Finite Group Databasebeta

← #37546626

Subgroups — #37546626

Order 23168.

340 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042009C211yesmaximal, characteristic, complemented
2#26041988C211yesmaximal, characteristic
3#26041687C211yesmaximal, characteristic, complemented
4#26041685C211yesmaximal, characteristic, complemented
5#26041683C211yesmaximal, characteristic, complemented
6#26041587C211yesmaximal, characteristic, complemented
7#26041585C211yesmaximal, characteristic, complemented
8#26041573C211yesmaximal, characteristic, complemented
9#26041567C211yesmaximal, characteristic
10#26041557C211yesmaximal, characteristic
11#26041408C211yesmaximal, characteristic, complemented
12#26041379C211yesmaximal, characteristic
13#26041376C211yesmaximal, characteristic
14#26041370C211yesmaximal, characteristic
15#474×C181C211yesmaximal, characteristic
16C2×#15668437C2211yescharacteristic
17#19153815—21no—
18#19153815C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153814—21no—
22#19153814—21no—
23#19153808C2211yescharacteristic
24#19153807C2211yescharacteristic
25#19153795—21no—
26#19153793—21no—
27#19153793—21no—
28C4×Dic362C2211yescharacteristic
29C4×Dic362C2211yescharacteristic
30#19153781—21no—
31#19153781C2211yescharacteristic
32#19153781C2211yescharacteristic
33D4×Dic181C2211yescharacteristic
34D4×Dic181C2211yescharacteristic
35C2×#15668428—21no—
36#19153750C2211yescharacteristic
37#19153749C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153748C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153747C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153727C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53C4×D4×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362D411yescharacteristic
63C2×Dic362C2311yescharacteristic
64C22×Dic181—21no—
65C22×Dic181—21no—
66#15668437Q814yescomplemented
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C4⋊Dic181—21no—
70C4⋊Dic181D411yescharacteristic
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84#15668428D411yescharacteristic
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C4×Dic181—21no—
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181C2311yescharacteristic
91C4×Dic181C2311yescharacteristic
92C2×Q8×C181—21no—
93C2×D4×C181C2311yescharacteristic
94C22×C4×C181—21no—
95C2×C8×C181C2311yescharacteristic
96C2×C8×C181C2311yescharacteristic
97C4⋊C4×C181—21no—
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C42×C181C2311yescharacteristic
103Dic362—21no—
104Dic362—22no—
105Dic362—81no—
106Dic362—81no—
107Dic362—81no—
108Dic362C2×Q812yes—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—42no—
114C2×Dic181—81no—
115C2×Dic181C2×D411yescharacteristic
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119#13089678—21no—
120#13089678C2×Q812yes—
121C23×C181—21no—
122Q8×C181—81no—
123D4×C181—21no—
124D4×C181C2×Q812yes—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—81no—
129C2×C4×C181C2×D411yescharacteristic
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2411yescharacteristic
132C8×C181—21no—
133C8×C181—21no—
134C22×C181—22no—
135C22×C181—41no—
136C22×C181—41no—
137C22×C181C22×D411yescharacteristic
138C4×C181—21no—
139C4×C181—21no—
140C4×C181—41no—
141C4×C181—41no—
142C4×C181—41no—
143C4×C181C22×Q811yescharacteristic
144C4×C1812−1+411yescharacteristic
145Dic181—22no—
146Dic181—21no—
147Dic181—21no—
148Dic181—21no—
149Dic181—41no—
150Dic181—41no—
151Dic181—41no—
152Dic181#13712yes—
153C2×C181—42no—
154C2×C181D4×Q811yescharacteristic
155C2×C181C2×#13711yescharacteristic
156C2×C181#57611yescharacteristic
157C181#338511yescharacteristic, complemented, Sylow
