Finite Group Databasebeta

← #37544909

Subgroups — #37544909

Order 23168.

368 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#26041826C211yesmaximal, characteristic
2#125×D181C211yesmaximal, characteristic, complemented
3#26041687C211yesmaximal, characteristic, complemented
4#26041686C211yesmaximal, characteristic, complemented
5#26041642C211yesmaximal, characteristic, complemented
6#26041642C211yesmaximal, characteristic, complemented
7#26041599C211yesmaximal, characteristic, complemented
8#26041596C211yesmaximal, characteristic, complemented
9#26041594C211yesmaximal, characteristic, complemented
10#26041591C211yesmaximal, characteristic, complemented
11#26041580C211yesmaximal, characteristic, complemented
12#26041579C211yesmaximal, characteristic, complemented
13#26041576C211yesmaximal, characteristic, complemented
14#26041335C211yesmaximal, characteristic, complemented
15#478×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17#19153815C2211yescharacteristic
18#19153812C2211yescharacteristic
19#19153808C2211yescharacteristic
20#19153807C2211yescharacteristic
21#19153807C2211yescharacteristic
22#19153806—21no—
23#19153806—21no—
24#19153802—21no—
25#19153791C2211yescharacteristic
26#19153789—21no—
27#19153787C2211yescharacteristic
28C4×D724C2211yescharacteristic
29C4×C22⋊D181—21no—
30C4⋊C4×D181—21no—
31C4⋊C4×D181—21no—
32C4⋊C4×D181C2211yescharacteristic
33C4⋊C4×D181C2211yescharacteristic
34C42×D181C2211yescharacteristic
35#19153750C2211yescharacteristic
36#19153749C2211yescharacteristic
37#19153749C2211yescharacteristic
38#19153748C2211yescharacteristic
39#19153747C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153738C2211yescharacteristic
43#19153738C2211yescharacteristic
44#19153737C2211yescharacteristic
45#19153737C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153728C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153726C2211yescharacteristic
51#19153726C2211yescharacteristic
52#125×C181C2211yescharacteristic
53C4×D4×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×D724—21no—
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69C2×C4×D181D411yescharacteristic
70C2×C4×D181D411yescharacteristic
71C2×C4×D181C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
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—
87#15668426—21no—
88#15668426—21no—
89C4×Dic181D411yescharacteristic
90C4×Dic181D411yescharacteristic
91C4×Dic181C2311yescharacteristic
92C2×D4×C181C2311yescharacteristic
93C22×C4×C181—21no—
94C2×C8×C181C2311yescharacteristic
95C2×C8×C181C2311yescharacteristic
96C4⋊C4×C181—21no—
97C4⋊C4×C181—21no—
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C42×C181C2311yescharacteristic
103C22×D181—21no—
104C22×D181C2×D411yescharacteristic
105Dic362—42no—
106D724—21no—
107D724—21no—
108D724—42no—
109C22⋊D181—42no—
110C22⋊D181—42no—
111C22⋊D181—42no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181C2×D411yescharacteristic
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C4×D181—22no—
119C4×D181—21no—
120C4×D181—21no—
121C4×D181—41no—
122C4×D181—42no—
123C4×D181—42no—
124C4×D181—42no—
125#13089678—21no—
126#13089678—21no—
127C23×C181—21no—
128D4×C181—21no—
129D4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—42no—
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2411yescharacteristic
137C8×C181—21no—
138C8×C181—21no—
139C22×C181—41no—
140C22×C181—41no—
141C22×C181—41no—
142C22×C181C22×D411yescharacteristic
143C2×D181—21no—
144C2×D181—21no—
145C2×D181—41no—
146C2×D181—41no—
147C2×D181—41no—
148C2×D181AGL(1,ℤ/8)12yes—
149C4×C181—21no—
150C4×C181—21no—
151C4×C181—41no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C1812−1+411yescharacteristic
155C4×C1812−1+411yescharacteristic
156Dic181—21no—
157Dic181—21no—
158Dic181—21no—
159Dic181—41no—
160Dic181—41no—
161Dic181—41no—
162C2×C181—81no—
163C2×C181#53611yescharacteristic
