Finite Group Databasebeta

← #37535391

Subgroups — #37535391

Order 23168.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#26041842C211yesmaximal, characteristic
2#26041251C212yesmaximal, complemented
3#26041072C211yesmaximal, characteristic, complemented
4#26041071C212yesmaximal, complemented
5#353×C181C211yesmaximal, characteristic, complemented
6C2×#15668448C2211yescharacteristic
7C2×D4×D181C411yescharacteristic, complemented
8#19153812C2211yescharacteristic
9#19153804C411yescharacteristic, complemented
10#19153803C411yescharacteristic, complemented
11#19153788—22no—
12#19153784—22no—
13#19153773C411yescharacteristic, complemented
14#19153772—22no—
15#19153731C2212yescomplemented
16#19153730—22no—
17#127×C181C2211yescharacteristic
18#99×C181C2212yescomplemented
19C23×D181—21no—
20#15668448—42no—
21C2×Dic362C2×C411yescharacteristic
22C2×D724C2×C411yescharacteristic
23C2×C22⋊D181—21no—
24C2×C22⋊D181—22no—
25C2×C22⋊D181D412yes—
26D4×D181—42no—
27C22×Dic181D412yes—
28C2×C4×D181C2×C411yescharacteristic
29#15668428—21no—
30#15668428—42no—
31C22⋊Dic181—22no—
32C22⋊Dic181—22no—
33#15668426—21no—
34#15668426—22no—
35C4×Dic181—42no—
36C4×Dic181C2×C411yescharacteristic
37C2×D4×C181—22no—
38C2×D4×C181C2×C411yescharacteristic
39C2×D4×C181C2311yescharacteristic
40C22⋊C4×C181—22no—
41C42×C181C2×C411yescharacteristic
42C22×D181—21no—
43C22×D181—42no—
44C22×D181—82no—
45C22×D181C22⋊C411yescharacteristic
46Dic362—41no—
47D724—41no—
48C22⋊D181—42no—
49C22⋊D181—42no—
50C22⋊D181—82no—
51C2×Dic181—22no—
52C2×Dic181—22no—
53C2×Dic181—21no—
54C2×Dic181—42no—
55C2×Dic181C22⋊C412yes—
56C2×Dic181C22⋊C411yescharacteristic
57C4×D181—22no—
58C23×C181—21no—
59C23×C181C2×D412yes—
60D4×C181—41no—
61D4×C181—41no—
62D4×C181—82no—
63C2×C4×C181—21no—
64C2×C4×C181—42no—
65C2×C4×C181C22×C411yescharacteristic
66C22×C181—22no—
67C22×C181—42no—
68C22×C181—42no—
69C22×C181—41no—
70C22×C181C2×C22⋊C411yescharacteristic
71C2×D181—21no—
72C2×D181—41no—
73C2×D181—42no—
74C2×D181—41no—
75C2×D181—84no—
76C4×C181—21no—
77C4×C181—41no—
78C4×C181—82no—
79Dic181—22no—
80Dic181—41no—
81Dic181—82no—
82Dic181#9912yes—
83C2×C181—21no—
84C2×C181—42no—
85C2×C181—81no—
86C2×C181C2×#9911yescharacteristic
87D181—41no—
88D181—81no—
89C181#212811yescharacteristic, complemented, Sylow
90#2128—1811nomaximal, Sylow
91#535—1811no—
92#410—1812no—
93#353—1811no—
94#353—1812no—
95#353D18111yescharacteristic, complemented
96C2×#41—1811no—
97C22×D4—1811no—
98#127—1811no—
99#127C2×D18111yescharacteristic
100#124—1811no—
101#124—1811no—
102#121—3622no—
103#120—3622no—
104#117—3622no—
105#117—1811no—
106#99—3622no—
107#99—1812no—
108#99C2×D18112yescomplemented
109C24—3621no—
110#41—7242no—
111C2×Q8—1811no—
112C2×D4—22no—
113C2×D4—3622no—
114C2×D4—3621no—
115C2×D4—1812no—
116C2×D4—1811no—
117C2×D4—7242no—
118C2×D4C4×D18111yescharacteristic
119C2×D4C22×D18111yescharacteristic
120C22×C4—1812no—
121C22×C4—1811no—
122C4⋊C4—3621no—
123C4⋊C4—7242no—
124C22⋊C4—22no—
125C22⋊C4—3621no—
126C22⋊C4—3622no—
127C22⋊C4—3622no—
128C22⋊C4—3622no—
129C42—1811no—
130C42—7242no—
131C42C4×D18111yescharacteristic
132C23—21no—
133C23—3621no—
134C23—14482no—
135C23—1811no—
136C23—7242no—
137C23D4×D18112yes—
138Q8—7241no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—14482no—
143D4—7241no—
144D4—7242no—
145D4—7242no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—3622no—
149C2×C4—3622no—
150C2×C4—3622no—
151C2×C4—3621no—
152C2×C4—1812no—
153C2×C4—1811no—
154C2×C4—7242no—
155C2×C4C2×C4×D18111yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—3621no—
161C22—14484no—
162C22—7242no—
163C22—7241no—
164C22—7241no—
165C22C22⋊C4×D18111yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—3622no—
170C4—14482no—
171C4—1812no—
172C4—7241no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—14481no—
177C2—7241no—
178C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC2×C4×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#353×C181C2
socleC2×C181C2×#99
prime radical#26041842C2
cosocleC2×D4×C181C23
hypercenter#353D181
hypocenterC181#2128
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #26041251C2semidirect
3. #26041072C2semidirect
4. #26041071C2semidirect
5. #353×C181C2semidirect
7. C2×D4×D181C4semidirect
9. #19153804C4semidirect
10. #19153803C4semidirect
13. #19153773C4semidirect
15. #19153731C22semidirect
18. #99×C181C22semidirect
89. C181#2128semidirect
95. #353D181semidirect
108. #99C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418422
#260412512
#260410722
#260410712
#353×C1812
#2128181