Finite Group Databasebeta

← #37541980

Subgroups — #37541980

Order 23168.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041220C211yesmaximal, characteristic, complemented
2#26041216C211yesmaximal, characteristic
3#26041213C211yesmaximal, characteristic, complemented
4#26041212C211yesmaximal, characteristic, complemented
5#26041212C211yesmaximal, characteristic, complemented
6#26041211C211yesmaximal, characteristic, complemented
7#26041203C211yesmaximal, characteristic
8#26041201C211yesmaximal, characteristic
9#26041200C211yesmaximal, characteristic, complemented
10#26041200C211yesmaximal, characteristic, complemented
11#26041192C211yesmaximal, characteristic
12#26041141C211yesmaximal, characteristic
13#26041138C211yesmaximal, characteristic
14#26041136C211yesmaximal, characteristic
15#400×C181C211yesmaximal, characteristic, complemented
16C22×Dic362C2211yescharacteristic
17C22×C4×D181C2211yescharacteristic
18C2×C4⋊Dic181C2211yescharacteristic
19C2×C4⋊Dic181C2211yescharacteristic
20C2×C4⋊Dic181C2211yescharacteristic
21C2×C4⋊Dic181C2211yescharacteristic
22C2×C4⋊Dic181C2211yescharacteristic
23C2×#15668428C2211yescharacteristic
24C2×#15668428C2211yescharacteristic
25C2×#15668428C2211yescharacteristic
26C2×#15668428C2211yescharacteristic
27C2×#15668428C2211yescharacteristic
28C2×#15668428C2211yescharacteristic
29C2×#15668426C2211yescharacteristic
30C2×#15668426C2211yescharacteristic
31C2×#15668426C2211yescharacteristic
32C2×#15668426C2211yescharacteristic
33C2×#15668426C2211yescharacteristic
34C2×#15668426C2211yescharacteristic
35#19153723C2211yescharacteristic
36#19153723C2211yescharacteristic
37#19153723C2211yescharacteristic
38#19153723C2211yescharacteristic
39#19153723C2211yescharacteristic
40#19153723C2211yescharacteristic
41#19153723C2211yescharacteristic
42#19153723C2211yescharacteristic
43#19153723C2211yescharacteristic
44C2×C4⋊C4×C181C2211yescharacteristic
45C2×C4⋊C4×C181C2211yescharacteristic
46#95×C181C2211yescharacteristic
47#95×C181C2211yescharacteristic
48#95×C181C2211yescharacteristic
49#95×C181C2211yescharacteristic
50#95×C181C2211yescharacteristic
51C23×D181C2311yescharacteristic
52C2×Dic362—41no—
53C2×Dic362—41no—
54C2×Dic362—41no—
55C22×Dic181C2311yescharacteristic
56C22×Dic181C2311yescharacteristic
57C22×Dic181C2311yescharacteristic
58C22×Dic181C2311yescharacteristic
59C22×Dic181C2311yescharacteristic
60C22×Dic181C2311yescharacteristic
61C22×Dic181C2311yescharacteristic
62C2×C4×D181—41no—
63C2×C4×D181—41no—
64C2×C4×D181Q814yescomplemented
65C4⋊Dic181—41no—
66C4⋊Dic181—41no—
67C4⋊Dic181—41no—
68C4⋊Dic181—41no—
69C4⋊Dic181—41no—
70#15668428—41no—
71#15668428—41no—
72#15668428—41no—
73#15668428—41no—
74#15668428—41no—
75#15668428—41no—
76#15668426—41no—
77#15668426—41no—
78#15668426—41no—
79#15668426—41no—
80#15668426—41no—
81#15668426—41no—
82C22×C4×C181C2311yescharacteristic
83C22×C4×C181C2311yescharacteristic
84C22×C4×C181C2311yescharacteristic
85C22×C4×C181C2311yescharacteristic
86C22×C4×C181C2311yescharacteristic
87C22×C4×C181C2311yescharacteristic
88C22×C4×C181C2311yescharacteristic
89C4⋊C4×C181—41no—
90C4⋊C4×C181—41no—
91C22×D181—21no—
92C22×D181—21no—
93C22×D181—21no—
94C22×D181—21no—
95C22×D181—21no—
96C22×D181—21no—
97C22×D181C2×Q812yes—
98Dic362—44no—
99C2×Dic181—21no—
100C2×Dic181—21no—
101C2×Dic181—21no—
102C2×Dic181—21no—
103C2×Dic181—21no—
104C2×Dic181—21no—
105C2×Dic181—21no—
106C2×Dic181—21no—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181C2×Q812yes—
120C4×D181—44no—
121C23×C181C2411yescharacteristic
122C2×C4×C181—21no—
123C2×C4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181C2×Q812yes—
143C22×C181C22×Q811yescharacteristic
144C22×C181H(5,𝔽2)11yescharacteristic
145C22×C181H(5,𝔽2)11yescharacteristic
146C22×C181H(5,𝔽2)11yescharacteristic
147C22×C181H(5,𝔽2)11yescharacteristic
148C22×C1812−1+411yescharacteristic
149C22×C1812−1+411yescharacteristic
150C2×D181—22no—
151C2×D181—22no—
152C2×D181—22no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—41no—
156C2×D181—41no—
157C4×C181—22no—
158C4×C181—41no—
159C4×C181—41no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181—41no—
164Dic181—22no—
165Dic181—41no—
166Dic181—41no—
167Dic181—41no—
