Finite Group Databasebeta

← #37535706

Subgroups — #37535706

Order 23168.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041911C211yesmaximal, characteristic, complemented
2#26041467C211yesmaximal, characteristic, complemented
3#26041465C211yesmaximal, characteristic, complemented
4#26041449C211yesmaximal, characteristic, complemented
5#26041071C211yesmaximal, characteristic, complemented
6#26041067C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C181C211yesmaximal, characteristic, complemented
8#19153861C2211yescharacteristic, complemented
9C22×C22⋊D181—21no—
10C2×D4×D181—21no—
11#19153844—21no—
12#19153812C2211yescharacteristic
13#19153789—21no—
14#19153788—21no—
15#19153786—21no—
16#19153786—21no—
17#19153785—21no—
18#19153784C2211yescharacteristic
19#19153776—21no—
20#19153742—21no—
21#19153731C2211yescharacteristic, complemented
22#19153730—21no—
23#19153730—21no—
24#120×C181C2211yescharacteristic
25#100×C181C2211yescharacteristic, complemented
26#99×C181C2211yescharacteristic, complemented
27C23×D181—21no—
28C23×D181D411yescharacteristic, complemented
29#15668449—21no—
30#15668448—41no—
31C2×D724—21no—
32C2×D724—41no—
33C2×D724D411yescharacteristic, complemented
34C2×C22⋊D181—21no—
35C2×C22⋊D181—21no—
36C2×C22⋊D181—21no—
37C2×C22⋊D181—21no—
38C2×C22⋊D181—21no—
39C2×C22⋊D181—21no—
40C2×C22⋊D181—21no—
41C2×C22⋊D181—44no—
42C2×C22⋊D181—41no—
43D4×D181—44no—
44D4×D181—41no—
45C22×Dic181—21no—
46C2×C4×D181—21no—
47#15668438—41no—
48D1448—41no—
49#15668428—21no—
50C22⋊Dic181—21no—
51C22⋊Dic181—41no—
52C22⋊Dic181D411yescharacteristic, complemented
53#15668426—21no—
54#15668426—21no—
55#15668426—41no—
56C4×Dic181D411yescharacteristic, complemented
57C24×C181D411yescharacteristic, complemented
58C2×D4×C181—21no—
59C2×D4×C181C2311yescharacteristic
60M16×C181—21no—
61C22⋊C4×C181—21no—
62C22⋊C4×C181—21no—
63C22⋊C4×C181D411yescharacteristic, complemented
64C22×D181—21no—
65C22×D181—21no—
66C22×D181—21no—
67C22×D181—21no—
68C22×D181—42no—
69C22×D181—42no—
70C22×D181—41no—
71C22×D181—42no—
72C22×D181—81no—
73Dic362—21no—
74D724—21no—
75D724—41no—
76D724—42no—
77D724—82no—
78C22⋊D181—24no—
79C22⋊D181—42no—
80C22⋊D181—42no—
81C22⋊D181—42no—
82C22⋊D181—41no—
83C22⋊D181—41no—
84C22⋊D181—42no—
85C22⋊D181—41no—
86C22⋊D181—82no—
87C2×Dic181—21no—
88C2×Dic181—21no—
89C2×Dic181—21no—
90C2×Dic181—21no—
91C2×Dic181—42no—
92C2×Dic181—41no—
93C4×D181—42no—
94C4×D181—41no—
95C23×C181—21no—
96C23×C181—41no—
97C23×C181—42no—
98C23×C181C2×D411yescharacteristic
99C23×C181C2×D411yescharacteristic
100D4×C181—41no—
101D4×C181—42no—
102C2×C4×C181—21no—
103C2×C4×C181—41no—
104C2×C4×C181C2×D411yescharacteristic
105C8×C181—41no—
106C22×C181—22no—
107C22×C181—21no—
108C22×C181—21no—
109C22×C181—41no—
110C22×C181—41no—
111C22×C181—41no—
112C22×C181—42no—
113C22×C181—41no—
114C22×C181—81no—
115C22×C181#12011yescharacteristic
116C2×D181—22no—
117C2×D181—41no—
118C2×D181—41no—
119C2×D181—42no—
120C2×D181—41no—
121C2×D181—41no—
122C2×D181—41no—
123C2×D181—41no—
124C2×D181—81no—
125C2×D181—84no—
126C2×D181—81no—
127C2×D181—81no—
128C4×C181—21no—
129C4×C181—41no—
130C4×C181—81no—
131Dic181—22no—
132Dic181—41no—
133Dic181—41no—
134Dic181—81no—
135C2×C181—21no—
136C2×C181—41no—
137C2×C181—41no—
138C2×C181—42no—
139C2×C181B(4,𝔽2)11yescharacteristic
140D181—42no—
141D181—81no—
142D181—81no—
143C181#219611yescharacteristic, complemented, Sylow
144#2196—1811nomaximal, Sylow
145D42—1811no—
