Finite Group Databasebeta

← #37544860

Subgroups — #37544860

Order 23168.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041834C211yesmaximal, characteristic, complemented
2#26041830C211yesmaximal, characteristic
3#26041675C211yesmaximal, characteristic, complemented
4#26041627C211yesmaximal, characteristic, complemented
5#26041626C211yesmaximal, characteristic, complemented
6#26041606C214yesmaximal, complemented
7#26041582C214yesmaximal, complemented
8#26041330C211yesmaximal, characteristic, complemented
9#500×C181C211yesmaximal, characteristic, complemented
10C2×#15668450—21no—
11#19153816C2211yescharacteristic
12#19153813C2211yescharacteristic
13#19153807C2212yes—
14#19153805—21no—
15#19153796—21no—
16#19153792—21no—
17#19153791—21no—
18#19153791C2212yes—
19Q8×Dic181—21no—
20#19153781—21no—
21#19153778—21no—
22#19153778C2212yes—
23#19153774C2212yes—
24C42×D181C2211yescharacteristic
25#19153747C2214yes—
26#19153745C2212yes—
27#19153738C2214yes—
28#19153736C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C4⋊Q8×C181C2211yescharacteristic
32C4⋊Q8×C181C2211yescharacteristic
33#107×C181C2214yes—
34C4×C8×C181C2211yescharacteristic
35#15668450—44no—
36C2×Dic362—21no—
37C2×D724—22no—
38C2×D724—21no—
39C2×D724—21no—
40C2×D724C2312yes—
41C2×C4×D181—22no—
42C2×C4×D181D411yescharacteristic
43C2×C4×D181D411yescharacteristic
44C2×C4×D181C2311yescharacteristic
45C2×#13089678C2312yes—
46C4⋊Dic181—22no—
47C4⋊Dic181—21no—
48C4⋊Dic181C2312yes—
49#15668428—22no—
50#15668426—22no—
51#15668426—22no—
52#15668426—22no—
53C4×Dic181—22no—
54C4×Dic181D411yescharacteristic
55C4×Dic181D411yescharacteristic
56C4×Dic181C2311yescharacteristic
57C2×Q8×C181—21no—
58C2×Q8×C181—21no—
59C2×C8×C181C2312yes—
60C4⋊C4×C181—21no—
61C4⋊C4×C181—21no—
62C4⋊C4×C181C2312yes—
63C4⋊C4×C181C2312yes—
64C42×C181C2311yescharacteristic
65C22×D181—22no—
66C22×D181C2×D411yescharacteristic
67Dic362—42no—
68D724—24no—
69D724—44no—
70D724—42no—
71D724—42no—
72C2×Dic181—22no—
73C2×Dic181C2×D411yescharacteristic
74C2×Dic181C2×D411yescharacteristic
75C2×Dic181C2×D411yescharacteristic
76C4×D181—22no—
77C4×D181—22no—
78C4×D181—22no—
79C4×D181—44no—
80#13089678—22no—
81Q8×C181—42no—
82Q8×C181—42no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2411yescharacteristic
88C8×C181—22no—
89C22×C181C22×D411yescharacteristic
90C2×D181—21no—
91C2×D181—21no—
92C2×D181—21no—
93C2×D181—42no—
94C2×D181—42no—
95C2×D181—42no—
96C4×C181—42no—
97C4×C181—42no—
98C4×C181#13512yes—
99C4×C181AGL(1,ℤ/8)12yes—
100C4×C1812−1+412yes—
101Dic181—21no—
102Dic181—21no—
103Dic181—21no—
104Dic181—42no—
105C2×C181#53611yescharacteristic
106C2×C181C2×#13511yescharacteristic
107C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
108D181—41no—
109D181—82no—
110C181#322811yescharacteristic, complemented, Sylow
111#3228—1811nomaximal, Sylow
112#533—1811no—
113#532—1811no—
114#500D18111yescharacteristic, complemented
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#125—1812no—
127#124—3621no—
128C22⋊Q8—3621no—
129C22⋊Q8—3621no—
130#121—3621no—
131#121—1812no—
132C4×Q8—3621no—
133C4×Q8—3621no—
134C4×D4—3621no—
135C4×D4—1812no—
136#117—1812no—
137C2×C42—1811no—
138#107C2×D18114yes—
139#106—1814no—
140C4⋊C8—1812no—
141#102—1814no—
142#102—1814no—
143C22⋊C8—1812no—
144C4×C8—1811no—
145C4×C8C2×D18111yescharacteristic
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×C4—1811no—
156C22×C4—1811no—
157C22×C4—1811no—
158C2×C8—1812no—
159C2×C8C22×D18112yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—3622no—
163C4⋊C4—3622no—
164C4⋊C4—3621no—
165C4⋊C4—1812no—
166C4⋊C4C22×D18112yes—
167C4⋊C4C22×D18112yes—
168C22⋊C4—3622no—
169C22⋊C4—3622no—
170C22⋊C4—3622no—
171C42—3622no—
172C42—1811no—
173C42—1811no—
174C42—1811no—
175C42C22×D18111yescharacteristic
176C23—3622no—
177C23—1811no—
178Q8—42no—
179Q8—42no—
180Q8—7242no—
181D4—3624no—
182D4—7244no—
183D4—7242no—
184D4—7242no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—3622no—
188C2×C4—3622no—
189C2×C4—3622no—
190C2×C4—3622no—
191C2×C4—1811no—
192C2×C4—1811no—
193C2×C4—1811no—
194C2×C4—7244no—
195C2×C4D4×D18111yescharacteristic
196C2×C4D4×D18111yescharacteristic
197C2×C4C23×D18111yescharacteristic
198C8—22no—
199C8—3622no—
200C22—3621no—
201C22—3621no—
202C22—3621no—
203C22—7242no—
204C22—7242no—
205C22—7242no—
206C22C2×D4×D18111yescharacteristic
207C4—42no—
208C4—42no—
209C4—3621no—
210C4—3621no—
211C4—3621no—
212C4—7242no—
213C4#1915383912yes—
214C4#1915384012yes—
215C4#1915386412yes—
216C2—14482no—
217C2—7241no—
218C2#2604185411yescharacteristic
219C2C2×#1915383911yescharacteristic
220C2C2×#1915384011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#500×C181C2
socleC22×C181C22×D4
prime radical#26041830C2
hypercenter#500D181
hypocenterC181#3228
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041675C2semidirect
4. #26041627C2semidirect
5. #26041626C2semidirect
6. #26041606C2semidirect
7. #26041582C2semidirect
8. #26041330C2semidirect
9. #500×C181C2semidirect
110. C181#3228semidirect
114. #500D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418302
#260416752
#260416272
#260416262
#260416062
#260415822
#260413302
#500×C1812
#3228181