Finite Group Databasebeta

← #37532921

Subgroups — #37532921

Order 23168.

250 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 61 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#26041625C211yesmaximal, characteristic, complemented
2#26041564C211yesmaximal, characteristic, complemented
3#26041484C211yesmaximal, characteristic, complemented
4#26041023C211yesmaximal, characteristic, complemented
5Q8⋊#13089678C211yesmaximal, characteristic, complemented
6D4⋊#13089678C211yesmaximal, characteristic, complemented
7#550×C181C211yesmaximal, characteristic
8C2×#15668437—21no—
9#19153816C2211yescharacteristic
10#19153747—21no—
11C4⋊#13089678C2211yescharacteristic
12C4⋊#13089678C2211yescharacteristic
13#19153740—21no—
14#19153739—21no—
15#19153737—22no—
16C4×#13089678C2211yescharacteristic
17C2×#41×C181—21no—
18#127×C181—21no—
19#127×C181C2211yescharacteristic
20#121×C181—22no—
21#121×C181—21no—
22C4×Q8×C181C2211yescharacteristic
23C4×D4×C181—21no—
24C4×D4×C181C2211yescharacteristic
25C2×Dic362—21no—
26#15668437—42no—
27C2×#13089678—21no—
28C2×#13089678—21no—
29C2×#13089678—21no—
30C4⋊Dic181—21no—
31C4⋊Dic181—21no—
32#41×C181—42no—
33#41×C181—42no—
34C2×Q8×C181D411yescharacteristic
35C2×D4×C181—22no—
36C2×D4×C181—22no—
37C2×D4×C181—21no—
38C2×D4×C181—21no—
39C2×D4×C181D411yescharacteristic
40C2×D4×C181D412yes—
41C22×C4×C181—22no—
42C22×C4×C181—21no—
43C4⋊C4×C181—21no—
44C4⋊C4×C181D411yescharacteristic
45C4⋊C4×C181D411yescharacteristic
46C22⋊C4×C181—22no—
47C22⋊C4×C181—21no—
48C42×C181—21no—
49C42×C181C2311yescharacteristic
50Dic362—41no—
51Dic362—41no—
52C2×Dic181—41no—
53#13089678—41no—
54#13089678—41no—
55#13089678—41no—
56C23×C181—21no—
57C23×C181—22no—
58Q8×C181—21no—
59Q8×C181SD1612yescomplemented
60D4×C181—21no—
61D4×C181—21no—
62D4×C181—42no—
63D4×C181—44no—
64D4×C181—41no—
65D4×C181—41no—
66D4×C181—41no—
67D4×C181—41no—
68D4×C181—82no—
69C2×C4×C181—21no—
70C2×C4×C181—21no—
71C2×C4×C181—21no—
72C2×C4×C181—44no—
73C2×C4×C181—41no—
74C2×C4×C181—41no—
75C2×C4×C181C2×D411yescharacteristic
76C2×C4×C181C2×D411yescharacteristic
77C2×C4×C181C2×D411yescharacteristic
78C22×C181—41no—
79C22×C181—42no—
80C22×C181—42no—
81C22×C181—41no—
82C22×C181—42no—
83C22×C181—41no—
84C22×C181#12011yescharacteristic
85C4×C181—22no—
86C4×C181—21no—
87C4×C181—41no—
88C4×C181—41no—
89C4×C181C2×SD1611yescharacteristic
90C4×C181#13511yescharacteristic
91C4×C181AGL(1,ℤ/8)11yescharacteristic
92C4×C181AGL(1,ℤ/8)11yescharacteristic
93Dic181—81no—
94C2×C181—81no—
95C2×C181—82no—
96C2×C181#44911yescharacteristic
97C2×C181#45011yescharacteristic
98C2×C181#45511yescharacteristic
99C181#164211yescharacteristic, complemented, Sylow
100#1642—1811nomaximal, Sylow
101#550D18111yescharacteristic
102#486—1811no—
103#475—1811no—
104#461—1811no—
105#331—1811no—
106Q8⋊C8—1811no—
107D4⋊C8—1811no—
108C2×#41—21no—
109C2×SD16—3621no—
110C4⋊Q8—1811no—
111#127—21no—
112#127C2×D18111yescharacteristic
113#121—22no—
114#121—21no—
115C4×Q8C2×D18111yescharacteristic
116C4×D4—21no—
117C4×D4C2×D18111yescharacteristic
118#106—3621no—
119C4⋊C8—1811no—
120C4⋊C8—1811no—
121#103—3621no—
122#103—3621no—
123#102—3622no—
124C4×C8—1811no—
125#41—42no—
126#41—42no—
127C2×Q8—3621no—
128C2×Q8C22⋊D18111yescharacteristic
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C22⋊D18111yescharacteristic
134C2×D4C22⋊D18112yes—
135C22×C4—22no—
136C22×C4—21no—
137SD16—7242no—
138C2×C8—3621no—
139C2×C8—3621no—
140C2×C8—3621no—
141C4⋊C4—21no—
142C4⋊C4—3621no—
143C4⋊C4—3621no—
144C4⋊C4C22⋊D18111yescharacteristic
145C4⋊C4C22⋊D18111yescharacteristic
146C22⋊C4—22no—
147C22⋊C4—21no—
148C42—21no—
149C42C22×D18111yescharacteristic
150C23—21no—
151C23—22no—
152Q8—21no—
153Q8—7241no—
154Q8—7241no—
155Q8#1566843712yescomplemented
156D4—21no—
157D4—21no—
158D4—42no—
159D4—44no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—7241no—
172C2×C4C2×C22⋊D18111yescharacteristic
173C2×C4C2×C22⋊D18111yescharacteristic
174C2×C4C2×C22⋊D18111yescharacteristic
175C8—7241no—
176C8—7241no—
177C8—7241no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22#1915378511yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—41no—
189C4—14481no—
190C4C2×#1566843711yescharacteristic
191C4#1915383611yescharacteristic
192C4#1915384111yescharacteristic
193C4#1915384211yescharacteristic
194C2—81no—
195C2—82no—
196C2#2604142411yescharacteristic
197C2#2604143011yescharacteristic
198C2#2604145711yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting, prime radical#550×C181C2
socleC22×C181#120
hypercenter#550D181
hypocenterC181#1642
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #26041625C2semidirect
2. #26041564C2semidirect
3. #26041484C2semidirect
4. #26041023C2semidirect
5. Q8⋊#13089678C2semidirect
6. D4⋊#13089678C2semidirect
59. Q8×C181SD16semidirect
99. C181#1642semidirect
155. Q8#15668437semidirect

Maximal subgroups

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

subgroupindex
#260416252
#260415642
#260414842
#260410232
Q8⋊#130896782
D4⋊#130896782
#550×C1812
#1642181