Finite Group Databasebeta

← #37549034

Subgroups — #37549034

Order 23168.

212 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 60 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#26042296C211yesmaximal, characteristic, complemented
2#26042295C211yesmaximal, characteristic, complemented
3Q8⋊#13089685C212yesmaximal, complemented
4#26042127C212yesmaximal, complemented
5#26041815C211yesmaximal, characteristic
6#19153883—21no—
7C4⋊#13089685C2212yes—
8C22⋊#13089685—21no—
9C22⋊#13089685—21no—
10C4×#13089685—21no—
11C4×#13089685C2212yes—
12#19153811—21no—
13#19153807C411yescharacteristic
14#19153805C411yescharacteristic
15#19153800—21no—
16#19153790C411yescharacteristic
17Q8×Dic181C2211yescharacteristic
18#19153778C2211yescharacteristic
19#19153776—21no—
20#19153774—21no—
21C2×C4×Dic181C2211yescharacteristic
22C22⋊Q8×C181C411yescharacteristic
23C2×#13089685—22no—
24C2×#13089685—22no—
25C2×D724C2×C411yescharacteristic
26C2×C22⋊D181—21no—
27C22×Dic181D411yescharacteristic
28C22×Dic181D411yescharacteristic
29C2×C4×D181—21no—
30C4⋊Dic181—21no—
31C4⋊Dic181—21no—
32C4⋊Dic181C2×C411yescharacteristic
33#15668428—21no—
34#15668426—21no—
35#15668426—21no—
36#15668426—21no—
37#15668426—21no—
38C4×Dic181—21no—
39C4×Dic181C2×C411yescharacteristic
40C4×Dic181D411yescharacteristic
41C4×Dic181D411yescharacteristic
42C4×Dic181C2311yescharacteristic
43C2×Q8×C181C2×C411yescharacteristic
44C22×C4×C181C2×C411yescharacteristic
45C4⋊C4×C181—21no—
46C4⋊C4×C181C2×C411yescharacteristic
47C22⋊C4×C181—21no—
48#13089685—42no—
49#13089685—42no—
50C22×D181—21no—
51D724—22no—
52C22⋊D181—42no—
53C2×Dic181—22no—
54C2×Dic181—21no—
55C2×Dic181—41no—
56C2×Dic181C22⋊C411yescharacteristic
57C2×Dic181C22⋊C411yescharacteristic
58C2×Dic181C2×D411yescharacteristic
59C2×Dic181C2×D411yescharacteristic
60C4×D181—42no—
61C23×C181C22⋊C411yescharacteristic
62Q8×C181—22no—
63C2×C4×C181—22no—
64C2×C4×C181—21no—
65C2×C4×C181—21no—
66C2×C4×C181C22⋊C411yescharacteristic
67C2×C4×C181C22×C411yescharacteristic
68C22×C181—21no—
69C22×C181—21no—
70C22×C181—21no—
71C22×C181C2×C22⋊C411yescharacteristic
72C2×D181—41no—
73C2×D181—41no—
74C2×D181—41no—
75C4×C181—21no—
76C4×C181—41no—
77C4×C181—41no—
78C4×C181#13112yes—
79Dic181—21no—
80Dic181—21no—
81Dic181—21no—
82Dic181—41no—
83Dic181#13512yes—
84C2×C181—41no—
85C2×C181#40811yescharacteristic
86C2×C181#41611yescharacteristic
87C2×C181#42111yescharacteristic
88D181—81no—
89C181#148211yescharacteristic, complemented, Sylow
90#1482—1811nomaximal, Sylow
91#529—1811no—
92#431—1811no—
93#431—1811no—
94Q8⋊C8—1812no—
95D4⋊C8—1812no—
96#126—3621no—
97#125—1811no—
98#124—1811no—
99#123—3621no—
100C22⋊Q8#1062438211yescharacteristic
101#121—1811no—
102C4×Q8—1811no—
103C4×D4—3621no—
104C4×D4—1811no—
105#117—3621no—
106C2×C42—1811no—
107C4⋊C8—3621no—
108C4⋊C8—1812no—
109C22⋊C8—3621no—
110C22⋊C8—3621no—
111C4×C8—3621no—
112C4×C8—1812no—
113C2×Q8C2×#1062438211yescharacteristic
114C2×D4—3621no—
115C2×D4—1811no—
116C22×C4—3621no—
117C22×C4—1811no—
118C22×C4—1811no—
119C22×C4C2×#1062438211yescharacteristic
120C2×C8—3622no—
121C2×C8—3622no—
122C4⋊C4—21no—
123C4⋊C4—3621no—
124C4⋊C4—3621no—
125C4⋊C4—3621no—
126C4⋊C4—1811no—
127C4⋊C4C2×#1062438211yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3621no—
130C22⋊C4—3621no—
131C22⋊C4—3621no—
132C22⋊C4—3621no—
133C42—3621no—
134C42—1811no—
135C42—1811no—
136C42—1811no—
137C42—1811no—
138C23—3621no—
139C23C22⋊#1062438211yescharacteristic
140Q8—22no—
141D4—3622no—
142D4—7242no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—3622no—
147C2×C4—3621no—
148C2×C4—1811no—
149C2×C4—1811no—
150C2×C4—1811no—
151C2×C4—1811no—
152C2×C4—7242no—
153C2×C4—7241no—
154C2×C4C22⋊#1062438211yescharacteristic
155C2×C4C22×#1062438211yescharacteristic
156C8—7242no—
157C8—7242no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—7241no—
162C22—7241no—
163C22—7241no—
164C22C2×C22⋊#1062438211yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—3621no—
169C4—3621no—
170C4—3621no—
171C4—1812no—
172C4—7241no—
173C4#1915390312yes—
174C2—41no—
175C2—14481no—
176C2#2604225211yescharacteristic
177C2#2604226911yescharacteristic
178C2#2604227811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC22⋊Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radical#19153790C4
cosocleC4×Dic181C23
hypercenterC22⋊Q8#10624382
hypocenterC181#1482
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26042296C2semidirect
2. #26042295C2semidirect
3. Q8⋊#13089685C2semidirect
4. #26042127C2semidirect
89. C181#1482semidirect

Maximal subgroups

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

subgroupindex
#260422962
#260422952
Q8⋊#130896852
#260421272
#260418152
#1482181