168Dic181—41no—
169Dic181—41no—
170Dic181—41no—
171C2×C181#54311yescharacteristic
172C2×C181#54411yescharacteristic
173C2×C181#54411yescharacteristic
174C2×C181#56011yescharacteristic
175C2×C181#56211yescharacteristic
176C2×C181#56211yescharacteristic
177C2×C181#56311yescharacteristic
178D181—42no—
179C181#284111yescharacteristic, complemented, Sylow
180#2841—1811nomaximal, Sylow
181#400—1811no—
182#400D18111yescharacteristic, complemented
183#399—1811no—
184#398—1811no—
185#398—1811no—
186#398—1811no—
187#398—1811no—
188#396—1811no—
189#396—1811no—
190#395—1811no—
191#395—1811no—
192#393—1811no—
193#381—1811no—
194#380—1811no—
195#380—1811no—
196C22×Q8—1811no—
197C23×C4—1811no—
198C2×C4⋊C4—1811no—
199C2×C4⋊C4—1811no—
200C2×C4⋊C4—1811no—
201C2×C4⋊C4—1811no—
202C2×C4⋊C4—1811no—
203C2×C4⋊C4—1811no—
204C2×C4⋊C4—1811no—
205C2×C4⋊C4—1811no—
206C2×C4⋊C4—1811no—
207C2×C4⋊C4—1811no—
208C2×C4⋊C4—1811no—
209C2×C4⋊C4C2×D18111yescharacteristic
210C2×C4⋊C4C2×D18111yescharacteristic
211C2×C22⋊C4—1811no—
212C2×C22⋊C4—1811no—
213C2×C22⋊C4—1811no—
214C2×C22⋊C4—1811no—
215C2×C22⋊C4—1811no—
216C2×C22⋊C4—1811no—
217#95—1811no—
218#95—1811no—
219#95—1811no—
220#95—1811no—
221#95—1811no—
222#95—1811no—
223#95—1811no—
224#95—1811no—
225#95—1811no—
226#95C2×D18111yescharacteristic
227#95C2×D18111yescharacteristic
228#95C2×D18111yescharacteristic
229#95C2×D18111yescharacteristic
230#95C2×D18111yescharacteristic
231C24—1811no—
232C2×Q8—7241no—
233C2×Q8—7241no—
234C2×Q8—7241no—
235C22×C4—1811no—
236C22×C4—1811no—
237C22×C4—1811no—
238C22×C4—1811no—
239C22×C4—1811no—
240C22×C4—1811no—
241C22×C4—1811no—
242C22×C4—1814no—
243C22×C4—7241no—
244C22×C4—7241no—
245C22×C4C22×D18111yescharacteristic
246C22×C4C22×D18111yescharacteristic
247C22×C4C22×D18111yescharacteristic
248C22×C4C22×D18111yescharacteristic
249C22×C4C22×D18111yescharacteristic
250C22×C4C22×D18111yescharacteristic
251C22×C4C22×D18111yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—7241no—
255C4⋊C4—7241no—
256C4⋊C4—7241no—
257C4⋊C4—7241no—
258C4⋊C4—7241no—
259C4⋊C4—7241no—
260C4⋊C4—7241no—
261C4⋊C4—7241no—
262C4⋊C4—7241no—
263C4⋊C4—7241no—
264C4⋊C4—7241no—
265C22⋊C4—7241no—
266C22⋊C4—7241no—
267C22⋊C4—7241no—
268C22⋊C4—7241no—
269C22⋊C4—7241no—
270C22⋊C4—7241no—
271C23—3621no—
272C23—3621no—
273C23—3621no—
274C23—3621no—
275C23—3621no—
276C23—3621no—
277C23—1812no—
278C23C23×D18111yescharacteristic
279Q8—7244no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—3621no—
301C2×C4—3621no—
302C2×C4—3621no—
303C2×C4—3621no—
304C2×C4—3621no—
305C2×C4—3621no—
306C2×C4—3621no—
307C2×C4—3621no—
308C2×C4—3621no—
309C2×C4—3621no—
310C2×C4—3621no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—3621no—
319C2×C4—3621no—
320C2×C4—1812no—
321C2×C4—7244no—
322C2×C4Q8×D18112yes—
323C22—3622no—
324C22—3622no—
325C22—3622no—
326C22—7241no—
327C22—7241no—
328C22—7241no—
329C22—7241no—
330C22C2×Q8×D18111yescharacteristic
331C22#1915386111yescharacteristic
332C22#1915386211yescharacteristic
333C22#1915386211yescharacteristic
334C22#1915386211yescharacteristic
335C22#1915386311yescharacteristic
336C22#1915386311yescharacteristic
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3622no—
345C4—7241no—
346C4—7241no—
347C4—7241no—
348C4—7241no—
349C4—7241no—
350C4—7241no—
351C2—7242no—
352C2#2604190411yescharacteristic
353C2#2604190911yescharacteristic
354C2#2604190911yescharacteristic
355C2#2604202011yescharacteristic
356C2#2604202311yescharacteristic
357C2#2604202611yescharacteristic
358C2#2604203111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#400×C181C2
prime radicalC23×D181C23
hypercenter#400D181
hypocenterC181#2841
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #26041220C2semidirect
3. #26041213C2semidirect
4. #26041212C2semidirect
5. #26041212C2semidirect
6. #26041211C2semidirect
9. #26041200C2semidirect
10. #26041200C2semidirect
15. #400×C181C2semidirect
64. C2×C4×D181Q8semidirect
179. C181#2841semidirect
182. #400D181semidirect

Maximal subgroups

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

subgroupindex
#260412202
#260412162
#260412132
#260412122
#260412112
#260412032
#260412012
#260412002
#260411922
#260411412
#260411382
#260411362
#400×C1812
#2841181