Finite Group Databasebeta

← #37543251

Subgroups — #37543251

Order 23168.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153847C211yesmaximal, characteristic
2C22×#15668439C211yesmaximal, characteristic, complemented
3#26041855C211yesmaximal, characteristic
4#26041501C214yesmaximal, complemented
5#26041489C214yesmaximal, complemented
6#26041306C211yesmaximal, characteristic, complemented
7#26041287C211yesmaximal, characteristic, complemented
8#26041273C211yesmaximal, characteristic
9#519×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2211yescharacteristic
11C22×Dic362C2211yescharacteristic
12#19153847—24no—
13C2×#15668439—24no—
14C2×#15668439C2212yes—
15C2×#15668439C2214yes—
16C2×#15668436C2212yes—
17C2×#15668433C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19#19153816C2212yes—
20#19153804C2212yes—
21#19153800—22no—
22#19153796—22no—
23C4⋊#13089678C2214yes—
24#19153739C2212yes—
25#19153739C2214yes—
26#19153736C2212yes—
27C22×Q8×C181C2211yescharacteristic
28C4×Q8×C181—22no—
29C4×Q8×C181C2214yes—
30#117×C181—22no—
31#117×C181C2211yescharacteristic
32#117×C181C2211yescharacteristic
33#15668449—22no—
34#15668449—22no—
35C2×Dic362—21no—
36C2×Dic362—22no—
37C2×Dic362—22no—
38C2×Dic362C2312yes—
39C2×Dic362C2311yescharacteristic
40C2×D724C2311yescharacteristic
41C2×C22⋊D181—21no—
42C22×Dic181—21no—
43C2×C4×D181—21no—
44#15668439—28no—
45#15668439—24no—
46#15668436—24no—
47#15668433—22no—
48C2×#13089678—22no—
49C2×#13089678C2312yes—
50C2×#13089678C2312yes—
51C4⋊Dic181—24no—
52#15668426—24no—
53C2×Q8×C181—21no—
54C2×Q8×C181—22no—
55C2×Q8×C181D414yes—
56C2×Q8×C181C2312yes—
57C22×C4×C181—21no—
58C22×C4×C181C2311yescharacteristic
59C4⋊C4×C181—24no—
60C4⋊C4×C181D412yes—
61C4⋊C4×C181C2312yes—
62C22⋊C4×C181—21no—
63C22⋊C4×C181D412yes—
64C42×C181—24no—
65C42×C181—21no—
66C42×C181C2312yes—
67C22×D181—21no—
68Dic362—22no—
69Dic362—21no—
70Dic362—22no—
71Dic362—21no—
72Dic362—42no—
73D724—21no—
74D724—21no—
75C22⋊D181—42no—
76C2×Dic181—22no—
77C2×Dic181—21no—
78C2×Dic181—42no—
79C4×D181—42no—
80#13089678—22no—
81#13089678—22no—
82C23×C181C2×D411yescharacteristic
83Q8×C181—24no—
84Q8×C181—22no—
85Q8×C181C2×D414yes—
86C2×C4×C181—24no—
87C2×C4×C181—22no—
88C2×C4×C181—22no—
89C2×C4×C181C2×D414yes—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D412yes—
92C2×C4×C181#4112yes—
93C2×C4×C181C2411yescharacteristic
94C22×C181—22no—
95C22×C181#12112yes—
96C22×C181C22×D411yescharacteristic
97C2×D181—41no—
98C2×D181—42no—
99C4×C181—24no—
100C4×C181—24no—
101C4×C181—42no—
102C4×C181#12112yes—
103C4×C181C22×D411yescharacteristic
104C4×C181C2×#4111yescharacteristic
105Dic181—41no—
106Dic181—42no—
107C2×C181—22no—
108C2×C181C2×#12111yescharacteristic
109C2×C181#57812yes—
110D181—81no—
111C181#304411yescharacteristic, complemented, Sylow
112#3044—1811nomaximal, Sylow
113C2×#137—1811no—
114C22×Q16—1811no—
115#536—1811no—
116#519D18111yescharacteristic, complemented
117#464—1814no—
118#462—1814no—
119#424—1811no—
120#419—1811no—
121#416—1811no—
122C2×#41—1811no—
123C22×Q8—1811no—
124C22×Q8C2×D18111yescharacteristic
125#137—3624no—
126C2×Q16—3624no—
127C2×Q16—1814no—
128C2×Q16—1812no—
129C2×SD16—1812no—
130C2×M16—1811no—
131C22×C8—1811no—
132C4⋊Q8—1812no—
133#124—1812no—
134#123—3622no—
135C22⋊Q8—3622no—
136C4×Q8—22no—
137C4×Q8C2×D18114yes—
138#117—22no—
139#117C2×D18111yescharacteristic
140#117C2×D18111yescharacteristic
141C4⋊C8—1814no—
142#103—1812no—
143#103—1814no—
144#102—1812no—
145#41—3622no—
146#41—3622no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3621no—
150C2×Q8—3622no—
151C2×Q8—3622no—
152C2×Q8—1812no—
153C2×Q8—1811no—
154C2×Q8D72414yes—
155C2×Q8C22×D18112yes—
156C2×D4—3621no—
157C2×D4—1811no—
158C22×C4—21no—
159C22×C4—3621no—
160C22×C4—3621no—
161C22×C4C22×D18111yescharacteristic
162Q16—3628no—
163Q16—3624no—
164SD16—3624no—
165M16—3622no—
166C2×C8—3622no—
167C2×C8—1812no—
168C2×C8—1812no—
169C4⋊C4—24no—
170C4⋊C4—3624no—
171C4⋊C4C22⋊D18112yes—
172C4⋊C4C22×D18112yes—
173C22⋊C4—21no—
174C22⋊C4—3624no—
175C22⋊C4C22⋊D18112yes—
176C42—24no—
177C42—21no—
178C42C22×D18112yes—
179C23—3621no—
180C23C2×C22⋊D18111yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—3622no—
184Q8—3621no—
185Q8—3622no—
186Q8—3621no—
187Q8—7242no—
188Q8C2×D72414yes—
189D4—3621no—
190D4—3621no—
191D4—7242no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3622no—
196C2×C4—3621no—
197C2×C4—7242no—
198C2×C4—7242no—
199C2×C4C2×C22⋊D18114yes—
200C2×C4C2×C22⋊D18111yescharacteristic
201C2×C4C2×D72412yes—
202C2×C4#1566844912yes—
203C2×C4C23×D18111yescharacteristic
204C8—3622no—
205C8—3622no—
206C22—22no—
207C22—7241no—
208C22—7242no—
209C22#1915379012yes—
210C22C22×C22⋊D18111yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—7241no—
215C4—7242no—
216C4#1915379012yes—
217C4C22×D72411yescharacteristic
218C4C2×#1566844911yescharacteristic
219C2—22no—
220C2—14481no—
221C2C2×#1915379011yescharacteristic
222C2#2604209412yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#519×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668449C22
hypercenter#519D181
hypocenterC181#3044
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. C22×#15668439C2semidirect
4. #26041501C2semidirect
5. #26041489C2semidirect
6. #26041306C2semidirect
7. #26041287C2semidirect
9. #519×C181C2semidirect
111. C181#3044semidirect
116. #519D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
C22×#156684392
#260418552
#260415012
#260414892
#260413062
#260412872
#260412732
#519×C1812
#3044181