Finite Group Databasebeta

← #37549172

Subgroups — #37549172

Order 23168.

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

normal subgroups only

This group has 234 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
1C4×#15668458C211yesmaximal, characteristic, complemented
2C2×#19153871C211yesmaximal, characteristic, complemented
3#26042136C212yesmaximal, complemented
4#26042134C212yesmaximal, complemented
5#26041810C211yesmaximal, characteristic
6C2×#15668458—21no—
7C22×#13089685—21no—
8#19153871—21no—
9#19153871C2212yes—
10#19153871C2212yes—
11C4×#13089685—21no—
12#19153809—21no—
13#19153807C2211yescharacteristic
14#19153803C2211yescharacteristic
15#19153797—21no—
16#19153795C411yescharacteristic
17#19153781C411yescharacteristic
18D4×Dic181—21no—
19D4×Dic181C411yescharacteristic
20#19153772—21no—
21C2×C4×Dic181C2211yescharacteristic
22#121×C181C411yescharacteristic
23#15668458—24no—
24C2×#13089685—24no—
25C2×#13089685—22no—
26C2×#13089685—22no—
27C2×Dic362C2×C411yescharacteristic
28C22×Dic181—21no—
29C22×Dic181D411yescharacteristic
30C22×Dic181D411yescharacteristic
31C4⋊Dic181—21no—
32C4⋊Dic181C2×C411yescharacteristic
33#15668428—21no—
34#15668428—21no—
35#15668428—21no—
36C22⋊Dic181—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39C22⋊Dic181—21no—
40C4×Dic181—21no—
41C4×Dic181C2×C411yescharacteristic
42C4×Dic181D411yescharacteristic
43C4×Dic181D411yescharacteristic
44C4×Dic181C2311yescharacteristic
45C2×D4×C181—21no—
46C2×D4×C181C2×C411yescharacteristic
47C22×C4×C181C2×C411yescharacteristic
48C4⋊C4×C181C2×C411yescharacteristic
49C22⋊C4×C181—21no—
50#13089685—24no—
51#13089685—24no—
52Dic362—41no—
53C2×Dic181—22no—
54C2×Dic181—22no—
55C2×Dic181—21no—
56C2×Dic181—21no—
57C2×Dic181—42no—
58C2×Dic181C22⋊C411yescharacteristic
59C2×Dic181C22⋊C411yescharacteristic
60C2×Dic181C2×D411yescharacteristic
61C2×Dic181C2×D411yescharacteristic
62C23×C181—21no—
63C23×C181C22⋊C411yescharacteristic
64D4×C181—41no—
65D4×C181—42no—
66C2×C4×C181—21no—
67C2×C4×C181—41no—
68C2×C4×C181C22⋊C411yescharacteristic
69C2×C4×C181C22×C411yescharacteristic
70C22×C181—21no—
71C22×C181—21no—
72C22×C181—41no—
73C22×C181—41no—
74C22×C181—41no—
75C22×C181#10412yes—
76C22×C181C2×C22⋊C411yescharacteristic
77C4×C181—21no—
78C4×C181—21no—
79C4×C181—41no—
80Dic181—21no—
81Dic181—21no—
82Dic181—21no—
83Dic181—41no—
84Dic181—41no—
85Dic181#10412yes—
86C2×C181—22no—
87C2×C181—81no—
88C2×C181#41311yescharacteristic
89C2×C181C2×#10411yescharacteristic
90C2×C181C2×#10411yescharacteristic
91C181#152411yescharacteristic, complemented, Sylow
92#1524—1811nomaximal, Sylow
93#529—1811no—
94C4×M16—1811no—
95C2×#97—1811no—
96#330—1812no—
97#329—1812no—
98C2×M16—3621no—
99C22×C8—3621no—
100#126—3621no—
101#125—1811no—
102#124—1811no—
103#123—3621no—
104C22⋊Q8—1811no—
105#121#1062438211yescharacteristic
106C4×Q8—1811no—
107C4×D4—3621no—
108C4×D4—1811no—
109#117—3621no—
110C2×C42—1811no—
111#97—3621no—
112#97—1812no—
113#97—1812no—
114C4×C8—3621no—
115C2×Q8—1811no—
116C2×D4—21no—
117C2×D4C2×#1062438211yescharacteristic
118C22×C4—3621no—
119C22×C4—1811no—
120C22×C4—1811no—
121C22×C4C2×#1062438211yescharacteristic
122M16—3624no—
123C2×C8—3624no—
124C2×C8—3622no—
125C2×C8—3622no—
126C4⋊C4—3621no—
127C4⋊C4—3621no—
128C4⋊C4—3621no—
129C4⋊C4—3621no—
130C4⋊C4—1811no—
131C4⋊C4C2×#1062438211yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3621no—
134C22⋊C4—3621no—
135C22⋊C4—3621no—
136C22⋊C4—3621no—
137C42—3621no—
138C42—1811no—
139C42—1811no—
140C42—1811no—
141C42—1811no—
142C23—21no—
143C23C22⋊#1062438211yescharacteristic
144Q8—7241no—
145D4—41no—
146D4—42no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3622no—
150C2×C4—3622no—
151C2×C4—3621no—
152C2×C4—3621no—
153C2×C4—1811no—
154C2×C4—1811no—
155C2×C4—1811no—
156C2×C4—1811no—
157C2×C4—7242no—
158C2×C4C22⋊#1062438211yescharacteristic
159C2×C4C22×#1062438211yescharacteristic
160C8—3624no—
161C8—3624no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22#1915388012yes—
168C22C2×C22⋊#1062438211yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—3621no—
173C4—3621no—
174C4—3621no—
175C4—1812no—
176C4—7241no—
177C4—7241no—
178C2—22no—
179C2—81no—
180C2#2604226611yescharacteristic
181C2C2×#1915388011yescharacteristic
182C2D181⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting, prime radical#121×C181C4
socleC22×C181C2×C22⋊C4
cosocleC4×Dic181C23
hypercenter#121#10624382
hypocenterC181#1524
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C4×#15668458C2semidirect
2. C2×#19153871C2semidirect
3. #26042136C2semidirect
4. #26042134C2semidirect
91. C181#1524semidirect

Maximal subgroups

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

subgroupindex
C4×#156684582
C2×#191538712
#260421362
#260421342
#260418102
#1524181