Finite Group Databasebeta

← #37544919

Subgroups — #37544919

Order 23168.

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

normal subgroups only

This group has 366 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#26041689C212yesmaximal, complemented
3#26041645C212yesmaximal, complemented
4#26041603C214yesmaximal, complemented
5#26041566C212yesmaximal, complemented
6#26041556C212yesmaximal, complemented
7#26041336C211yesmaximal, characteristic, complemented
8#533×C181C211yesmaximal, characteristic, complemented
9C2×#15668449—21no—
10#19153816C2211yescharacteristic
11#19153813C2211yescharacteristic
12#19153804—21no—
13#19153790—21no—
14#19153790C2212yes—
15C4×D724—21no—
16C4×D724C2212yes—
17#19153749C2214yes—
18#19153747C2214yes—
19C4⋊#13089678C2212yes—
20#19153736C2214yes—
21#19153736C2214yes—
22C22⋊#13089678C2212yes—
23#19153727C2212yes—
24C4⋊Q8×C181C2211yescharacteristic
25C4⋊Q8×C181C2211yescharacteristic
26#125×C181C2212yes—
27C22⋊Q8×C181—21no—
28C22⋊Q8×C181—21no—
29C4×Q8×C181—21no—
30C4×Q8×C181—21no—
31#117×C181C2212yes—
32C2×C42×C181C2211yescharacteristic
33#15668449—44no—
34C2×Dic362—21no—
35C2×D724—21no—
36C2×D724—21no—
37C2×D724C2312yes—
38C2×C22⋊D181—22no—
39C2×C4×D181—22no—
40C2×#13089678C2314yes—
41C4⋊Dic181—21no—
42C4⋊Dic181C2312yes—
43#15668426—22no—
44#15668426—22no—
45C2×Q8×C181—21no—
46C2×Q8×C181—21no—
47C22×C4×C181D411yescharacteristic
48C22×C4×C181D411yescharacteristic
49C22×C4×C181C2311yescharacteristic
50C4⋊C4×C181—22no—
51C4⋊C4×C181—22no—
52C4⋊C4×C181—21no—
53C4⋊C4×C181—21no—
54C4⋊C4×C181C2312yes—
55C4⋊C4×C181C2312yes—
56C22⋊C4×C181—22no—
57C42×C181—22no—
58C42×C181D411yescharacteristic
59C42×C181D411yescharacteristic
60C42×C181C2311yescharacteristic
61C42×C181C2311yescharacteristic
62C22×D181—22no—
63Dic362—42no—
64D724—24no—
65D724—42no—
66D724—42no—
67C22⋊D181—44no—
68C2×Dic181—22no—
69C4×D181—44no—
70#13089678—24no—
71C23×C181C2×D411yescharacteristic
72Q8×C181—42no—
73Q8×C181—42no—
74C2×C4×C181—22no—
75C2×C4×C181—22no—
76C2×C4×C181—22no—
77C2×C4×C181—22no—
78C2×C4×C181—41no—
79C2×C4×C181C2×D411yescharacteristic
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181C2×D411yescharacteristic
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2411yescharacteristic
85C22×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181C22×D411yescharacteristic
89C2×D181—42no—
90C2×D181—42no—
91C2×D181—42no—
92C4×C181—21no—
93C4×C181—21no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181—42no—
97C4×C181—42no—
98C4×C181AGL(1,ℤ/8)12yes—
99C4×C1812−1+412yes—
100Dic181—42no—
101C2×C181—41no—
102C2×C181#53611yescharacteristic
103C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C181#57511yescharacteristic
105D181—82no—
106C181#323511yescharacteristic, complemented, Sylow
107#3235—1811nomaximal, Sylow
108#533D18111yescharacteristic, complemented
109#532—1811no—
110#501—1812no—
111#490—1812no—
112#482—1814no—
113#476—1812no—
114#474—1812no—
115#432—1811no—
116C2×#41—3621no—
117C4⋊Q8—1811no—
118C4⋊Q8C2×D18111yescharacteristic
119C4⋊Q8C2×D18111yescharacteristic
120#127—1811no—
121#125C2×D18112yes—
122#124—3621no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125#121—3621no—
126#121—1812no—
127C4×Q8—21no—
128C4×Q8—21no—
129C4×D4—3621no—
130C4×D4—1812no—
131#117C2×D18112yes—
132C2×C42C2×D18111yescharacteristic
133#107—1814no—
134#106—1814no—
135C4⋊C8—1812no—
136#102—1814no—
137#102—1814no—
138C22⋊C8—1812no—
139#97—1812no—
140#41—7244no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—3621no—
144C2×D4—3622no—
145C2×D4—3621no—
146C2×D4—3621no—
147C2×D4—1812no—
148C22×C4—3622no—
149C22×C4C22⋊D18111yescharacteristic
150C22×C4C22⋊D18111yescharacteristic
151C22×C4C22×D18111yescharacteristic
152C2×C8—1814no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3621no—
158C4⋊C4—1812no—
159C4⋊C4C22×D18112yes—
160C4⋊C4C22×D18112yes—
161C22⋊C4—22no—
162C22⋊C4—3622no—
163C22⋊C4—3622no—
164C42—22no—
165C42C22⋊D18111yescharacteristic
166C42C22⋊D18111yescharacteristic
167C42C22×D18111yescharacteristic
168C42C22×D18111yescharacteristic
169C23—3622no—
170C23C2×C22⋊D18111yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—7242no—
174D4—3624no—
175D4—7242no—
176D4—7242no—
177D4—7244no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—3622no—
184C2×C4—7244no—
185C2×C4C2×C22⋊D18111yescharacteristic
186C2×C4C2×C22⋊D18111yescharacteristic
187C2×C4C2×C22⋊D18111yescharacteristic
188C2×C4C2×C22⋊D18111yescharacteristic
189C2×C4C2×C22⋊D18111yescharacteristic
190C2×C4C23×D18111yescharacteristic
191C8—3624no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—7242no—
196C22—7242no—
197C22—7242no—
198C22C22×C22⋊D18111yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—42no—
205C4—7242no—
206C4#1915384112yes—
207C4#1915386412yes—
208C2—41no—
209C2—14482no—
210C2#2604185311yescharacteristic
211C2C2×#1915384111yescharacteristic
212C2#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#3235
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. #26041689C2semidirect
3. #26041645C2semidirect
4. #26041603C2semidirect
5. #26041566C2semidirect
6. #26041556C2semidirect
7. #26041336C2semidirect
8. #533×C181C2semidirect
106. C181#3235semidirect
108. #533D181semidirect

Maximal subgroups

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

subgroupindex
#260418292
#260416892
#260416452
#260416032
#260415662
#260415562
#260413362
#533×C1812
#3235181