146B(4,𝔽2)—1811no—
147B(4,𝔽2)—1811no—
148#453—1811no—
149#353—1811no—
150AΣL(1,𝔽16)—1811no—
151AΣL(1,𝔽16)D18111yescharacteristic, complemented
152H(5,𝔽2)—1811no—
153C22×D4—3621no—
154C22×D4—3621no—
155AGL(1,ℤ/8)—3621no—
156#127—1811no—
157#121—3621no—
158#121—3621no—
159#120—3621no—
160#120—3621no—
161#120—3621no—
162#120—1811no—
163#120C2×D18111yescharacteristic
164C4×D4—3621no—
165#104—3621no—
166#100C2×D18111yescharacteristic, complemented
167#99—3621no—
168#99—3621no—
169#99—1811no—
170#99C2×D18111yescharacteristic, complemented
171C24—3621no—
172C24—1811no—
173C24D72411yescharacteristic, complemented
174#41—3621no—
175#41—7241no—
176C2×D4—21no—
177C2×D4—3621no—
178C2×D4—3621no—
179C2×D4—3621no—
180C2×D4—3621no—
181C2×D4—3621no—
182C2×D4—3621no—
183C2×D4—3621no—
184C2×D4—3621no—
185C2×D4—1811no—
186C2×D4—7241no—
187C2×D4—7241no—
188C2×D4—7244no—
189C2×D4—7244no—
190C2×D4—7241no—
191C2×D4C22×D18111yescharacteristic
192C22×C4—3621no—
193C22×C4—3621no—
194SD16—7241no—
195D8—7241no—
196M16—21no—
197C4⋊C4—3621no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C22⋊C4—3621no—
203C22⋊C4—1811no—
204C22⋊C4—7241no—
205C22⋊C4—7241no—
206C22⋊C4D72411yescharacteristic, complemented
207C42—1811no—
208C23—21no—
209C23—41no—
210C23—42no—
211C23—3621no—
212C23—3621no—
213C23—3621no—
214C23—3621no—
215C23—14481no—
216C23—7242no—
217C23—7241no—
218C23—7242no—
219C23—7242no—
220C23D4×D18111yescharacteristic
221C23C2×D72411yescharacteristic
222Q8—3621no—
223D4—41no—
224D4—42no—
225D4—3621no—
226D4—3624no—
227D4—14482no—
228D4—14482no—
229D4—7241no—
230D4—7242no—
231D4—7242no—
232D4—7242no—
233D4—7242no—
234D4—7241no—
235D4—7241no—
236D4—7242no—
237D4—7241no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—3621no—
241C2×C4—3621no—
242C2×C4—3621no—
243C2×C4—3621no—
244C2×C4—7241no—
245C2×C4—7242no—
246C2×C4—7242no—
247C2×C4—7241no—
248C2×C4D4×D18111yescharacteristic
249C8—41no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—41no—
258C22—81no—
259C22—3622no—
260C22—14481no—
261C22—14481no—
262C22—14484no—
263C22—14481no—
264C22—7241no—
265C22—7241no—
266C22—7241no—
267C22—7242no—
268C22—7241no—
269C22—7241no—
270C22—7241no—
271C22#1915378611yescharacteristic
272C4—21no—
273C4—41no—
274C4—81no—
275C4—3622no—
276C4—14481no—
277C4—7241no—
278C4—7241no—
279C2—21no—
280C2—41no—
281C2—41no—
282C2—42no—
283C2—14481no—
284C2—14481no—
285C2—7242no—
286C2#2604146711yescharacteristic
subgroupG/H
centerC2#26041467
commutator, cosocleC2×D4×C181C23
FrattiniC2×D4C22×D181
FittingAΣL(1,𝔽16)×C181C2
socleC2×C181B(4,𝔽2)
hypercenterAΣL(1,𝔽16)D181
hypocenterC181#2196
LCSG ▷ C2×D4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #26041911C2semidirect
2. #26041467C2semidirect
3. #26041465C2semidirect
4. #26041449C2semidirect
5. #26041071C2semidirect
6. #26041067C2semidirect
7. AΣL(1,𝔽16)×C181C2semidirect
8. #19153861C22semidirect
21. #19153731C22semidirect
25. #100×C181C22semidirect
26. #99×C181C22semidirect
28. C23×D181D4semidirect
33. C2×D724D4semidirect
52. C22⋊Dic181D4semidirect
56. C4×Dic181D4semidirect
57. C24×C181D4semidirect
63. C22⋊C4×C181D4semidirect
143. C181#2196semidirect
151. AΣL(1,𝔽16)D181semidirect
166. #100C2×D181semidirect
170. #99C2×D181semidirect
173. C24D724semidirect
206. C22⋊C4D724semidirect

Maximal subgroups

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

subgroupindex
#260419112
#260414672
#260414652
#260414492
#260410712
#260410672
AΣL(1,𝔽16)×C1812
#2196181