Finite Group Databasebeta

← #37546236

Subgroups — #37546236

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#26041969C211yesmaximal, characteristic, complemented
2#26041685C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041620C211yesmaximal, characteristic, complemented
5#26041600C211yesmaximal, characteristic, complemented
6#26041595C211yesmaximal, characteristic, complemented
7#26041579C211yesmaximal, characteristic, complemented
8#26041561C211yesmaximal, characteristic, complemented
9#26041523C211yesmaximal, characteristic, complemented
10#26041506C211yesmaximal, characteristic, complemented
11#26041379C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041322C211yesmaximal, characteristic, complemented
14#26041315C211yesmaximal, characteristic, complemented
15#555×C181C211yesmaximal, characteristic
16C2×#15668437C2211yescharacteristic
17C2×#15668433C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19#19153809—21no—
20#19153808C2211yescharacteristic
21#19153806—21no—
22#19153797—21no—
23#19153795C2211yescharacteristic
24#19153795C2211yescharacteristic
25C4×Dic362C2211yescharacteristic
26#19153775C2211yescharacteristic
27#19153772—21no—
28C2×C4⋊Dic181C2211yescharacteristic
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153749C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153747C2211yescharacteristic
37#19153747C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153739C2211yescharacteristic
40#19153739C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153737C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#124×C181—21no—
50#124×C181—21no—
51#121×C181—21no—
52#121×C181C2211yescharacteristic
53#121×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55C4×D4×C181C2211yescharacteristic
56#117×C181—21no—
57#117×C181C2211yescharacteristic
58#117×C181C2211yescharacteristic
59C2×Dic362C2311yescharacteristic
60C22×Dic181—21no—
61#15668437—22no—
62#15668433—22no—
63C2×#13089678—22no—
64C2×#13089678C2311yescharacteristic
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C4⋊Dic181—21no—
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71C4⋊Dic181C2311yescharacteristic
72#15668428—21no—
73#15668428—21no—
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77C22⋊Dic181—21no—
78C22⋊Dic181—21no—
79C22⋊Dic181—21no—
80C4×Dic181—21no—
81C4×Dic181—21no—
82C2×Q8×C181—21no—
83C2×D4×C181—21no—
84C2×D4×C181—21no—
85C2×D4×C181D411yescharacteristic
86C2×D4×C181C2311yescharacteristic
87C22×C4×C181—21no—
88C22×C4×C181C2311yescharacteristic
89C22×C4×C181C2311yescharacteristic
90C4⋊C4×C181—21no—
91C4⋊C4×C181D411yescharacteristic
92C4⋊C4×C181C2311yescharacteristic
93C4⋊C4×C181C2311yescharacteristic
94C4⋊C4×C181C2311yescharacteristic
95C22⋊C4×C181—21no—
96C22⋊C4×C181—21no—
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
105Dic362—21no—
106Dic362—21no—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—42no—
112#13089678—21no—
113#13089678—21no—
114#13089678—21no—
115#13089678#4112yes—
116C23×C181—21no—
117C23×C181C2×D411yescharacteristic
118C23×C181C2×D411yescharacteristic
119Q8×C181—81no—
120D4×C181—21no—
121D4×C181—21no—
122D4×C181—41no—
123D4×C181—81no—
124D4×C181—81no—
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×C181—41no—
145C22×C181—41no—
146C22×C181—41no—
147C22×C181C22×D411yescharacteristic
148C4×C181—21no—
149C4×C181—21no—
150C4×C181—21no—
151C4×C181—21no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181—41no—
155C4×C181C2×#4111yescharacteristic
156C4×C1812−1+411yescharacteristic
157Dic181—41no—
158Dic181—41no—
159Dic181—41no—
160Dic181—41no—
161C2×C181—41no—
162C2×C181—41no—
163C2×C181—81no—
164C2×C181#54711yescharacteristic
165C2×C181#57511yescharacteristic
166C2×C181#57711yescharacteristic
167C181#334811yescharacteristic, complemented, Sylow
168#3348—1811nomaximal, Sylow
169#555D18111yescharacteristic
170#552—1811no—
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⋊Q8C2×D18111yescharacteristic
188#126—3621no—
189#125—3621no—
190#125—1811no—
191#124—21no—
192#124—21no—
193#123—3621no—
194C22⋊Q8—1811no—
195C22⋊Q8—1811no—
196#121—21no—
197#121C2×D18111yescharacteristic
198#121C2×D18111yescharacteristic
199C4×Q8—1811no—
200C4×D4C2×D18111yescharacteristic
201C4×D4C2×D18111yescharacteristic
202#117—21no—
203#117—3621no—
204#117—1811no—
205#117C2×D18111yescharacteristic
206#117C2×D18111yescharacteristic
207C2×C4⋊C4—1811no—
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—21no—
228C2×Q8—1811no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4C22⋊D18111yescharacteristic
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—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
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—21no—
262C22⋊C4—21no—
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—21no—
274C23C2×C22⋊D18111yescharacteristic
275C23C2×C22⋊D18111yescharacteristic
276Q8—81no—
277Q8—3621no—
278Q8—3621no—
279D4—21no—
280D4—21no—
281D4—41no—
282D4—81no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—3621no—
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—41no—
313C22—41no—
314C22—41no—
315C22C22×C22⋊D18111yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—7241no—
324C4—7241no—
325C4—7241no—
326C4—7241no—
327C4#41×D18111yescharacteristic
328C4#1915386311yescharacteristic
329C2—41no—
330C2—41no—
331C2—81no—
332C2#2604192811yescharacteristic
333C2#2604207711yescharacteristic
334C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#555×C181C2
socleC22×C181C22×D4
hypercenter#555D181
hypocenterC181#3348
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #26041969C2semidirect
2. #26041685C2semidirect
3. #26041620C2semidirect
4. #26041620C2semidirect
5. #26041600C2semidirect
6. #26041595C2semidirect
7. #26041579C2semidirect
8. #26041561C2semidirect
9. #26041523C2semidirect
10. #26041506C2semidirect
11. #26041379C2semidirect
12. #26041351C2semidirect
13. #26041322C2semidirect
14. #26041315C2semidirect
167. C181#3348semidirect

Maximal subgroups

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

subgroupindex
#260419692
#260416852
#260416202
#260416002
#260415952
#260415792
#260415612
#260415232
#260415062
#260413792
#260413512
#260413222
#260413152
#555×C1812
#3348181