Finite Group Databasebeta

← #37545186

Subgroups — #37545186

Order 23168.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041900C211yesmaximal, characteristic, complemented
2#26041569C211yesmaximal, characteristic, complemented
3#26041566C211yesmaximal, characteristic, complemented
4#26041564C211yesmaximal, characteristic, complemented
5#26041561C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041530C211yesmaximal, characteristic
8#26041510C211yesmaximal, characteristic, complemented
9#26041506C211yesmaximal, characteristic, complemented
10#26041501C211yesmaximal, characteristic, complemented
11#26041498C211yesmaximal, characteristic, complemented
12#26041494C211yesmaximal, characteristic, complemented
13#26041491C211yesmaximal, characteristic, complemented
14#26041306C211yesmaximal, characteristic, complemented
15#542×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668437C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668434C2211yescharacteristic
20C2×#15668433C2211yescharacteristic
21C22×#13089678C2211yescharacteristic
22#19153816C2211yescharacteristic
23#19153811—21no—
24#19153804C2211yescharacteristic
25#19153802—21no—
26#19153798—21no—
27#19153795C2211yescharacteristic
28#19153794—21no—
29#19153790C2211yescharacteristic
30C4×Dic362C2211yescharacteristic
31C4×D724C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36C4⋊#13089678C2211yescharacteristic
37C4⋊#13089678C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39C4⋊#13089678C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153739C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153737C2211yescharacteristic
45#19153737C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#126×C181—21no—
50#124×C181—21no—
51#124×C181C2211yescharacteristic
52#123×C181—21no—
53C22⋊Q8×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58#117×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C4×D181—21no—
64#15668439—41no—
65#15668437—41no—
66#15668436—41no—
67#15668434—41no—
68#15668433—41no—
69C2×#13089678—41no—
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80#15668426—21no—
81#15668426—21no—
82#15668426—21no—
83#15668426—21no—
84C4×Dic181—21no—
85C2×Q8×C181—21no—
86C2×Q8×C181C2311yescharacteristic
87C2×D4×C181—21no—
88C2×D4×C181C2311yescharacteristic
89C22×C4×C181—21no—
90C22×C4×C181C2311yescharacteristic
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×C181—21no—
100C22⋊C4×C181—21no—
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C22⋊C4×C181D411yescharacteristic
104C22⋊C4×C181D411yescharacteristic
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×C181—21no—
124C23×C181C2×D411yescharacteristic
125Q8×C181—21no—
126Q8×C181—21no—
127Q8×C181—81no—
128D4×C181—21no—
129D4×C181—21no—
130D4×C181—81no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—81no—
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2×D411yescharacteristic
142C2×C4×C181C2411yescharacteristic
143C22×C181—21no—
144C22×C181—21no—
145C22×C181—21no—
146C22×C181—41no—
147C22×C181—41no—
148C22×C181—41no—
149C22×C181C22×D411yescharacteristic
150C2×D181—41no—
151C2×D181—41no—
152C2×D181—41no—
153C4×C181—21no—
154C4×C181—21no—
155C4×C181—21no—
156C4×C181—21no—
157C4×C181—21no—
158C4×C181—41no—
159C4×C181—41no—
160C4×C181—41no—
161C4×C181H(5,𝔽2)11yescharacteristic
162C4×C1812−1+411yescharacteristic
163Dic181—41no—
164Dic181—41no—
165Dic181—41no—
166C2×C181—41no—
167C2×C181—81no—
168C2×C181#53711yescharacteristic
169C2×C181#57711yescharacteristic
170C2×C181#57711yescharacteristic
171D181—81no—
172C181#326311yescharacteristic, complemented, Sylow
173#3263—1811nomaximal, Sylow
174#542—1811no—
175#542D18111yescharacteristic, complemented
176#476—1811no—
177#476—1811no—
178#475—1811no—
179#475—1811no—
180#470—1811no—
181#469—1811no—