164C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C181#57511yescharacteristic
166D181—22no—
167D181—81no—
168C181#323411yescharacteristic, complemented, Sylow
169#3234—1811nomaximal, Sylow
170#532—1811no—
171C2×#125—1811no—
172#501—1811no—
173#501—1811no—
174#490—1811no—
175#490—1811no—
176#481—1811no—
177#481—1811no—
178#480—1811no—
179#480—1811no—
180#478—1811no—
181#478—1811no—
182#478—1811no—
183#478D18111yescharacteristic, complemented
184#432—1811no—
185C2×#41—3621no—
186C4⋊Q8—1811no—
187#127—1811no—
188#125—3621no—
189#125—3621no—
190#125—1811no—
191#125—1811no—
192#125—1811no—
193#125C2×D18111yescharacteristic
194#124—3621no—
195#121—3621no—
196#121—1811no—
197#121—1811no—
198C4×D4—3621no—
199C4×D4—1811no—
200C4×D4C2×D18111yescharacteristic
201C2×C4⋊C4—3621no—
202C2×C4⋊C4—3621no—
203C2×C4⋊C4—1811no—
204C2×C4⋊C4—1811no—
205C2×C42—1811no—
206#107—1811no—
207#107—1811no—
208#107—1811no—
209#107C2×D18111yescharacteristic
210#106—1811no—
211#106—1811no—
212#106—1811no—
213#106C2×D18111yescharacteristic
214C4⋊C8—1811no—
215C4⋊C8C2×D18111yescharacteristic
216#102—1811no—
217#102—1811no—
218#102—1811no—
219#102—1811no—
220#102—1811no—
221#102—1811no—
222#102C2×D18111yescharacteristic
223#102C2×D18111yescharacteristic
224C22⋊C8—1811no—
225C22⋊C8—1811no—
226#97—1811no—
227#97—1811no—
228#41—7244no—
229C2×Q8—3621no—
230C2×D4—3621no—
231C2×D4—3621no—
232C2×D4—3621no—
233C2×D4—3621no—
234C2×D4—1811no—
235C2×D4C22×D18111yescharacteristic
236C22×C4—21no—
237C22×C4—3621no—
238C22×C4—3621no—
239C22×C4—3621no—
240C22×C4—1811no—
241C22×C4—1811no—
242C22×C4—1811no—
243C2×C8—1811no—
244C2×C8—1811no—
245C2×C8C22×D18111yescharacteristic
246C2×C8C22×D18111yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—3621no—
258C4⋊C4—1811no—
259C4⋊C4—1811no—
260C4⋊C4—1811no—
261C4⋊C4C22×D18111yescharacteristic
262C4⋊C4C22×D18111yescharacteristic
263C4⋊C4C22×D18111yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C42—1811no—
269C42—1811no—
270C42—1811no—
271C42C22×D18111yescharacteristic
272C23—21no—
273C23—3621no—
274C23—1811no—
275Q8—7242no—
276D4—21no—
277D4—21no—
278D4—3621no—
279D4—3621no—
280D4—7242no—
281D4—7242no—
282D4—7242no—
283D4—7242no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—3622no—
289C2×C4—3621no—
290C2×C4—3621no—
291C2×C4—3621no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—1811no—
295C2×C4—1811no—
296C2×C4—1811no—
297C2×C4—7241no—
298C2×C4—7242no—
299C2×C4—7242no—
300C2×C4—7242no—
301C2×C4D4×D18111yescharacteristic
302C2×C4D4×D18111yescharacteristic
303C2×C4C23×D18111yescharacteristic
304C8—21no—
305C8—21no—
306C8—3621no—
307C8—3621no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—3621no—
312C22—3621no—
313C22—1812no—
314C22—7241no—
315C22—7241no—
316C22—7241no—
317C22C2×D4×D18111yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3621no—
324C4—3621no—
325C4—3621no—
326C4—7241no—
327C4—7241no—
328C4—7241no—
329C4#1915386311yescharacteristic
330C4#1915386411yescharacteristic
331C2—81no—
332C2—3622no—
333C2—14481no—
334C2#2604185211yescharacteristic
335C2AGL(1,ℤ/8)×D18111yescharacteristic
336C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#478×C181C2
socleC22×C181C22×D4
prime radical#26041826C2
hypercenter#478D181
hypocenterC181#3234
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
2. #125×D181C2semidirect
3. #26041687C2semidirect
4. #26041686C2semidirect
5. #26041642C2semidirect
6. #26041642C2semidirect
7. #26041599C2semidirect
8. #26041596C2semidirect
9. #26041594C2semidirect
10. #26041591C2semidirect
11. #26041580C2semidirect
12. #26041579C2semidirect
13. #26041576C2semidirect
14. #26041335C2semidirect
15. #478×C181C2semidirect
168. C181#3234semidirect
183. #478D181semidirect

Maximal subgroups

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

subgroupindex
#260418262
#125×D1812
#260416872
#260416862
#260416422
#260415992
#260415962
#260415942
#260415912
#260415802
#260415792
#260415762
#260413352
#478×C1812
#3234181