Finite Group Databasebeta

← #37546166

Subgroups — #37546166

Order 23168.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041970C211yesmaximal, characteristic, complemented
2#26041934C211yesmaximal, characteristic
3#26041688C211yesmaximal, characteristic, complemented
4#26041613C211yesmaximal, characteristic, complemented
5#26041608C211yesmaximal, characteristic, complemented
6#26041602C211yesmaximal, characteristic, complemented
7#26041591C211yesmaximal, characteristic, complemented
8#26041579C211yesmaximal, characteristic, complemented
9#26041574C211yesmaximal, characteristic, complemented
10#26041522C211yesmaximal, characteristic, complemented
11#26041505C211yesmaximal, characteristic, complemented
12#26041352C211yesmaximal, characteristic, complemented
13#26041320C211yesmaximal, characteristic, complemented
14#107×D181C211yesmaximal, characteristic, complemented
15#440×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×#15668432C2211yescharacteristic
18C2×C8×D181C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153810—21no—
21#19153808C2211yescharacteristic
22#19153806—21no—
23#19153801—21no—
24#19153798—21no—
25#19153798—21no—
26#19153795—21no—
27#19153792C2211yescharacteristic
28#19153792C2211yescharacteristic
29#19153787C2211yescharacteristic
30#19153787C2211yescharacteristic
31C4×D724C2211yescharacteristic
32#19153774C2211yescharacteristic
33#19153773—21no—
34C4⋊C4×D181C2211yescharacteristic
35C4⋊C4×D181C2211yescharacteristic
36C4⋊C4×D181C2211yescharacteristic
37#19153750C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153748C2211yescharacteristic
43#19153748C2211yescharacteristic
44#19153747C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153740C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153728C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×D724D411yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C2×C4×D181—21no—
65C2×C4×D181—21no—
66C2×C4×D181—21no—
67C2×C4×D181C2311yescharacteristic
68C2×C4×D181C2311yescharacteristic
69#15668432—22no—
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C8×D181—22no—
73C4⋊Dic181—21no—
74C4⋊Dic181D411yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426D411yescharacteristic
92#15668426D411yescharacteristic
93C4×Dic181—21no—
94C2×Q8×C181C2311yescharacteristic
95C2×D4×C181C2311yescharacteristic
96C22×C4×C181—21no—
97SD16×C181—22no—
98C2×C8×C181C2311yescharacteristic
99C2×C8×C181C2311yescharacteristic
100C4⋊C4×C181—21no—
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C42×C181—21no—
105C42×C181C2311yescharacteristic
106C22×D181C2×D411yescharacteristic
107C22×D181C2×D411yescharacteristic
108Dic362—42no—
109D724—22no—
110C22⋊D181—42no—
111C22⋊D181—42no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C4×D181—21no—
119C4×D181—21no—
120C4×D181—21no—
121C4×D181—21no—
122C4×D181—42no—
123C4×D181—42no—
124C4×D181—42no—
125#13089678—21no—
126#13089678—21no—
127C23×C181—21no—
128Q8×C181—21no—
129Q8×C181—21no—
130D4×C181—21no—
131D4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—42no—
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2411yescharacteristic
139C8×C181—21no—
140C8×C181#4112yescomplemented
141C22×C181—41no—
142C22×C181—41no—
143C22×C181—41no—
144C22×C181C22×D411yescharacteristic
145C2×D181—21no—
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—21no—
149C2×D181—21no—
150C2×D181AGL(1,ℤ/8)12yes—
151C4×C181—21no—
152C4×C181—21no—
153C4×C181—41no—
154C4×C181—41no—
155C4×C181—41no—
156C4×C181C2×#4111yescharacteristic
157C4×C1812−1+411yescharacteristic
158Dic181—21no—
159Dic181—21no—
160Dic181—41no—
161Dic181—41no—
162Dic181—41no—
163Dic181—41no—
164C2×C181—81no—
165C2×C181#54711yescharacteristic
166C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C181#57811yescharacteristic
168D181—22no—
169D181—41no—
