Finite Group Databasebeta

← #37544806

Subgroups — #37544806

Order 23168.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041891C211yesmaximal, characteristic, complemented
2#26041855C211yesmaximal, characteristic
3#26041673C211yesmaximal, characteristic, complemented
4#26041659C211yesmaximal, characteristic, complemented
5#26041648C211yesmaximal, characteristic, complemented
6#26041634C211yesmaximal, characteristic, complemented
7#26041502C212yesmaximal, complemented
8#26041490C212yesmaximal, complemented
9#26041443C212yesmaximal
10#26041437C212yesmaximal, complemented
11#433×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×Dic362C2211yescharacteristic
14C2×Dic724C2212yes—
15C2×Dic724C2212yes—
16C2×Dic724C2212yes—
17C2×#15668438C2212yes—
18#19153816C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153816C2211yescharacteristic
21#19153809—22no—
22#19153808C2211yescharacteristic
23#19153806—22no—
24#19153804C2211yescharacteristic
25#19153804C2211yescharacteristic
26#19153800—22no—
27#19153796—22no—
28#19153795C2212yes—
29C4×Dic362C2212yes—
30#19153741C2212yes—
31#19153741C2212yes—
32#19153741C2212yes—
33#19153738C2212yes—
34#117×C181C2211yescharacteristic
35C4⋊C8×C181C2212yes—
36C22⋊C8×C181C2212yes—
37#97×C181C2211yescharacteristic
38C4×C8×C181C2211yescharacteristic
39#15668449—42no—
40C2×Dic362—21no—
41C2×Dic362—42no—
42C2×Dic362C2311yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×Dic362C2311yescharacteristic
45C2×Dic362C2312yes—
46C2×D724C2311yescharacteristic
47C2×C22⋊D181—21no—
48C22×Dic181—21no—
49C2×C4×D181—21no—
50Dic724—42no—
51Dic724—42no—
52Dic724—42no—
53#15668438—42no—
54C4⋊Dic181—22no—
55C4⋊Dic181—22no—
56C4⋊Dic181—22no—
57C4⋊Dic181C2312yes—
58#15668428—22no—
59#15668428—22no—
60#15668428—22no—
61C22⋊Dic181—22no—
62#15668426—22no—
63#15668426—22no—
64C4×Dic181—22no—
65C22×C4×C181C2311yescharacteristic
66C2×C8×C181C2312yes—
67C2×C8×C181C2312yes—
68C4⋊C4×C181D412yes—
69C22⋊C4×C181D412yes—
70C42×C181C2311yescharacteristic
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73Dic362—22no—
74Dic362—22no—
75Dic362—22no—
76Dic362—22no—
77Dic362—22no—
78Dic362—42no—
79D724—22no—
80C22⋊D181—42no—
81C2×Dic181—22no—
82C2×Dic181—22no—
83C2×Dic181—21no—
84C2×Dic181—21no—
85C2×Dic181—21no—
86C2×Dic181—42no—
87C4×D181—42no—
88C23×C181C2×D411yescharacteristic
89C2×C4×C181—22no—
90C2×C4×C181C2×D412yes—
91C2×C4×C181C2×D412yes—
92C2×C4×C181C2×D411yescharacteristic
93C2×C4×C181C2411yescharacteristic
94C8×C181—22no—
95C8×C181—22no—
96C22×C181—21no—
97C22×C181—22no—
98C22×C181C22×D411yescharacteristic
99C2×D181—41no—
100C2×D181—42no—
101C4×C181—21no—
102C4×C181—22no—
103C4×C181—22no—
104C4×C181H(5,𝔽2)12yes—
105Dic181—41no—
106Dic181—41no—
107Dic181—41no—
108Dic181—42no—
109Dic181—42no—
110C2×C181—41no—
111C2×C181#53511yescharacteristic
112C2×C181#57812yes—
113D181—81no—
