Finite Group Databasebeta

← #37544012

Subgroups — #37544012

Order 23168.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041884C211yesmaximal, characteristic
2#26041603C212yesmaximal, complemented
3#26041586C212yesmaximal, complemented
4#26041479C212yesmaximal, complemented
5#26041418C212yesmaximal, complemented
6#26041377C212yesmaximal, complemented
7C4×#15668436C212yesmaximal, complemented
8#26041340C211yesmaximal, characteristic, complemented
9#519×C181C211yesmaximal, characteristic, complemented
10C2×#15668436C2212yes—
11C2×#15668436C2212yes—
12#19153813C2211yescharacteristic
13#19153808C2211yescharacteristic
14#19153801—22no—
15#19153790C2212yes—
16#19153789—22no—
17C4×D724C2212yes—
18#19153749C2212yes—
19#19153747C2212yes—
20C4⋊#13089678C2212yes—
21#19153740C2212yes—
22#19153740C2212yes—
23#19153736C2212yes—
24#19153736C2212yes—
25C22⋊#13089678C2212yes—
26#19153727C2211yescharacteristic
27C4×#13089678C2211yescharacteristic
28C22×Q8×C181C2211yescharacteristic
29C4×Q8×C181—22no—
30C4×Q8×C181C2212yes—
31C4×Q8×C181C2212yes—
32#117×C181—22no—
33#117×C181C2211yescharacteristic
34#117×C181C2211yescharacteristic
35C2×D724—22no—
36C2×D724C2312yes—
37C2×C22⋊D181—22no—
38C2×C4×D181—22no—
39#15668436—24no—
40#15668436—24no—
41C2×#13089678C2312yes—
42C2×#13089678C2312yes—
43C4⋊Dic181C2312yes—
44#15668428—22no—
45#15668426—22no—
46#15668426—22no—
47C2×Q8×C181—24no—
48C2×Q8×C181—21no—
49C2×Q8×C181C2312yes—
50C22×C4×C181—21no—
51C22×C4×C181C2311yescharacteristic
52C4⋊C4×C181—22no—
53C4⋊C4×C181—22no—
54C4⋊C4×C181D412yes—
55C4⋊C4×C181C2312yes—
56C22⋊C4×C181—21no—
57C22⋊C4×C181D412yes—
58C42×C181—22no—
59C42×C181—22no—
60C42×C181—21no—
61C42×C181C2311yescharacteristic
62C42×C181C2311yescharacteristic
63C22×D181—22no—
64D724—22no—
65D724—22no—
66D724—82no—
67C22⋊D181—82no—
68C2×Dic181—22no—
69C4×D181—82no—
70#13089678—22no—
71#13089678—22no—
72C23×C181C2×D411yescharacteristic
73Q8×C181—24no—
74Q8×C181—22no—
75Q8×C181#4114yes—
76C2×C4×C181—24no—
77C2×C4×C181—22no—
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181C2×D412yes—
81C2×C4×C181C2×D412yes—
82C2×C4×C181C2×D411yescharacteristic
83C2×C4×C181C2411yescharacteristic
84C22×C181—21no—
85C22×C181—22no—
86C22×C181C22×D411yescharacteristic
87C2×D181—42no—
88C2×D181—42no—
89C2×D181—42no—
90C4×C181—24no—
91C4×C181—21no—
92C4×C181—22no—
93C4×C181—22no—
94C4×C181—42no—
95C4×C181C2×#4112yes—
96Dic181—42no—
97C2×C181—41no—
98C2×C181#52511yescharacteristic
99C2×C181#57712yes—
100D181—82no—
101C181#312611yescharacteristic, complemented, Sylow
102#3126—1811nomaximal, Sylow
103#540—1811no—
104#519D18111yescharacteristic, complemented
105#482—1812no—
106#479—1812no—
107#460—1812no—
108#448—1812no—
109#440—1812no—
110C4×SD16—1812no—
111#433—1811no—
112C22×Q8C2×D18111yescharacteristic
113C2×SD16—1812no—
114C2×SD16—1812no—
115#127—1811no—
116#125—1811no—
117#123—3622no—
118#121—3622no—
119#121—1812no—
120C4×Q8—22no—
121C4×Q8C2×D18112yes—
122C4×Q8C2×D18112yes—
123C4×D4—1812no—
124#117—22no—
125#117C2×D18111yescharacteristic
126#117C2×D18111yescharacteristic
127#107—1812no—
128#106—1812no—
129C4⋊C8—1812no—
130#103—1812no—
131#103—1812no—
132#102—1812no—
133#102—1812no—
134C22⋊C8—1812no—
135#97—1811no—
136C4×C8—1811no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D18112yes—
140C2×D4—3622no—
141C2×D4—3622no—
142C2×D4—1812no—
143C22×C4—21no—
144C22×C4—3622no—
145C22×C4C22×D18111yescharacteristic
146SD16—3624no—
147SD16—3624no—
148C2×C8—1812no—
149C2×C8—1812no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3622no—
153C4⋊C4—1812no—
154C4⋊C4C22⋊D18112yes—
155C4⋊C4C22×D18112yes—
156C22⋊C4—21no—
157C22⋊C4—3622no—
158C22⋊C4—3622no—
159C22⋊C4C22⋊D18112yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D18111yescharacteristic
164C42C22×D18111yescharacteristic
165C23—3622no—
166C23C2×C22⋊D18111yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1566844914yes—
170D4—3622no—
171D4—3622no—
172D4—14482no—
173D4—14482no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3622no—
179C2×C4—14482no—
180C2×C4C2×C22⋊D18112yes—
181C2×C4C2×C22⋊D18112yes—
182C2×C4C2×C22⋊D18111yescharacteristic
183C2×C4C23×D18111yescharacteristic
184C8—3622no—
185C8—3622no—
186C22—21no—
187C22—22no—
188C22—7242no—
189C22—7242no—
190C22—7242no—
191C22C22×C22⋊D18111yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—7242no—
198C4C2×#1566844912yes—
199C2—41no—
200C2—14482no—
201C2#2604179011yescharacteristic
202C2#2604208712yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#519×C181C2
socleC22×C181C22×D4
prime radical#26041884C2
hypercenter#519D181
hypocenterC181#3126
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #26041603C2semidirect
3. #26041586C2semidirect
4. #26041479C2semidirect
5. #26041418C2semidirect
6. #26041377C2semidirect
7. C4×#15668436C2semidirect
8. #26041340C2semidirect
9. #519×C181C2semidirect
101. C181#3126semidirect
104. #519D181semidirect

Maximal subgroups

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

subgroupindex
#260418842
#260416032
#260415862
#260414792
#260414182
#260413772
C4×#156684362
#260413402
#519×C1812
#3126181