Finite Group Databasebeta

← #37546530

Subgroups — #37546530

Order 23168.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042011C211yesmaximal, characteristic
2Q8×Dic362C211yesmaximal, characteristic, complemented
3#26041640C211yesmaximal, characteristic, complemented
4#26041638C211yesmaximal, characteristic, complemented
5#26041624C211yesmaximal, characteristic
6#26041572C211yesmaximal, characteristic, complemented
7#26041560C211yesmaximal, characteristic, complemented
8#26041557C211yesmaximal, characteristic, complemented
9#26041503C211yesmaximal, characteristic, complemented
10#26041477C211yesmaximal, characteristic
11#26041408C211yesmaximal, characteristic, complemented
12#26041380C211yesmaximal, characteristic, complemented
13#26041377C211yesmaximal, characteristic, complemented
14Q16×Dic181C211yesmaximal, characteristic, complemented
15#464×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17#19153816—21no—
18#19153816C2211yescharacteristic
19#19153815—21no—
20#19153815—21no—
21#19153815C2211yescharacteristic
22#19153815C2211yescharacteristic
23#19153812C2211yescharacteristic
24#19153805C2211yescharacteristic
25#19153802—21no—
26#19153800—21no—
27#19153798—21no—
28Q8×Dic181C2211yescharacteristic
29Q8×Dic181C2211yescharacteristic
30Q8×Dic181C2211yescharacteristic
31C4×Dic362—21no—
32C4×Dic362C2211yescharacteristic
33C4×D724C2211yescharacteristic
34D4×Dic181C2211yescharacteristic
35#19153776—21no—
36#19153750C2211yescharacteristic
37#19153750C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153745C2211yescharacteristic
40#19153745C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153727C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54#124×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362D411yescharacteristic
63C2×D724C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C22×Dic181—21no—
67C2×C4×D181—21no—
68#15668436—22no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181D411yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82#15668428D411yescharacteristic
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87C4×Dic181—21no—
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181C2311yescharacteristic
91C4×Dic181C2311yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×Q8×C181C2311yescharacteristic
94C2×D4×C181C2311yescharacteristic
95Q16×C181—22no—
96SD16×C181—22no—
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105C22×D181—21no—
106Dic362—22no—
107Dic362—81no—
108Dic362—81no—
109Dic362—81no—
110D724—21no—
111D724—21no—
112C22⋊D181—42no—
113C22⋊D181—81no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—81no—
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C4×D181—81no—
123#13089678—21no—
124#13089678—21no—
125C23×C181—21no—
126Q8×C181—21no—
127Q8×C181—21no—
128Q8×C181—21no—
129Q8×C181#4112yes—
130D4×C181—21no—
131D4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C8×C181—21no—
139C8×C181—21no—
140C22×C181—41no—
141C22×C181—41no—
142C22×C181—41no—
143C22×C181C22×D411yescharacteristic
144C2×D181—41no—
145C2×D181—41no—
146C2×D181—41no—
147C4×C181—22no—
148C4×C181—21no—
149C4×C181—21no—
150C4×C181—41no—
151C4×C181—41no—
152C4×C181C2×#4111yescharacteristic
153C4×C1812−1+411yescharacteristic
154Dic181—21no—
155Dic181—21no—
156Dic181—21no—
157Dic181—41no—
158Dic181—41no—
159Dic181—41no—
160Dic181#13712yes—
161C2×C181—81no—
162C2×C181#54811yescharacteristic
163C2×C181C2×#13711yescharacteristic
164C2×C181#57611yescharacteristic
165D181—81no—
166C181#337511yescharacteristic, complemented, Sylow
167#3375—1811nomaximal, Sylow
168#559—1811no—
169Q82—1811no—
170#489—1811no—
171#489—1811no—
