Finite Group Databasebeta

← #37532686

Subgroups — #37532686

Order 23168.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#26041876C211yesmaximal, characteristic
2#26041437C211yesmaximal, characteristic, complemented
3#26041420C211yesmaximal, characteristic
4#26041199C211yesmaximal, characteristic, complemented
5#26041017C211yesmaximal, characteristic, complemented
6#26041014C211yesmaximal, characteristic
7#324×C181C211yesmaximal, characteristic, complemented
8C22×Dic362C2211yescharacteristic
9C2×Dic724—21no—
10C2×#15668438—21no—
11#19153811—21no—
12#19153806—21no—
13#19153801—21no—
14#19153799—21no—
15#19153795C2211yescharacteristic
16#19153794—21no—
17#19153790C2211yescharacteristic
18C4×C22⋊D181—21no—
19C2×#15668428—21no—
20#19153741—21no—
21#19153741—21no—
22#19153723—21no—
23#19153723C2211yescharacteristic
24C2×C4⋊C4×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26C22⋊C8×C181C2211yescharacteristic
27C2×Dic362—21no—
28C2×Dic362—21no—
29C2×Dic362—42no—
30C2×Dic362D411yescharacteristic
31C2×D724D411yescharacteristic
32C2×C22⋊D181—21no—
33C22×Dic181—21no—
34C22×Dic181—21no—
35C2×C4×D181—21no—
36Dic724—42no—
37#15668438—42no—
38C4⋊Dic181D411yescharacteristic
39C4⋊Dic181D411yescharacteristic
40#15668428—21no—
41#15668428—21no—
42#15668428—21no—
43#15668428—21no—
44#15668428—42no—
45C22⋊Dic181—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—21no—
49#15668426—21no—
50C4×Dic181—21no—
51C22×C4×C181D411yescharacteristic
52C22×C4×C181D411yescharacteristic
53C22×C4×C181C2311yescharacteristic
54C2×C8×C181—21no—
55C2×C8×C181—21no—
56C4⋊C4×C181—21no—
57C4⋊C4×C181—21no—
58C22×D181—21no—
59Dic362—22no—
60Dic362—22no—
61Dic362—41no—
62Dic362—41no—
63Dic362—41no—
64D724—41no—
65C22⋊D181—41no—
66C22⋊D181—41no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—41no—
73C2×Dic181—41no—
74C2×Dic181—41no—
75C2×Dic181—41no—
76C4×D181—81no—
77C23×C181C2×D411yescharacteristic
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181—21no—
82C2×C4×C181—41no—
83C2×C4×C181—41no—
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C8×C181—41no—
87C8×C181—41no—
88C22×C181—21no—
89C22×C181—21no—
90C22×C181#12011yescharacteristic
91C22×C181#13511yescharacteristic
92C22×C181#13711yescharacteristic
93C2×D181—41no—
94C2×D181—41no—
95C2×D181—41no—
96C4×C181—21no—
97C4×C181—21no—
98C4×C181—41no—
99C4×C181—41no—
100Dic181—42no—
101Dic181—41no—
102Dic181—41no—
103Dic181—41no—
104Dic181—42no—
105C2×C181—21no—
106C2×C181—21no—
107C2×C181#45211yescharacteristic
108C2×C181#45611yescharacteristic
109C2×C181B(4,𝔽2)11yescharacteristic
110D181—81no—
111C181#160311yescharacteristic, complemented, Sylow
112#1603—1811nomaximal, Sylow
113#539—1811no—
114#451—1811no—
115#448—1811no—
116#395—1811no—
117#328—1811no—
118#327—1811no—
119#324D18111yescharacteristic, complemented
120C22×Q8—1811no—
121C2×Q16—3621no—
122C2×SD16—3621no—
123#126—3621no—
124#125—3621no—
125#123—3621no—
126#123—3621no—
127C22⋊Q8—3621no—
128C22⋊Q8—1811no—
129#121—1811no—
130C4×D4—3621no—
131C2×C4⋊C4—3621no—
132C2×C4⋊C4C2×D18111yescharacteristic
133#103—3621no—
134#103—3621no—
135C22⋊C8C2×D18111yescharacteristic
136C22⋊C8C2×D18111yescharacteristic
137#95—3621no—
138#95—1811no—
139C2×Q8—3621no—
140C2×Q8—3621no—
141C2×Q8—1811no—
142C2×Q8—7242no—
143C2×D4—3621no—
144C2×D4—1811no—
145C22×C4—3621no—
146C22×C4—3621no—
147C22×C4—3621no—
148C22×C4D72411yescharacteristic
149C22×C4D72411yescharacteristic
150C22×C4C22×D18111yescharacteristic
151Q16—7242no—
152SD16—7242no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3621no—
158C4⋊C4—3621no—
159C4⋊C4—3621no—
160C4⋊C4—3621no—
161C4⋊C4—1811no—
162C4⋊C4—1811no—
163C4⋊C4—7242no—
164C22⋊C4—3621no—
165C22⋊C4—3621no—
166C22⋊C4—3621no—
167C22⋊C4—3621no—
168C22⋊C4—3621no—
169C42—3621no—
170C23—3621no—
171C23C2×D72411yescharacteristic
172Q8—3622no—
173Q8—3622no—
174Q8—7241no—
175Q8—7241no—
176Q8—7241no—
177D4—7241no—
178D4—7241no—
179D4—7241no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3622no—
187C2×C4—3622no—
188C2×C4—3621no—
189C2×C4—3621no—
190C2×C4—3621no—
191C2×C4—14481no—
192C2×C4—7241no—
193C2×C4—7241no—
194C2×C4—7241no—
195C2×C4—7241no—
196C2×C4D4×D18111yescharacteristic
197C2×C4D4×D18111yescharacteristic
198C8—41no—
199C8—41no—
200C22—21no—
201C22—21no—
202C22—7241no—
203C22—7241no—
204C22—7241no—
205C22#1915378611yescharacteristic
206C22#1915383811yescharacteristic
207C22#1915384911yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—7242no—
213C4—7241no—
214C4—7241no—
215C4—7241no—
216C4—7242no—
217C2—21no—
218C2—21no—
219C2—14481no—
220C2#2604144311yescharacteristic
221C2#2604146411yescharacteristic
222C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#324×C181C2
socleC22×C181#120
prime radical#19153790C22
hypercenter#324D181
hypocenterC181#1603
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #26041437C2semidirect
4. #26041199C2semidirect
5. #26041017C2semidirect
7. #324×C181C2semidirect
111. C181#1603semidirect
119. #324D181semidirect

Maximal subgroups

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

subgroupindex
#260418762
#260414372
#260414202
#260411992
#260410172
#260410142
#324×C1812
#1603181