Finite Group Databasebeta

← #37546781

Subgroups — #37546781

Order 23168.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042002C211yesmaximal, characteristic, complemented
2#26041674C211yesmaximal, characteristic, complemented
3#26041666C211yesmaximal, characteristic
4#26041661C211yesmaximal, characteristic, complemented
5#26041501C211yesmaximal, characteristic, complemented
6#26041498C211yesmaximal, characteristic, complemented
7#26041494C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041484C211yesmaximal, characteristic, complemented
10#26041476C211yesmaximal, characteristic, complemented
11#26041410C211yesmaximal, characteristic, complemented
12#26041389C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14C4×#15668437C211yesmaximal, characteristic, complemented
15#559×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668437C2211yescharacteristic
22C2×#15668436C2211yescharacteristic
23C2×#15668434C2211yescharacteristic
24C2×#15668434C2211yescharacteristic
25#19153816C2211yescharacteristic
26#19153811—21no—
27#19153811—21no—
28#19153804C2211yescharacteristic
29#19153804C2211yescharacteristic
30#19153792—21no—
31C4×Dic362C2211yescharacteristic
32C4×Dic362C2211yescharacteristic
33C4×D724C2211yescharacteristic
34#19153774—21no—
35#19153747C2211yescharacteristic
36C4⋊#13089678C2211yescharacteristic
37C4⋊#13089678C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153739C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153736C2211yescharacteristic
45#19153727C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48#127×C181C2211yescharacteristic
49#124×C181—21no—
50#124×C181C2211yescharacteristic
51#124×C181C2211yescharacteristic
52#123×C181—21no—
53#123×C181—21no—
54C4×Q8×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4×D4×C181—21no—
57C4×D4×C181C2211yescharacteristic
58C4×D4×C181C2211yescharacteristic
59C2×Dic362C2311yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C4×D181—21no—
63#15668439—22no—
64#15668439—41no—
65#15668437—22no—
66#15668437—41no—
67#15668437—41no—
68#15668437—41no—
69#15668436—41no—
70#15668434—22no—
71#15668434—41no—
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668426—21no—
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86C4×Dic181—21no—
87C4×Dic181—21no—
88C2×Q8×C181D411yescharacteristic
89C2×Q8×C181C2311yescharacteristic
90C2×D4×C181—21no—
91C2×D4×C181—21no—
92C2×D4×C181C2311yescharacteristic
93C2×D4×C181C2311yescharacteristic
94C22×C4×C181—21no—
95C22×C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181D411yescharacteristic
98C4⋊C4×C181D411yescharacteristic
99C4⋊C4×C181D411yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C42×C181—21no—
108C42×C181C2311yescharacteristic
109C42×C181C2311yescharacteristic
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112Dic362—21no—
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116D724—21no—
117D724—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C4×D181—42no—
122#13089678—21no—
123#13089678—21no—
124#13089678—21no—
125#13089678#4112yes—
126C23×C181—21no—
127C23×C181—21no—
128Q8×C181—21no—
129Q8×C181—21no—
130Q8×C181—41no—
131D4×C181—21no—
132D4×C181—21no—
133D4×C181—21no—
134D4×C181—21no—
135D4×C181—42no—
136D4×C181—81no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—81no—
140C2×C4×C181—81no—
141C2×C4×C181C2×D411yescharacteristic
142C2×C4×C181C2×D411yescharacteristic
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2411yescharacteristic
148C22×C181—41no—
149C22×C181—41no—
150C22×C181—41no—
151C22×C181—41no—
152C22×C181—41no—
153C22×C181—41no—
154C22×C181C22×D411yescharacteristic
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—21no—
164C4×C181—41no—
165C4×C181—41no—
166C4×C181C2×#4111yescharacteristic
167C4×C181H(5,𝔽2)11yescharacteristic
168Dic181—41no—
169Dic181—41no—
170Dic181—41no—
171C2×C181—81no—
172C2×C181—81no—
173C2×C181#55011yescharacteristic
174C2×C181#57511yescharacteristic
