Finite Group Databasebeta

← #37546176

Subgroups — #37546176

Order 23168.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041986C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041612C211yesmaximal, characteristic, complemented
5#26041609C211yesmaximal, characteristic, complemented
6#26041607C211yesmaximal, characteristic, complemented
7#26041583C211yesmaximal, characteristic, complemented
8#26041564C211yesmaximal, characteristic, complemented
9#26041535C211yesmaximal, characteristic, complemented
10#26041518C211yesmaximal, characteristic, complemented
11#26041384C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic
13#26041322C211yesmaximal, characteristic, complemented
14C2×#19153747C211yesmaximal, characteristic, complemented
15#552×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
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#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153747C2211yescharacteristic
37#19153747C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153739C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153739C2211yescharacteristic
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×Dic362C2311yescharacteristic
61C22×Dic181—21no—
62C22×Dic181—21no—
63#15668439—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×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
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×C181C2×D411yescharacteristic
119C23×C181C2×D411yescharacteristic
120Q8×C181—21no—
121Q8×C181—21no—
122D4×C181—22no—
123C2×C4×C181—21no—
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—42no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2411yescharacteristic
137C22×C181—21no—
138C22×C181—21no—
139C22×C181—21no—
140C22×C181—21no—
141C22×C181—21no—
142C22×C181#13712yes—
143C22×C181C22×D411yescharacteristic
144C4×C181—21no—
145C4×C181—21no—
146C4×C181—21no—
147C4×C181—21no—
148C4×C181—41no—
149C4×C181—41no—
150C4×C181—41no—
151C4×C181—41no—
152C4×C181C2×#4111yescharacteristic
153C4×C1812−1+411yescharacteristic
154Dic181—41no—
155Dic181—41no—
156Dic181—41no—
157Dic181—41no—
158C2×C181—22no—
159C2×C181—41no—
160C2×C181#54711yescharacteristic
161C2×C181C2×#13711yescharacteristic
162C2×C181#57711yescharacteristic
163C181#334411yescharacteristic, complemented, Sylow
164#3344—1811nomaximal, Sylow
165#554—1811no—
166#552D18111yescharacteristic
167#501—1811no—
168#485—1811no—
169#484—1811no—
170#483—1811no—
171#483—1811no—
172#479—1811no—
173#475—1811no—
174#470—1811no—
175#467—1811no—
176#441—1811no—
177#435—1811no—
178#428—1811no—
179C2×#106—1811no—
180C2×Q16—1811no—
181C2×M16—1811no—
182C22×C8—1811no—
183C4⋊Q8—1811no—
184#126—21no—
185#125—21no—
186#125—3621no—
187#125C2×D18111yescharacteristic
188#123—21no—
189C22⋊Q8—3621no—
190C22⋊Q8—3621no—
191C22⋊Q8—1811no—
192C22⋊Q8—1811no—
193C22⋊Q8C2×D18111yescharacteristic
194C22⋊Q8C2×D18111yescharacteristic
195C4×Q8—1811no—
196C4×Q8C2×D18111yescharacteristic
197C4×D4—3621no—
198C4×D4C2×D18111yescharacteristic
199#117—21no—
200#117C2×D18111yescharacteristic
201C2×C4⋊C4—1811no—
202C2×C4⋊C4—1811no—
203C2×C4⋊C4C2×D18111yescharacteristic
204#107—1811no—
205#107—1811no—
206#107—1811no—
207#106—1811no—
208#106—1811no—
209#106—1811no—
210#106—1811no—
211#106—1811no—
212#106—1811no—
213C4⋊C8—1811no—
214#103—1811no—
215#103—1811no—
216#103—1811no—
217#103—1811no—
218#103—1811no—
219#103—1811no—
220C22⋊C8—1811no—
221C22⋊C8—1811no—
222#97—1811no—
223C2×Q8—3621no—
224C2×Q8—1811no—
225C2×Q8C22×D18111yescharacteristic
226C2×D4C22⋊D18111yescharacteristic
227C22×C4—21no—
228C22×C4—3621no—
229C22×C4—3621no—
230C22×C4C22×D18111yescharacteristic
231C22×C4C22×D18111yescharacteristic
232Q16—3622no—
233M16—3622no—
234C2×C8—3622no—
235C2×C8—1811no—
236C2×C8—1811no—
237C2×C8—1811no—
238C2×C8—1811no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
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—3621no—
262C22⋊C4—3621no—
263C22⋊C4C22⋊D18111yescharacteristic
264C22⋊C4C22⋊D18111yescharacteristic
265C42—21no—
266C42—21no—
267C42—3621no—
268C42C22×D18111yescharacteristic
269C23C2×C22⋊D18111yescharacteristic
270C23C2×C22⋊D18111yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—3621no—
274Q8—3621no—
275Q8—14481no—
276D4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—3621no—
287C2×C4—3621no—
288C2×C4—3621no—
289C2×C4—3621no—
290C2×C4—14481no—
291C2×C4—14481no—
292C2×C4C2×C22⋊D18111yescharacteristic
293C2×C4C2×C22⋊D18111yescharacteristic
294C2×C4C2×C22⋊D18111yescharacteristic
295C2×C4C2×C22⋊D18111yescharacteristic
296C2×C4C23×D18111yescharacteristic
297C8—3621no—
298C8—3621no—
299C8—3621no—
300C8—1812no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22#1915384612yes—
307C22C22×C22⋊D18111yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—7241no—
317C4—7241no—
318C4—7241no—
319C4—7241no—
320C4#41×D18111yescharacteristic
321C4#1915386311yescharacteristic
322C2—22no—
323C2—41no—
324C2#2604192811yescharacteristic
325C2C2×#1915384611yescharacteristic
326C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#552×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181D4
hypercenter#552D181
hypocenterC181#3344
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. #26041620C2semidirect
4. #26041612C2semidirect
5. #26041609C2semidirect
6. #26041607C2semidirect
7. #26041583C2semidirect
8. #26041564C2semidirect
9. #26041535C2semidirect
10. #26041518C2semidirect
11. #26041384C2semidirect
13. #26041322C2semidirect
14. C2×#19153747C2semidirect
163. C181#3344semidirect

Maximal subgroups

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

subgroupindex
#260419862
#260416892
#260416202
#260416122
#260416092
#260416072
#260415832
#260415642
#260415352
#260415182
#260413842
#260413512
#260413222
C2×#191537472
#552×C1812
#3344181