Finite Group Databasebeta

← #37549103

Subgroups — #37549103

Order 23168.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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#26042297C211yesmaximal, characteristic
2C4×C4⋊#10624382C211yesmaximal, characteristic
3#26042133C212yesmaximal
4#26042132C212yesmaximal
5#26041833C211yesmaximal, characteristic
6C2×C4⋊#10624382—21no—
7C2×C4×#10624382—21no—
8#19153883—21no—
9#19153873C2212yes—
10C4×#13089685—21no—
11(C2×#10624382)⋊C4C2212yes—
12#19153815C411yescharacteristic, complemented
13#19153814C411yescharacteristic, complemented
14#19153806—21no—
15#19153796C2211yescharacteristic
16#19153794—21no—
17#19153792C2211yescharacteristic
18C4×Dic362—21no—
19C4×Dic362C411yescharacteristic, complemented
20#19153773—21no—
21C42×D181C2211yescharacteristic
22C4×Q8×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24C2×#13089685—22no—
25C4⋊#10624382—24no—
26C4×#10624382—42no—
27C2×Dic362—21no—
28C2×Dic362C2×C411yescharacteristic
29C2×C4×D181D411yescharacteristic
30C2×C4×D181D411yescharacteristic
31C2×C4×D181C2311yescharacteristic
32C4⋊Dic181—21no—
33C4⋊Dic181C2×C411yescharacteristic
34#15668428—21no—
35#15668428—21no—
36#15668428—21no—
37#15668428—21no—
38#15668428—21no—
39#15668426—21no—
40#15668426—21no—
41C4×Dic181—21no—
42C4×Dic181C2×C411yescharacteristic
43C4×Dic181D411yescharacteristic
44C4×Dic181D411yescharacteristic
45C2×Q8×C181C2×C411yescharacteristic
46C4⋊C4×C181—21no—
47C4⋊C4×C181C2×C411yescharacteristic
48C42×C181—21no—
49C42×C181C2×C411yescharacteristic
50C2×#10624382—24no—
51C2×#10624382—42no—
52C2×#10624382—42no—
53#13089685—42no—
54C22×D181C2×D411yescharacteristic
55Dic362—42no—
56Dic362—41no—
57C2×Dic181—21no—
58C2×Dic181—21no—
59C2×Dic181C22⋊C411yescharacteristic
60C2×Dic181C22⋊C411yescharacteristic
61C2×Dic181C2×D411yescharacteristic
62C4×D181—22no—
63C4×D181—41no—
64C4×D181SD1612yes—
65C4×D181Q1612yes—
66Q8×C181—41no—
67C2×C4×C181—21no—
68C2×C4×C181—21no—
69C2×C4×C181C22⋊C411yescharacteristic
70C2×C4×C181C22⋊C411yescharacteristic
71C2×C4×C181C22×C411yescharacteristic
72C22×C181C2×C22⋊C411yescharacteristic
73C2×D181—21no—
74C2×D181—21no—
75C2×D181C2×SD1611yescharacteristic
76C2×D181C2×Q1611yescharacteristic
77#10624382—44no—
78C4×C181—21no—
79C4×C181—21no—
80C4×C181—41no—
81C4×C181—41no—
82C4×C181#10312yes—
83Dic181—21no—
84Dic181—41no—
85Dic181—41no—
86Dic181#10312yes—
87Dic181#10412yes—
88C2×C181#41011yescharacteristic
89C2×C181C2×#10311yescharacteristic
90C2×C181C2×#10411yescharacteristic
91D181—21no—
92D181—21no—
93C181#150311yescharacteristic, complemented, Sylow
94#1503—1811nomaximal, Sylow
95#533—1811no—
96#431—1811no—
97C4×C4⋊C4—1811no—
98#328—1812no—
99#328—1812no—
100C4⋊Q8—1811no—
101C4⋊Q8—1811no—
102#125—3621no—
103C22⋊Q8—3621no—
104C22⋊Q8—1811no—
105C22⋊Q8—1811no—
106C4×Q8—3621no—
107C4×Q8—1811no—
108C4×Q8#1062438211yescharacteristic, complemented
109#117—3621no—
110C2×C4⋊C4—3621no—
111C2×C42—3621no—
112C2×C42—1811no—
113C4⋊C8—3621no—
114C22⋊C8—1812no—
115C4×C8—3621no—
116#95—1812no—
117C2×Q8—3621no—
118C2×Q8—1811no—
119C2×Q8C2×#1062438211yescharacteristic
120C22×C4—3622no—
121C22×C4—1811no—
122C22×C4—1811no—
123C22×C4—1811no—
124C2×C8—3622no—
125C4⋊C4—21no—
126C4⋊C4—3621no—
127C4⋊C4—3621no—
128C4⋊C4—3621no—
129C4⋊C4—3621no—
130C4⋊C4—3621no—
131C4⋊C4—3624no—
132C4⋊C4—3621no—
133C4⋊C4—1811no—
134C4⋊C4C2×#1062438211yescharacteristic
135C22⋊C4—3621no—
136C22⋊C4—3621no—
137C42—21no—
138C42—3621no—
139C42—1811no—
140C42—1811no—
141C42—1811no—
142C42—7242no—
143C42C2×#1062438211yescharacteristic
144C23—1811no—
145Q8—41no—
146Q8—7241no—
147Q8—7242no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—3622no—
151C2×C4—3624no—
152C2×C4—3621no—
153C2×C4—3621no—
154C2×C4—1811no—
155C2×C4—1811no—
156C2×C4—1811no—
157C2×C4—1812no—
158C2×C4—1812no—
159C2×C4—7241no—
160C2×C4—7242no—
161C2×C4—7242no—
162C2×C4C22⋊#1062438211yescharacteristic
163C2×C4C22⋊#1062438211yescharacteristic
164C2×C4C22×#1062438211yescharacteristic
165C8—7242no—
166C22—3621no—
167C22—3621no—
168C22—1811no—
169C22—1811no—
170C22C2×C22⋊#1062438211yescharacteristic
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—3621no—
176C4—1812no—
177C4—1812no—
178C4—7244no—
179C4—7241no—
180C4—7241no—
181C4#1915387912yes—
182C2—3621no—
183C2—3621no—
184C2#2604225811yescharacteristic
185C2C2×#1915387911yescharacteristic
186C2D181⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4×Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×D181C2×D4
cosocleC2×C4×D181C23
hypercenterC4×Q8#10624382
hypocenterC181#1503
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
12. #19153815C4semidirect
13. #19153814C4semidirect
19. C4×Dic362C4semidirect
22. C4×Q8×C181C4semidirect
93. C181#1503semidirect
108. C4×Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422972
C4×C4⋊#106243822
#260421332
#260421322
#260418332
#1503181