Finite Group Databasebeta

← #37546161

Subgroups — #37546161

Order 23168.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041933C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041612C211yesmaximal, characteristic, complemented
4#26041607C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041592C211yesmaximal, characteristic, complemented
7#26041576C211yesmaximal, characteristic, complemented
8#26041574C211yesmaximal, characteristic, complemented
9#26041527C211yesmaximal, characteristic, complemented
10#26041510C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041322C211yesmaximal, characteristic, complemented
14C2×#19153749C211yesmaximal, characteristic, complemented
15#552×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668436C2211yescharacteristic
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#19153749C2211yescharacteristic
33#19153749C2211yescharacteristic
34#19153749C2211yescharacteristic
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
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#15668436—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×Q8×C181C2311yescharacteristic
84C2×D4×C181D411yescharacteristic
85C22×C4×C181—21no—
86C22×C4×C181C2311yescharacteristic
87C22×C4×C181C2311yescharacteristic
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
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×C181D411yescharacteristic
102C22⋊C4×C181D411yescharacteristic
103C42×C181—21no—
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×C181C2×D411yescharacteristic
123C23×C181C2×D411yescharacteristic
124Q8×C181—21no—
125Q8×C181—21no—
126D4×C181—22no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—42no—
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2411yescharacteristic
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—21no—
144C22×C181—21no—
145C22×C181—21no—
146C22×C181AGL(1,ℤ/8)12yes—
147C22×C181C22×D411yescharacteristic
148C2×D181—41no—
149C2×D181—41no—
150C2×D181—41no—
151C4×C181—21no—
152C4×C181—21no—
153C4×C181—21no—
154C4×C181—21no—
155C4×C181—41no—
156C4×C181—41no—
157C4×C181—41no—
158C4×C181—41no—
159C4×C181C2×#4111yescharacteristic
160C4×C1812−1+411yescharacteristic
161Dic181—41no—
162Dic181—41no—
163Dic181—41no—
164C2×C181—22no—
165C2×C181—41no—
166C2×C181#54711yescharacteristic
167C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C181#57811yescharacteristic
169D181—81no—
170C181#334311yescharacteristic, complemented, Sylow
171#3343—1811nomaximal, Sylow
172#552D18111yescharacteristic, complemented
173#547—1811no—
174#501—1811no—
175#484—1811no—
176#483—1811no—
177#482—1811no—
178#480—1811no—
179#478—1811no—
180#477—1811no—
181#468—1811no—
182#465—1811no—
183#440—1811no—
184#435—1811no—
185#428—1811no—
186C2×#107—1811no—
187C2×#41—3621no—
188C2×SD16—1811no—
189C2×M16—1811no—
190C22×C8—1811no—
191C4⋊Q8—1811no—
192#126—21no—
193#125—21no—
194#125C2×D18111yescharacteristic
195#123—21no—
196#123—3621no—
197#123—3621no—
198C22⋊Q8—3621no—
199C22⋊Q8C2×D18111yescharacteristic
200C22⋊Q8C2×D18111yescharacteristic
201#121—1811no—
202#121—1811no—
203C4×Q8C2×D18111yescharacteristic
204C4×D4—1811no—
205C4×D4C2×D18111yescharacteristic
206#117—21no—
207#117C2×D18111yescharacteristic
208C2×C4⋊C4—1811no—
209C2×C4⋊C4—1811no—
210C2×C4⋊C4C2×D18111yescharacteristic
211#107—1811no—
212#107—1811no—
213#107—1811no—
214#107—1811no—
215#107—1811no—
216#107—1811no—
217#106—1811no—
218#106—1811no—
219#106—1811no—
220C4⋊C8—1811no—
221#103—1811no—
222#103—1811no—
223#103—1811no—
224#102—1811no—
225#102—1811no—
226#102—1811no—
227C22⋊C8—1811no—
228C22⋊C8—1811no—
229#97—1811no—
230#41—7244no—
231C2×Q8—3621no—
232C2×Q8C22×D18111yescharacteristic
233C2×D4—3621no—
234C2×D4—3621no—
235C2×D4—1811no—
236C2×D4C22⋊D18111yescharacteristic
237C22×C4—21no—
238C22×C4—3621no—
239C22×C4—3621no—
240C22×C4—3621no—
241C22×C4C22×D18111yescharacteristic
242C22×C4C22×D18111yescharacteristic
243SD16—3622no—
244M16—3622no—
245C2×C8—3622no—
246C2×C8—1811no—
247C2×C8—1811no—
248C2×C8—1811no—
249C2×C8—1811no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—3621no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—1811no—
260C4⋊C4—1811no—
261C4⋊C4—1811no—
262C4⋊C4C22⋊D18111yescharacteristic
263C4⋊C4C22×D18111yescharacteristic
264C4⋊C4C22×D18111yescharacteristic
265C4⋊C4C22×D18111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C22⋊C4C22⋊D18111yescharacteristic
273C22⋊C4C22⋊D18111yescharacteristic
274C42—21no—
275C42—21no—
276C42C22×D18111yescharacteristic
277C23—3621no—
278C23C2×C22⋊D18111yescharacteristic
279C23C2×C22⋊D18111yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—7242no—
283D4—22no—
284D4—3621no—
285D4—3621no—
286D4—7242no—
287D4—7242no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3621no—
300C2×C4—7242no—
301C2×C4—7242no—
302C2×C4—7242no—
303C2×C4C2×C22⋊D18111yescharacteristic
304C2×C4C2×C22⋊D18111yescharacteristic
305C2×C4C2×C22⋊D18111yescharacteristic
306C2×C4C2×C22⋊D18111yescharacteristic
307C2×C4C23×D18111yescharacteristic
308C8—3621no—
309C8—3621no—
310C8—3621no—
311C8—1812no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—7241no—
318C22—7241no—
319C22—7241no—
320C22#1915384112yes—
321C22C22×C22⋊D18111yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—7241no—
331C4—7241no—
332C4—7241no—
333C4#41×D18111yescharacteristic
334C4#1915386311yescharacteristic
335C2—22no—
336C2—41no—
337C2—14481no—
338C2#2604192811yescharacteristic
339C2C2×#1915384111yescharacteristic
340C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#552×C181C2
socleC22×C181C22×D4
prime radical#26041933C2
hypercenter#552D181
hypocenterC181#3343
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. #26041612C2semidirect
4. #26041607C2semidirect
5. #26041603C2semidirect
6. #26041592C2semidirect
7. #26041576C2semidirect
8. #26041574C2semidirect
9. #26041527C2semidirect
10. #26041510C2semidirect
11. #26041377C2semidirect
12. #26041351C2semidirect
13. #26041322C2semidirect
14. C2×#19153749C2semidirect
15. #552×C181C2semidirect
121. #13089678#41semidirect
170. C181#3343semidirect
172. #552D181semidirect

Maximal subgroups

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

subgroupindex
#260419332
#260416892
#260416122
#260416072
#260416032
#260415922
#260415762
#260415742
#260415272
#260415102
#260413772
#260413512
#260413222
C2×#191537492
#552×C1812
#3343181