Finite Group Databasebeta

← #37534768

Subgroups — #37534768

Order 23168.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19153846C212yesmaximal, complemented
2C2×#19153743C212yesmaximal, complemented
3#26041184C211yesmaximal, characteristic, complemented
4#26041048C211yesmaximal, characteristic, complemented
5#537×C181C211yesmaximal, characteristic
6C22×Dic362C2211yescharacteristic
7#19153846—24no—
8#19153846—24no—
9C2×#15668439—22no—
10C2×#15668437—22no—
11C2×#15668433C2212yes—
12C2×C4×Dic181C2211yescharacteristic
13#19153743—24no—
14#19153723—22no—
15C2×#41×C181C2212yes—
16C22⋊Q8×C181—22no—
17#121×C181—22no—
18#121×C181—22no—
19C2×C4⋊C4×C181C2211yescharacteristic
20C2×Dic362—22no—
21C2×Dic362—22no—
22C2×Dic362—21no—
23C2×Dic362D412yescomplemented
24C22×Dic181—21no—
25C22×Dic181—21no—
26#15668439—28no—
27#15668437—28no—
28#15668433—22no—
29#15668433D414yescomplemented
30C2×#13089678—22no—
31C4×Dic181—22no—
32#41×C181—22no—
33#41×C181—22no—
34#41×C181—22no—
35#41×C181—22no—
36C2×Q8×C181D412yes—
37C2×D4×C181—22no—
38C2×D4×C181—22no—
39C2×D4×C181D412yes—
40C22×C4×C181—22no—
41C22×C4×C181D412yes—
42C22×C4×C181C2311yescharacteristic
43C4⋊C4×C181—22no—
44C4⋊C4×C181—22no—
45C22⋊C4×C181—24no—
46Dic362—24no—
47Dic362—42no—
48C2×Dic181—22no—
49C2×Dic181—22no—
50C2×Dic181—41no—
51C2×Dic181—41no—
52C2×Dic181—41no—
53C2×Dic181—41no—
54#13089678—24no—
55C23×C181—22no—
56C23×C181C2×D411yescharacteristic
57Q8×C181—22no—
58Q8×C181—22no—
59D4×C181—22no—
60D4×C181—22no—
61D4×C181—42no—
62D4×C181—42no—
63D4×C181—82no—
64C2×C4×C181—22no—
65C2×C4×C181—22no—
66C2×C4×C181—42no—
67C2×C4×C181—42no—
68C2×C4×C181—42no—
69C2×C4×C181C2×D411yescharacteristic
70C2×C4×C181C2×D412yes—
71C2×C4×C181C2×D412yes—
72C2×C4×C181#4111yescharacteristic
73C22×C181—21no—
74C22×C181—21no—
75C22×C181—42no—
76C22×C181—42no—
77C22×C181—42no—
78C22×C181#12011yescharacteristic
79C22×C181#12111yescharacteristic
80C22×C181#12711yescharacteristic
81C4×C181—42no—
82C4×C181—42no—
83C4×C181#12012yes—
84C4×C181#12112yes—
85Dic181—42no—
86Dic181—42no—
87C2×C181—21no—
88C2×C181—21no—
89C2×C181—82no—
90C2×C181#39211yescharacteristic
91C2×C181#45511yescharacteristic
92C2×C181#45611yescharacteristic
93C181#200711yescharacteristic, complemented, Sylow
94#2007—1811nomaximal, Sylow
95C2×#137—1812no—
96#537D18111yescharacteristic
97C2×#104—1812no—
98#391—1811no—
99#343—1811no—
100C2×#41C2×D18112yes—
101C22×Q8—1811no—
102#137—3624no—
103#137—3624no—
104C2×Q16—3622no—
105C2×SD16—3622no—
106C2×M16—1812no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D18111yescharacteristic
111C2×C42—1811no—
112#104—3624no—
113#95—3622no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—3621no—
119C2×Q8—3622no—
120C2×Q8—3622no—
121C2×Q8—1812no—
122C2×Q8C22⋊D18112yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D18112yes—
126C22×C4—22no—
127C22×C4—3621no—
128C22×C4—3621no—
129C22×C4C22⋊D18112yes—
130C22×C4C22×D18111yescharacteristic
131Q16—3628no—
132SD16—3628no—
133M16—3622no—
134M16—1814no—
135C2×C8—3622no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—3622no—
140C23—22no—
141C23C2×C22⋊D18111yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—3624no—
145Q8—7242no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—3622no—
157C2×C4—3622no—
158C2×C4—7241no—
159C2×C4—7241no—
160C2×C4—7241no—
161C2×C4—7241no—
162C2×C4D4×D18111yescharacteristic
163C2×C4D4×D18112yes—
164C2×C4C2×C22⋊D18112yes—
165C2×C4#1566844811yescharacteristic
166C8—3624no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1915378511yescharacteristic
173C22#1915378711yescharacteristic
174C22#1915381211yescharacteristic
175C4—42no—
176C4—42no—
177C4—7242no—
178C4—7242no—
179C4#1915378412yes—
180C4#1915378812yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2604118711yescharacteristic
185C2#2604145711yescharacteristic
186C2#2604146211yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting, prime radical#537×C181C2
socleC22×C181#120
hypercenter#537D181
hypocenterC181#2007
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#19153846C2semidirect
2. C2×#19153743C2semidirect
3. #26041184C2semidirect
4. #26041048C2semidirect
23. C2×Dic362D4semidirect
29. #15668433D4semidirect
93. C181#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#191538462
C2×#191537432
#260411842
#260410482
#537×C1812
#2007181