Finite Group Databasebeta

← #37546777

Subgroups — #37546777

Order 23168.

362 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#26042011C211yesmaximal, characteristic
2#26042007C211yesmaximal, characteristic, complemented
3#26041674C211yesmaximal, characteristic, complemented
4#26041672C211yesmaximal, characteristic, complemented
5#26041650C211yesmaximal, characteristic
6#26041504C211yesmaximal, characteristic, complemented
7#26041487C211yesmaximal, characteristic, complemented
8#26041485C211yesmaximal, characteristic, complemented
9#26041482C211yesmaximal, characteristic, complemented
10#26041478C211yesmaximal, characteristic
11#26041408C211yesmaximal, characteristic, complemented
12#26041377C211yesmaximal, characteristic, complemented
13#26041376C211yesmaximal, characteristic, complemented
14#26041371C211yesmaximal, characteristic, complemented
15#464×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668437C2211yescharacteristic
22C2×#15668436C2211yescharacteristic
23#19153816C2211yescharacteristic
24#19153814C2211yescharacteristic
25#19153814C2211yescharacteristic
26#19153812C2211yescharacteristic
27#19153808—21no—
28#19153806—21no—
29#19153806—21no—
30#19153805C2211yescharacteristic
31#19153802—21no—
32#19153800—21no—
33#19153798—21no—
34Q8×Dic181C2211yescharacteristic
35C4×Dic362—21no—
36C4×Dic362C2211yescharacteristic
37#19153781C2211yescharacteristic
38#19153781C2211yescharacteristic
39C4×D724C2211yescharacteristic
40D4×Dic181C2211yescharacteristic
41#19153776—21no—
42#19153749C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153740C2211yescharacteristic
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×Dic362D411yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67#15668439—22no—
68#15668439—41no—
69#15668438—41no—
70#15668438—41no—
71#15668437—22no—
72#15668437—41no—
73#15668436—22no—
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181—21no—
79C4⋊Dic181D411yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428—21no—
87#15668428D411yescharacteristic
88#15668428D411yescharacteristic
89C22⋊Dic181—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97C4×Dic181C2311yescharacteristic
98C2×Q8×C181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181C2311yescharacteristic
101Q16×C181—41no—
102SD16×C181—41no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C22⋊C4×C181—21no—
108C22⋊C4×C181—21no—
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112Dic362—22no—
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117D724—21no—
118D724—21no—
119C22⋊D181—42no—
120C22⋊D181—81no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—81no—
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C4×D181—81no—
130#13089678—21no—
131#13089678#4112yes—
132C23×C181—21no—
133Q8×C181—21no—
134Q8×C181—21no—
135Q8×C181—21no—
136Q8×C181—21no—
137D4×C181—21no—
138D4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181C2×D411yescharacteristic
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2411yescharacteristic
145C8×C181—21no—
146C8×C181—21no—
147C22×C181—41no—
148C22×C181—41no—
149C22×C181—41no—
150C22×C181C22×D411yescharacteristic
151C2×D181—41no—
152C2×D181—41no—
153C2×D181—41no—
154C4×C181—21no—
155C4×C181—21no—
156C4×C181—41no—
157C4×C181—41no—
158C4×C181—41no—
159C4×C181C2×#4111yescharacteristic
160C4×C181H(5,𝔽2)11yescharacteristic
161Dic181—21no—
162Dic181—21no—
163Dic181—21no—
164Dic181—41no—
165Dic181—41no—
166Dic181—41no—
167Dic181#13712yes—
168C2×C181—81no—
169C2×C181#55011yescharacteristic
170C2×C181C2×#13711yescharacteristic
171C2×C181#57711yescharacteristic
172D181—81no—
173C181#340011yescharacteristic, complemented, Sylow
174#3400—1811nomaximal, Sylow
175#559—1811no—
176#557—1811no—
177#497—1811no—
178#497—1811no—
179#492—1811no—
