Finite Group Databasebeta

← #37546217

Subgroups — #37546217

Order 23168.

360 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#26041992C211yesmaximal, characteristic, complemented
2#26041973C211yesmaximal, characteristic, complemented
3#26041686C211yesmaximal, characteristic, complemented
4#26041606C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041603C211yesmaximal, characteristic, complemented
7#26041594C211yesmaximal, characteristic, complemented
8#26041582C211yesmaximal, characteristic, complemented
9#26041558C211yesmaximal, characteristic, complemented
10#26041533C211yesmaximal, characteristic, complemented
11#26041515C211yesmaximal, characteristic, complemented
12#26041390C211yesmaximal, characteristic, complemented
13#26041350C211yesmaximal, characteristic, complemented
14#26041318C211yesmaximal, characteristic, complemented
15#427×C181C211yesmaximal, characteristic, complemented
16C2×D1448C2211yescharacteristic
17C2×#15668432C2211yescharacteristic
18#19153815C2211yescharacteristic
19#19153810—21no—
20#19153807C2211yescharacteristic
21#19153804—21no—
22#19153802—21no—
23#19153798—21no—
24#19153791C2211yescharacteristic
25#19153791C2211yescharacteristic
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153789—21no—
29#19153789—21no—
30C4×D724—21no—
31#19153778C2211yescharacteristic
32#19153778C2211yescharacteristic
33C4×C22⋊D181C2211yescharacteristic
34#19153774C2211yescharacteristic
35#19153773—21no—
36C4⋊C4×D181C2211yescharacteristic
37#19153750C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153747C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153738C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153726C2211yescharacteristic
52C22×C8×C181C2211yescharacteristic
53#117×C181C2211yescharacteristic
54#117×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58#106×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×D724—21no—
61C2×D724—21no—
62C2×D724C2311yescharacteristic
63C2×D724C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181D411yescharacteristic
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69C2×C4×D181—21no—
70C2×C4×D181C2311yescharacteristic
71D1448—22no—
72#15668432—22no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C4⋊Dic181—21no—
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82C22⋊Dic181D411yescharacteristic
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426D411yescharacteristic
91C4×Dic181—21no—
92C4×Dic181C2311yescharacteristic
93C22×C4×C181C2311yescharacteristic
94C2×C8×C181—22no—
95C2×C8×C181C2311yescharacteristic
96C2×C8×C181C2311yescharacteristic
97C4⋊C4×C181C2311yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C42×C181—21no—
104C42×C181—21no—
105C22×D181—21no—
106C22×D181—21no—
107C22×D181C2×D411yescharacteristic
108Dic362—81no—
109D724—21no—
110D724—21no—
111D724—21no—
112D724—21no—
113D724—42no—
114D724—81no—
115C22⋊D181—41no—
116C22⋊D181—81no—
117C22⋊D181—81no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C2×Dic181C2×D411yescharacteristic
123C4×D181—21no—
124C4×D181—21no—
125C4×D181—42no—
126C4×D181—81no—
127C4×D181—81no—
128#13089678—21no—
129#13089678—21no—
130C23×C181C2×D411yescharacteristic
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×C181C2×D411yescharacteristic
138C2×C4×C181C2411yescharacteristic
139C8×C181—21no—
140C8×C181#4112yes—
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—21no—
144C22×C181C22×D411yescharacteristic
145C2×D181—21no—
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—41no—
149C2×D181—41no—
150C2×D181—41no—
151C2×D181—41no—
152C2×D181—41no—
153C2×D181—41no—
154C4×C181—21no—
155C4×C181—41no—
156C4×C181—41no—
157C4×C181—41no—
158C4×C181—41no—
159C4×C181C2×#4111yescharacteristic
160C4×C1812−1+411yescharacteristic
161Dic181—21no—
162Dic181—21no—
163Dic181—21no—
164Dic181—41no—
165Dic181—41no—
166C2×C181—41no—
167C2×C181#54711yescharacteristic
168C2×C181#57511yescharacteristic
169C2×C181#57611yescharacteristic
