Finite Group Databasebeta

← #37543250

Subgroups — #37543250

Order 23168.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19153849C211yesmaximal, characteristic
2C22×#15668439C211yesmaximal, characteristic, complemented
3#26041853C211yesmaximal, characteristic
4#26041756C211yesmaximal, characteristic, complemented
5#26041504C214yesmaximal, complemented
6#26041488C214yesmaximal, complemented
7#26041307C211yesmaximal, characteristic, complemented
8#26041273C211yesmaximal, characteristic
9#419×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2211yescharacteristic
11C22×Dic362C2211yescharacteristic
12#19153849—24no—
13C2×Dic724C2212yes—
14C2×#15668439—24no—
15C2×#15668439C2214yes—
16C2×#15668438C2212yes—
17C22×#13089678C2211yescharacteristic
18#19153814C2212yes—
19#19153805C2212yes—
20#19153799—22no—
21#19153792—22no—
22C4×Dic362—22no—
23#19153781C2214yes—
24#19153775C2211yescharacteristic
25#19153772—22no—
26#19153745C2214yes—
27#19153739C2212yes—
28#19153736C2212yes—
29C22×Q8×C181C2211yescharacteristic
30C2×M16×C181C2211yescharacteristic
31#117×C181C2211yescharacteristic
32#103×C181C2214yes—
33#15668449—22no—
34#15668449—22no—
35C2×Dic362—21no—
36C2×Dic362—22no—
37C2×Dic362D414yes—
38C2×Dic362C2311yescharacteristic
39C2×Dic362C2312yes—
40C2×D724C2311yescharacteristic
41C2×C22⋊D181—21no—
42C22×Dic181—21no—
43C2×C4×D181—21no—
44Dic724—24no—
45#15668439—28no—
46#15668438—24no—
47C2×#13089678—22no—
48C2×#13089678C2312yes—
49C4⋊Dic181D412yes—
50#15668428—24no—
51#15668428—24no—
52C22⋊Dic181D412yes—
53#15668426—24no—
54C4×Dic181—24no—
55C4×Dic181C2312yes—
56C2×Q8×C181—21no—
57C2×Q8×C181—22no—
58C2×Q8×C181—22no—
59C2×Q8×C181C2312yes—
60C22×C4×C181—21no—
61C22×C4×C181C2311yescharacteristic
62M16×C181—22no—
63C2×C8×C181C2312yes—
64C4⋊C4×C181C2312yes—
65C22⋊C4×C181—21no—
66C42×C181—21no—
67C22×D181—21no—
68Dic362—24no—
69Dic362—22no—
70Dic362—21no—
71Dic362—21no—
72Dic362C2×D414yes—
73D724—21no—
74D724—21no—
75C22⋊D181—42no—
76C2×Dic181—24no—
77C2×Dic181—22no—
78C2×Dic181—21no—
79C2×Dic181C2×D414yes—
80C4×D181—42no—
81#13089678—22no—
82C23×C181C2×D411yescharacteristic
83Q8×C181—22no—
84Q8×C181—22no—
85Q8×C181—42no—
86C2×C4×C181—22no—
87C2×C4×C181—22no—
88C2×C4×C181—42no—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2×D412yes—
91C2×C4×C181#4112yes—
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181—22no—
95C22×C181#12112yes—
96C22×C181C22×D411yescharacteristic
97C2×D181—41no—
98C2×D181—42no—
99C4×C181—42no—
100C4×C181—42no—
101C4×C181#12112yes—
102C4×C181C22×D411yescharacteristic
103C4×C181C2×#4111yescharacteristic
104Dic181—24no—
105Dic181—24no—
106Dic181—41no—
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#519—1811no—
117#464—1814no—
118#462—1814no—
119#424—1811no—
120#419D18111yescharacteristic, complemented
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×M16C2×D18111yescharacteristic
131C22×C8—1811no—
132C4⋊Q8—1812no—
133#124—1812no—
134#123—3622no—
135C22⋊Q8—3622no—
136C4×Q8—3622no—
137C4×Q8—1814no—
138#117—3622no—
139#117—1811no—
140#117C2×D18111yescharacteristic
141C4⋊C8—1814no—
142#103—1812no—
143#103C2×D18114yes—
144#102—1812no—
145#41—3622no—
146#41—3622no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—3621no—
151C2×Q8—3622no—
152C2×Q8—1814no—
153C2×Q8—1811no—
154C2×Q8—1812no—
155C2×Q8C22×D18112yes—
156C2×D4—3621no—
157C2×D4—1811no—
158C22×C4—21no—
159C22×C4—3621no—
160C22×C4—3621no—
161C22×C4C22×D18111yescharacteristic
162Q16—3624no—
163Q16—3628no—
164SD16—3624no—
165M16—22no—
166C2×C8—3622no—
167C2×C8—1812no—
168C2×C8C22×D18112yes—
169C4⋊C4—3624no—
170C4⋊C4—3624no—
171C4⋊C4—1812no—
172C4⋊C4C22×D18112yes—
173C22⋊C4—21no—
174C22⋊C4—3624no—
175C22⋊C4—1812no—
176C42—21no—
177C42—3624no—
178C42—1812no—
179C23—3621no—
180C23D4×D18111yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—3624no—
185Q8—3622no—
186Q8—3621no—
187Q8—3621no—
188Q8—1814no—
189D4—3621no—
190D4—3621no—
191D4—7242no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—3624no—
196C2×C4—3622no—
197C2×C4—3621no—
198C2×C4—1814no—
199C2×C4—7242no—
200C2×C4D4×D18111yescharacteristic
201C2×C4D4×D18112yes—
202C2×C4#1566844912yes—
203C2×C4C23×D18111yescharacteristic
204C8—22no—
205C8—3622no—
206C22—22no—
207C22—7241no—
208C22—7242no—
209C22#1915378912yes—
210C22C2×D4×D18111yescharacteristic
211C4—42no—
212C4—42no—
213C4—3624no—
214C4—3624no—
215C4—7241no—
216C4#1915378912yes—
217C4C2×D4×D18111yescharacteristic
218C4C2×#1566844911yescharacteristic
219C2—22no—
220C2—14481no—
221C2C2×#1915378911yescharacteristic
222C2#2604209512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#419×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668449C22
hypercenter#419D181
hypocenterC181#3044
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#15668439C2semidirect
4. #26041756C2semidirect
5. #26041504C2semidirect
6. #26041488C2semidirect
7. #26041307C2semidirect
9. #419×C181C2semidirect
111. C181#3044semidirect
120. #419D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C22×#156684392
#260418532
#260417562
#260415042
#260414882
#260413072
#260412732
#419×C1812
#3044181