Finite Group Databasebeta

← #37545852

Subgroups — #37545852

Order 23168.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041970C211yesmaximal, characteristic, complemented
2#26041951C211yesmaximal, characteristic
3#26041644C211yesmaximal, characteristic, complemented
4#26041605C211yesmaximal, characteristic, complemented
5#26041599C211yesmaximal, characteristic, complemented
6#26041586C211yesmaximal, characteristic, complemented
7#26041528C211yesmaximal, characteristic, complemented
8#26041511C211yesmaximal, characteristic, complemented
9#26041479C211yesmaximal, characteristic, complemented
10#26041440C211yesmaximal, characteristic, complemented
11#26041440C211yesmaximal, characteristic, complemented
12#26041352C211yesmaximal, characteristic, complemented
13#26041279C211yesmaximal, characteristic, complemented
14#26041274C211yesmaximal, characteristic, complemented
15#441×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668432C2211yescharacteristic
21C2×C8×D181C2211yescharacteristic
22#19153813—21no—
23#19153813C2211yescharacteristic
24#19153810—21no—
25#19153808C2211yescharacteristic
26#19153806—21no—
27#19153801—21no—
28#19153792C2211yescharacteristic
29#19153792C2211yescharacteristic
30#19153791—21no—
31#19153791—21no—
32#19153791C2211yescharacteristic
33#19153791C2211yescharacteristic
34C4×D724—21no—
35C4×D724C2211yescharacteristic
36#19153774C2211yescharacteristic
37#19153773—21no—
38C4⋊C4×D181C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153728C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—22no—
61#15668450—41no—
62#15668450—41no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—21no—
68C2×D724—21no—
69C2×D724D411yescharacteristic
70C2×D724C2311yescharacteristic
71C2×D724C2311yescharacteristic
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181C2311yescharacteristic
76C2×C4×D181C2311yescharacteristic
77#15668436—41no—
78#15668436—41no—
79#15668432—41no—
80C2×#13089678C2311yescharacteristic
81C2×#13089678C2311yescharacteristic
82C8×D181—41no—
83C4⋊Dic181D411yescharacteristic
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90#15668428—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426D411yescharacteristic
97#15668426D411yescharacteristic
98C4×Dic181—21no—
99C2×Q8×C181C2311yescharacteristic
100C2×Q8×C181C2311yescharacteristic
101Q16×C181—22no—
102C2×C8×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C42×C181—21no—
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181—21no—
113C22×D181C2×D411yescharacteristic
114C22×D181C2×D411yescharacteristic
115D724—21no—
116D724—21no—
117D724—21no—
118D724—21no—
119D724—41no—
120D724—42no—
121D724—42no—
122D724—42no—
123D724—81no—
124D724—81no—
125D724—81no—
126C2×Dic181—21no—
127C2×Dic181—21no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—21no—
133C4×D181—21no—
134C4×D181—42no—
135C4×D181—42no—
136C4×D181—42no—
137#13089678—21no—
138#13089678—21no—
139Q8×C181—21no—
140Q8×C181—21no—
141Q8×C181—21no—
142Q8×C181#4112yes—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2411yescharacteristic
150C8×C181—21no—
151C8×C181—21no—
152C22×C181C22×D411yescharacteristic
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—21no—
157C2×D181—21no—
158C2×D181—21no—
159C2×D181—41no—
160C2×D181—41no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C4×C181—22no—
166C4×C181—21no—
167C4×C181—21no—
168C4×C181—41no—
169C4×C181—41no—
170C4×C181—41no—
171C4×C181C2×#4111yescharacteristic
172C4×C181H(5,𝔽2)11yescharacteristic
173Dic181—21no—
174Dic181—21no—
175Dic181—41no—
176Dic181—41no—
177C2×C181#54611yescharacteristic
178C2×C181#57511yescharacteristic
179C2×C181#57711yescharacteristic
180D181—41no—
181D181—41no—
182D181—81no—
183D181—81no—
184C181#331711yescharacteristic, complemented, Sylow
185#3317—1811nomaximal, Sylow
186#552—1811no—
187#550—1811no—
188#490—1811no—
189#482—1811no—
190#481—1811no—
191#479—1811no—
192#468—1811no—
193#465—1811no—
