402
#519×C1812
#3121181