158#3385—1811nomaximal, Sylow
159Q82—1811no—
160#554—1811no—
161#501—1811no—
162#501—1811no—
163#500—1811no—
164#479—1811no—
165#479—1811no—
166#477—1811no—
167#476—1811no—
168#474—1811no—
169#474D18111yescharacteristic
170#446—1811no—
171#440—1811no—
172#440—1811no—
173C4×SD16—1811no—
174C2×SD16—1811no—
175C4⋊Q8—3621no—
176C4⋊Q8—3621no—
177C4⋊Q8—3621no—
178C4⋊Q8—1811no—
179C4⋊Q8—1811no—
180C4⋊Q8—1811no—
181C4⋊Q8C2×D18111yescharacteristic
182#125—1811no—
183#125—1811no—
184C22⋊Q8—3621no—
185C22⋊Q8—3621no—
186C22⋊Q8—3621no—
187C4×Q8—3621no—
188C4×Q8—1811no—
189C4×Q8—1811no—
190C4×Q8—1811no—
191C4×Q8—1811no—
192C4×D4—1811no—
193C4×D4—1811no—
194C4×D4C2×D18111yescharacteristic
195C2×C4⋊C4—3621no—
196#107—1811no—
197#107—1811no—
198#107—1811no—
199#107—1811no—
200#107C2×D18111yescharacteristic
201#107C2×D18111yescharacteristic
202#106—1811no—
203#106—1811no—
204#106—1811no—
205C4⋊C8—1811no—
206C4⋊C8—1811no—
207C4⋊C8C2×D18111yescharacteristic
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#102—1811no—
212#102C2×D18111yescharacteristic
213#102C2×D18111yescharacteristic
214#97—1811no—
215#97—1811no—
216C4×C8—1811no—
217C2×Q8—21no—
218C2×Q8—3621no—
219C2×Q8—3621no—
220C2×Q8—3621no—
221C2×Q8—1811no—
222C2×Q8—1811no—
223C2×D4C22×D18111yescharacteristic
224C22×C4—21no—
225C22×C4—3621no—
226C22×C4—3621no—
227SD16—1814no—
228C2×C8—1811no—
229C2×C8—1811no—
230C2×C8C22×D18111yescharacteristic
231C2×C8C22×D18111yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—3621no—
234C4⋊C4—3621no—
235C4⋊C4—3621no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—1811no—
244C4⋊C4—1811no—
245C4⋊C4—1811no—
246C4⋊C4—1811no—
247C4⋊C4—1811no—
248C4⋊C4—1811no—
249C4⋊C4C22×D18111yescharacteristic
250C4⋊C4C22×D18111yescharacteristic
251C4⋊C4C22×D18111yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—3621no—
254C22⋊C4—3621no—
255C42—3621no—
256C42—3621no—
257C42—3621no—
258C42—1811no—
259C42—1811no—
260C42C22×D18111yescharacteristic
261C23—21no—
262Q8—81no—
263Q8—3622no—
264Q8—3621no—
265Q8—14481no—
266Q8—14481no—
267Q8—14481no—
268Q8—1812no—
269D4—21no—
270D4Q8×D18112yes—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—81no—
275C2×C4—3621no—
276C2×C4—3621no—
277C2×C4—3621no—
278C2×C4—3621no—
279C2×C4—14481no—
280C2×C4—1811no—
281C2×C4—1811no—
282C2×C4—1811no—
283C2×C4—1811no—
284C2×C4—7242no—
285C2×C4D4×D18111yescharacteristic
286C2×C4D4×D18111yescharacteristic
287C2×C4C23×D18111yescharacteristic
288C8—21no—
289C8—21no—
290C8—3621no—
291C8—1812no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22C2×D4×D18111yescharacteristic
296C4—21no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—3622no—
302C4—3621no—
303C4—3621no—
304C4—3621no—
305C4—1812no—
306C4—7241no—
307C4—7241no—
308C4—7241no—
309C4C2×Q8×D18111yescharacteristic
310C4#1915386311yescharacteristic
311C2—42no—
312C2#2604194211yescharacteristic
313C2#137×D18111yescharacteristic
314C2#2604208011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#474×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181C23
hypercenter#474D181
hypocenterC181#3385
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #26042009C2semidirect
3. #26041687C2semidirect
4. #26041685C2semidirect
5. #26041683C2semidirect
6. #26041587C2semidirect
7. #26041585C2semidirect
8. #26041573C2semidirect
11. #26041408C2semidirect
66. #15668437Q8semidirect
157. C181#3385semidirect

Maximal subgroups

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

subgroupindex
#260420092
#260419882
#260416872
#260416852
#260416832
#260415872
#260415852
#260415732
#260415672
#260415572
#260414082
#260413792
#260413762
#260413702
#474×C1812
#3385181