Finite Group Databasebeta

← #37546230

Subgroups — #37546230

Order 23168.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041994C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041617C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041597C211yesmaximal, characteristic, complemented
7#26041576C211yesmaximal, characteristic, complemented
8#26041564C211yesmaximal, characteristic, complemented
9#26041527C211yesmaximal, characteristic, complemented
10#26041510C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041319C211yesmaximal, characteristic, complemented
14#26041315C211yesmaximal, characteristic, complemented
15#552×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17C2×#15668433C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153802—21no—
21#19153802—21no—
22#19153790C2211yescharacteristic
23#19153790C2211yescharacteristic
24#19153787—21no—
25C4×D724C2211yescharacteristic
26#19153775C2211yescharacteristic
27#19153775C2211yescharacteristic
28#19153774—21no—
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153749C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153747C2211yescharacteristic
37#19153747C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48#126×C181—21no—
49#125×C181—21no—
50#125×C181C2211yescharacteristic
51#123×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53C22⋊Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55C4×D4×C181C2211yescharacteristic
56#117×C181—21no—
57#117×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×D724C2311yescharacteristic
61C2×C22⋊D181—21no—
62C2×C22⋊D181—21no—
63C2×C4×D181—21no—
64#15668436—22no—
65#15668433—22no—
66C2×#13089678—22no—
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C4⋊Dic181—21no—
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75C22⋊Dic181—21no—
76C22⋊Dic181—21no—
77#15668426—21no—
78#15668426—21no—
79#15668426—21no—
80C4×Dic181—21no—
81C4×Dic181—21no—
82C2×Q8×C181C2311yescharacteristic
83C2×D4×C181D411yescharacteristic
84C22×C4×C181—21no—
85C22×C4×C181C2311yescharacteristic
86C22×C4×C181C2311yescharacteristic
87C4⋊C4×C181—21no—
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
91C4⋊C4×C181—21no—
92C4⋊C4×C181—21no—
93C4⋊C4×C181D411yescharacteristic
94C4⋊C4×C181C2311yescharacteristic
95C4⋊C4×C181C2311yescharacteristic
96C4⋊C4×C181C2311yescharacteristic
97C22⋊C4×C181—21no—
98C22⋊C4×C181—21no—
99C22⋊C4×C181—21no—
100C22⋊C4×C181D411yescharacteristic
101C22⋊C4×C181D411yescharacteristic
102C42×C181—21no—
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105C22×D181—21no—
106Dic362—81no—
107D724—21no—
108D724—21no—
109C22⋊D181—81no—
110C22⋊D181—81no—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C4×D181—42no—
115#13089678—21no—
116#13089678—21no—
117#13089678—21no—
118#13089678#4112yes—
119C23×C181C2×D411yescharacteristic
120C23×C181C2×D411yescharacteristic
121Q8×C181—21no—
122Q8×C181—21no—
123D4×C181—41no—
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—42no—
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C22×C181—21no—
139C22×C181—21no—
140C22×C181—21no—
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—21no—
144C22×C181C22×D411yescharacteristic
145C2×D181—41no—
146C2×D181—41no—
147C2×D181—41no—
148C4×C181—21no—
149C4×C181—21no—
150C4×C181—21no—
151C4×C181—21no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181—41no—
155C4×C181—41no—
156C4×C181C2×#4111yescharacteristic
157C4×C1812−1+411yescharacteristic
158Dic181—41no—
159Dic181—41no—
160Dic181—41no—
161C2×C181—41no—
162C2×C181—41no—
163C2×C181#54711yescharacteristic
164C2×C181#57511yescharacteristic
165C2×C181#57711yescharacteristic
166D181—81no—
167C181#334811yescharacteristic, complemented, Sylow
168#3348—1811nomaximal, Sylow
169#555—1811no—
