Finite Group Databasebeta

← #37532259

Subgroups — #37532259

Order 23168.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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#26041336C211yesmaximal, characteristic, complemented
2#26041145C211yesmaximal, characteristic, complemented
3#26041016C212yesmaximal, complemented
4#26041013C212yesmaximal
5#529×C181C211yesmaximal, characteristic
6C2×C4⋊Dic181—21no—
7C4⋊#13089678—21no—
8C22⋊#13089678C2212yes—
9#19153727—21no—
10#19153723—21no—
11#19153723C2212yes—
12#126×C181—21no—
13#125×C181C411yescharacteristic, complemented
14#124×C181C411yescharacteristic, complemented
15#123×C181—21no—
16C22⋊Q8×C181C2211yescharacteristic
17#121×C181C2211yescharacteristic
18C4×Q8×C181C411yescharacteristic, complemented
19C4×D4×C181—21no—
20C4×D4×C181C411yescharacteristic, complemented
21#117×C181—21no—
22C2×C42×C181C2211yescharacteristic
23C22×Dic181—22no—
24C2×#13089678—22no—
25C4⋊Dic181—42no—
26C2×Q8×C181C2×C411yescharacteristic
27C2×D4×C181—21no—
28C2×D4×C181C2×C411yescharacteristic
29C22×C4×C181—21no—
30C22×C4×C181D411yescharacteristic
31C22×C4×C181D411yescharacteristic
32C22×C4×C181C2311yescharacteristic
33C4⋊C4×C181—21no—
34C4⋊C4×C181—21no—
35C4⋊C4×C181—21no—
36C4⋊C4×C181—21no—
37C4⋊C4×C181C2×C411yescharacteristic
38C4⋊C4×C181C2×C411yescharacteristic
39C22⋊C4×C181—21no—
40C22⋊C4×C181—21no—
41C22⋊C4×C181—21no—
42C22⋊C4×C181—21no—
43C22⋊C4×C181—21no—
44C42×C181—21no—
45C42×C181C2×C411yescharacteristic
46C42×C181C2×C411yescharacteristic
47C42×C181D411yescharacteristic
48C42×C181D411yescharacteristic
49C2×Dic181—42no—
50C2×Dic181—42no—
51C2×Dic181—42no—
52#13089678—42no—
53C23×C181—21no—
54C23×C181C2×D411yescharacteristic
55Q8×C181—41no—
56D4×C181—41no—
57D4×C181—41no—
58D4×C181—41no—
59C2×C4×C181—21no—
60C2×C4×C181—21no—
61C2×C4×C181—21no—
62C2×C4×C181—21no—
63C2×C4×C181—21no—
64C2×C4×C181—41no—
65C2×C4×C181—41no—
66C2×C4×C181—42no—
67C2×C4×C181C22⋊C411yescharacteristic
68C2×C4×C181C22⋊C411yescharacteristic
69C2×C4×C181C22⋊C411yescharacteristic
70C2×C4×C181C22⋊C411yescharacteristic
71C2×C4×C181C22×C411yescharacteristic
72C2×C4×C181C2×D411yescharacteristic
73C22×C181—21no—
74C22×C181—21no—
75C22×C181—41no—
76C22×C181—41no—
77C22×C181—41no—
78C22×C181C2×C22⋊C411yescharacteristic
79C22×C181AGL(1,ℤ/8)11yescharacteristic
80C22×C181#13711yescharacteristic
81C4×C181—21no—
82C4×C181—21no—
83C4×C181—21no—
84C4×C181—21no—
85C4×C181—21no—
86C4×C181—21no—
87C4×C181—41no—
88C4×C181—41no—
89C4×C181—41no—
90Dic181—82no—
91C2×C181—21no—
92C2×C181—21no—
93C2×C181—81no—
94C2×C181#41011yescharacteristic
95C2×C181#41711yescharacteristic
96C2×C181#42111yescharacteristic
97C181#151211yescharacteristic, complemented, Sylow
98#1512—1811nomaximal, Sylow
99#529D18111yescharacteristic
100#432—1811no—
101#382—1811no—
102#328—1812no—
103#327—1812no—
104#126—21no—
105#125Dic18111yescharacteristic, complemented
106#124Dic18111yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D18111yescharacteristic
109#121C2×D18111yescharacteristic
110C4×Q8Dic18111yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic18111yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—3621no—
115C2×C42C2×D18111yescharacteristic
116C4⋊C8—3621no—
117C22⋊C8—1812no—
118#97—3621no—
119#95—3621no—
120#95—1812no—
121C2×Q8C2×Dic18111yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic18111yescharacteristic
124C22×C4—21no—
125C22×C4—3622no—
126C22×C4C22⋊D18111yescharacteristic
127C22×C4C22⋊D18111yescharacteristic
128C22×C4C22×D18111yescharacteristic
129C2×C8—3622no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—7242no—
135C4⋊C4C2×Dic18111yescharacteristic
136C4⋊C4C2×Dic18111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic18111yescharacteristic
144C42C2×Dic18111yescharacteristic
145C42C22⋊D18111yescharacteristic
146C42C22⋊D18111yescharacteristic
147C23—21no—
148C23C2×C22⋊D18111yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—7242no—
162C2×C4—7242no—
163C2×C4—7242no—
164C2×C4C22⋊Dic18111yescharacteristic
165C2×C4C22⋊Dic18111yescharacteristic
166C2×C4C22⋊Dic18111yescharacteristic
167C2×C4C22⋊Dic18111yescharacteristic
168C2×C4C22×Dic18111yescharacteristic
169C2×C4C2×C22⋊D18111yescharacteristic
170C8—7242no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic18111yescharacteristic
177C22#1915384111yescharacteristic
178C22#1915384611yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—14482no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#2604125311yescharacteristic
193C2#2604128011yescharacteristic
194C2#2604129611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic181
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#529×C181C2
socleC22×C181C2×C22⋊C4
prime radical#121×C181C22
cosocleC22×C4×C181C23
hypercenter#529D181
hypocenterC181#1512
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #26041336C2semidirect
2. #26041145C2semidirect
3. #26041016C2semidirect
13. #125×C181C4semidirect
14. #124×C181C4semidirect
18. C4×Q8×C181C4semidirect
20. C4×D4×C181C4semidirect
97. C181#1512semidirect
105. #125Dic181semidirect
106. #124Dic181semidirect
110. C4×Q8Dic181semidirect
112. C4×D4Dic181semidirect

Maximal subgroups

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

subgroupindex
#260413362
#260411452
#260410162
#260410132
#529×C1812
#1512181