Finite Group Databasebeta

← #37532740

Subgroups — #37532740

Order 23168.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#26041841C211yesmaximal, characteristic
2#26041597C212yesmaximal, complemented
3#26041012C212yesmaximal, complemented
4#26041003C211yesmaximal, characteristic, complemented
5#399×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181—21no—
7#19153800—22no—
8#19153790—21no—
9#19153790C2212yes—
10#19153786—22no—
11#19153747—22no—
12#19153736—22no—
13C22⋊#13089678C2212yescomplemented
14C2×C4⋊C4×C181—21no—
15C2×C4⋊C4×C181C2211yescharacteristic, complemented
16C2×C22⋊C4×C181—21no—
17C2×C22⋊C4×C181C2211yescharacteristic
18#95×C181C2211yescharacteristic, complemented
19C23×D181—21no—
20C2×D724—21no—
21C2×D724D412yes—
22C2×C22⋊D181—22no—
23C2×C22⋊D181—22no—
24C2×C22⋊D181—42no—
25C2×C22⋊D181—42no—
26C22×Dic181—21no—
27C2×#13089678—22no—
28C4⋊Dic181—21no—
29C4⋊Dic181D412yes—
30#15668426—22no—
31#15668426—22no—
32C24×C181D411yescharacteristic
33C22×C4×C181—21no—
34C22×C4×C181—21no—
35C22×C4×C181D411yescharacteristic
36C22×C4×C181C2311yescharacteristic
37C4⋊C4×C181—22no—
38C4⋊C4×C181—42no—
39C22⋊C4×C181—22no—
40C22⋊C4×C181—42no—
41C22×D181—22no—
42C22×D181—42no—
43C22×D181—81no—
44C22×D181—81no—
45D724—42no—
46D724—81no—
47C22⋊D181—44no—
48C22⋊D181—42no—
49C22⋊D181—42no—
50C2×Dic181—22no—
51C2×Dic181—42no—
52#13089678—42no—
53C23×C181—21no—
54C23×C181—21no—
55C23×C181—41no—
56C23×C181—41no—
57C23×C181SD1612yes—
58C23×C181C2×D411yescharacteristic
59C2×C4×C181—22no—
60C2×C4×C181—21no—
61C2×C4×C181—41no—
62C2×C4×C181—41no—
63C2×C4×C181—41no—
64C2×C4×C181—41no—
65C2×C4×C181—41no—
66C2×C4×C181—41no—
67C2×C4×C181—41no—
68C2×C4×C181C2×D412yes—
69C22×C181—21no—
70C22×C181—21no—
71C22×C181—22no—
72C22×C181—42no—
73C22×C181—41no—
74C22×C181—41no—
75C22×C181—81no—
76C22×C181#12011yescharacteristic
77C22×C181C2×SD1611yescharacteristic
78C22×C181AGL(1,ℤ/8)11yescharacteristic
79C2×D181—42no—
80C2×D181—42no—
81C2×D181—42no—
82C2×D181—84no—
83C4×C181—22no—
84C4×C181—42no—
85C4×C181—41no—
86C4×C181—81no—
87Dic181—42no—
88C2×C181—21no—
89C2×C181—21no—
90C2×C181—42no—
91C2×C181#45011yescharacteristic
92C2×C181#45311yescharacteristic
93C2×C181#45811yescharacteristic
94D181—82no—
95C181#161311yescharacteristic, complemented, Sylow
96#1613—1811nomaximal, Sylow
97#534—1811no—
98#481—1812no—
99#399D18111yescharacteristic, complemented
100#327—1812no—
101#323—1811no—
102C22×D4—3621no—
103#123—3622no—
104#121—3621no—
105#121—1812no—
106#120—3622no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D18111yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D18111yescharacteristic
111#106—3622no—
112#102—3622no—
113C22⋊C8—1812no—
114#95C2×D18111yescharacteristic, complemented
115C24—3621no—
116C24C22⋊D18111yescharacteristic
117C2×D4—3621no—
118C2×D4—3622no—
119C2×D4—3622no—
120C2×D4—1812no—
121C2×D4—7242no—
122C2×D4—7242no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—3621no—
126C22×C4C22⋊D18111yescharacteristic
127C22×C4C22×D18111yescharacteristic
128C2×C8—3622no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—3621no—
132C4⋊C4—1812no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—3622no—
136C22⋊C4—3622no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—3622no—
142C23—14481no—
143C23—14481no—
144C23—7242no—
145C23#1566843612yes—
146C23C2×C22⋊D18111yescharacteristic
147D4—14481no—
148D4—7242no—
149D4—7244no—
150D4—7242no—
151D4—7242no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3622no—
162C2×C4—7242no—
163C2×C4D4×D18112yes—
164C8—7242no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—14484no—
173C22—7242no—
174C22—7242no—
175C22—7242no—
176C22#1915378411yescharacteristic
177C22C2×#1566843611yescharacteristic
178C22#1915384111yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—7242no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—14482no—
188C2#2604142911yescharacteristic
189C2#2604144611yescharacteristic
190C2#2604146811yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#399×C181C2
socleC22×C181#120
prime radical#26041841C2
hypercenter#399D181
hypocenterC181#1613
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #26041597C2semidirect
3. #26041012C2semidirect
4. #26041003C2semidirect
5. #399×C181C2semidirect
13. C22⋊#13089678C22semidirect
15. C2×C4⋊C4×C181C22semidirect
18. #95×C181C22semidirect
95. C181#1613semidirect
99. #399D181semidirect
108. C2×C4⋊C4C2×D181semidirect
114. #95C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418412
#260415972
#260410122
#260410032
#399×C1812
#1613181