Finite Group Databasebeta

← #37543522

Subgroups — #37543522

Order 23168.

376 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×C4×Dic362C211yesmaximal, characteristic
2#26041583C214yesmaximal, complemented
3#26041571C212yesmaximal, complemented
4#26041561C212yesmaximal, complemented
5#26041322C212yesmaximal
6#26041303C211yesmaximal, characteristic
7#26041287C212yesmaximal
8#533×C181C211yesmaximal, characteristic
9C22×Dic362C2211yescharacteristic
10C2×#15668433C2212yes—
11C4×Dic362—21no—
12C4×Dic362—21no—
13C4×Dic362C2212yes—
14C4×Dic362C2212yes—
15C2×C4⋊Dic181C2211yescharacteristic
16C2×#15668428—21no—
17C2×C4×Dic181—21no—
18#19153749C2214yes—
19#19153747C2214yes—
20C4⋊#13089678C2212yes—
21C4⋊#13089678C2212yes—
22#19153739C2214yes—
23#19153739C2214yes—
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
33C2×Dic362—21no—
34C2×Dic362—22no—
35C2×Dic362Q814yescomplemented
36C2×Dic362C2312yes—
37C22×Dic181—21no—
38C22×Dic181—21no—
39#15668433—24no—
40C2×#13089678C2314yes—
41C4⋊Dic181—21no—
42C4⋊Dic181C2312yes—
43#15668428—22no—
44#15668428—22no—
45C4×Dic181—22no—
46C4×Dic181—22no—
47C2×Q8×C181—21no—
48C2×Q8×C181—21no—
49C22×C4×C181D411yescharacteristic
50C22×C4×C181D411yescharacteristic
51C22×C4×C181C2311yescharacteristic
52C4⋊C4×C181—22no—
53C4⋊C4×C181—22no—
54C4⋊C4×C181—21no—
55C4⋊C4×C181—21no—
56C4⋊C4×C181C2312yes—
57C4⋊C4×C181C2312yes—
58C22⋊C4×C181—22no—
59C42×C181—22no—
60C42×C181D411yescharacteristic
61C42×C181D411yescharacteristic
62C42×C181C2311yescharacteristic
63C42×C181C2311yescharacteristic
64Dic362—24no—
65Dic362—22no—
66Dic362C2×Q814yes—
67C2×Dic181—24no—
68C2×Dic181—22no—
69C2×Dic181—22no—
70C2×Dic181—42no—
71#13089678—24no—
72C23×C181C2×D411yescharacteristic
73Q8×C181—42no—
74Q8×C181—42no—
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×C181C2×Q812yes—
85C2×C4×C181#4112yes—
86C2×C4×C181C2411yescharacteristic
87C22×C181—21no—
88C22×C181—21no—
89C22×C181C22⋊Q812yes—
90C22×C181C22×D411yescharacteristic
91C4×C181—21no—
92C4×C181—21no—
93C4×C181—21no—
94C4×C181—42no—
95C4×C181—42no—
96C4×C181C22⋊Q812yes—
97C4×C181#13712yes—
98C4×C181C22×Q811yescharacteristic
99C4×C181C2×#4111yescharacteristic
100Dic181—24no—
101Dic181—42no—
102C2×C181—22no—
103C2×C181C2×C22⋊Q811yescharacteristic
104C2×C181C2×#13711yescharacteristic
105C2×C181#57511yescharacteristic
106C181#307811yescharacteristic, complemented, Sylow
107#3078—1811nomaximal, Sylow
108#533D18111yescharacteristic
109C2×C4×Q8—1811no—
110#479—1814no—
111#477—1812no—
112#475—1812no—
113#428—1812no—
114#423—1811no—
115#419—1812no—
116C22×Q8—1811no—
117C2×M16—1812no—
118C4⋊Q8C2×D18111yescharacteristic
119C4⋊Q8C2×D18111yescharacteristic
120#125C2×D18112yes—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—3621no—
126C4×Q8—3621no—
127C4×Q8—1812no—
128C4×Q8—1812no—
129#117C2×D18112yes—
130C2×C4⋊C4—3621no—
131C2×C4⋊C4—1811no—
132C2×C42—3621no—
133C2×C42C2×D18111yescharacteristic
134#107—1814no—
135#106—1814no—
136C4⋊C8—1812no—
137C4⋊C8—1812no—
138#103—1814no—
139#103—1814no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—3621no—
143C2×Q8—3622no—
144C2×Q8—1814no—
145C2×Q8—1812no—
146C22×C4—3621no—
147C22×C4—3621no—
148C22×C4C22⋊D18111yescharacteristic
149C22×C4C22⋊D18111yescharacteristic
150C22×C4C22×D18111yescharacteristic
151M16—3624no—
152C2×C8—1814no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3622no—
158C4⋊C4—3622no—
159C4⋊C4—3621no—
160C4⋊C4—1812no—
161C4⋊C4C22×D18112yes—
162C4⋊C4C22×D18112yes—
163C22⋊C4—22no—
164C42—22no—
165C42—3622no—
166C42—3622no—
167C42C22⋊D18111yescharacteristic
168C42C22⋊D18111yescharacteristic
169C42C22×D18111yescharacteristic
170C42C22×D18111yescharacteristic
171C23C2×C22⋊D18111yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—3624no—
175Q8—3622no—
176Q8—1814no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—3624no—
182C2×C4—3622no—
183C2×C4—3622no—
184C2×C4—7242no—
185C2×C4C2×C22⋊D18111yescharacteristic
186C2×C4C2×C22⋊D18111yescharacteristic
187C2×C4C2×C22⋊D18111yescharacteristic
188C2×C4C2×C22⋊D18111yescharacteristic
189C2×C4C2×C22⋊D18111yescharacteristic
190C2×C4Q8×D18112yes—
191C2×C4#1566845012yes—
192C2×C4C23×D18111yescharacteristic
193C8—3624no—
194C22—21no—
195C22—21no—
196C22#1915379212yes—
197C22C22×C22⋊D18111yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—42no—
202C4—42no—
203C4—3624no—
204C4—7242no—
205C4#1915379212yes—
206C4#1915384612yes—
207C4C2×Q8×D18111yescharacteristic
208C4C2×#1566845011yescharacteristic
209C2—22no—
210C2C2×#1915379211yescharacteristic
211C2C2×#1915384611yescharacteristic
212C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#533×C181C2
socleC22×C181C22×D4
prime radicalC23×C181C2×D4
hypercenter#533D181
hypocenterC181#3078
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. #26041583C2semidirect
3. #26041571C2semidirect
4. #26041561C2semidirect
35. C2×Dic362Q8semidirect
106. C181#3078semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic3622
#260415832
#260415712
#260415612
#260413222
#260413032
#260412872
#533×C1812
#3078181