180#464—1811no—
181#464D18111yescharacteristic, complemented
182#462—1811no—
183#461—1811no—
184#461—1811no—
185#460—1811no—
186#446—1811no—
187#440—1811no—
188#440—1811no—
189C4×Q16—1811no—
190C2×Q16—1811no—
191C2×Q16—1811no—
192C2×Q16C2×D18111yescharacteristic
193C2×SD16—1811no—
194C2×SD16—1811no—
195C2×SD16—1811no—
196C2×SD16—1811no—
197C2×SD16—1811no—
198C2×SD16C2×D18111yescharacteristic
199C4⋊Q8—1811no—
200C4⋊Q8—1811no—
201C4⋊Q8—1811no—
202#127—1811no—
203#125—3621no—
204#125—3621no—
205#125—3621no—
206#124—3621no—
207#124—1811no—
208#124C2×D18111yescharacteristic
209#123—3621no—
210#123—3621no—
211C4×Q8—3621no—
212C4×Q8—1811no—
213C4×Q8—1811no—
214C4×Q8—1811no—
215C4×Q8—1811no—
216C4×Q8C2×D18111yescharacteristic
217C4×D4—3621no—
218C4×D4—1811no—
219C4×D4—1811no—
220#107—1811no—
221C4⋊C8—1811no—
222C4⋊C8—1811no—
223C4⋊C8C2×D18111yescharacteristic
224#103—1811no—
225#103—1811no—
226#103—1811no—
227#103C2×D18111yescharacteristic
228#102—1811no—
229#102C2×D18111yescharacteristic
230#97—1811no—
231#97—1811no—
232C4×C8—1811no—
233C2×Q8—1811no—
234C2×Q8—1811no—
235C2×Q8—1811no—
236C2×Q8C22×D18111yescharacteristic
237C2×Q8C22×D18111yescharacteristic
238C2×D4—3621no—
239C2×D4—3621no—
240C2×D4—1811no—
241C2×D4C22×D18111yescharacteristic
242C22×C4—3621no—
243C22×C4—3621no—
244Q16—41no—
245Q16—3622no—
246Q16—7241no—
247SD16—41no—
248SD16—3622no—
249SD16—3622no—
250SD16—7241no—
251SD16—7241no—
252SD16—7241no—
253C2×C8—1811no—
254C2×C8—1811no—
255C2×C8C22×D18111yescharacteristic
256C2×C8C22×D18111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—3621no—
259C4⋊C4—3621no—
260C4⋊C4—3621no—
261C4⋊C4—3621no—
262C4⋊C4—3621no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—1811no—
268C4⋊C4—1811no—
269C4⋊C4—1811no—
270C4⋊C4—1811no—
271C4⋊C4C22×D18111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C42—21no—
279C42—3621no—
280C42—3621no—
281C42—3621no—
282C42—1811no—
283C42—1811no—
284C42C22×D18111yescharacteristic
285C23—21no—
286C23—3621no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—3622no—
292Q8—3621no—
293Q8—3621no—
294Q8—3621no—
295Q8—3621no—
296D4—21no—
297D4—21no—
298D4—3621no—
299D4—3621no—
300D4—14481no—
301D4—7242no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—3621no—
306C2×C4—3621no—
307C2×C4—3621no—
308C2×C4—14481no—
309C2×C4—14481no—
310C2×C4—1811no—
311C2×C4—1811no—
312C2×C4—1811no—
313C2×C4—1811no—
314C2×C4D4×D18111yescharacteristic
315C2×C4D4×D18111yescharacteristic
316C2×C4C23×D18111yescharacteristic
317C8—21no—
318C8—21no—
319C8—3621no—
320C8—1812no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—7241no—
325C22—7241no—
326C22—7241no—
327C22C2×D4×D18111yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3621no—
334C4—3621no—
335C4—3621no—
336C4—1812no—
337C4—7241no—
338C4—7241no—
339C4—7241no—
340C4#41×D18111yescharacteristic
341C4#1915386111yescharacteristic
342C2—81no—
343C2—14481no—
344C2#2604194711yescharacteristic
345C2#137×D18111yescharacteristic
346C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radical#19153812C22
hypercenter#464D181
hypocenterC181#3400
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. #26042007C2semidirect
3. #26041674C2semidirect
4. #26041672C2semidirect
6. #26041504C2semidirect
7. #26041487C2semidirect
8. #26041485C2semidirect
9. #26041482C2semidirect
11. #26041408C2semidirect
12. #26041377C2semidirect
13. #26041376C2semidirect
14. #26041371C2semidirect
15. #464×C181C2semidirect
173. C181#3400semidirect
181. #464D181semidirect

Maximal subgroups

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

subgroupindex
#260420112
#260420072
#260416742
#260416722
#260416502
#260415042
#260414872
#260414852
#260414822
#260414782
#260414082
#260413772
#260413762
#260413712
#464×C1812
#3400181