170#552D18111yescharacteristic, complemented
171#501—1811no—
172#485—1811no—
173#485—1811no—
174#482—1811no—
175#481—1811no—
176#478—1811no—
177#475—1811no—
178#468—1811no—
179#465—1811no—
180#440—1811no—
181#435—1811no—
182#428—1811no—
183#427—1811no—
184C2×SD16—1811no—
185C2×M16—1811no—
186C22×C8—1811no—
187C4⋊Q8—1811no—
188#126—21no—
189#125—21no—
190#125C2×D18111yescharacteristic
191#124—3621no—
192#124—3621no—
193#123—21no—
194C22⋊Q8C2×D18111yescharacteristic
195C22⋊Q8C2×D18111yescharacteristic
196#121—3621no—
197#121—1811no—
198#121—1811no—
199C4×Q8C2×D18111yescharacteristic
200C4×D4—1811no—
201C4×D4C2×D18111yescharacteristic
202#117—21no—
203#117—3621no—
204#117—1811no—
205#117—1811no—
206#117C2×D18111yescharacteristic
207C2×C4⋊C4C2×D18111yescharacteristic
208#107—1811no—
209#107—1811no—
210#107—1811no—
211#107—1811no—
212#106—1811no—
213#106—1811no—
214#106—1811no—
215#106—1811no—
216#106—1811no—
217C4⋊C8—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#102—1811no—
222#102—1811no—
223#102—1811no—
224C22⋊C8—1811no—
225C22⋊C8—1811no—
226#97—1811no—
227C2×Q8—3621no—
228C2×Q8C22×D18111yescharacteristic
229C2×D4—3621no—
230C2×D4—3621no—
231C2×D4—1811no—
232C2×D4C22⋊D18111yescharacteristic
233C22×C4—21no—
234C22×C4—3621no—
235C22×C4C22×D18111yescharacteristic
236C22×C4C22×D18111yescharacteristic
237SD16—3622no—
238M16—3622no—
239C2×C8—3622no—
240C2×C8—1811no—
241C2×C8—1811no—
242C2×C8—1811no—
243C2×C8—1811no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—3621no—
251C4⋊C4—1811no—
252C4⋊C4—1811no—
253C4⋊C4—1811no—
254C4⋊C4C22⋊D18111yescharacteristic
255C4⋊C4C22×D18111yescharacteristic
256C4⋊C4C22×D18111yescharacteristic
257C4⋊C4C22×D18111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4C22⋊D18111yescharacteristic
267C22⋊C4C22⋊D18111yescharacteristic
268C42—21no—
269C42—21no—
270C42—3621no—
271C42—3621no—
272C42C22×D18111yescharacteristic
273C23—3621no—
274C23C2×C22⋊D18111yescharacteristic
275C23C2×C22⋊D18111yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—14481no—
279D4—41no—
280D4—3621no—
281D4—3621no—
282D4—14481no—
283D4—14481no—
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—42no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—7242no—
297C2×C4C2×C22⋊D18111yescharacteristic
298C2×C4C2×C22⋊D18111yescharacteristic
299C2×C4C2×C22⋊D18111yescharacteristic
300C2×C4C2×C22⋊D18111yescharacteristic
301C2×C4C23×D18111yescharacteristic
302C8—3621no—
303C8—3621no—
304C8—3621no—
305C8—1812no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—7241no—
313C22—7241no—
314C22—7241no—
315C22C22×C22⋊D18111yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—7241no—
325C4—7241no—
326C4—7241no—
327C4#41×D18111yescharacteristic
328C4#1915386311yescharacteristic
329C2—41no—
330C2—41no—
331C2—14481no—
332C2#2604192811yescharacteristic
333C2#2604207711yescharacteristic
334C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#552×C181C2
socleC22×C181C22×D4
prime radical#26041994C2
hypercenter#552D181
hypocenterC181#3348
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26041689C2semidirect
3. #26041617C2semidirect
4. #26041617C2semidirect
5. #26041603C2semidirect
6. #26041597C2semidirect
7. #26041576C2semidirect
8. #26041564C2semidirect
9. #26041527C2semidirect
10. #26041510C2semidirect
11. #26041377C2semidirect
12. #26041351C2semidirect
13. #26041319C2semidirect
14. #26041315C2semidirect
15. #552×C181C2semidirect
167. C181#3348semidirect
170. #552D181semidirect

Maximal subgroups

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

subgroupindex
#260419942
#260416892
#260416172
#260416032
#260415972
#260415762
#260415642
#260415272
#260415102
#260413772
#260413512
#260413192
#260413152
#552×C1812
#3348181