Finite Group Databasebeta

← #37544797

Subgroups — #37544797

Order 23168.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#26041890C211yesmaximal, characteristic, complemented
2#26041854C211yesmaximal, characteristic
3#26041671C211yesmaximal, characteristic, complemented
4#26041647C211yesmaximal, characteristic, complemented
5#26041634C211yesmaximal, characteristic, complemented
6#26041503C212yesmaximal, complemented
7#26041487C212yesmaximal, complemented
8#26041440C212yesmaximal
9#26041433C212yesmaximal, complemented
10#26041338C211yesmaximal, characteristic, complemented
11#494×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×Q8×D181C2211yescharacteristic
14C2×#15668439C2212yes—
15C2×#15668436C2212yes—
16#19153814C2211yescharacteristic
17#19153811—22no—
18#19153808C2211yescharacteristic
19#19153806—22no—
20#19153805C2211yescharacteristic
21#19153804C2211yescharacteristic
22#19153801—22no—
23#19153794—22no—
24#19153792C2212yes—
25Q8×Dic181C2212yes—
26#19153773C2211yescharacteristic
27#19153745C2212yes—
28#19153741C2212yes—
29#19153740C2212yes—
30#19153740C2212yes—
31#19153738C2212yes—
32#19153728C2212yes—
33#19153727C2211yescharacteristic
34C2×Q16×C181C2212yes—
35C2×Q16×C181C2212yes—
36C4⋊Q8×C181C2211yescharacteristic
37C4⋊Q8×C181C2211yescharacteristic
38C4×C8×C181C2211yescharacteristic
39#15668450—42no—
40C2×Dic362—21no—
41C2×Dic362C2311yescharacteristic
42Q8×D181—42no—
43C2×D724—21no—
44C2×D724C2311yescharacteristic
45C2×C4×D181—21no—
46C2×C4×D181—21no—
47C2×C4×D181C2311yescharacteristic
48#15668439—42no—
49#15668436—42no—
50C2×#13089678C2312yes—
51C4⋊Dic181—22no—
52C4⋊Dic181C2312yes—
53#15668428—22no—
54#15668428—22no—
55#15668428—22no—
56#15668428D412yes—
57#15668426—22no—
58#15668426—22no—
59#15668426—22no—
60#15668426D412yes—
61C4×Dic181—22no—
62C4×Dic181C2311yescharacteristic
63C2×Q8×C181C2311yescharacteristic
64C2×Q8×C181C2311yescharacteristic
65C2×Q8×C181C2312yes—
66Q16×C181—42no—
67Q16×C181—42no—
68C2×C8×C181C2312yes—
69C4⋊C4×C181—22no—
70C4⋊C4×C181—22no—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181C2×D411yescharacteristic
74Dic362—22no—
75Dic362—42no—
76D724—22no—
77D724—42no—
78C2×Dic181—22no—
79C2×Dic181—21no—
80C2×Dic181C2×D412yes—
81C2×Dic181C2×D411yescharacteristic
82C4×D181—22no—
83C4×D181—42no—
84C4×D181—42no—
85#13089678—22no—
86Q8×C181—22no—
87Q8×C181—22no—
88Q8×C181—22no—
89Q8×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—21no—
92C2×C4×C181—21no—
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2411yescharacteristic
95C8×C181—22no—
96C22×C181C22×D411yescharacteristic
97C2×D181—21no—
98C2×D181—22no—
99C2×D181—41no—
100C2×D181—42no—
101C4×C181—22no—
102C4×C181—42no—
103C4×C181—41no—
104C4×C181—41no—
105C4×C181H(5,𝔽2)12yes—
106Dic181—21no—
107Dic181—22no—
108Dic181—41no—
109Dic181—42no—
110C2×C181#53511yescharacteristic
111C2×C181#57812yes—
112D181—41no—
113D181—81no—
114C181#322311yescharacteristic, complemented, Sylow
115#3223—1811nomaximal, Sylow
116#541—1811no—
117#536—1811no—
118#497—1811no—
119#494D18111yescharacteristic, complemented
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×Q16C2×D18112yes—
131C2×Q16C2×D18112yes—
132C2×SD16—1812no—
133C4⋊Q8—1811no—
134C4⋊Q8C2×D18111yescharacteristic
135C4⋊Q8C2×D18111yescharacteristic
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#117—1811no—
146C4⋊C8—1812no—
147#103—1812no—
148#103—1812no—
149#103—1812no—
150#102—1812no—
151C22⋊C8—1812no—
152#97—1811no—
153C4×C8C2×D18111yescharacteristic
154#41—7242no—
155C2×Q8—3621no—
156C2×Q8—1811no—
157C2×Q8—7242no—
158C2×Q8C22×D18111yescharacteristic
159C2×Q8C22×D18111yescharacteristic
160C2×Q8C22×D18112yes—
161C2×D4—3621no—
162C2×D4—1811no—
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4—1811no—
166Q16—42no—
167Q16—42no—
168Q16—7242no—
169SD16—7242no—
170C2×C8—1812no—
171C2×C8C22×D18112yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—3622no—
175C4⋊C4—3622no—
176C4⋊C4—3622no—
177C4⋊C4—3622no—
178C4⋊C4—1812no—
179C4⋊C4—1812no—
180C22⋊C4—3622no—
181C22⋊C4—3622no—
182C22⋊C4—3622no—
183C22⋊C4—1812no—
184C42—3622no—
185C42—1811no—
186C42C22×D18111yescharacteristic
187C23—3621no—
188C23—1811no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—22no—
193Q8—3622no—
194Q8—7242no—
195D4—3622no—
196D4—7242no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—3622no—
201C2×C4—3622no—
202C2×C4—3621no—
203C2×C4—1812no—
204C2×C4—1811no—
205C2×C4—7242no—
206C2×C4—7242no—
207C2×C4D4×D18112yes—
208C2×C4C23×D18111yescharacteristic
209C8—22no—
210C8—3622no—
211C22—3622no—
212C22—3621no—
213C22—7242no—
214C22—7241no—
215C22C2×D4×D18111yescharacteristic
216C4—22no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—3622no—
221C4—3621no—
222C4—7242no—
223C4—7241no—
224C4#1915386112yes—
225C2—14481no—
226C2—7241no—
227C2#2604184211yescharacteristic
228C2#2604209212yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#494×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668450C22
hypercenter#494D181
hypocenterC181#3223
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #26041890C2semidirect
3. #26041671C2semidirect
4. #26041647C2semidirect
5. #26041634C2semidirect
6. #26041503C2semidirect
7. #26041487C2semidirect
9. #26041433C2semidirect
10. #26041338C2semidirect
11. #494×C181C2semidirect
114. C181#3223semidirect
119. #494D181semidirect

Maximal subgroups

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

subgroupindex
#260418902
#260418542
#260416712
#260416472
#260416342
#260415032
#260414872
#260414402
#260414332
#260413382
#494×C1812
#3223181