175C2×C181#57711yescharacteristic
176D181—81no—
177C181#340111yescharacteristic, complemented, Sylow
178#3401—1811nomaximal, Sylow
179#559D18111yescharacteristic, complemented
180#556—1811no—
181#497—1811no—
182#496—1811no—
183#495—1811no—
184#464—1811no—
185#464—1811no—
186#463—1811no—
187#463—1811no—
188#461—1811no—
189#460—1811no—
190#446—1811no—
191#442—1811no—
192#441—1811no—
193C4×SD16—1811no—
194C2×Q16—1811no—
195C2×Q16—1811no—
196C2×SD16—1811no—
197C2×SD16—1811no—
198C2×SD16—1811no—
199C2×SD16—1811no—
200C2×SD16—1811no—
201C2×D8—1811no—
202C2×D8—1811no—
203C4⋊Q8—1811no—
204#127C2×D18111yescharacteristic
205#126—3621no—
206#126—3621no—
207#124—21no—
208#124—1811no—
209#124—1811no—
210#124C2×D18111yescharacteristic
211#124C2×D18111yescharacteristic
212#123—21no—
213#123—21no—
214C22⋊Q8—3621no—
215C4×Q8—1811no—
216C4×Q8—1811no—
217C4×Q8C2×D18111yescharacteristic
218C4×Q8C2×D18111yescharacteristic
219C4×D4—21no—
220C4×D4—1811no—
221C4×D4C2×D18111yescharacteristic
222C4×D4C2×D18111yescharacteristic
223#117—3621no—
224#106—1811no—
225C4⋊C8—1811no—
226C4⋊C8—1811no—
227C4⋊C8—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#102—1811no—
232#102—1811no—
233#102—1811no—
234#97—1811no—
235#97—1811no—
236C4×C8—1811no—
237C2×Q8—1811no—
238C2×Q8—1811no—
239C2×Q8C22⋊D18111yescharacteristic
240C2×Q8C22×D18111yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—1811no—
244C2×D4C22×D18111yescharacteristic
245C2×D4C22×D18111yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—3621no—
249Q16—3622no—
250Q16—7241no—
251SD16—3622no—
252SD16—7241no—
253SD16—7241no—
254SD16—7241no—
255SD16—7241no—
256D8—3622no—
257D8—7241no—
258C2×C8—1811no—
259C2×C8—1811no—
260C2×C8—1811no—
261C2×C8—1811no—
262C4⋊C4—21no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
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—21no—
277C22⋊C4—21no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C42—21no—
284C42—3621no—
285C42—3621no—
286C42C22×D18111yescharacteristic
287C42C22×D18111yescharacteristic
288C42C22×D18111yescharacteristic
289C23—21no—
290C23—21no—
291C23—3621no—
292Q8—21no—
293Q8—21no—
294Q8—41no—
295Q8—3621no—
296Q8—3621no—
297Q8—3621no—
298Q8—3621no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—81no—
305D4—3621no—
306D4—3621no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—7242no—
315C2×C4C2×C22⋊D18111yescharacteristic
316C2×C4C2×C22⋊D18111yescharacteristic
317C2×C4C2×C22⋊D18111yescharacteristic
318C2×C4C2×C22⋊D18111yescharacteristic
319C2×C4C2×C22⋊D18111yescharacteristic
320C2×C4C2×C22⋊D18111yescharacteristic
321C2×C4C23×D18111yescharacteristic
322C8—3621no—
323C8—3621no—
324C8—3621no—
325C8—1812no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—7241no—
333C22—7241no—
334C22—7241no—
335C22C22×C22⋊D18111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—7241no—
345C4—7241no—
346C4—7241no—
347C4#41×D18111yescharacteristic
348C4#1915386211yescharacteristic
349C2—81no—
350C2—81no—
351C2—14481no—
352C2#2604194811yescharacteristic
353C2#2604207711yescharacteristic
354C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#559×C181C2
socleC22×C181C22×D4
prime radical#26041666C2
hypercenter#559D181
hypocenterC181#3401
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #26042002C2semidirect
2. #26041674C2semidirect
4. #26041661C2semidirect
5. #26041501C2semidirect
6. #26041498C2semidirect
7. #26041494C2semidirect
8. #26041491C2semidirect
9. #26041484C2semidirect
10. #26041476C2semidirect
11. #26041410C2semidirect
12. #26041389C2semidirect
13. #26041384C2semidirect
14. C4×#15668437C2semidirect
15. #559×C181C2semidirect
177. C181#3401semidirect
179. #559D181semidirect

Maximal subgroups

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

subgroupindex
#260420022
#260416742
#260416662
#260416612
#260415012
#260414982
#260414942
#260414912
#260414842
#260414762
#260414102
#260413892
#260413842
C4×#156684372
#559×C1812
#3401181