Finite Group Databasebeta

← #37533036

Subgroups — #37533036

Order 23168.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#26042027C211yesmaximal, characteristic
2#26041671C211yesmaximal, characteristic, complemented
3#26041641C211yesmaximal, characteristic, complemented
4#26041021C211yesmaximal, characteristic, complemented
5#26041018C211yesmaximal, characteristic, complemented
6#26041018C211yesmaximal, characteristic, complemented
7#501×C181C211yesmaximal, characteristic, complemented
8C2×#15668439—21no—
9C2×#15668436—21no—
10#19153804C2211yescharacteristic
11#19153804C2211yescharacteristic
12#19153801—21no—
13#19153801—21no—
14#19153794—21no—
15#19153794—21no—
16#19153791—21no—
17#19153791—21no—
18#19153739—21no—
19#19153736—21no—
20#19153727C2211yescharacteristic
21#19153727C2211yescharacteristic
22C4⋊Q8×C181C2211yescharacteristic
23#125×C181C2211yescharacteristic
24#107×C181—21no—
25#106×C181—21no—
26#97×C181C2211yescharacteristic
27C2×Dic362D411yescharacteristic
28C2×Dic362D411yescharacteristic
29C2×D724—21no—
30C2×D724—21no—
31C2×D724D411yescharacteristic
32C2×D724D411yescharacteristic
33C2×C4×D181—21no—
34C2×C4×D181—21no—
35#15668439—42no—
36#15668436—42no—
37C2×#13089678—21no—
38C2×#13089678—21no—
39#15668428—21no—
40#15668428—21no—
41#15668426—21no—
42#15668426—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47C2×Q8×C181—21no—
48C2×C8×C181—21no—
49C4⋊C4×C181—21no—
50C4⋊C4×C181—21no—
51C4⋊C4×C181—21no—
52C4⋊C4×C181—21no—
53C4⋊C4×C181D411yescharacteristic
54C4⋊C4×C181D411yescharacteristic
55C42×C181C2311yescharacteristic
56C22×D181—21no—
57C22×D181—21no—
58Dic362—41no—
59Dic362—41no—
60D724—41no—
61D724—41no—
62D724—81no—
63D724—81no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C4×D181—81no—
67C4×D181—81no—
68#13089678—22no—
69#13089678—41no—
70Q8×C181—41no—
71Q8×C181—41no—
72C2×C4×C181—21no—
73C2×C4×C181—21no—
74C2×C4×C181—41no—
75C2×C4×C181C2×D411yescharacteristic
76C2×C4×C181C2×D411yescharacteristic
77C2×C4×C181C2×D411yescharacteristic
78C8×C181—22no—
79C22×C181#12011yescharacteristic
80C2×D181—41no—
81C2×D181—41no—
82C2×D181—41no—
83C2×D181—41no—
84C2×D181—41no—
85C2×D181—41no—
86C4×C181—21no—
87C4×C181—21no—
88C4×C181—21no—
89C4×C181—41no—
90C4×C181—41no—
91C4×C181—81no—
92Dic181—41no—
93Dic181—41no—
94C2×C181#45311yescharacteristic
95C2×C181#45511yescharacteristic
96C2×C181#45611yescharacteristic
97D181—81no—
98D181—81no—
99C181#166111yescharacteristic, complemented, Sylow
100#1661—1811nomaximal, Sylow
101#562—1811no—
102#501D18111yescharacteristic, complemented
103#497—1811no—
104#489—1811no—
105#330—1811no—
106#329—1811no—
107#329—1811no—
108C2×Q16—3621no—
109C2×SD16—3621no—
110C4⋊Q8C2×D18111yescharacteristic
111#125C2×D18111yescharacteristic
112#124—1811no—
113#124—1811no—
114#123—3621no—
115#123—3621no—
116C22⋊Q8—3621no—
117C22⋊Q8—3621no—
118#121—3621no—
119#121—3621no—
120#107—21no—
121#106—21no—
122#103—3621no—
123#102—3621no—
124#97—1811no—
125#97—1811no—
126#97C2×D18111yescharacteristic
127C2×Q8—21no—
128C2×Q8—1811no—
129C2×Q8—1811no—
130C2×D4—3621no—
131C2×D4—3621no—
132C2×D4—1811no—
133C2×D4—1811no—
134C22×C4—3621no—
135C22×C4—3621no—
136Q16—7242no—
137SD16—7242no—
138C2×C8—21no—
139C2×C8—3621no—
140C2×C8—3621no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—3621no—
146C4⋊C4—3621no—
147C4⋊C4C22⋊D18111yescharacteristic
148C4⋊C4C22⋊D18111yescharacteristic
149C22⋊C4—3621no—
150C22⋊C4—3621no—
151C22⋊C4—3621no—
152C22⋊C4—3621no—
153C22⋊C4—3621no—
154C22⋊C4—3621no—
155C42C22×D18111yescharacteristic
156C23—3621no—
157C23—3621no—
158Q8—41no—
159Q8—41no—
160Q8—7241no—
161Q8—7241no—
162D4—14481no—
163D4—14481no—
164D4—7241no—
165D4—7241no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—3621no—
170C2×C4—3621no—
171C2×C4—14481no—
172C2×C4—14481no—
173C2×C4D4×D18111yescharacteristic
174C2×C4D4×D18111yescharacteristic
175C2×C4C2×C22⋊D18111yescharacteristic
176C8—22no—
177C8—3622no—
178C8—7241no—
179C22—7241no—
180C22—7241no—
181C22—7241no—
182C22—7241no—
183C22—7241no—
184C22—7241no—
185C22#1915378411yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—81no—
192C4—7241no—
193C4—7241no—
194C2—14481no—
195C2—14481no—
196C2#2604144511yescharacteristic
197C2#2604145511yescharacteristic
198C2#2604146111yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#501×C181C2
socleC22×C181#120
prime radical#26042027C2
hypercenter#501D181
hypocenterC181#1661
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #26041671C2semidirect
3. #26041641C2semidirect
4. #26041021C2semidirect
5. #26041018C2semidirect
6. #26041018C2semidirect
7. #501×C181C2semidirect
99. C181#1661semidirect
102. #501D181semidirect

Maximal subgroups

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

subgroupindex
#260420272
#260416712
#260416412
#260410212
#260410182
#501×C1812
#1661181