Finite Group Databasebeta

← #37535392

Subgroups — #37535392

Order 23168.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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#26041843C211yesmaximal, characteristic
2#26041253C211yesmaximal, characteristic, complemented
3#26041071C214yesmaximal, complemented
4#410×C181C211yesmaximal, characteristic, complemented
5C22×D724C411yescharacteristic, complemented
6#19153812C2212yes—
7#19153805C412yescomplemented
8#19153790—21no—
9#19153790—21no—
10#19153784—21no—
11#19153784—21no—
12#19153775—21no—
13#19153775C411yescharacteristic, complemented
14#19153731—21no—
15#19153731C2212yescomplemented
16C2×#41×C181C2211yescharacteristic
17#117×C181—21no—
18#99×C181—21no—
19#99×C181C2212yescomplemented
20C23×D181—21no—
21C2×D724—21no—
22C2×D724—41no—
23C2×D724—41no—
24C2×D724C2×C412yes—
25C2×C22⋊D181—22no—
26C2×C22⋊D181—22no—
27C4⋊Dic181—21no—
28C4⋊Dic181—41no—
29C22⋊Dic181—21no—
30C22⋊Dic181—22no—
31#15668426—22no—
32#15668426—22no—
33C4×Dic181—41no—
34C4×Dic181C2×C412yes—
35#41×C181—41no—
36#41×C181—41no—
37C2×Q8×C181C2×C411yescharacteristic
38C2×D4×C181D411yescharacteristic
39C2×D4×C181D411yescharacteristic
40C2×D4×C181C2311yescharacteristic
41C22×C4×C181C2×C411yescharacteristic
42C22×C4×C181D411yescharacteristic
43C22×C4×C181D411yescharacteristic
44C4⋊C4×C181—41no—
45C22⋊C4×C181—22no—
46C42×C181—41no—
47C22×D181—22no—
48C22×D181—42no—
49C22×D181—82no—
50D724—42no—
51D724—41no—
52D724—41no—
53C22⋊D181—82no—
54C22⋊D181—82no—
55C2×Dic181—22no—
56C2×Dic181—42no—
57C23×C181C22⋊C411yescharacteristic
58C23×C181C2×D411yescharacteristic
59C23×C181C2×D411yescharacteristic
60Q8×C181—41no—
61D4×C181—41no—
62D4×C181—41no—
63D4×C181—41no—
64C2×C4×C181—21no—
65C2×C4×C181—21no—
66C2×C4×C181—21no—
67C2×C4×C181—21no—
68C2×C4×C181—21no—
69C2×C4×C181—21no—
70C2×C4×C181—42no—
71C2×C4×C181C22⋊C411yescharacteristic
72C2×C4×C181C22⋊C411yescharacteristic
73C2×C4×C181C22⋊C411yescharacteristic
74C2×C4×C181C22×C411yescharacteristic
75C22×C181—21no—
76C22×C181—21no—
77C22×C181—21no—
78C22×C181—41no—
79C22×C181—41no—
80C22×C181—41no—
81C22×C181C2×C22⋊C411yescharacteristic
82C2×D181—42no—
83C2×D181—42no—
84C2×D181—42no—
85C2×D181—84no—
86C4×C181—21no—
87C4×C181—21no—
88C4×C181—21no—
89C4×C181—82no—
90C4×C181#9911yescharacteristic
91C4×C181#9911yescharacteristic
92Dic181—42no—
93Dic181—82no—
94C2×C181—21no—
95C2×C181—41no—
96C2×C181—41no—
97C2×C181—41no—
98C2×C181C2×#9911yescharacteristic
99D181—82no—
100C181#212811yescharacteristic, complemented, Sylow
101#2128—1811nomaximal, Sylow
102#535—1811no—
103#410—1811no—
104#410D18111yescharacteristic, complemented
105#353—1814no—
106C2×#41C2×D18111yescharacteristic
107C22×D4—1811no—
108#127—1812no—
109#124—1812no—
110#121—3621no—
111#121—3621no—
112#120—3621no—
113#120—3621no—
114#117—21no—
115#117—3621no—
116#117—1811no—
117#99—21no—
118#99—3621no—
119#99—1812no—
120#99C2×D18112yescomplemented
121C24—3621no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D18111yescharacteristic
125C2×D4—3622no—
126C2×D4—3622no—
127C2×D4—3621no—
128C2×D4—1812no—
129C2×D4—7241no—
130C2×D4—7241no—
131C2×D4C22⋊D18111yescharacteristic
132C2×D4D72411yescharacteristic
133C2×D4C22×D18111yescharacteristic
134C22×C4C4×D18111yescharacteristic
135C22×C4C22⋊D18111yescharacteristic
136C22×C4D72411yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—3621no—
139C4⋊C4—7241no—
140C22⋊C4—22no—
141C22⋊C4—3621no—
142C22⋊C4—3622no—
143C22⋊C4—3622no—
144C22⋊C4—3622no—
145C42—41no—
146C42—1812no—
147C42—7241no—
148C23—3622no—
149C23—14482no—
150C23—7242no—
151C23#1566842611yescharacteristic
152C23C2×C22⋊D18111yescharacteristic
153C23C2×D72411yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—14482no—
159D4—14482no—
160D4—7242no—
161D4—7241no—
162D4—7241no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—3622no—
171C2×C4—7242no—
172C2×C4#1566842611yescharacteristic
173C2×C4#1566842611yescharacteristic
174C2×C4#1566842611yescharacteristic
175C2×C4C2×C4×D18111yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—14484no—
183C22—7242no—
184C22—7242no—
185C22—7242no—
186C22C2×#1566842611yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—14482no—
192C4—7242no—
193C4#1915373011yescharacteristic
194C4#1915373011yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—14482no—
200C2C2×#1915373011yescharacteristic
subgroupG/H
centerC2C2×#19153730
commutatorC2×C4×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#410×C181C2
socleC2×C181C2×#99
prime radical#26041843C2
cosocleC2×D4×C181C23
hypercenter#410D181
hypocenterC181#2128
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26041253C2semidirect
3. #26041071C2semidirect
4. #410×C181C2semidirect
5. C22×D724C4semidirect
7. #19153805C4semidirect
13. #19153775C4semidirect
15. #19153731C22semidirect
19. #99×C181C22semidirect
100. C181#2128semidirect
104. #410D181semidirect
120. #99C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418432
#260412532
#260410712
#410×C1812
#2128181