Finite Group Databasebeta

← #37544670

Subgroups — #37544670

Order 23168.

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

normal subgroups only

This group has 384 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#26041834C211yesmaximal, characteristic
2C4⋊Q8×D181C211yesmaximal, characteristic
3#26041659C211yesmaximal, characteristic, complemented
4#26041658C211yesmaximal, characteristic, complemented
5#26041631C211yesmaximal, characteristic, complemented
6#26041488C214yesmaximal, complemented
7#26041436C214yesmaximal
8#26041330C211yesmaximal, characteristic
9#487×C181C211yesmaximal, characteristic, complemented
10C2×Q8×D181C2212yes—
11C2×Dic724C2214yes—
12C2×#15668439C2214yes—
13#19153816C2211yescharacteristic
14#19153816C2211yescharacteristic
15#19153815—21no—
16#19153815—21no—
17#19153814C2212yes—
18#19153807—21no—
19#19153796C2212yes—
20#19153792—21no—
21Q8×Dic181—21no—
22#19153781C2212yes—
23#19153774—21no—
24C4⋊C4×D181—21no—
25C4⋊C4×D181—21no—
26C42×D181C2211yescharacteristic
27#19153745C2212yes—
28#19153739C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C4⋊Q8×C181C2211yescharacteristic
32C4⋊Q8×C181C2211yescharacteristic
33#103×C181C2214yes—
34C4×C8×C181C2211yescharacteristic
35C2×Dic362—22no—
36C2×Dic362C2312yes—
37C2×Dic362C2312yes—
38Q8×D181—44no—
39C2×C4×D181—22no—
40C2×C4×D181D411yescharacteristic
41C2×C4×D181D411yescharacteristic
42C2×C4×D181C2311yescharacteristic
43Dic724—44no—
44#15668439—44no—
45C2×#13089678C2312yes—
46C4⋊Dic181—22no—
47C4⋊Dic181—21no—
48C4⋊Dic181—21no—
49C4⋊Dic181—21no—
50C4⋊Dic181—21no—
51#15668428—22no—
52#15668428—22no—
53#15668428—22no—
54#15668426—22no—
55C4×Dic181—22no—
56C4×Dic181D411yescharacteristic
57C4×Dic181D411yescharacteristic
58C4×Dic181C2311yescharacteristic
59C2×Q8×C181—21no—
60C2×Q8×C181C2312yes—
61C2×C8×C181C2312yes—
62C4⋊C4×C181—21no—
63C4⋊C4×C181—21no—
64C4⋊C4×C181—21no—
65C4⋊C4×C181C2312yes—
66C42×C181C2311yescharacteristic
67C22×D181C2×D411yescharacteristic
68Dic362—24no—
69Dic362—24no—
70Dic362—44no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181C2×D411yescharacteristic
74C2×Dic181C2×D411yescharacteristic
75C2×Dic181C2×D411yescharacteristic
76C4×D181—22no—
77C4×D181—22no—
78C4×D181—44no—
79C4×D181Q1614yescomplemented
80#13089678—22no—
81Q8×C181—24no—
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×D181C2×Q1612yes—
93C4×C181—42no—
94C4×C181—42no—
95C4×C181C2×Q1612yes—
96C4×C181#13712yes—
97C4×C181H(5,𝔽2)12yes—
98Dic181—21no—
99Dic181—21no—
100Dic181—42no—
101Dic181—42no—
102Dic181C2×Q1612yes—
103C2×C181#53511yescharacteristic
104C2×C181C22×Q1611yescharacteristic
105C2×C181C2×#13711yescharacteristic
106D181—22no—
107C181#320911yescharacteristic, complemented, Sylow
108#3209—1811nomaximal, Sylow
109#533—1811no—
110C2×C4⋊Q8—1811no—
111#494—1811no—
112#494—1811no—
113#487—1811no—
114#487D18111yescharacteristic, complemented
115#462—1814no—
116#451—1814no—
117#431—1811no—
118C22×Q8—1812no—
119C2×Q16—1814no—
120C2×Q16—1814no—
121C4⋊Q8—3621no—
122C4⋊Q8—3621no—
123C4⋊Q8—1812no—
124C4⋊Q8—1811no—
125C4⋊Q8—1811no—
126C4⋊Q8C2×D18111yescharacteristic
127C4⋊Q8C2×D18111yescharacteristic
128#125—3621no—
129C22⋊Q8—3621no—
130C22⋊Q8—1812no—
131C4×Q8—3621no—
132C4×Q8—1812no—
133#117—3621no—
134C2×C4⋊C4—3621no—
135C2×C4⋊C4—3621no—
136C2×C42—1811no—
137C4⋊C8—1812no—
138#103—1814no—
139#103C2×D18114yes—
140C22⋊C8—1812no—
141C4×C8—1811no—
142C4×C8C2×D18111yescharacteristic
143C2×Q8—21no—
144C2×Q8—3622no—
145C2×Q8—1812no—
146C2×Q8—1812no—
147C2×Q8—7244no—
148C2×Q8C22×D18112yes—
149C22×C4—3622no—
150C22×C4—1811no—
151C22×C4—1811no—
152C22×C4—1811no—
153Q16—7244no—
154Q16—7244no—
155C2×C8—1812no—
156C2×C8C22×D18112yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—3622no—
161C4⋊C4—3622no—
162C4⋊C4—3622no—
163C4⋊C4—3622no—
164C4⋊C4—3621no—
165C4⋊C4—3621no—
166C4⋊C4—3621no—
167C4⋊C4—3621no—
168C4⋊C4C22×D18112yes—
169C22⋊C4—3622no—
170C42—3622no—
171C42—1811no—
172C42—1811no—
173C42—1811no—
174C42C22×D18111yescharacteristic
175C23—1811no—
176Q8—24no—
177Q8—42no—
178Q8—3624no—
179Q8—3624no—
180Q8—7244no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—3622no—
184C2×C4—3622no—
185C2×C4—3622no—
186C2×C4—3622no—
187C2×C4—1811no—
188C2×C4—1811no—
189C2×C4—1811no—
190C2×C4—1814no—
191C2×C4—7244no—
192C2×C4D4×D18111yescharacteristic
193C2×C4D4×D18111yescharacteristic
194C2×C4C23×D18111yescharacteristic
195C8—22no—
196C8—3622no—
197C22—3621no—
198C22—3621no—
199C22—1812no—
200C22C2×D4×D18111yescharacteristic
201C4—42no—
202C4—42no—
203C4—3621no—
204C4—3621no—
205C4—1812no—
206C4—7242no—
207C4—7242no—
208C4Q16×D18112yes—
209C4#1915384512yes—
210C4#1915386212yes—
211C2—3622no—
212C2#2604184711yescharacteristic
213C2C2×Q16×D18111yescharacteristic
214C2C2×#1915384511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#487×C181C2
socleC22×C181C22×D4
prime radicalC22×D181C2×D4
hypercenter#487D181
hypocenterC181#3209
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #26041659C2semidirect
4. #26041658C2semidirect
5. #26041631C2semidirect
6. #26041488C2semidirect
9. #487×C181C2semidirect
79. C4×D181Q16semidirect
107. C181#3209semidirect
114. #487D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
C4⋊Q8×D1812
#260416592
#260416582
#260416312
#260414882
#260414362
#260413302
#487×C1812
#3209181