172#486—1811no—
173#477—1811no—
174#474—1811no—
175#474—1811no—
176#464—1811no—
177#464D18111yescharacteristic, complemented
178#460—1811no—
179#446—1811no—
180#440—1811no—
181#440—1811no—
182C4×Q16—1811no—
183C2×Q16C2×D18111yescharacteristic
184C2×SD16—1811no—
185C2×SD16C2×D18111yescharacteristic
186C4⋊Q8—3621no—
187C4⋊Q8—3621no—
188C4⋊Q8—3621no—
189C4⋊Q8—1811no—
190C4⋊Q8—1811no—
191C4⋊Q8—1811no—
192#127—1811no—
193#124—3621no—
194#124—1811no—
195#124C2×D18111yescharacteristic
196#123—3621no—
197#123—3621no—
198C4×Q8—3621no—
199C4×Q8—1811no—
200C4×Q8—1811no—
201C4×Q8—1811no—
202C4×Q8—1811no—
203C4×Q8C2×D18111yescharacteristic
204C4×D4—3621no—
205C4×D4—1811no—
206C4×D4—1811no—
207#107—1811no—
208#107—1811no—
209#107—1811no—
210C4⋊C8—1811no—
211C4⋊C8—1811no—
212C4⋊C8C2×D18111yescharacteristic
213#103—1811no—
214#103—1811no—
215#103—1811no—
216#103C2×D18111yescharacteristic
217#102—1811no—
218#102—1811no—
219#102—1811no—
220#102—1811no—
221#102—1811no—
222#102C2×D18111yescharacteristic
223#97—1811no—
224#97—1811no—
225C4×C8—1811no—
226C2×Q8—3621no—
227C2×Q8—3621no—
228C2×Q8—3621no—
229C2×Q8—1811no—
230C2×Q8C22×D18111yescharacteristic
231C2×Q8C22×D18111yescharacteristic
232C2×D4—3621no—
233C2×D4—3621no—
234C2×D4—1811no—
235C2×D4C22×D18111yescharacteristic
236C22×C4—3621no—
237C22×C4—3621no—
238Q16—22no—
239SD16—22no—
240SD16—3622no—
241C2×C8—1811no—
242C2×C8—1811no—
243C2×C8C22×D18111yescharacteristic
244C2×C8C22×D18111yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—1811no—
253C4⋊C4—1811no—
254C4⋊C4—1811no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4—1811no—
258C4⋊C4C22×D18111yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C42—21no—
266C42—3621no—
267C42—3621no—
268C42—3621no—
269C42—1811no—
270C42—1811no—
271C42C22×D18111yescharacteristic
272C23—21no—
273C23—3621no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—3622no—
278Q8—14481no—
279Q8—14481no—
280Q8—14481no—
281Q8#1566844812yes—
282D4—21no—
283D4—21no—
284D4—3621no—
285D4—3621no—
286D4—14481no—
287D4—7242no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3621no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—14481no—
295C2×C4—14481no—
296C2×C4—1811no—
297C2×C4—1811no—
298C2×C4—1811no—
299C2×C4—1811no—
300C2×C4D4×D18111yescharacteristic
301C2×C4D4×D18111yescharacteristic
302C2×C4C23×D18111yescharacteristic
303C8—21no—
304C8—21no—
305C8—3621no—
306C8—3621no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—7241no—
311C22—7241no—
312C22—7241no—
313C22C2×D4×D18111yescharacteristic
314C4—22no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—3621no—
320C4—3621no—
321C4—3621no—
322C4—1812no—
323C4—7241no—
324C4—7241no—
325C4—7241no—
326C4C2×#1566844811yescharacteristic
327C4#1915386311yescharacteristic
328C2—81no—
329C2—14481no—
330C2#2604193511yescharacteristic
331C2#137×D18111yescharacteristic
332C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radical#19153812C22
hypercenter#464D181
hypocenterC181#3375
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. Q8×Dic362C2semidirect
3. #26041640C2semidirect
4. #26041638C2semidirect
6. #26041572C2semidirect
7. #26041560C2semidirect
8. #26041557C2semidirect
9. #26041503C2semidirect
11. #26041408C2semidirect
12. #26041380C2semidirect
13. #26041377C2semidirect
14. Q16×Dic181C2semidirect
15. #464×C181C2semidirect
166. C181#3375semidirect
177. #464D181semidirect

Maximal subgroups

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

subgroupindex
#260420112
Q8×Dic3622
#260416402
#260416382
#260416242
#260415722
#260415602
#260415572
#260415032
#260414772
#260414082
#260413802
#260413772
Q16×Dic1812
#464×C1812
#3375181