Finite Group Databasebeta

← #37544857

Subgroups — #37544857

Order 23168.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041682C211yesmaximal, characteristic, complemented
3#26041675C211yesmaximal, characteristic, complemented
4#26041627C211yesmaximal, characteristic, complemented
5#26041627C211yesmaximal, characteristic, complemented
6#26041603C214yesmaximal, complemented
7#26041576C214yesmaximal, complemented
8#26041332C211yesmaximal, characteristic, complemented
9#533×C181C211yesmaximal, characteristic, complemented
10C2×#15668449—21no—
11#19153816C2211yescharacteristic
12#19153813C2211yescharacteristic
13#19153804—21no—
14#19153790—21no—
15#19153790C2212yes—
16C4×D724—21no—
17C4×D724C2212yes—
18#19153749C2214yes—
19#19153747C2214yes—
20C4⋊#13089678C2212yes—
21#19153736C2214yes—
22#19153736C2214yes—
23C22⋊#13089678C2212yes—
24C4×#13089678C2211yescharacteristic
25C4×#13089678C2211yescharacteristic
26C4⋊Q8×C181C2211yescharacteristic
27C4⋊Q8×C181C2211yescharacteristic
28#125×C181C2212yes—
29C22⋊Q8×C181—21no—
30C22⋊Q8×C181—21no—
31C4×Q8×C181—21no—
32C4×Q8×C181—21no—
33#117×C181C2212yes—
34C2×C42×C181C2211yescharacteristic
35#15668449—44no—
36C2×Dic362—21no—
37C2×D724—21no—
38C2×D724—21no—
39C2×D724C2312yes—
40C2×C22⋊D181—22no—
41C2×C4×D181—22no—
42C2×#13089678C2312yes—
43C2×#13089678C2312yes—
44C4⋊Dic181—21no—
45C4⋊Dic181C2312yes—
46#15668426—22no—
47#15668426—22no—
48C2×Q8×C181—21no—
49C2×Q8×C181—21no—
50C22×C4×C181D411yescharacteristic
51C22×C4×C181D411yescharacteristic
52C22×C4×C181C2311yescharacteristic
53C4⋊C4×C181—22no—
54C4⋊C4×C181—22no—
55C4⋊C4×C181—21no—
56C4⋊C4×C181—21no—
57C4⋊C4×C181C2312yes—
58C4⋊C4×C181C2312yes—
59C22⋊C4×C181—22no—
60C42×C181—22no—
61C42×C181D411yescharacteristic
62C42×C181D411yescharacteristic
63C42×C181C2311yescharacteristic
64C42×C181C2311yescharacteristic
65C22×D181—22no—
66Dic362—42no—
67D724—24no—
68D724—42no—
69D724—42no—
70C22⋊D181—44no—
71C2×Dic181—22no—
72C4×D181—44no—
73#13089678—22no—
74#13089678—22no—
75C23×C181C2×D411yescharacteristic
76Q8×C181—42no—
77Q8×C181—42no—
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2411yescharacteristic
89C22×C181—21no—
90C22×C181—21no—
91C22×C181—21no—
92C22×C181C22×D411yescharacteristic
93C2×D181—42no—
94C2×D181—42no—
95C2×D181—42no—
96C4×C181—21no—
97C4×C181—21no—
98C4×C181—21no—
99C4×C181—42no—
100C4×C181—42no—
101C4×C181#13512yes—
102C4×C181AGL(1,ℤ/8)12yes—
103C4×C1812−1+412yes—
104Dic181—42no—
105C2×C181—41no—
106C2×C181#53611yescharacteristic
107C2×C181C2×#13511yescharacteristic
108C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
109D181—82no—
110C181#322811yescharacteristic, complemented, Sylow
111#3228—1811nomaximal, Sylow
112#533D18111yescharacteristic, complemented
113#532—1811no—
114#500—1811no—
115#498—1811no—
116#486—1811no—
117#486—1811no—
118#482—1814no—
119#478—1814no—
120#431—1811no—
121C2×#41—3621no—
122C4⋊Q8—1811no—
123C4⋊Q8C2×D18111yescharacteristic
124C4⋊Q8C2×D18111yescharacteristic
125#127—1811no—
126#125C2×D18112yes—
127#124—3621no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130#121—3621no—
131#121—1812no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×D4—3621no—
135C4×D4—1812no—
136#117C2×D18112yes—
137C2×C42C2×D18111yescharacteristic
138#107—1814no—
139#106—1814no—
140C4⋊C8—1812no—
141#102—1814no—
142#102—1814no—
143C22⋊C8—1812no—
144C4×C8—1811no—
145C4×C8—1811no—
146#41—7244no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—3621no—
150C2×D4—3622no—
151C2×D4—3621no—
152C2×D4—3621no—
153C2×D4—1812no—
154C22×C4—3622no—
155C22×C4C22⋊D18111yescharacteristic
156C22×C4C22⋊D18111yescharacteristic
157C22×C4C22×D18111yescharacteristic
158C2×C8—1812no—
159C2×C8—1812no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—3621no—
165C4⋊C4—1812no—
166C4⋊C4C22×D18112yes—
167C4⋊C4C22×D18112yes—
168C22⋊C4—22no—
169C22⋊C4—3622no—
170C22⋊C4—3622no—
171C42—22no—
172C42C22⋊D18111yescharacteristic
173C42C22⋊D18111yescharacteristic
174C42C22×D18111yescharacteristic
175C42C22×D18111yescharacteristic
176C23—3622no—
177C23C2×C22⋊D18111yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—7242no—
181D4—3624no—
182D4—7244no—
183D4—7242no—
184D4—7242no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—3622no—
191C2×C4—7244no—
192C2×C4C2×C22⋊D18111yescharacteristic
193C2×C4C2×C22⋊D18111yescharacteristic
194C2×C4C2×C22⋊D18111yescharacteristic
195C2×C4C2×C22⋊D18111yescharacteristic
196C2×C4C2×C22⋊D18111yescharacteristic
197C2×C4C23×D18111yescharacteristic
198C8—3622no—
199C8—3622no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—7242no—
204C22—7242no—
205C22—7242no—
206C22C22×C22⋊D18111yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—7242no—
213C4#1915383612yes—
214C4#1915384112yes—
215C4#1915386412yes—
216C2—41no—
217C2—14482no—
218C2#2604185311yescharacteristic
219C2C2×#1915383611yescharacteristic
220C2C2×#1915384111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#533×C181C2
socleC22×C181C22×D4
prime radical#26041829C2
hypercenter#533D181
hypocenterC181#3228
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. #26041682C2semidirect
3. #26041675C2semidirect
4. #26041627C2semidirect
5. #26041627C2semidirect
6. #26041603C2semidirect
7. #26041576C2semidirect
8. #26041332C2semidirect
9. #533×C181C2semidirect
110. C181#3228semidirect
112. #533D181semidirect

Maximal subgroups

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

subgroupindex
#260418292
#260416822
#260416752
#260416272
#260416032
#260415762
#260413322
#533×C1812
#3228181