114C181#322311yescharacteristic, complemented, Sylow
115#3223—1811nomaximal, Sylow
116#541—1811no—
117#536—1811no—
118#497—1811no—
119#494—1811no—
120#491—1811no—
121#488—1811no—
122#464—1812no—
123#462—1812no—
124#452—1812no—
125#451—1812no—
126#433D18111yescharacteristic, complemented
127C2×#41—1811no—
128C22×Q8—1811no—
129C2×Q16—1812no—
130C2×Q16—1812no—
131C2×Q16—1812no—
132C2×SD16—1812no—
133C4⋊Q8—1811no—
134C4⋊Q8—1811no—
135C4⋊Q8—1811no—
136#126—3622no—
137#125—3622no—
138#125—1811no—
139#124—1811no—
140#124—1811no—
141#123—3622no—
142C22⋊Q8—3622no—
143C22⋊Q8—1812no—
144C4×Q8—1812no—
145#117C2×D18111yescharacteristic
146C4⋊C8C2×D18112yes—
147#103—1812no—
148#103—1812no—
149#103—1812no—
150#102—1812no—
151C22⋊C8C2×D18112yes—
152#97C2×D18111yescharacteristic
153C4×C8C2×D18111yescharacteristic
154#41—7242no—
155C2×Q8—3621no—
156C2×Q8—1811no—
157C2×Q8—1811no—
158C2×Q8—1811no—
159C2×Q8—1812no—
160C2×Q8—7242no—
161C2×D4—3621no—
162C2×D4—1811no—
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4C22×D18111yescharacteristic
166Q16—7242no—
167Q16—7242no—
168Q16—7242no—
169SD16—7242no—
170C2×C8C22×D18112yes—
171C2×C8C22×D18112yes—
172C4⋊C4—3622no—
173C4⋊C4—3622no—
174C4⋊C4—3622no—
175C4⋊C4—3622no—
176C4⋊C4—3622no—
177C4⋊C4—3622no—
178C4⋊C4—1812no—
179C4⋊C4D72412yes—
180C22⋊C4—3622no—
181C22⋊C4—3622no—
182C22⋊C4—3622no—
183C22⋊C4D72412yes—
184C42—3622no—
185C42C22×D18111yescharacteristic
186C42C22×D18111yescharacteristic
187C23—3621no—
188C23C2×D72411yescharacteristic
189Q8—3622no—
190Q8—3622no—
191Q8—3622no—
192Q8—3622no—
193Q8—3622no—
194Q8—7242no—
195D4—3622no—
196D4—7242no—
197C2×C4—22no—
198C2×C4—3622no—
199C2×C4—3622no—
200C2×C4—3621no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—7242no—
204C2×C4—7242no—
205C2×C4C2×D72412yes—
206C2×C4C2×D72412yes—
207C2×C4C2×D72411yescharacteristic
208C2×C4C23×D18111yescharacteristic
209C8—22no—
210C8—22no—
211C22—22no—
212C22—21no—
213C22—7242no—
214C22—7241no—
215C22C22×D72411yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—7242no—
220C4—7241no—
221C4—7241no—
222C4—7241no—
223C4—7242no—
224C4#1915386212yes—
225C2—41no—
226C2—14481no—
227C2#2604184811yescharacteristic
228C2#2604209612yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#433×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668449C22
hypercenter#433D181
hypocenterC181#3223
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #26041891C2semidirect
3. #26041673C2semidirect
4. #26041659C2semidirect
5. #26041648C2semidirect
6. #26041634C2semidirect
7. #26041502C2semidirect
8. #26041490C2semidirect
10. #26041437C2semidirect
11. #433×C181C2semidirect
114. C181#3223semidirect
126. #433D181semidirect

Maximal subgroups

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

subgroupindex
#260418912
#260418552
#260416732
#260416592
#260416482
#260416342
#260415022
#260414902
#260414432
#260414372
#433×C1812
#3223181