182#465—1811no—
183#465—1811no—
184#464—1811no—
185#464—1811no—
186#463—1811no—
187#463—1811no—
188#424—1811no—
189C2×Q16—1811no—
190C2×SD16—1811no—
191C2×SD16—1811no—
192C2×D8—1811no—
193C2×M16—1811no—
194C22×C8—1811no—
195C4⋊Q8—1811no—
196C4⋊Q8C2×D18111yescharacteristic
197#126—21no—
198#126—3621no—
199#124—21no—
200#124—3621no—
201#124—1811no—
202#124C2×D18111yescharacteristic
203#123—21no—
204#123—3621no—
205C22⋊Q8—21no—
206C22⋊Q8—3621no—
207C22⋊Q8—1811no—
208C22⋊Q8C2×D18111yescharacteristic
209#121—1811no—
210#121C2×D18111yescharacteristic
211C4×Q8—1811no—
212C4×Q8C2×D18111yescharacteristic
213C4×D4—1811no—
214C4×D4C2×D18111yescharacteristic
215#117C2×D18111yescharacteristic
216#106—1811no—
217#106—1811no—
218#106—1811no—
219#106—1811no—
220C4⋊C8—1811no—
221C4⋊C8—1811no—
222C4⋊C8—1811no—
223C4⋊C8—1811no—
224#103—1811no—
225#103—1811no—
226#103—1811no—
227#103—1811no—
228#102—1811no—
229#102—1811no—
230#102—1811no—
231#102—1811no—
232C2×Q8—21no—
233C2×Q8—3621no—
234C2×Q8—1811no—
235C2×Q8C22×D18111yescharacteristic
236C2×D4—21no—
237C2×D4—3621no—
238C2×D4—1811no—
239C2×D4C22×D18111yescharacteristic
240C22×C4—21no—
241C22×C4—3621no—
242C22×C4C22×D18111yescharacteristic
243Q16—7241no—
244SD16—7241no—
245SD16—7241no—
246D8—7241no—
247M16—7241no—
248C2×C8—1811no—
249C2×C8—1811no—
250C2×C8—1811no—
251C2×C8—1811no—
252C2×C8—7241no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—3621no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—1811no—
260C4⋊C4—1811no—
261C4⋊C4C22⋊D18111yescharacteristic
262C4⋊C4C22⋊D18111yescharacteristic
263C4⋊C4C22×D18111yescharacteristic
264C4⋊C4C22×D18111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C22⋊C4—3621no—
273C22⋊C4—3621no—
274C22⋊C4—3621no—
275C22⋊C4C22⋊D18111yescharacteristic
276C22⋊C4C22⋊D18111yescharacteristic
277C42—21no—
278C42—3621no—
279C42C22×D18111yescharacteristic
280C42C22×D18111yescharacteristic
281C23—21no—
282C23—3621no—
283C23C2×C22⋊D18111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—3621no—
288Q8—3621no—
289Q8—14481no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—3621no—
294D4—3621no—
295D4—14481no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—3621no—
303C2×C4—3621no—
304C2×C4—3621no—
305C2×C4—14481no—
306C2×C4C2×C22⋊D18111yescharacteristic
307C2×C4C2×C22⋊D18111yescharacteristic
308C2×C4C2×C22⋊D18111yescharacteristic
309C2×C4C2×C22⋊D18111yescharacteristic
310C2×C4C2×C22⋊D18111yescharacteristic
311C2×C4C23×D18111yescharacteristic
312C8—3621no—
313C8—3621no—
314C8—3621no—
315C8—3621no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—7241no—
323C22—7241no—
324C22—7241no—
325C22C22×C22⋊D18111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—7241no—
335C4—7241no—
336C4—7241no—
337C4#1915386211yescharacteristic
338C4#1915386311yescharacteristic
339C2—41no—
340C2—81no—
341C2—14481no—
342C2#2604185711yescharacteristic
343C2#2604208711yescharacteristic
344C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#542×C181C2
socleC22×C181C22×D4
prime radical#26041530C2
hypercenter#542D181
hypocenterC181#3263
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #26041900C2semidirect
2. #26041569C2semidirect
3. #26041566C2semidirect
4. #26041564C2semidirect
5. #26041561C2semidirect
6. #26041535C2semidirect
8. #26041510C2semidirect
9. #26041506C2semidirect
10. #26041501C2semidirect
11. #26041498C2semidirect
12. #26041494C2semidirect
13. #26041491C2semidirect
14. #26041306C2semidirect
15. #542×C181C2semidirect
172. C181#3263semidirect
175. #542D181semidirect

Maximal subgroups

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

subgroupindex
#260419002
#260415692
#260415662
#260415642
#260415612
#260415352
#260415302
#260415102
#260415062
#260415012
#260414982
#260414942
#260414912
#260413062
#542×C1812
#3263181