Finite Group Databasebeta

← #37546185

Subgroups — #37546185

Order 23168.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041933C211yesmaximal, characteristic, complemented
2#26041689C211yesmaximal, characteristic, complemented
3#26041603C211yesmaximal, characteristic, complemented
4#26041597C211yesmaximal, characteristic, complemented
5#26041595C211yesmaximal, characteristic, complemented
6#26041590C211yesmaximal, characteristic, complemented
7#26041576C211yesmaximal, characteristic, complemented
8#26041569C211yesmaximal, characteristic, complemented
9#26041530C211yesmaximal, characteristic, complemented
10#26041513C211yesmaximal, characteristic, complemented
11#26041389C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041322C211yesmaximal, characteristic, complemented
14C2×#19153747C211yesmaximal, characteristic, complemented
15#553×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668434C2211yescharacteristic
18C2×#15668433C2211yescharacteristic
19C22×#13089678C2211yescharacteristic
20#19153816C2211yescharacteristic
21#19153800—21no—
22#19153800—21no—
23#19153796—21no—
24#19153790C2211yescharacteristic
25#19153790C2211yescharacteristic
26C4×D724C2211yescharacteristic
27C2×C4⋊Dic181C2211yescharacteristic
28C2×C4⋊Dic181C2211yescharacteristic
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153747C2211yescharacteristic
37#19153747C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153737C2211yescharacteristic
40#19153737C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48#126×C181—21no—
49#125×C181C2211yescharacteristic
50#123×C181—21no—
51#121×C181—21no—
52#121×C181C2211yescharacteristic
53#121×C181C2211yescharacteristic
54C4×D4×C181—21no—
55C4×D4×C181C2211yescharacteristic
56C4×D4×C181C2211yescharacteristic
57#117×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59#15668448—44no—
60C2×Dic362—21no—
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C22×Dic181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67#15668434—22no—
68#15668433—22no—
69C2×#13089678—22no—
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76C4⋊Dic181—21no—
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80#15668426—21no—
81#15668426—21no—
82#15668426—21no—
83C2×D4×C181—21no—
84C2×D4×C181—21no—
85C2×D4×C181D411yescharacteristic
86C2×D4×C181C2311yescharacteristic
87C22×C4×C181—21no—
88C22×C4×C181—21no—
89C22×C4×C181C2311yescharacteristic
90C22×C4×C181C2311yescharacteristic
91C4⋊C4×C181—21no—
92C4⋊C4×C181—21no—
93C4⋊C4×C181—21no—
94C4⋊C4×C181D411yescharacteristic
95C4⋊C4×C181C2311yescharacteristic
96C4⋊C4×C181C2311yescharacteristic
97C4⋊C4×C181C2311yescharacteristic
98C22⋊C4×C181—21no—
99C22⋊C4×C181—21no—
100C22⋊C4×C181—21no—
101C22⋊C4×C181—21no—
102C22⋊C4×C181D411yescharacteristic
103C22⋊C4×C181D411yescharacteristic
104C42×C181—21no—
105C42×C181C2311yescharacteristic
106C22×D181—21no—
107Dic362—42no—
108D724—21no—
109D724—21no—
110C22⋊D181—42no—
111C22⋊D181—42no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—42no—
116C2×Dic181—42no—
117C4×D181—42no—
118#13089678—21no—
119#13089678—21no—
120#13089678—21no—
121#13089678#4112yescomplemented
122C23×C181—21no—
123C23×C181C2×D411yescharacteristic
124C23×C181C2×D411yescharacteristic
125D4×C181—22no—
126D4×C181—21no—
127D4×C181—21no—
128D4×C181—42no—
129D4×C181—81no—
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—81no—
138C2×C4×C181—81no—
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2×D411yescharacteristic
142C2×C4×C181C2×D411yescharacteristic
143C2×C4×C181C2411yescharacteristic
144C22×C181—21no—
145C22×C181—21no—
146C22×C181—21no—
147C22×C181—21no—
148C22×C181—21no—
149C22×C181—41no—
150C22×C181—41no—
151C22×C181—41no—
152C22×C181AGL(1,ℤ/8)12yes—
153C22×C181C22×D411yescharacteristic
154C2×D181—41no—
155C2×D181—41no—
156C2×D181—41no—
157C4×C181—21no—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181—41no—
164C4×C181C2×#4111yescharacteristic
165C4×C1812−1+411yescharacteristic
166Dic181—41no—
167Dic181—41no—
168Dic181—41no—
169C2×C181—22no—
170C2×C181—41no—
171C2×C181—81no—
172C2×C181#54711yescharacteristic
173C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C181#57711yescharacteristic