194#460—1811no—
195#452—1811no—
196#452—1811no—
197#441D18111yescharacteristic, complemented
198#435—1811no—
199#417—1811no—
200#416—1811no—
201C2×#41—1811no—
202C2×#41—1811no—
203C2×Q16C2×D18111yescharacteristic
204C2×SD16—1811no—
205C2×SD16—1811no—
206C2×M16—1811no—
207C22×C8—1811no—
208#127—3621no—
209#127—1811no—
210#126—3621no—
211#125—3621no—
212#125—1811no—
213#123—3621no—
214C22⋊Q8—1811no—
215C22⋊Q8—1811no—
216#121—3621no—
217#121—3621no—
218#121—1811no—
219#121—1811no—
220C4×Q8C2×D18111yescharacteristic
221C4×Q8C2×D18111yescharacteristic
222C4×D4—3621no—
223C4×D4—1811no—
224#117—3621no—
225#117—1811no—
226C2×C4⋊C4—1811no—
227#107—1811no—
228#106—1811no—
229#106C2×D18111yescharacteristic
230C4⋊C8—1811no—
231#103—1811no—
232#103—1811no—
233#103C2×D18111yescharacteristic
234#103C2×D18111yescharacteristic
235#102—1811no—
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241C22⋊C8—1811no—
242C22⋊C8—1811no—
243#97C2×D18111yescharacteristic
244#41—3622no—
245#41—3622no—
246#41—7241no—
247#41—7241no—
248C2×Q8C22×D18111yescharacteristic
249C2×Q8C22×D18111yescharacteristic
250C2×D4—3621no—
251C2×D4—3621no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—3621no—
256C2×D4—1811no—
257C2×D4—1811no—
258C2×D4—1811no—
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4—3621no—
262C22×C4—1811no—
263C22×C4—1811no—
264Q16—22no—
265SD16—7241no—
266SD16—7241no—
267M16—7241no—
268C2×C8—1811no—
269C2×C8—1811no—
270C2×C8—7241no—
271C2×C8C22×D18111yescharacteristic
272C2×C8C22×D18111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—3621no—
279C4⋊C4—3621no—
280C4⋊C4—1811no—
281C4⋊C4—1811no—
282C4⋊C4—1811no—
283C4⋊C4C22×D18111yescharacteristic
284C4⋊C4C22×D18111yescharacteristic
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—1811no—
291C22⋊C4—1811no—
292C42—21no—
293C42—21no—
294C42—3621no—
295C42C22×D18111yescharacteristic
296C23—3621no—
297C23—3621no—
298C23—1811no—
299C23—1811no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8#1566844912yes—
304D4—3621no—
305D4—3621no—
306D4—3621no—
307D4—3621no—
308D4—14481no—
309D4—14481no—
310D4—14481no—
311D4—7241no—
312D4—7242no—
313D4—7242no—
314D4—7242no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—3621no—
320C2×C4—3621no—
321C2×C4—3621no—
322C2×C4—3621no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—1811no—
326C2×C4—1811no—
327C2×C4—7242no—
328C2×C4—7242no—
329C2×C4—7242no—
330C2×C4D4×D18111yescharacteristic
331C2×C4D4×D18111yescharacteristic
332C2×C4C23×D18111yescharacteristic
333C8—21no—
334C8—21no—
335C8—3621no—
336C8—3621no—
337C22—3621no—
338C22—3621no—
339C22—3621no—
340C22—3621no—
341C22—3621no—
342C22—3621no—
343C22—7241no—
344C22—7241no—
345C22—7241no—
346C22—7241no—
347C22—7241no—
348C22—7241no—
349C22C2×D4×D18111yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3621no—
357C4—3621no—
358C4—7241no—
359C4—7241no—
360C4C2×#1566844911yescharacteristic
361C4#1915386211yescharacteristic
362C2—14481no—
363C2—14481no—
364C2—7241no—
365C2—7241no—
366C2#2604192511yescharacteristic
367C2#2604207811yescharacteristic
368C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#441×C181C2
socleC22×C181C22×D4
prime radical#26041951C2
hypercenter#441D181
hypocenterC181#3317
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #26041970C2semidirect
3. #26041644C2semidirect
4. #26041605C2semidirect
5. #26041599C2semidirect
6. #26041586C2semidirect
7. #26041528C2semidirect
8. #26041511C2semidirect
9. #26041479C2semidirect
10. #26041440C2semidirect
11. #26041440C2semidirect
12. #26041352C2semidirect
13. #26041279C2semidirect
14. #26041274C2semidirect
15. #441×C181C2semidirect
184. C181#3317semidirect
197. #441D181semidirect

Maximal subgroups

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

subgroupindex
#260419702
#260419512
#260416442
#260416052
#260415992
#260415862
#260415282
#260415112
#260414792
#260414402
#260413522
#260412792
#260412742
#441×C1812
#3317181