Finite Group Databasebeta

← #37544798

Subgroups — #37544798

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#26041855C211yesmaximal, characteristic
2#26041671C211yesmaximal, characteristic, complemented
3#26041658C211yesmaximal, characteristic, complemented
4#26041649C211yesmaximal, characteristic, complemented
5#26041636C211yesmaximal, characteristic, complemented
6#26041501C212yesmaximal, complemented
7#26041489C212yesmaximal, complemented
8#26041439C212yesmaximal
9#26041434C212yesmaximal, complemented
10#26041340C211yesmaximal, characteristic, complemented
11#541×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×Dic362C2211yescharacteristic
14C2×#15668439C2212yes—
15C2×#15668439C2212yes—
16C2×#15668439C2212yes—
17C2×#15668436C2212yes—
18#19153816C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153804C2211yescharacteristic
21#19153804C2211yescharacteristic
22#19153800—22no—
23#19153796—22no—
24C4⋊#13089678C2212yes—
25#19153739C2212yes—
26#19153739C2212yes—
27#19153739C2212yes—
28#19153736C2212yes—
29C22⋊#13089678C2212yes—
30#19153727C2211yescharacteristic
31C4×#13089678C2211yescharacteristic
32C4⋊Q8×C181C2211yescharacteristic
33#126×C181—22no—
34#125×C181—22no—
35#125×C181C2211yescharacteristic
36C22⋊Q8×C181C2212yes—
37C4×Q8×C181C2212yes—
38#117×C181C2211yescharacteristic
39#15668449—42no—
40C2×Dic362—21no—
41C2×Dic362—42no—
42C2×Dic362C2311yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×Dic362C2311yescharacteristic
45C2×D724C2311yescharacteristic
46C2×C22⋊D181—21no—
47C22×Dic181—21no—
48C2×C4×D181—21no—
49#15668439—42no—
50#15668439—42no—
51#15668439—42no—
52#15668436—42no—
53C2×#13089678C2312yes—
54C2×#13089678C2312yes—
55C4⋊Dic181—22no—
56C4⋊Dic181—22no—
57#15668426—22no—
58#15668426—22no—
59C2×Q8×C181C2312yes—
60C22×C4×C181C2311yescharacteristic
61C4⋊C4×C181—22no—
62C4⋊C4×C181—22no—
63C4⋊C4×C181—22no—
64C4⋊C4×C181—22no—
65C4⋊C4×C181D412yes—
66C4⋊C4×C181C2312yes—
67C22⋊C4×C181—22no—
68C22⋊C4×C181D412yes—
69C42×C181—22no—
70C42×C181C2311yescharacteristic
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73Dic362—22no—
74Dic362—22no—
75Dic362—22no—
76Dic362—42no—
77D724—22no—
78C22⋊D181—42no—
79C2×Dic181—21no—
80C2×Dic181—21no—
81C2×Dic181—21no—
82C2×Dic181—42no—
83C4×D181—42no—
84#13089678—22no—
85#13089678—22no—
86C23×C181C2×D411yescharacteristic
87Q8×C181—22no—
88Q8×C181—22no—
89C2×C4×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—22no—
92C2×C4×C181C2×D412yes—
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2×D411yescharacteristic
95C2×C4×C181C2411yescharacteristic
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×C181—42no—
105C4×C181—42no—
106C4×C181H(5,𝔽2)12yes—
107Dic181—41no—
108Dic181—41no—
109Dic181—41no—
110C2×C181—41no—
111C2×C181#53511yescharacteristic
112C2×C181#57812yes—
113D181—81no—
114C181#322311yescharacteristic, complemented, Sylow
115#3223—1811nomaximal, Sylow
116#541D18111yescharacteristic, complemented
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#433—1811no—
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⋊Q8C2×D18111yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D18111yescharacteristic
139#124—1811no—
140#124—1811no—
141#123—3622no—
142C22⋊Q8—3622no—
143C22⋊Q8C2×D18112yes—
144C4×Q8C2×D18112yes—
145#117C2×D18111yescharacteristic
146C4⋊C8—1812no—
147#103—1812no—
148#103—1812no—
149#103—1812no—
150#102—1812no—
151C22⋊C8—1812no—
152#97—1811no—
153C4×C8—1811no—
154#41—7242no—
155C2×Q8—3621no—
156C2×Q8—1811no—
157C2×Q8—1811no—
158C2×Q8—1811no—
159C2×Q8—7242no—
160C2×Q8C22×D18112yes—
161C2×D4—3621no—
162C2×D4—1811no—
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4C22×D18111yescharacteristic
166Q16—7242no—
167Q16—7242no—
168Q16—7242no—
169SD16—7242no—
170C2×C8—1812no—
171C2×C8—1812no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—3622no—
177C4⋊C4—3622no—
178C4⋊C4C22⋊D18112yes—
179C4⋊C4C22×D18112yes—
180C22⋊C4—22no—
181C22⋊C4—3622no—
182C22⋊C4—3622no—
183C22⋊C4C22⋊D18112yes—
184C42—22no—
185C42C22×D18111yescharacteristic
186C42C22×D18111yescharacteristic
187C23—3621no—
188C23C2×C22⋊D18111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—3622no—
192Q8—3622no—
193Q8—3622no—
194Q8—7242no—
195D4—3622no—
196D4—7242no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—3621no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—7242no—
204C2×C4—7242no—
205C2×C4C2×C22⋊D18112yes—
206C2×C4C2×C22⋊D18112yes—
207C2×C4C2×C22⋊D18111yescharacteristic
208C2×C4C23×D18111yescharacteristic
209C8—3622no—
210C8—3622no—
211C22—22no—
212C22—21no—
213C22—7242no—
214C22—7241no—
215C22C22×C22⋊D18111yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—7241no—
222C4—7241no—
223C4—7241no—
224C4#1915386212yes—
225C2—41no—
226C2—14481no—
227C2#2604184311yescharacteristic
228C2#2604209412yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#541×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668449C22
hypercenter#541D181
hypocenterC181#3223
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #26041671C2semidirect
3. #26041658C2semidirect
4. #26041649C2semidirect
5. #26041636C2semidirect
6. #26041501C2semidirect
7. #26041489C2semidirect
9. #26041434C2semidirect
10. #26041340C2semidirect
11. #541×C181C2semidirect
114. C181#3223semidirect
116. #541D181semidirect

Maximal subgroups

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

subgroupindex
#260418552
#260416712
#260416582
#260416492
#260416362
#260415012
#260414892
#260414392
#260414342
#260413402
#541×C1812
#3223181