Finite Group Databasebeta

← #37535387

Subgroups — #37535387

Order 23168.

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

normal subgroups only

This group has 286 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#99×D181C212yesmaximal, complemented
3#26041073C212yesmaximal, complemented
4#26041072C211yesmaximal, characteristic, complemented
5#353×C181C211yesmaximal, characteristic, complemented
6C2×#15668448C411yescharacteristic, complemented
7C2×D4×D181C2211yescharacteristic
8#19153812C411yescharacteristic, complemented
9#19153804C411yescharacteristic, complemented
10#19153803C2211yescharacteristic
11#19153788—22no—
12#19153784—22no—
13#19153773C411yescharacteristic, complemented
14C22⋊C4×D181—22no—
15#19153731C2212yes—
16#19153730—22no—
17#127×C181C2211yescharacteristic
18#99×C181C2212yes—
19C23×D181D412yes—
20#15668448—42no—
21C2×Dic362C2×C411yescharacteristic, complemented
22C2×D724C2×C411yescharacteristic
23C2×C22⋊D181—21no—
24C2×C22⋊D181—22no—
25C2×C22⋊D181D412yes—
26D4×D181—42no—
27C22×Dic181—21no—
28C2×C4×D181—42no—
29C2×C4×D181C2×C411yescharacteristic
30#15668428—21no—
31C22⋊Dic181—22no—
32C22⋊Dic181—22no—
33#15668426—21no—
34#15668426—22no—
35#15668426—42no—
36C4×Dic181C2×C411yescharacteristic, complemented
37C2×D4×C181—22no—
38C2×D4×C181C2×C411yescharacteristic, complemented
39C2×D4×C181C2311yescharacteristic
40C22⋊C4×C181—22no—
41C42×C181C2×C411yescharacteristic, complemented
42C22×D181—22no—
43C22×D181—22no—
44C22×D181—82no—
45C22×D181C22⋊C412yes—
46C22×D181C22⋊C411yescharacteristic
47Dic362—41no—
48D724—41no—
49C22⋊D181—42no—
50C22⋊D181—42no—
51C22⋊D181—82no—
52C2×Dic181—21no—
53C2×Dic181—21no—
54C2×Dic181—42no—
55C2×Dic181—42no—
56C2×Dic181C22⋊C411yescharacteristic
57C4×D181—22no—
58C4×D181—84no—
59C23×C181—21no—
60C23×C181C2×D412yes—
61D4×C181—41no—
62D4×C181—41no—
63D4×C181—82no—
64C2×C4×C181—21no—
65C2×C4×C181—42no—
66C2×C4×C181C22×C411yescharacteristic
67C22×C181—22no—
68C22×C181—42no—
69C22×C181—42no—
70C22×C181—41no—
71C22×C181C2×C22⋊C411yescharacteristic
72C2×D181—22no—
73C2×D181—41no—
74C2×D181—44no—
75C2×D181—42no—
76C2×D181—82no—
77C2×D181#9912yes—
78C4×C181—21no—
79C4×C181—41no—
80C4×C181—82no—
81Dic181—21no—
82Dic181—41no—
83Dic181—41no—
84Dic181—82no—
85C2×C181—21no—
86C2×C181—42no—
87C2×C181—81no—
88C2×C181C2×#9911yescharacteristic
89D181—22no—
90D181—42no—
91C181#212711yescharacteristic, complemented, Sylow
92#2127—1811nomaximal, Sylow
93#535—1811no—
94C2×#99—1812no—
95#354—1812no—
96#353—1811no—
97#353D18111yescharacteristic, complemented
98C2×#41—1811no—
99C22×D4—1811no—
100#127—1811no—
101#127C2×D18111yescharacteristic
102#124—1811no—
103#124—1811no—
104#121—3622no—
105#120—3622no—
106#117—1811no—
107C2×C22⋊C4—3622no—
108#99—3622no—
109#99—1812no—
110#99C2×D18112yes—
111C24—1812no—
112#41—7242no—
113C2×Q8—1811no—
114C2×D4—22no—
115C2×D4—3622no—
116C2×D4—3621no—
117C2×D4—1812no—
118C2×D4—1811no—
119C2×D4—7242no—
120C2×D4C4×D18111yescharacteristic, complemented
121C2×D4C22×D18111yescharacteristic
122C22×C4—3621no—
123C22×C4—1811no—
124C22×C4—7242no—
125C4⋊C4—3621no—
126C22⋊C4—22no—
127C22⋊C4—3621no—
128C22⋊C4—3622no—
129C22⋊C4—3622no—
130C22⋊C4—3622no—
131C22⋊C4—7242no—
132C42—1811no—
133C42C4×D18111yescharacteristic, complemented
134C23—21no—
135C23—3622no—
136C23—3622no—
137C23—14482no—
138C23—1812no—
139C23—1811no—
140C23D4×D18112yes—
141Q8—7241no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—14482no—
146D4—7241no—
147D4—7242no—
148D4—7242no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—3622no—
152C2×C4—3621no—
153C2×C4—3621no—
154C2×C4—14484no—
155C2×C4—1811no—
156C2×C4—7242no—
157C2×C4—7242no—
158C2×C4C2×C4×D18111yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—3622no—
164C22—14482no—
165C22—1812no—
166C22—7241no—
167C22—7242no—
168C22—7244no—
169C22C22⋊C4×D18111yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—3621no—
174C4—14482no—
175C4—7241no—
176C4—7241no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—3622no—
181C2—7242no—
182C2#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#2127
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. #99×D181C2semidirect
3. #26041073C2semidirect
4. #26041072C2semidirect
5. #353×C181C2semidirect
6. C2×#15668448C4semidirect
8. #19153812C4semidirect
9. #19153804C4semidirect
13. #19153773C4semidirect
21. C2×Dic362C2×C4semidirect
36. C4×Dic181C2×C4semidirect
38. C2×D4×C181C2×C4semidirect
41. C42×C181C2×C4semidirect
91. C181#2127semidirect
97. #353D181semidirect
120. C2×D4C4×D181semidirect
133. C42C4×D181semidirect

Maximal subgroups

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

subgroupindex
#260418422
#99×D1812
#260410732
#260410722
#353×C1812
#2127181