Finite Group Databasebeta

← #37544728

Subgroups — #37544728

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#26041829C211yesmaximal, characteristic
2#26041671C212yesmaximal, complemented
3#26041641C212yesmaximal, complemented
4#26041489C212yesmaximal, complemented
5#26041479C212yesmaximal, complemented
6#26041439C214yesmaximal, complemented
7#26041336C211yesmaximal, characteristic, complemented
8#533×C181C211yesmaximal, characteristic, complemented
9C2×#15668449C2212yes—
10C2×#15668439C2214yes—
11C2×#15668436C2214yes—
12#19153816C2211yescharacteristic
13#19153813C2211yescharacteristic
14#19153804C2212yes—
15#19153790—21no—
16#19153790—21no—
17C4×D724—21no—
18C4×D724—21no—
19C4⋊#13089678C2212yes—
20#19153739C2214yes—
21#19153736C2214yes—
22C22⋊#13089678C2212yes—
23#19153727C2212yes—
24C4⋊Q8×C181C2211yescharacteristic
25C4⋊Q8×C181C2211yescharacteristic
26#125×C181—21no—
27C22⋊Q8×C181—21no—
28C22⋊Q8×C181C2212yes—
29C4×Q8×C181—21no—
30C4×Q8×C181C2212yes—
31#117×C181—21no—
32C2×C42×C181C2211yescharacteristic
33#15668449—44no—
34C2×Dic362C2312yes—
35C2×D724—21no—
36C2×D724—21no—
37C2×D724C2312yes—
38C2×C22⋊D181—22no—
39C2×C4×D181—22no—
40#15668439—44no—
41#15668436—44no—
42C2×#13089678C2314yes—
43C4⋊Dic181—21no—
44C4⋊Dic181—21no—
45#15668426—22no—
46#15668426—22no—
47C2×Q8×C181—21no—
48C2×Q8×C181C2312yes—
49C22×C4×C181D411yescharacteristic
50C22×C4×C181D411yescharacteristic
51C22×C4×C181C2311yescharacteristic
52C4⋊C4×C181—21no—
53C4⋊C4×C181—21no—
54C4⋊C4×C181—21no—
55C4⋊C4×C181—22no—
56C4⋊C4×C181—22no—
57C4⋊C4×C181C2312yes—
58C22⋊C4×C181—22no—
59C42×C181—22no—
60C42×C181D411yescharacteristic
61C42×C181D411yescharacteristic
62C42×C181C2311yescharacteristic
63C42×C181C2311yescharacteristic
64C22×D181—22no—
65Dic362—24no—
66D724—24no—
67D724—42no—
68D724—42no—
69C22⋊D181—44no—
70C2×Dic181—22no—
71C4×D181—44no—
72#13089678—24no—
73C23×C181C2×D411yescharacteristic
74Q8×C181—22no—
75Q8×C181—22no—
76Q8×C181—42no—
77C2×C4×C181—22no—
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181—22no—
81C2×C4×C181—41no—
82C2×C4×C181C2×D411yescharacteristic
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2411yescharacteristic
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—42no—
100C4×C181—42no—
101C4×C181#13712yes—
102C4×C181H(5,𝔽2)12yes—
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#533D18111yescharacteristic, complemented
112#532—1811no—
113#497—1812no—
114#489—1812no—
115#462—1812no—
116#460—1812no—
117#452—1814no—
118#432—1811no—
119C2×#41—1812no—
120C2×Q16—1814no—
121C2×SD16—1814no—
122C4⋊Q8—1811no—
123C4⋊Q8C2×D18111yescharacteristic
124C4⋊Q8C2×D18111yescharacteristic
125#127—1811no—
126#125—21no—
127#124—1812no—
128C22⋊Q8—21no—
129C22⋊Q8C2×D18112yes—
130#121—3621no—
131#121—3621no—
132C4×Q8—21no—
133C4×Q8C2×D18112yes—
134C4×D4—3621no—
135C4×D4—3621no—
136#117—21no—
137C2×C42C2×D18111yescharacteristic
138C4⋊C8—1812no—
139#103—1814no—
140#102—1814no—
141C22⋊C8—1812no—
142#97—1812no—
143#41—7244no—
144C2×Q8—21no—
145C2×Q8—1812no—
146C2×Q8C22×D18112yes—
147C2×D4—3622no—
148C2×D4—3621no—
149C2×D4—3621no—
150C2×D4—1812no—
151C22×C4—3622no—
152C22×C4C22⋊D18111yescharacteristic
153C22×C4C22⋊D18111yescharacteristic
154C22×C4C22×D18111yescharacteristic
155Q16—7244no—
156SD16—7244no—
157C2×C8—1814no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—3621no—
164C4⋊C4—3621no—
165C4⋊C4C22×D18112yes—
166C22⋊C4—22no—
167C22⋊C4—3622no—
168C22⋊C4—3622no—
169C42—22no—
170C42C22⋊D18111yescharacteristic
171C42C22⋊D18111yescharacteristic
172C42C22×D18111yescharacteristic
173C42C22×D18111yescharacteristic
174C23—3622no—
175C23C2×C22⋊D18111yescharacteristic
176Q8—22no—
177Q8—22no—
178Q8—42no—
179Q8—3624no—
180D4—3624no—
181D4—7242no—
182D4—7244no—
183D4—7242no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—3622no—
190C2×C4—7244no—
191C2×C4C2×C22⋊D18111yescharacteristic
192C2×C4C2×C22⋊D18111yescharacteristic
193C2×C4C2×C22⋊D18111yescharacteristic
194C2×C4C2×C22⋊D18111yescharacteristic
195C2×C4C2×C22⋊D18111yescharacteristic
196C2×C4C23×D18111yescharacteristic
197C8—3624no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—7242no—
202C22—7242no—
203C22—7242no—
204C22C22×C22⋊D18111yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—7242no—
212C4#1915384712yes—
213C4#1915386212yes—
214C2—41no—
215C2—14482no—
216C2#2604184311yescharacteristic
217C2C2×#1915384711yescharacteristic
218C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#533×C181C2
socleC22×C181C22×D4
prime radical#26041829C2
hypercenter#533D181
hypocenterC181#3216
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #26041671C2semidirect
3. #26041641C2semidirect
4. #26041489C2semidirect
5. #26041479C2semidirect
6. #26041439C2semidirect
7. #26041336C2semidirect
8. #533×C181C2semidirect
109. C181#3216semidirect
111. #533D181semidirect

Maximal subgroups

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

subgroupindex
#260418292
#260416712
#260416412
#260414892
#260414792
#260414392
#260413362
#533×C1812
#3216181