175D181—81no—
176C181#334511yescharacteristic, complemented, Sylow
177#3345—1811nomaximal, Sylow
178#553D18111yescharacteristic, complemented
179#547—1811no—
180#501—1811no—
181#482—1811no—
182#481—1811no—
183#481—1811no—
184#480—1811no—
185#478—1811no—
186#476—1811no—
187#469—1811no—
188#466—1811no—
189#442—1811no—
190#435—1811no—
191#428—1811no—
192C2×#106—1811no—
193C2×#41—3621no—
194C2×D8—1811no—
195C2×M16—1811no—
196C22×C8—1811no—
197C4⋊Q8—1811no—
198#126—21no—
199#125C2×D18111yescharacteristic
200#123—21no—
201#123—3621no—
202#123—3621no—
203C22⋊Q8—3621no—
204#121—21no—
205#121—1811no—
206#121—1811no—
207#121C2×D18111yescharacteristic
208#121C2×D18111yescharacteristic
209C4×D4—21no—
210C4×D4—1811no—
211C4×D4C2×D18111yescharacteristic
212C4×D4C2×D18111yescharacteristic
213#117C2×D18111yescharacteristic
214C2×C4⋊C4—1811no—
215C2×C4⋊C4—1811no—
216C2×C4⋊C4C2×D18111yescharacteristic
217#107—1811no—
218#107—1811no—
219#107—1811no—
220#106—1811no—
221#106—1811no—
222#106—1811no—
223#106—1811no—
224#106—1811no—
225#106—1811no—
226C4⋊C8—1811no—
227#102—1811no—
228#102—1811no—
229#102—1811no—
230#102—1811no—
231#102—1811no—
232#102—1811no—
233C22⋊C8—1811no—
234C22⋊C8—1811no—
235#97—1811no—
236#41—7244no—
237C2×Q8—3621no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—3621no—
241C2×D4—3621no—
242C2×D4—1811no—
243C2×D4C22⋊D18111yescharacteristic
244C2×D4C22×D18111yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3621no—
248C22×C4—3621no—
249C22×C4—3621no—
250C22×C4C22×D18111yescharacteristic
251C22×C4C22×D18111yescharacteristic
252D8—3622no—
253M16—3622no—
254C2×C8—3622no—
255C2×C8—1811no—
256C2×C8—1811no—
257C2×C8—1811no—
258C2×C8—1811no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—3621no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—1811no—
266C4⋊C4—1811no—
267C4⋊C4—1811no—
268C4⋊C4C22⋊D18111yescharacteristic
269C4⋊C4C22×D18111yescharacteristic
270C4⋊C4C22×D18111yescharacteristic
271C4⋊C4C22×D18111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4C22⋊D18111yescharacteristic
280C22⋊C4C22⋊D18111yescharacteristic
281C42—21no—
282C42C22×D18111yescharacteristic
283C23—21no—
284C23—3621no—
285C23C2×C22⋊D18111yescharacteristic
286C23C2×C22⋊D18111yescharacteristic
287Q8—7242no—
288D4—22no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—81no—
293D4—3621no—
294D4—3621no—
295D4—7242no—
296D4—7242no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—3621no—
307C2×C4—3621no—
308C2×C4—3621no—
309C2×C4—7242no—
310C2×C4—7242no—
311C2×C4—7242no—
312C2×C4C2×C22⋊D18111yescharacteristic
313C2×C4C2×C22⋊D18111yescharacteristic
314C2×C4C2×C22⋊D18111yescharacteristic
315C2×C4C2×C22⋊D18111yescharacteristic
316C2×C4C23×D18111yescharacteristic
317C8—3621no—
318C8—3621no—
319C8—3621no—
320C8—1812no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—7241no—
330C22—7241no—
331C22—7241no—
332C22#1915384112yes—
333C22C22×C22⋊D18111yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—7241no—
342C4—7241no—
343C4—7241no—
344C4#41×D18111yescharacteristic
345C4#1915386311yescharacteristic
346C2—22no—
347C2—41no—
348C2—81no—
349C2—14481no—
350C2#2604192811yescharacteristic
351C2C2×#1915384111yescharacteristic
352C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#553×C181C2
socleC22×C181C22×D4
hypercenter#553D181
hypocenterC181#3345
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #26041933C2semidirect
2. #26041689C2semidirect
3. #26041603C2semidirect
4. #26041597C2semidirect
5. #26041595C2semidirect
6. #26041590C2semidirect
7. #26041576C2semidirect
8. #26041569C2semidirect
9. #26041530C2semidirect
10. #26041513C2semidirect
11. #26041389C2semidirect
12. #26041351C2semidirect
13. #26041322C2semidirect
14. C2×#19153747C2semidirect
15. #553×C181C2semidirect
121. #13089678#41semidirect
176. C181#3345semidirect
178. #553D181semidirect

Maximal subgroups

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

subgroupindex
#260419332
#260416892
#260416032
#260415972
#260415952
#260415902
#260415762
#260415692
#260415302
#260415132
#260413892
#260413512
#260413222
C2×#191537472
#553×C1812
#3345181