Finite Group Databasebeta

← #37549186

Subgroups — #37549186

Order 23168.

216 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 58 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#26042301C211yesmaximal, characteristic, complemented
2C4×#15668459C211yesmaximal, characteristic, complemented
3#26042136C212yesmaximal, complemented
4#26042134C212yesmaximal, complemented
5#26041811C211yesmaximal, characteristic
6C2×#15668459—21no—
7C4⋊#13089685—21no—
8#19153873—21no—
9#19153871C2212yes—
10#19153871C2212yes—
11C4×#13089685—21no—
12#19153809—21no—
13#19153807C2211yescharacteristic
14#19153803C2211yescharacteristic
15#19153798—21no—
16#19153796C411yescharacteristic
17#19153787C411yescharacteristic
18C4×Dic362C411yescharacteristic
19C4×C22⋊D181—21no—
20#19153775—21no—
21C42×D181C2211yescharacteristic
22C4×D4×C181C411yescharacteristic
23#15668459—24no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C2×Dic362C2×C411yescharacteristic
27C2×C22⋊D181—21no—
28C2×C4×D181C2×C411yescharacteristic
29C2×C4×D181D411yescharacteristic
30C2×C4×D181D411yescharacteristic
31C4⋊Dic181—21no—
32C4⋊Dic181—21no—
33C4⋊Dic181C2×C411yescharacteristic
34#15668428—21no—
35#15668428—21no—
36C22⋊Dic181—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39#15668426—21no—
40C4×Dic181—21no—
41C4×Dic181D411yescharacteristic
42C4×Dic181D411yescharacteristic
43C4×Dic181C2311yescharacteristic
44C2×D4×C181C2×C411yescharacteristic
45C22×C4×C181—21no—
46C4⋊C4×C181C2×C411yescharacteristic
47C22⋊C4×C181—21no—
48C42×C181C2×C411yescharacteristic
49#13089685—24no—
50#13089685—42no—
51C22×D181C22⋊C411yescharacteristic
52Dic362—41no—
53C22⋊D181—42no—
54C2×Dic181—21no—
55C2×Dic181—21no—
56C2×Dic181C22⋊C411yescharacteristic
57C2×Dic181C2×D411yescharacteristic
58C2×Dic181C2×D411yescharacteristic
59C4×D181—22no—
60C4×D181—41no—
61C4×D181—41no—
62C23×C181—21no—
63D4×C181—41no—
64C2×C4×C181—21no—
65C2×C4×C181—42no—
66C2×C4×C181C22⋊C411yescharacteristic
67C2×C4×C181C22⋊C411yescharacteristic
68C2×C4×C181C22×C411yescharacteristic
69C22×C181—41no—
70C22×C181—41no—
71C22×C181—41no—
72C22×C181C2×C22⋊C411yescharacteristic
73C2×D181—21no—
74C2×D181—21no—
75C2×D181—21no—
76C4×C181—21no—
77C4×C181—21no—
78C4×C181—41no—
79C4×C181#10412yes—
80Dic181—21no—
81Dic181—21no—
82Dic181—21no—
83Dic181—41no—
84Dic181—41no—
85C2×C181—81no—
86C2×C181#41311yescharacteristic
87C2×C181C2×#10411yescharacteristic
88C2×C181#42111yescharacteristic
89D181—41no—
90C181#152911yescharacteristic, complemented, Sylow
91#1529—1811nomaximal, Sylow
92#529—1811no—
93#432—1811no—
94C4×M16—1811no—
95#330—1812no—
96#329—1812no—
97C2×M16—3621no—
98#126—3621no—
99#125—1811no—
100#124—1811no—
101#123—3621no—
102C22⋊Q8—1811no—
103#121—1811no—
104C4×Q8—1811no—
105C4×D4—3621no—
106C4×D4#1062438211yescharacteristic
107#117—3621no—
108C2×C42—1811no—
109C4⋊C8—3621no—
110C22⋊C8—3621no—
111#97—1812no—
112#97—1812no—
113C4×C8—3621no—
114C2×Q8—1811no—
115C2×D4—3621no—
116C2×D4C2×#1062438211yescharacteristic
117C22×C4—21no—
118C22×C4—1811no—
119C22×C4—1811no—
120C22×C4—1811no—
121M16—3624no—
122C2×C8—3622no—
123C2×C8—3622no—
124C4⋊C4—3621no—
125C4⋊C4—3621no—
126C4⋊C4—3621no—
127C4⋊C4—3621no—
128C4⋊C4—1811no—
129C4⋊C4C2×#1062438211yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—3621no—
132C22⋊C4—3621no—
133C22⋊C4—3621no—
134C22⋊C4—3621no—
135C42—3621no—
136C42—1811no—
137C42—1811no—
138C42—1811no—
139C42C2×#1062438211yescharacteristic
140C23—21no—
141C23—1811no—
142Q8—7241no—
143D4—41no—
144D4—7242no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—3622no—
148C2×C4—3621no—
149C2×C4—3621no—
150C2×C4—1811no—
151C2×C4—1811no—
152C2×C4—1811no—
153C2×C4—7241no—
154C2×C4—7241no—
155C2×C4C22⋊#1062438211yescharacteristic
156C2×C4C22⋊#1062438211yescharacteristic
157C2×C4C22×#1062438211yescharacteristic
158C8—3624no—
159C8—7242no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—3621no—
164C22—3621no—
165C22—3621no—
166C22C2×C22⋊#1062438211yescharacteristic
167C4—21no—
168C4—21no—
169C4—41no—
170C4—3621no—
171C4—3621no—
172C4—3621no—
173C4—7241no—
174C4—7241no—
175C4#1915388012yes—
176C2—81no—
177C2—7241no—
178C2#2604226511yescharacteristic
179C2C2×#1915388011yescharacteristic
180C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4×D4×C181C4
socleC22×C181C2×C22⋊C4
prime radical#19153787C4
cosocleC4×Dic181C23
hypercenterC4×D4#10624382
hypocenterC181#1529
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26042301C2semidirect
2. C4×#15668459C2semidirect
3. #26042136C2semidirect
4. #26042134C2semidirect
90. C181#1529semidirect

Maximal subgroups

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

subgroupindex
#260423012
C4×#156684592
#260421362
#260421342
#260418112
#1529181