Finite Group Databasebeta

← #37549150

Subgroups — #37549150

Order 23168.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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, complemented
2#26042223C211yesmaximal, characteristic, complemented
3#26042130C214yesmaximal, complemented
4#26041827C211yesmaximal, characteristic
5C2×C4⋊#10624382—21no—
6#19153883—21no—
7#19153873C2212yes—
8C4×#13089685—21no—
9(C2×#10624382)⋊C4—21no—
10(C2×#10624382)⋊C4C2212yes—
11C2×#15668448—21no—
12#19153816C411yescharacteristic, complemented
13#19153803—21no—
14#19153787—21no—
15#19153787C2212yes—
16D4×Dic181—21no—
17D4×Dic181C412yescomplemented
18C42×D181C2211yescharacteristic
19#127×C181C411yescharacteristic, complemented
20C22×#10624382—22no—
21C2×#13089685—22no—
22C4⋊#10624382—42no—
23#15668448—42no—
24#15668448—42no—
25C2×Dic362—21no—
26C2×C22⋊D181—22no—
27C22×Dic181—22no—
28C2×C4×D181D411yescharacteristic
29C2×C4×D181D411yescharacteristic
30C2×C4×D181C2311yescharacteristic
31C4⋊Dic181—21no—
32C4⋊Dic181C2×C412yes—
33C22⋊Dic181—22no—
34C22⋊Dic181—22no—
35C4×Dic181C2×C411yescharacteristic
36C4×Dic181D411yescharacteristic
37C4×Dic181D411yescharacteristic
38C2×D4×C181—21no—
39C2×D4×C181—21no—
40C2×D4×C181C2×C412yes—
41C42×C181C2×C411yescharacteristic
42C2×#10624382—42no—
43C2×#10624382—42no—
44C2×#10624382—42no—
45#13089685—42no—
46C22×D181C2×D411yescharacteristic
47Dic362—42no—
48C22⋊D181—42no—
49C22⋊D181—42no—
50C2×Dic181—22no—
51C2×Dic181—44no—
52C2×Dic181C22⋊C411yescharacteristic
53C2×Dic181C22⋊C411yescharacteristic
54C2×Dic181C2×D411yescharacteristic
55C4×D181—41no—
56C4×D181—41no—
57C4×D181D812yes—
58C4×D181SD1612yes—
59C23×C181—22no—
60D4×C181—42no—
61D4×C181—42no—
62D4×C181—42no—
63C2×C4×C181C22⋊C411yescharacteristic
64C2×C4×C181C22⋊C411yescharacteristic
65C2×C4×C181C22×C411yescharacteristic
66C22×C181—42no—
67C22×C181—42no—
68C22×C181—42no—
69C22×C181C2×C22⋊C411yescharacteristic
70C2×D181—21no—
71C2×D181—21no—
72C2×D181C2×D811yescharacteristic
73C2×D181C2×SD1611yescharacteristic
74#10624382—82no—
75C4×C181—21no—
76C4×C181—21no—
77C4×C181#10212yes—
78Dic181—21no—
79Dic181—21no—
80Dic181—42no—
81Dic181#10212yes—
82C2×C181—82no—
83C2×C181#41011yescharacteristic
84C2×C181C2×#10211yescharacteristic
85C2×C181#42111yescharacteristic
86D181—21no—
87D181—21no—
88C181#151711yescharacteristic, complemented, Sylow
89#1517—1811nomaximal, Sylow
90#532—1811no—
91#431—1811no—
92#382—1811no—
93#327—1814no—
94C2×#41—3621no—
95C4⋊Q8—1811no—
96#127#1062438211yescharacteristic, complemented
97#124—3621no—
98#121—3621no—
99#121—1812no—
100C4×D4—3621no—
101C4×D4—1812no—
102C2×C4⋊C4—3621no—
103C2×C42—1811no—
104C4⋊C8—3621no—
105C22⋊C8—1812no—
106C4×C8—3621no—
107#95—3621no—
108#95—1812no—
109#41—7242no—
110#41—7242no—
111C2×Q8—3621no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—3622no—
115C2×D4C2×#1062438212yes—
116C22×C4—3622no—
117C22×C4—3622no—
118C22×C4—1811no—
119C22×C4—1811no—
120C22×C4—1811no—
121C2×C8—3622no—
122C4⋊C4—3621no—
123C4⋊C4—1812no—
124C4⋊C4—7242no—
125C22⋊C4—3622no—
126C22⋊C4—3622no—
127C42—1811no—
128C42—1811no—
129C42—1811no—
130C42C2×#1062438211yescharacteristic
131C23—22no—
132C23—1811no—
133Q8—7242no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—7242no—
138D4—7242no—
139C2×C4—3622no—
140C2×C4—1811no—
141C2×C4—1811no—
142C2×C4—1811no—
143C2×C4—1812no—
144C2×C4—1812no—
145C2×C4—7241no—
146C2×C4—7241no—
147C2×C4—7242no—
148C2×C4—7242no—
149C2×C4—7242no—
150C2×C4—7244no—
151C2×C4C22⋊#1062438211yescharacteristic
152C2×C4C22⋊#1062438211yescharacteristic
153C2×C4C22×#1062438211yescharacteristic
154C8—7242no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22—3621no—
159C22—3621no—
160C22—1811no—
161C22—1811no—
162C22C2×C22⋊#1062438211yescharacteristic
163C4—21no—
164C4—21no—
165C4—3621no—
166C4—3621no—
167C4—14482no—
168C4—1812no—
169C4—7242no—
170C4#1915387812yes—
171C2—82no—
172C2—3621no—
173C2—3621no—
174C2#2604225911yescharacteristic
175C2C2×#1915387811yescharacteristic
176C2#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#1517
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26042297C2semidirect
2. #26042223C2semidirect
3. #26042130C2semidirect
12. #19153816C4semidirect
17. D4×Dic181C4semidirect
19. #127×C181C4semidirect
88. C181#1517semidirect
96. #127#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422972
#260422232
#260421302
#260418272
#1517181