170D181—41no—
171D181—81no—
172D181—81no—
173C181#334711yescharacteristic, complemented, Sylow
174#3347—1811nomaximal, Sylow
175#555—1811no—
176#553—1811no—
177#501—1811no—
178#482—1811no—
179#482—1811no—
180#482—1811no—
181#480—1811no—
182#478—1811no—
183#474—1811no—
184#469—1811no—
185#466—1811no—
186#442—1811no—
187#435—1811no—
188#428—1811no—
189#427D18111yescharacteristic, complemented
190C2×D8—1811no—
191C2×M16—1811no—
192C22×C8C2×D18111yescharacteristic
193C4⋊Q8—1811no—
194#126—3621no—
195#125—1811no—
196#124—3621no—
197#124—3621no—
198#123—3621no—
199#121—3621no—
200#121—3621no—
201#121—1811no—
202#121—1811no—
203#121—1811no—
204#121—1811no—
205C4×D4—3621no—
206C4×D4—1811no—
207C4×D4—1811no—
208C4×D4—1811no—
209#117—3621no—
210#117—1811no—
211#117C2×D18111yescharacteristic
212#117C2×D18111yescharacteristic
213C2×C4⋊C4—1811no—
214#107—1811no—
215#107—1811no—
216#107—1811no—
217#107C2×D18111yescharacteristic
218#107C2×D18111yescharacteristic
219#106—1811no—
220#106—1811no—
221#106C2×D18111yescharacteristic
222#106C2×D18111yescharacteristic
223C4⋊C8—1811no—
224#102—1811no—
225#102—1811no—
226#102—1811no—
227#102—1811no—
228#102—1811no—
229#102—1811no—
230C22⋊C8—1811no—
231C22⋊C8—1811no—
232#97—1811no—
233C2×Q8—3621no—
234C2×D4—3621no—
235C2×D4—3621no—
236C2×D4—3621no—
237C2×D4—3621no—
238C2×D4—1811no—
239C2×D4—1811no—
240C2×D4—1811no—
241C22×C4—3621no—
242C22×C4—3621no—
243C22×C4—3621no—
244C22×C4—1811no—
245C22×C4C22×D18111yescharacteristic
246D8—3622no—
247M16—3622no—
248C2×C8—22no—
249C2×C8—1811no—
250C2×C8—1811no—
251C2×C8C22×D18111yescharacteristic
252C2×C8C22×D18111yescharacteristic
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4—1811no—
260C4⋊C4C22×D18111yescharacteristic
261C4⋊C4C22×D18111yescharacteristic
262C4⋊C4C22×D18111yescharacteristic
263C4⋊C4C22×D18111yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C22⋊C4—3621no—
273C22⋊C4—1811no—
274C22⋊C4—1811no—
275C42—21no—
276C42—21no—
277C42—3621no—
278C42—1811no—
279C23—3621no—
280C23—3621no—
281C23—1811no—
282C23D4×D18111yescharacteristic
283Q8—14481no—
284D4—3621no—
285D4—3621no—
286D4—3621no—
287D4—3621no—
288D4—14481no—
289D4—14481no—
290D4—14481no—
291D4—7242no—
292D4—7241no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—3621no—
300C2×C4—3621no—
301C2×C4—3621no—
302C2×C4—3621no—
303C2×C4—14481no—
304C2×C4—14481no—
305C2×C4—1811no—
306C2×C4—1811no—
307C2×C4—1811no—
308C2×C4—7242no—
309C2×C4D4×D18111yescharacteristic
310C2×C4C23×D18111yescharacteristic
311C8—21no—
312C8—3621no—
313C8—3621no—
314C8#1566844912yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—3621no—
319C22—3621no—
320C22—3621no—
321C22—7241no—
322C22—7241no—
323C22—7241no—
324C22—7241no—
325C22—7241no—
326C22—7241no—
327C22C2×D4×D18111yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3621no—
334C4—3621no—
335C4—3621no—
336C4—7241no—
337C4—7241no—
338C4C2×#1566844911yescharacteristic
339C4#1915386411yescharacteristic
340C2—41no—
341C2—14481no—
342C2—14481no—
343C2—7241no—
344C2#2604193011yescharacteristic
345C2#2604207611yescharacteristic
346C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#427×C181C2
socleC22×C181C22×D4
hypercenter#427D181
hypocenterC181#3347
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #26041992C2semidirect
2. #26041973C2semidirect
3. #26041686C2semidirect
4. #26041606C2semidirect
5. #26041603C2semidirect
6. #26041603C2semidirect
7. #26041594C2semidirect
8. #26041582C2semidirect
9. #26041558C2semidirect
10. #26041533C2semidirect
11. #26041515C2semidirect
12. #26041390C2semidirect
13. #26041350C2semidirect
14. #26041318C2semidirect
15. #427×C181C2semidirect
173. C181#3347semidirect
189. #427D181semidirect

Maximal subgroups

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

subgroupindex
#260419922
#260419732
#260416862
#260416062
#260416032
#260415942
#260415822
#260415582
#260415332
#260415152
#260413902
#260413502
#260413182
#427×C1812
#3347181