Finite Group Databasebeta

← #37549885

Subgroups — #37549885

Order 23168.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×#19153880C214yesmaximal, complemented
2C4×#15668459C211yesmaximal, characteristic, complemented
3C4×C4⋊#10624382C211yesmaximal, characteristic, complemented
4#26041827C211yesmaximal, characteristic
5C2×#15668459C2212yes—
6C2×C4⋊#10624382—21no—
7C2×C4×#10624382C2212yes—
8#19153880—28no—
9#19153871—21no—
10C4×#13089685—21no—
11(C2×#10624382)⋊C4—21no—
12C2×#15668448C2212yes—
13#19153816C411yescharacteristic, complemented
14#19153803C412yescomplemented
15#19153787—21no—
16#19153787—21no—
17D4×Dic181—21no—
18D4×Dic181—21no—
19C42×D181C2211yescharacteristic
20#127×C181C411yescharacteristic, complemented
21C22×#10624382—22no—
22#15668459—22no—
23#15668459D414yescomplemented
24C2×#13089685—22no—
25C4⋊#10624382—24no—
26C4×#10624382—24no—
27#15668448—24no—
28#15668448—24no—
29C2×Dic362C2×C412yes—
30C2×C22⋊D181—22no—
31C22×Dic181—22no—
32C2×C4×D181D411yescharacteristic
33C2×C4×D181D411yescharacteristic
34C2×C4×D181C2311yescharacteristic
35C4⋊Dic181—21no—
36C4⋊Dic181—21no—
37C22⋊Dic181—22no—
38C22⋊Dic181—22no—
39C4×Dic181C2×C411yescharacteristic
40C4×Dic181D411yescharacteristic
41C4×Dic181D411yescharacteristic
42C2×D4×C181—21no—
43C2×D4×C181—21no—
44C2×D4×C181C2×C412yes—
45C42×C181C2×C411yescharacteristic
46C2×#10624382—24no—
47C2×#10624382—42no—
48C2×#10624382—42no—
49#13089685—24no—
50C22×D181C2×D411yescharacteristic
51Dic362—24no—
52C22⋊D181—42no—
53C22⋊D181—42no—
54C2×Dic181—22no—
55C2×Dic181—44no—
56C2×Dic181C22⋊C411yescharacteristic
57C2×Dic181C22⋊C411yescharacteristic
58C2×Dic181C2×D411yescharacteristic
59C4×D181—22no—
60C4×D181—41no—
61C4×D181C2×D412yes—
62C4×D181#4112yes—
63C23×C181—22no—
64D4×C181—24no—
65D4×C181—42no—
66D4×C181—42no—
67C2×C4×C181C22⋊C411yescharacteristic
68C2×C4×C181C22⋊C411yescharacteristic
69C2×C4×C181C22×C411yescharacteristic
70C22×C181—42no—
71C22×C181—42no—
72C22×C181—42no—
73C22×C181C2×C22⋊C411yescharacteristic
74C2×D181—21no—
75C2×D181—21no—
76C2×D181#12411yescharacteristic
77C2×D181#12711yescharacteristic
78#10624382—44no—
79C4×C181—21no—
80C4×C181#10412yescomplemented
81C4×C181C4×D412yes—
82Dic181—21no—
83Dic181—21no—
84Dic181—42no—
85Dic181#12112yes—
86C2×C181—82no—
87C2×C181#39011yescharacteristic
88C2×C181C2×#10411yescharacteristic
89C2×C181#42111yescharacteristic
90D181—21no—
91D181—21no—
92C181#198011yescharacteristic, complemented, Sylow
93#1980—1811nomaximal, Sylow
94#532—1811no—
95C2×#104—1814no—
96C4×M16—1811no—
97C4×C4⋊C4—1811no—
98C2×#41—1812no—
99C2×M16—1812no—
100C4⋊Q8—1811no—
101#127#1062438211yescharacteristic, complemented
102#124—1812no—
103#121—3621no—
104#121—3621no—
105C4×D4—3621no—
106C4×D4—3621no—
107C2×C4⋊C4—3621no—
108C2×C42—1812no—
109C2×C42—1811no—
110#104—3628no—
111#97—3621no—
112C4×C8—3621no—
113#95—3621no—
114#41—3624no—
115#41—3624no—
116C2×Q8—1812no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—3622no—
120C2×D4C2×#1062438212yes—
121C22×C4—3622no—
122C22×C4—3622no—
123C22×C4—1811no—
124C22×C4—1811no—
125C22×C4—1811no—
126M16—3622no—
127M16—1814no—
128C2×C8—3622no—
129C4⋊C4—3624no—
130C4⋊C4—3621no—
131C4⋊C4—3621no—
132C22⋊C4—3622no—
133C22⋊C4—3622no—
134C42—3624no—
135C42—1811no—
136C42—1811no—
137C42—1811no—
138C42C2×#1062438211yescharacteristic
139C23—22no—
140C23—1811no—
141Q8—3624no—
142D4—24no—
143D4—42no—
144D4—42no—
145D4—7242no—
146D4—7242no—
147C2×C4—3622no—
148C2×C4—3624no—
149C2×C4—3622no—
150C2×C4—1811no—
151C2×C4—1811no—
152C2×C4—1811no—
153C2×C4—1812no—
154C2×C4—1812no—
155C2×C4—7241no—
156C2×C4—7244no—
157C2×C4—7242no—
158C2×C4—7242no—
159C2×C4C22⋊#1062438211yescharacteristic
160C2×C4C22⋊#1062438211yescharacteristic
161C2×C4C22×#1062438211yescharacteristic
162C8—3624no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—3621no—
167C22—3621no—
168C22—1811no—
169C22—1811no—
170C22C2×C22⋊#1062438211yescharacteristic
171C4—21no—
172C4—3621no—
173C4—3621no—
174C4—1812no—
175C4—7244no—
176C4—7242no—
177C4#1915388012yescomplemented
178C4D4×#1062438212yes—
179C2—82no—
180C2—3621no—
181C2—3621no—
182C2#2604223111yescharacteristic
183C2C2×#1915388011yescharacteristic
184C2#2604227711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#127×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041827C2
cosocleC2×C4×D181C23
hypercenter#127#10624382
hypocenterC181#1980
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#19153880C2semidirect
2. C4×#15668459C2semidirect
3. C4×C4⋊#10624382C2semidirect
13. #19153816C4semidirect
14. #19153803C4semidirect
20. #127×C181C4semidirect
23. #15668459D4semidirect
80. C4×C181#104semidirect
92. C181#1980semidirect
101. #127#10624382semidirect
177. C4#19153880semidirect

Maximal subgroups

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

subgroupindex
C2×#191538802
C4×#156684592
C4×C4⋊#106243822
#260418272
#1980181