170C181#334311yescharacteristic, complemented, Sylow
171#3343—1811nomaximal, Sylow
172#552—1811no—
173#547—1811no—
174#501—1811no—
175#484—1811no—
176#483—1811no—
177#482—1811no—
178#480—1811no—
179#478—1811no—
180#477—1811no—
181#468—1811no—
182#465—1811no—
183#440D18111yescharacteristic, complemented
184#435—1811no—
185#428—1811no—
186C2×#107—1811no—
187C2×#41—3621no—
188C2×SD16C2×D18111yescharacteristic
189C2×M16—1811no—
190C22×C8—1811no—
191C4⋊Q8—1811no—
192#126—3621no—
193#125—3621no—
194#125—1811no—
195#123—3621no—
196#123—3621no—
197#123—3621no—
198C22⋊Q8—3621no—
199C22⋊Q8—1811no—
200C22⋊Q8—1811no—
201#121—1811no—
202#121—1811no—
203C4×Q8C2×D18111yescharacteristic
204C4×D4—1811no—
205C4×D4C2×D18111yescharacteristic
206#117—3621no—
207#117—1811no—
208C2×C4⋊C4—1811no—
209C2×C4⋊C4—1811no—
210C2×C4⋊C4—1811no—
211#107—1811no—
212#107—1811no—
213#107—1811no—
214#107—1811no—
215#107—1811no—
216#107C2×D18111yescharacteristic
217#106—1811no—
218#106—1811no—
219#106—1811no—
220C4⋊C8—1811no—
221#103—1811no—
222#103—1811no—
223#103C2×D18111yescharacteristic
224#102—1811no—
225#102—1811no—
226#102C2×D18111yescharacteristic
227C22⋊C8—1811no—
228C22⋊C8—1811no—
229#97C2×D18111yescharacteristic
230#41—7244no—
231C2×Q8—3621no—
232C2×Q8C22×D18111yescharacteristic
233C2×D4—3621no—
234C2×D4—3621no—
235C2×D4—1811no—
236C2×D4C22×D18111yescharacteristic
237C22×C4—21no—
238C22×C4—3621no—
239C22×C4—3621no—
240C22×C4—3621no—
241C22×C4—1811no—
242C22×C4—1811no—
243SD16—22no—
244M16—3622no—
245C2×C8—3622no—
246C2×C8—1811no—
247C2×C8—1811no—
248C2×C8C22×D18111yescharacteristic
249C2×C8C22×D18111yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—1811no—
260C4⋊C4—1811no—
261C4⋊C4—1811no—
262C4⋊C4—1811no—
263C4⋊C4—1811no—
264C4⋊C4C22×D18111yescharacteristic
265C4⋊C4C22×D18111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—3621no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C22⋊C4—1811no—
273C22⋊C4—1811no—
274C42—21no—
275C42—3621no—
276C42C22×D18111yescharacteristic
277C23—21no—
278C23—1811no—
279C23—1811no—
280Q8—21no—
281Q8—21no—
282Q8—7242no—
283D4—21no—
284D4—21no—
285D4—3622no—
286D4—7242no—
287D4—7242no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3621no—
300C2×C4—1811no—
301C2×C4—1811no—
302C2×C4—7242no—
303C2×C4—7242no—
304C2×C4—7242no—
305C2×C4D4×D18111yescharacteristic
306C2×C4D4×D18111yescharacteristic
307C2×C4C23×D18111yescharacteristic
308C8—21no—
309C8—3621no—
310C8—3621no—
311C8#1566844912yescomplemented
312C22—41no—
313C22—41no—
314C22—41no—
315C22—3621no—
316C22—3621no—
317C22—3621no—
318C22—3621no—
319C22—3621no—
320C22—1812no—
321C22C2×D4×D18111yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3621no—
328C4—3621no—
329C4—7241no—
330C4—7241no—
331C4—7241no—
332C4—7241no—
333C4C2×#1566844911yescharacteristic
334C4#1915386311yescharacteristic
335C2—81no—
336C2—3622no—
337C2—7241no—
338C2#2604193411yescharacteristic
339C2AGL(1,ℤ/8)×D18111yescharacteristic
340C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#440×C181C2
socleC22×C181C22×D4
prime radical#26041934C2
hypercenter#440D181
hypocenterC181#3343
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #26041970C2semidirect
3. #26041688C2semidirect
4. #26041613C2semidirect
5. #26041608C2semidirect
6. #26041602C2semidirect
7. #26041591C2semidirect
8. #26041579C2semidirect
9. #26041574C2semidirect
10. #26041522C2semidirect
11. #26041505C2semidirect
12. #26041352C2semidirect
13. #26041320C2semidirect
14. #107×D181C2semidirect
15. #440×C181C2semidirect
140. C8×C181#41semidirect
170. C181#3343semidirect
183. #440D181semidirect
311. C8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419702
#260419342
#260416882
#260416132
#260416082
#260416022
#260415912
#260415792
#260415742
#260415222
#260415052
#260413522
#260413202
#107×D1812
#440×C1812
#3343181