Finite Group Databasebeta

← #37544734

Subgroups — #37544734

Order 23168.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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#26041836C211yesmaximal, characteristic, complemented
2#26041829C211yesmaximal, characteristic
3#26041673C212yesmaximal, complemented
4#26041639C212yesmaximal, complemented
5#26041490C212yesmaximal, complemented
6#26041480C212yesmaximal, complemented
7#26041443C214yesmaximal, complemented
8#432×C181C211yesmaximal, characteristic, complemented
9C2×#15668449C2212yes—
10C2×Dic724C2214yes—
11C2×#15668438C2214yes—
12#19153816C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153816C2211yescharacteristic
15#19153813C2211yescharacteristic
16#19153808—21no—
17#19153804C2212yes—
18#19153795—21no—
19#19153795C2212yes—
20#19153790—21no—
21#19153790—21no—
22C4×Dic362—21no—
23C4×Dic362C2212yes—
24C4×D724—21no—
25C4×D724—21no—
26#19153775—21no—
27#19153741C2214yes—
28#19153738C2214yes—
29C2×C42×C181C2211yescharacteristic
30C4⋊C8×C181C2212yes—
31C22⋊C8×C181C2212yes—
32#97×C181C2212yes—
33#15668449—44no—
34C2×Dic362—21no—
35C2×Dic362C2312yes—
36C2×Dic362C2312yes—
37C2×D724—21no—
38C2×D724—21no—
39C2×D724C2312yes—
40C2×C22⋊D181—22no—
41C2×C4×D181—22no—
42Dic724—44no—
43#15668438—44no—
44C4⋊Dic181—21no—
45C4⋊Dic181—21no—
46C4⋊Dic181—21no—
47C4⋊Dic181—21no—
48C4⋊Dic181—21no—
49C4⋊Dic181C2312yes—
50#15668428—22no—
51#15668428—22no—
52C22⋊Dic181—22no—
53#15668426—22no—
54#15668426—22no—
55C4×Dic181—22no—
56C22×C4×C181D411yescharacteristic
57C22×C4×C181D411yescharacteristic
58C22×C4×C181C2311yescharacteristic
59C2×C8×C181C2314yes—
60C42×C181D411yescharacteristic
61C42×C181D411yescharacteristic
62C42×C181C2311yescharacteristic
63C42×C181C2311yescharacteristic
64C22×D181—22no—
65Dic362—22no—
66Dic362—22no—
67Dic362—24no—
68Dic362—42no—
69D724—24no—
70D724—42no—
71D724—42no—
72C22⋊D181—44no—
73C2×Dic181—22no—
74C2×Dic181—22no—
75C2×Dic181—22no—
76C4×D181—44no—
77C23×C181C2×D411yescharacteristic
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181—41no—
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181C2×D411yescharacteristic
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2411yescharacteristic
87C8×C181—24no—
88C22×C181—21no—
89C22×C181—21no—
90C22×C181—21no—
91C22×C181C22×D411yescharacteristic
92C2×D181—42no—
93C2×D181—42no—
94C2×D181—42no—
95C4×C181—21no—
96C4×C181—21no—
97C4×C181—21no—
98C4×C181—21no—
99C4×C181#13712yes—
100C4×C181H(5,𝔽2)12yes—
101Dic181—42no—
102Dic181—42no—
103Dic181—42no—
104C2×C181—41no—
105C2×C181#53511yescharacteristic
106C2×C181C2×#13711yescharacteristic
107C2×C181#57511yescharacteristic
108D181—82no—
109C181#321611yescharacteristic, complemented, Sylow
110#3216—1811nomaximal, Sylow
111#533—1811no—
112#532—1811no—
113#497—1812no—
114#489—1812no—
115#462—1812no—
116#460—1812no—
117#452—1814no—
118#432D18111yescharacteristic, complemented
119C2×#41—1812no—
120C2×Q16—1814no—
121C2×SD16—1814no—
122C4⋊Q8—1811no—
123C4⋊Q8—1811no—
124C4⋊Q8—1811no—
125#127—1811no—
126#125—3621no—
127#124—1812no—
128C22⋊Q8—3621no—
129C22⋊Q8—1812no—
130#121—3621no—
131#121—3621no—
132C4×Q8—3621no—
133C4×Q8—1812no—
134C4×D4—3621no—
135C4×D4—3621no—
136#117—3621no—
137C2×C42C2×D18111yescharacteristic
138C4⋊C8C2×D18112yes—
139#103—1814no—
140#102—1814no—
141C22⋊C8C2×D18112yes—
142#97C2×D18112yes—
143#41—7244no—
144C2×Q8—3621no—
145C2×Q8—1812no—
146C2×Q8—1812no—
147C2×D4—3622no—
148C2×D4—3621no—
149C2×D4—3621no—
150C2×D4—1812no—
151C22×C4—3622no—
152C22×C4D72411yescharacteristic
153C22×C4D72411yescharacteristic
154C22×C4C22×D18111yescharacteristic
155Q16—7244no—
156SD16—7244no—
157C2×C8C22×D18114yes—
158C4⋊C4—3621no—
159C4⋊C4—3621no—
160C4⋊C4—3621no—
161C4⋊C4—3621no—
162C4⋊C4—3621no—
163C4⋊C4—3622no—
164C4⋊C4—3622no—
165C4⋊C4—1812no—
166C22⋊C4—3622no—
167C22⋊C4—3622no—
168C22⋊C4—3622no—
169C42—3622no—
170C42D72411yescharacteristic
171C42D72411yescharacteristic
172C42C22×D18111yescharacteristic
173C42C22×D18111yescharacteristic
174C23—3622no—
175C23C2×D72411yescharacteristic
176Q8—3622no—
177Q8—3622no—
178Q8—3624no—
179Q8—7242no—
180D4—3624no—
181D4—7242no—
182D4—7244no—
183D4—7242no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—3622no—
188C2×C4—3622no—
189C2×C4—3622no—
190C2×C4—7244no—
191C2×C4C2×D72411yescharacteristic
192C2×C4C2×D72411yescharacteristic
193C2×C4C2×D72411yescharacteristic
194C2×C4C2×D72411yescharacteristic
195C2×C4C2×D72411yescharacteristic
196C2×C4C23×D18111yescharacteristic
197C8—24no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—7242no—
202C22—7242no—
203C22—7242no—
204C22C22×D72411yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—7242no—
210C4—7242no—
211C4—7242no—
212C4#1915384912yes—
213C4#1915386212yes—
214C2—41no—
215C2—14482no—
216C2#2604184811yescharacteristic
217C2C2×#1915384911yescharacteristic
218C2#2604207911yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#432×C181C2
socleC22×C181C22×D4
prime radical#26041829C2
hypercenter#432D181
hypocenterC181#3216
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
3. #26041673C2semidirect
4. #26041639C2semidirect
5. #26041490C2semidirect
6. #26041480C2semidirect
7. #26041443C2semidirect
8. #432×C181C2semidirect
109. C181#3216semidirect
118. #432D181semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260418292
#260416732
#260416392
#260414902
#260414802
#260414432
#432×C1812
#3216181