Finite Group Databasebeta

← #37532663

Subgroups — #37532663

Order 23168.

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

normal subgroups only

This group has 266 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#26041910C211yesmaximal, characteristic, complemented
2#26041432C212yesmaximal
3#26041196C211yesmaximal, characteristic
4#26041017C212yesmaximal, complemented
5#324×C181C211yesmaximal, characteristic, complemented
6C22×D724C2211yescharacteristic
7C2×#15668438—22no—
8#19153815—22no—
9#19153807—22no—
10#19153795—21no—
11#19153795C2212yes—
12#19153775—21no—
13C2×#15668426—21no—
14C2×#15668426—21no—
15#19153738—22no—
16#19153723C2211yescharacteristic
17C2×C4⋊C4×C181C2211yescharacteristic
18C22⋊C8×C181C2212yes—
19C23×D181—21no—
20C2×Dic362—21no—
21C2×Dic362D412yes—
22C2×D724—22no—
23C2×D724—42no—
24C22×Dic181—21no—
25#15668438—44no—
26C4⋊Dic181—21no—
27C4⋊Dic181—21no—
28C4⋊Dic181—21no—
29C4⋊Dic181D412yes—
30#15668428—22no—
31#15668428—22no—
32C22⋊Dic181—22no—
33#15668426—42no—
34#15668426—42no—
35C4×Dic181—22no—
36C22×C4×C181D411yescharacteristic
37C22×C4×C181D411yescharacteristic
38C22×C4×C181C2311yescharacteristic
39C2×C8×C181—22no—
40C4⋊C4×C181—22no—
41C22×D181—22no—
42C22×D181—41no—
43C22×D181—41no—
44C22×D181—41no—
45C22×D181—41no—
46C22×D181—41no—
47C22×D181—41no—
48Dic362—42no—
49Dic362—81no—
50D724—24no—
51D724—42no—
52C2×Dic181—22no—
53C2×Dic181—22no—
54C2×Dic181—41no—
55C2×Dic181—41no—
56C2×Dic181—41no—
57C23×C181C2×D411yescharacteristic
58C2×C4×C181—21no—
59C2×C4×C181—21no—
60C2×C4×C181—21no—
61C2×C4×C181—41no—
62C2×C4×C181—41no—
63C2×C4×C181SD1612yescomplemented
64C2×C4×C181C2×D412yes—
65C8×C181—42no—
66C22×C181—21no—
67C22×C181—21no—
68C22×C181#12011yescharacteristic
69C22×C181C2×SD1611yescharacteristic
70C22×C181#13711yescharacteristic
71C2×D181—42no—
72C2×D181—42no—
73C2×D181—42no—
74C2×D181—81no—
75C2×D181—81no—
76C2×D181—81no—
77C2×D181—81no—
78C4×C181—22no—
79C4×C181—22no—
80C4×C181—41no—
81Dic181—42no—
82Dic181—42no—
83Dic181—81no—
84C2×C181—21no—
85C2×C181—21no—
86C2×C181#44811yescharacteristic
87C2×C181#45411yescharacteristic
88C2×C181B(4,𝔽2)11yescharacteristic
89D181—82no—
90C181#159911yescharacteristic, complemented, Sylow
91#1599—1811nomaximal, Sylow
92#544—1811no—
93#450—1812no—
94#394—1811no—
95#328—1812no—
96#324D18111yescharacteristic, complemented
97C22×D4—1811no—
98C2×SD16—3622no—
99C4⋊Q8—3622no—
100#125—3622no—
101C22⋊Q8—3621no—
102C22⋊Q8—1812no—
103#117—3621no—
104C2×C4⋊C4C2×D18111yescharacteristic
105C2×C22⋊C4—3621no—
106C2×C22⋊C4—3621no—
107#102—3622no—
108C22⋊C8C2×D18112yes—
109#95—1811no—
110C24—3621no—
111C2×Q8—3621no—
112C2×Q8—1812no—
113C2×D4—3622no—
114C2×D4—7242no—
115C22×C4—3621no—
116C22×C4D72411yescharacteristic
117C22×C4D72411yescharacteristic
118C22×C4C22×D18111yescharacteristic
119SD16—7244no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—3621no—
123C4⋊C4—3621no—
124C4⋊C4—3621no—
125C4⋊C4—3622no—
126C4⋊C4—3622no—
127C4⋊C4—1812no—
128C22⋊C4—3622no—
129C22⋊C4—7242no—
130C22⋊C4—7242no—
131C42—3622no—
132C23—3622no—
133C23—7241no—
134C23—7241no—
135C23—7241no—
136C23—7241no—
137C23—7241no—
138C23—7241no—
139C23C2×D72411yescharacteristic
140Q8—14481no—
141Q8—7242no—
142D4—3624no—
143D4—7242no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3622no—
150C2×C4—3622no—
151C2×C4—7241no—
152C2×C4—7241no—
153C2×C4—7241no—
154C2×C4#1566843812yescomplemented
155C2×C4D4×D18112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—14481no—
160C22—14481no—
161C22—14481no—
162C22—14481no—
163C22—7242no—
164C22—7242no—
165C22—7242no—
166C22#1915378611yescharacteristic
167C22C2×#1566843811yescharacteristic
168C22#1915384911yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—14481no—
173C4—7242no—
174C4—7242no—
175C2—21no—
176C2—21no—
177C2—14482no—
178C2#2604142011yescharacteristic
179C2#2604145411yescharacteristic
180C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#324×C181C2
socleC22×C181#120
prime radicalC22×D724C22
hypercenter#324D181
hypocenterC181#1599
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #26041910C2semidirect
4. #26041017C2semidirect
5. #324×C181C2semidirect
63. C2×C4×C181SD16semidirect
90. C181#1599semidirect
96. #324D181semidirect
154. C2×C4#15668438semidirect

Maximal subgroups

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

subgroupindex
#260419102
#260414322
#260411962
#260410172
#324×C1812
#1599181