Finite Group Databasebeta

← #37546206

Subgroups — #37546206

Order 23168.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041986C211yesmaximal, characteristic, complemented
2#26041689C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041614C211yesmaximal, characteristic, complemented
5#26041595C211yesmaximal, characteristic, complemented
6#26041590C211yesmaximal, characteristic, complemented
7#26041583C211yesmaximal, characteristic, complemented
8#26041559C211yesmaximal, characteristic, complemented
9#26041523C211yesmaximal, characteristic, complemented
10#26041506C211yesmaximal, characteristic, complemented
11#26041379C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041322C211yesmaximal, characteristic, complemented
14C2×#19153749C211yesmaximal, characteristic, complemented
15#553×C181C211yesmaximal, characteristic
16C2×#15668437C2211yescharacteristic
17C2×#15668433C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153807—21no—
21#19153795C2211yescharacteristic
22#19153795C2211yescharacteristic
23#19153793—21no—
24#19153793—21no—
25C4×Dic362C2211yescharacteristic
26D4×Dic181—21no—
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#19153739C2211yescharacteristic
40#19153739C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153737C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48#126×C181—21no—
49#125×C181C2211yescharacteristic
50#123×C181—21no—
51#121×C181—21no—
52#121×C181C2211yescharacteristic
53#121×C181C2211yescharacteristic
54C4×D4×C181—21no—
55C4×D4×C181C2211yescharacteristic
56C4×D4×C181C2211yescharacteristic
57#117×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C22×Dic181—21no—
62C22×Dic181—21no—
63#15668437—22no—
64#15668433—22no—
65C2×#13089678—22no—
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C4⋊Dic181—21no—
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80C22⋊Dic181—21no—
81C4×Dic181—21no—
82C2×D4×C181—21no—
83C2×D4×C181—21no—
84C2×D4×C181D411yescharacteristic
85C2×D4×C181C2311yescharacteristic
86C22×C4×C181—21no—
87C22×C4×C181—21no—
88C22×C4×C181C2311yescharacteristic
89C22×C4×C181C2311yescharacteristic
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×C181—21no—
101C22⋊C4×C181D411yescharacteristic
102C22⋊C4×C181D411yescharacteristic
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105Dic362—21no—
106Dic362—21no—
107Dic362—81no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181—81no—
113C2×Dic181—81no—
114#13089678—21no—
115#13089678—21no—
116#13089678—21no—
117#13089678#4112yes—
118C23×C181—21no—
119C23×C181C2×D411yescharacteristic
120C23×C181C2×D411yescharacteristic
121D4×C181—22no—
122D4×C181—21no—
123D4×C181—21no—
124D4×C181—42no—
125D4×C181—81no—
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—21no—
133C2×C4×C181—81no—
134C2×C4×C181—81no—
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2411yescharacteristic
140C22×C181—21no—
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—21no—
144C22×C181—21no—
145C22×C181—41no—
146C22×C181—41no—
147C22×C181—41no—
148C22×C181#13712yes—
149C22×C181C22×D411yescharacteristic
150C4×C181—21no—
151C4×C181—21no—
152C4×C181—21no—
153C4×C181—21no—
154C4×C181—41no—
155C4×C181—41no—
156C4×C181—41no—
157C4×C181C2×#4111yescharacteristic
158C4×C1812−1+411yescharacteristic
159Dic181—41no—
160Dic181—41no—
161Dic181—41no—
162Dic181—41no—
163C2×C181—22no—
164C2×C181—41no—
165C2×C181—81no—
166C2×C181#54711yescharacteristic
167C2×C181C2×#13711yescharacteristic
168C2×C181#57611yescharacteristic
169C181#334611yescharacteristic, complemented, Sylow
170#3346—1811nomaximal, Sylow
171#554—1811no—
172#553D18111yescharacteristic
