Finite Group Databasebeta

← #37533697

Subgroups — #37533697

Order 23168.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19153863C211yesmaximal, characteristic
2#26041295C212yesmaximal, complemented
3#26041289C212yesmaximal, complemented
4#26041253C211yesmaximal, characteristic, complemented
5#413×C181C211yesmaximal, characteristic, complemented
6#19153863—24no—
7#19153863—24no—
8C2×#15668449C412yescomplemented
9C2×#15668449C2211yescharacteristic
10C2×#15668448—21no—
11C2×#15668448—22no—
12C22×Dic362C412yescomplemented
13C22×Dic362C2211yescharacteristic
14C2×Q8×D181—21no—
15#19153775C2212yes—
16#19153742—24no—
17#19153741—24no—
18#19153731—22no—
19C2×#41×C181C2211yescharacteristic
20C2×M16×C181C2212yes—
21#101×C181—21no—
22#100×C181—21no—
23#15668449—24no—
24#15668449—24no—
25#15668449—22no—
26#15668449D414yes—
27#15668448—216no—
28#15668448—28no—
29C2×Dic362—24no—
30C2×Dic362—24no—
31C2×Dic362—22no—
32C2×Dic362—21no—
33C2×Dic362—22no—
34C2×Dic362C2×C411yescharacteristic
35C2×Dic362C2×C412yes—
36C2×Dic362D414yes—
37Q8×D181—28no—
38C2×D724C2×C411yescharacteristic
39C2×C22⋊D181—22no—
40C2×C22⋊D181—21no—
41C22×Dic181—22no—
42C22×Dic181—21no—
43C2×C4×D181—22no—
44C2×C4×D181—21no—
45C4⋊Dic181—22no—
46C22⋊Dic181—22no—
47C4×Dic181—22no—
48#41×C181—22no—
49#41×C181—22no—
50C2×Q8×C181D411yescharacteristic
51C2×D4×C181D411yescharacteristic
52C2×D4×C181D412yes—
53C22×C4×C181C2×C412yes—
54C22×C4×C181C2311yescharacteristic
55M16×C181—22no—
56M16×C181—22no—
57C2×C8×C181—22no—
58C22×D181—21no—
59Dic362—28no—
60Dic362—21no—
61Dic362—22no—
62Dic362—24no—
63Dic362C22⋊C412yes—
64Dic362C22⋊C414yes—
65D724—21no—
66D724C22⋊C412yes—
67C22⋊D181—28no—
68C22⋊D181—24no—
69C2×Dic181—28no—
70C2×Dic181—22no—
71C2×Dic181—21no—
72C2×Dic181—24no—
73C2×Dic181—42no—
74C4×D181—28no—
75C4×D181—24no—
76C23×C181C22⋊C412yes—
77C23×C181C2×D411yescharacteristic
78Q8×C181—22no—
79D4×C181—24no—
80D4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181C22⋊C412yes—
84C2×C4×C181C22×C411yescharacteristic
85C2×C4×C181C2×D412yes—
86C2×C4×C181C2×D412yes—
87C2×C4×C181C2×D411yescharacteristic
88C8×C181—42no—
89C22×C181—22no—
90C22×C181—41no—
91C22×C181—42no—
92C22×C181C2×C22⋊C411yescharacteristic
93C22×C181#12012yes—
94C2×D181—22no—
95C2×D181—42no—
96C4×C181—22no—
97C4×C181C2×C22⋊C411yescharacteristic
98C4×C181C2×C22⋊C411yescharacteristic
99C4×C181#12012yes—
100Dic181—24no—
101Dic181—22no—
102Dic181—82no—
103C2×C181—22no—
104C2×C181—21no—
105C2×C181—42no—
106C2×C181#37911yescharacteristic
107D181—42no—
108C181#179611yescharacteristic, complemented, Sylow
109#1796—1811nomaximal, Sylow
110C2×2−1+4—1811no—
111#421—1812no—
112#419—1812no—
113#413D18111yescharacteristic, complemented
114#410—1811no—
1152−1+4—3624no—
1162−1+4—3624no—
117C2×#41—3621no—
118C2×#41—3622no—
119C2×#41—1812no—
120C2×#41—1811no—
121C2×#41C2×D18111yescharacteristic
122C22×Q8—3621no—
123C22×Q8—1812no—
124C22×Q8—1811no—
125C2×M16C2×D18112yes—
126#117—1812no—
127#104—3624no—
128#103—3624no—
129#101—21no—
130#100—21no—
131#99—3622no—
132#41—22no—
133#41—22no—
134#41—3624no—
135#41—3624no—
136#41—36216no—
137#41—3628no—
138#41—3622no—
139#41—1814no—
140C2×Q8—3624no—
141C2×Q8—3624no—
142C2×Q8—3628no—
143C2×Q8—3622no—
144C2×Q8—3622no—
145C2×Q8—3621no—
146C2×Q8—1811no—
147C2×Q8—1812no—
148C2×Q8—1814no—
149C2×Q8C22⋊D18111yescharacteristic
150C2×D4—3622no—
151C2×D4—3621no—
152C2×D4—1811no—
153C2×D4C22⋊D18111yescharacteristic
154C2×D4D72412yes—
155C22×C4—3622no—
156C22×C4—3621no—
157C22×C4—3622no—
158C22×C4—3621no—
159C22×C4C4×D18112yes—
160C22×C4C22×D18111yescharacteristic
161M16—22no—
162M16—22no—
163C2×C8—22no—
164C4⋊C4—3622no—
165C22⋊C4—3622no—
166C42—3622no—
167C23—3621no—
168C23#1566842612yes—
169C23C2×D72411yescharacteristic
170Q8—22no—
171Q8—3628no—
172Q8—3621no—
173Q8—3622no—
174Q8—3624no—
175Q8—1812no—
176Q8—1814no—
177D4—24no—
178D4—22no—
179D4—3628no—
180D4—3624no—
181D4—3621no—
182D4—1812no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—3628no—
186C2×C4—3628no—
187C2×C4—3621no—
188C2×C4—3622no—
189C2×C4—3624no—
190C2×C4—3624no—
191C2×C4—7242no—
192C2×C4#1566842612yes—
193C2×C4C2×C4×D18111yescharacteristic
194C2×C4D4×D18112yes—
195C2×C4D4×D18112yes—
196C2×C4C2×C22⋊D18111yescharacteristic
197C8—42no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—3622no—
202C22—7242no—
203C22C2×#1566842611yescharacteristic
204C22#1915378612yes—
205C4—22no—
206C4—3624no—
207C4—3622no—
208C4—14482no—
209C4C22⋊C4×D18111yescharacteristic
210C4C22⋊C4×D18111yescharacteristic
211C4#1915378412yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—7242no—
216C2#2604113411yescharacteristic
subgroupG/H
centerC2#26041134
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#413×C181C2
socleC2×C181#379
prime radicalC2×#19153863C2
cosocleC22×C4×C181C23
hypercenter#413D181
hypocenterC181#1796
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #26041295C2semidirect
3. #26041289C2semidirect
4. #26041253C2semidirect
5. #413×C181C2semidirect
8. C2×#15668449C4semidirect
12. C22×Dic362C4semidirect
108. C181#1796semidirect
113. #413D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538632
#260412952
#260412892
#260412532
#413×C1812
#1796181