Finite Group Databasebeta

← #37534767

Subgroups — #37534767

Order 23168.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#19153847C212yesmaximal, complemented
2#26041860C211yesmaximal, characteristic
3C2×#19153744C212yesmaximal, complemented
4#26041048C211yesmaximal, characteristic, complemented
5#391×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C2212yes—
7#19153847—24no—
8#19153847—24no—
9C2×#15668439—22no—
10C2×#15668436—22no—
11C2×#15668433C2212yes—
12#19153796—22no—
13#19153791—22no—
14#19153790—22no—
15#19153744—24no—
16C22×Q8×C181C2211yescharacteristic
17C2×C4⋊C4×C181C2211yescharacteristic
18C2×C42×C181C2211yescharacteristic
19#95×C181—22no—
20#15668449—22no—
21#15668449—22no—
22#15668449—22no—
23#15668449—22no—
24C2×Dic362D412yes—
25C2×D724—22no—
26C2×D724D412yes—
27C2×C22⋊D181—22no—
28C2×C4×D181—22no—
29#15668439—28no—
30#15668436—28no—
31#15668433—22no—
32#15668433D414yescomplemented
33C2×#13089678—22no—
34C4⋊Dic181—22no—
35#15668426—24no—
36C2×Q8×C181—21no—
37C2×Q8×C181—22no—
38C2×Q8×C181—22no—
39C2×Q8×C181D412yescomplemented
40C22×C4×C181—21no—
41C22×C4×C181—21no—
42C22×C4×C181D412yes—
43C22×C4×C181C2311yescharacteristic
44C4⋊C4×C181—22no—
45C42×C181—22no—
46C22×D181—22no—
47Dic362—22no—
48Dic362—22no—
49D724—22no—
50D724—22no—
51D724—82no—
52C22⋊D181—42no—
53C22⋊D181—42no—
54C2×Dic181—22no—
55C4×D181—42no—
56C4×D181—42no—
57#13089678—24no—
58C23×C181C2×D411yescharacteristic
59Q8×C181—24no—
60Q8×C181—42no—
61C2×C4×C181—22no—
62C2×C4×C181—22no—
63C2×C4×C181—22no—
64C2×C4×C181—41no—
65C2×C4×C181—41no—
66C2×C4×C181—41no—
67C2×C4×C181—41no—
68C2×C4×C181—42no—
69C2×C4×C181C2×D412yes—
70C2×C4×C181C2×D412yes—
71C2×C4×C181C2×D411yescharacteristic
72C2×C4×C181#4111yescharacteristic
73C22×C181—21no—
74C22×C181—21no—
75C22×C181#12011yescharacteristic
76C22×C181#12111yescharacteristic
77C22×C181#12711yescharacteristic
78C2×D181—42no—
79C2×D181—42no—
80C2×D181—42no—
81C4×C181—42no—
82C4×C181—42no—
83C4×C181—42no—
84C4×C181#12012yes—
85C4×C181#12112yes—
86Dic181—42no—
87C2×C181—21no—
88C2×C181—21no—
89C2×C181#39211yescharacteristic
90C2×C181#45511yescharacteristic
91C2×C181#45611yescharacteristic
92D181—82no—
93C181#200711yescharacteristic, complemented, Sylow
94#2007—1811nomaximal, Sylow
95C2×#137—1812no—
96#537—1811no—
97C2×#104—1812no—
98#391D18111yescharacteristic, complemented
99#343—1811no—
100C2×#41—1812no—
101C22×Q8C2×D18111yescharacteristic
102#137—3624no—
103#137—3624no—
104C2×Q16—3622no—
105C2×SD16—3622no—
106C2×M16—1812no—
107C22⋊Q8—3622no—
108#121—3622no—
109#121—3622no—
110C2×C4⋊C4C2×D18111yescharacteristic
111C2×C42C2×D18111yescharacteristic
112#104—3624no—
113#95—22no—
114#41—3622no—
115#41—3622no—
116#41—3622no—
117#41—3622no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—1812no—
122C2×Q8D72412yescomplemented
123C2×D4—3622no—
124C2×D4—3622no—
125C2×D4—1812no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—3622no—
129C22×C4C22⋊D18112yes—
130C22×C4C22×D18111yescharacteristic
131Q16—3628no—
132SD16—3628no—
133M16—3622no—
134M16—1814no—
135C2×C8—3622no—
136C4⋊C4—22no—
137C4⋊C4—3622no—
138C22⋊C4—3624no—
139C42—22no—
140C23—3622no—
141C23C2×C22⋊D18111yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—3622no—
145Q8—3622no—
146D4—3622no—
147D4—3622no—
148D4—14482no—
149D4—7242no—
150D4—7242no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—3622no—
160C2×C4—7242no—
161C2×C4—7242no—
162C2×C4D4×D18112yes—
163C2×C4D4×D18112yes—
164C2×C4C2×D72411yescharacteristic
165C2×C4#1566844911yescharacteristic
166C8—3624no—
167C22—21no—
168C22—21no—
169C22—7242no—
170C22—7242no—
171C22—7242no—
172C22#1915378411yescharacteristic
173C22#1915379011yescharacteristic
174C22#1915381211yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—7242no—
179C4#1915378612yes—
180C4#1915378912yes—
181C2—21no—
182C2—21no—
183C2—14482no—
184C2#2604118811yescharacteristic
185C2#2604145611yescharacteristic
186C2#2604146111yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#391×C181C2
socleC22×C181#120
prime radical#26041860C2
hypercenter#391D181
hypocenterC181#2007
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#19153847C2semidirect
3. C2×#19153744C2semidirect
4. #26041048C2semidirect
5. #391×C181C2semidirect
32. #15668433D4semidirect
39. C2×Q8×C181D4semidirect
93. C181#2007semidirect
98. #391D181semidirect
122. C2×Q8D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
#260418602
C2×#191537442
#260410482
#391×C1812
#2007181