173#501—1811no—
174#485—1811no—
175#484—1811no—
176#481—1811no—
177#480—1811no—
178#479—1811no—
179#474—1811no—
180#468—1811no—
181#465—1811no—
182#440—1811no—
183#435—1811no—
184#428—1811no—
185C2×#107—1811no—
186C2×SD16—1811no—
187C2×M16—1811no—
188C22×C8—1811no—
189C4⋊Q8—1811no—
190#126—21no—
191#125—3621no—
192#125C2×D18111yescharacteristic
193#123—21no—
194C22⋊Q8—3621no—
195C22⋊Q8—3621no—
196C22⋊Q8—1811no—
197C22⋊Q8—1811no—
198#121—21no—
199#121C2×D18111yescharacteristic
200#121C2×D18111yescharacteristic
201C4×Q8—1811no—
202C4×D4—21no—
203C4×D4—3621no—
204C4×D4C2×D18111yescharacteristic
205C4×D4C2×D18111yescharacteristic
206#117C2×D18111yescharacteristic
207C2×C4⋊C4—1811no—
208C2×C4⋊C4—1811no—
209C2×C4⋊C4C2×D18111yescharacteristic
210#107—1811no—
211#107—1811no—
212#107—1811no—
213#107—1811no—
214#107—1811no—
215#107—1811no—
216#106—1811no—
217#106—1811no—
218#106—1811no—
219C4⋊C8—1811no—
220#103—1811no—
221#103—1811no—
222#103—1811no—
223#102—1811no—
224#102—1811no—
225#102—1811no—
226C22⋊C8—1811no—
227C22⋊C8—1811no—
228#97—1811no—
229C2×Q8—3621no—
230C2×Q8—1811no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4C22⋊D18111yescharacteristic
234C2×D4C22×D18111yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—3621no—
238C22×C4—3621no—
239C22×C4C22×D18111yescharacteristic
240C22×C4C22×D18111yescharacteristic
241SD16—3622no—
242M16—3622no—
243C2×C8—3622no—
244C2×C8—1811no—
245C2×C8—1811no—
246C2×C8—1811no—
247C2×C8—1811no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4—1811no—
260C4⋊C4C22⋊D18111yescharacteristic
261C4⋊C4C22×D18111yescharacteristic
262C4⋊C4C22×D18111yescharacteristic
263C4⋊C4C22×D18111yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4C22⋊D18111yescharacteristic
271C22⋊C4C22⋊D18111yescharacteristic
272C42—21no—
273C42—3621no—
274C42C22×D18111yescharacteristic
275C23—21no—
276C23C2×C22⋊D18111yescharacteristic
277C23C2×C22⋊D18111yescharacteristic
278Q8—3621no—
279Q8—3621no—
280Q8—14481no—
281D4—22no—
282D4—21no—
283D4—21no—
284D4—42no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—3621no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—14481no—
300C2×C4—14481no—
301C2×C4C2×C22⋊D18111yescharacteristic
302C2×C4C2×C22⋊D18111yescharacteristic
303C2×C4C2×C22⋊D18111yescharacteristic
304C2×C4C2×C22⋊D18111yescharacteristic
305C2×C4C23×D18111yescharacteristic
306C8—3621no—
307C8—3621no—
308C8—3621no—
309C8—1812no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22#1915384612yes—
319C22C22×C22⋊D18111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—7241no—
328C4—7241no—
329C4—7241no—
330C4—7241no—
331C4#41×D18111yescharacteristic
332C4#1915386311yescharacteristic
333C2—22no—
334C2—41no—
335C2—81no—
336C2#2604192811yescharacteristic
337C2C2×#1915384611yescharacteristic
338C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#553×C181C2
socleC22×C181C22×D4
hypercenter#553D181
hypocenterC181#3346
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #26041986C2semidirect
2. #26041689C2semidirect
3. #26041620C2semidirect
4. #26041614C2semidirect
5. #26041595C2semidirect
6. #26041590C2semidirect
7. #26041583C2semidirect
8. #26041559C2semidirect
9. #26041523C2semidirect
10. #26041506C2semidirect
11. #26041379C2semidirect
12. #26041351C2semidirect
13. #26041322C2semidirect
14. C2×#19153749C2semidirect
169. C181#3346semidirect

Maximal subgroups

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

subgroupindex
#260419862
#260416892
#260416202
#260416142
#260415952
#260415902
#260415832
#260415592
#260415232
#260415062
#260413792
#260413512
#260413222
C2×#191537492
#553×C1812
#3346181