Finite Group Databasebeta

← #37543252

Subgroups — #37543252

Order 23168.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×Dic724C211yesmaximal, characteristic, complemented
3#26041855C211yesmaximal, characteristic
4#26041756C211yesmaximal, characteristic, complemented
5#26041502C214yesmaximal, complemented
6#26041490C214yesmaximal, complemented
7#26041289C211yesmaximal, characteristic, complemented
8#26041276C211yesmaximal, characteristic
9#424×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2211yescharacteristic
11C22×Dic362C2211yescharacteristic
12C22×Dic362C2211yescharacteristic
13#19153849—24no—
14C2×Dic724—24no—
15C2×Dic724C2212yes—
16C2×Dic724C2214yes—
17C2×#15668438C2212yes—
18#19153816C2212yes—
19#19153804C2212yes—
20#19153800—22no—
21#19153796—22no—
22C4×Dic362C2214yes—
23#19153781—22no—
24#19153775C2211yescharacteristic
25#19153772—22no—
26#19153741C2212yes—
27#19153741C2214yes—
28#19153738C2212yes—
29C2×M16×C181C2211yescharacteristic
30C22×C8×C181C2211yescharacteristic
31#117×C181C2211yescharacteristic
32C4⋊C8×C181C2214yes—
33#15668449—22no—
34#15668449—22no—
35C2×Dic362—21no—
36C2×Dic362—21no—
37C2×Dic362—22no—
38C2×Dic362—22no—
39C2×Dic362—22no—
40C2×Dic362D414yes—
41C2×Dic362C2312yes—
42C2×Dic362C2311yescharacteristic
43C2×Dic362C2312yes—
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C22×Dic181—21no—
47C22×Dic181—21no—
48C2×C4×D181—21no—
49Dic724—28no—
50Dic724—24no—
51#15668438—24no—
52C4⋊Dic181—24no—
53C4⋊Dic181C2312yes—
54#15668428—24no—
55C22⋊Dic181—21no—
56#15668426—24no—
57C4×Dic181—24no—
58C4×Dic181—21no—
59C22×C4×C181C2311yescharacteristic
60M16×C181—22no—
61C2×C8×C181—22no—
62C2×C8×C181C2312yes—
63C2×C8×C181C2312yes—
64C4⋊C4×C181D412yes—
65C22⋊C4×C181D412yes—
66C42×C181C2312yes—
67C22×D181—21no—
68Dic362—24no—
69Dic362—22no—
70Dic362—22no—
71Dic362—21no—
72Dic362—22no—
73Dic362—21no—
74Dic362—42no—
75Dic362C2×D414yes—
76D724—21no—
77D724—21no—
78C22⋊D181—42no—
79C2×Dic181—24no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82C2×Dic181—22no—
83C2×Dic181—21no—
84C2×Dic181—42no—
85C4×D181—42no—
86C23×C181C2×D411yescharacteristic
87C2×C4×C181C2×D412yes—
88C2×C4×C181C2×D414yes—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181#4112yes—
91C2×C4×C181C2411yescharacteristic
92C8×C181—22no—
93C8×C181—22no—
94C22×C181—22no—
95C22×C181#12112yes—
96C22×C181C22×D411yescharacteristic
97C2×D181—41no—
98C2×D181—42no—
99C4×C181—24no—
100C4×C181#12112yes—
101C4×C181C22×D411yescharacteristic
102C4×C181C2×#4111yescharacteristic
103Dic181—24no—
104Dic181—41no—
105Dic181—42no—
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#519—1811no—
117#464—1814no—
118#462—1814no—
119#424D18111yescharacteristic, complemented
120#419—1811no—
121#416—1811no—
122C2×#41—1811no—
123C22×Q8—1811no—
124C22×Q8—1811no—
125#137—3624no—
126C2×Q16—3624no—
127C2×Q16—1814no—
128C2×Q16—1812no—
129C2×SD16—1812no—
130C2×M16C2×D18111yescharacteristic
131C22×C8C2×D18111yescharacteristic
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⋊C8C2×D18114yes—
142#103—1812no—
143#103—1814no—
144#102—1812no—
145#41—3622no—
146#41—3622no—
147C2×Q8—3621no—
148C2×Q8—3621no—
149C2×Q8—3622no—
150C2×Q8—3622no—
151C2×Q8—3622no—
152C2×Q8—1814no—
153C2×Q8—1812no—
154C2×Q8—1811no—
155C2×Q8—1812no—
156C2×D4—3621no—
157C2×D4—1811no—
158C22×C4—3621no—
159C22×C4—3621no—
160C22×C4—3621no—
161C22×C4C22×D18111yescharacteristic
162Q16—3628no—
163Q16—3624no—
164SD16—3624no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D18112yes—
168C2×C8C22×D18112yes—
169C4⋊C4—3624no—
170C4⋊C4—3624no—
171C4⋊C4—1812no—
172C4⋊C4D72412yes—
173C22⋊C4—3621no—
174C22⋊C4—3624no—
175C22⋊C4D72412yes—
176C42—3624no—
177C42—3621no—
178C42C22×D18112yes—
179C23—3621no—
180C23C2×D72411yescharacteristic
181Q8—3624no—
182Q8—3622no—
183Q8—3622no—
184Q8—3621no—
185Q8—3622no—
186Q8—3621no—
187Q8—1814no—
188Q8—7242no—
189D4—3621no—
190D4—3621no—
191D4—7242no—
192C2×C4—3624no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—3622no—
196C2×C4—3621no—
197C2×C4—7242no—
198C2×C4—7242no—
199C2×C4D4×D18112yes—
200C2×C4C2×D72414yes—
201C2×C4C2×D72411yescharacteristic
202C2×C4#1566845012yes—
203C2×C4C23×D18111yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—7241no—
208C22—7242no—
209C22#1915379112yes—
210C22C22×D72411yescharacteristic
211C4—24no—
212C4—3624no—
213C4—7241no—
214C4—7242no—
215C4—7242no—
216C4#1915379112yes—
217C4C2×D4×D18111yescharacteristic
218C4C2×#1566845011yescharacteristic
219C2—22no—
220C2—14481no—
221C2C2×#1915379111yescharacteristic
222C2#2604209612yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#424×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668449C22
hypercenter#424D181
hypocenterC181#3044
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic724C2semidirect
4. #26041756C2semidirect
5. #26041502C2semidirect
6. #26041490C2semidirect
7. #26041289C2semidirect
9. #424×C181C2semidirect
111. C181#3044semidirect
119. #424D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C22×Dic7242
#260418552
#260417562
#260415022
#260414902
#260412892
#260412762
#424×C1812
#3044181