Finite Group Databasebeta

← #37534104

Subgroups — #37534104

Order 23168.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#26041825C211yesmaximal, characteristic
2#26041295C212yesmaximal, complemented
3#26041253C211yesmaximal, characteristic, complemented
4#26041040C212yesmaximal, complemented
5#568×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C2211yescharacteristic
7#19153814C411yescharacteristic, complemented
8#19153812C411yescharacteristic, complemented
9#19153805C411yescharacteristic, complemented
10#19153803C411yescharacteristic, complemented
11#19153788—22no—
12C4×C22⋊D181—22no—
13#19153775C2212yes—
14C2×C4×Dic181C2211yescharacteristic
15#19153742—24no—
16#19153731—22no—
17C2×#41×C181C2211yescharacteristic
18#131×C181—24no—
19C2×M16×C181—22no—
20C2×M16×C181C2212yes—
21#15668449—22no—
22#15668449—22no—
23C2×Dic362C2×C411yescharacteristic
24C2×D724C2×C411yescharacteristic
25C2×C22⋊D181—22no—
26C2×C22⋊D181—21no—
27C22×Dic181—21no—
28C2×C4×D181—21no—
29C4⋊Dic181—22no—
30#15668428—22no—
31C22⋊Dic181—22no—
32C22⋊Dic181—22no—
33#15668426—22no—
34C4×Dic181—21no—
35C4×Dic181—22no—
36C4×Dic181C2×C411yescharacteristic
37C4×Dic181C2×C411yescharacteristic
38#41×C181—22no—
39#41×C181—22no—
40C2×Q8×C181C2×C411yescharacteristic
41C2×D4×C181C2×C411yescharacteristic
42C2×D4×C181D412yes—
43C22×C4×C181D412yes—
44C22×C4×C181C2311yescharacteristic
45M16×C181—24no—
46M16×C181—22no—
47M16×C181D414yescomplemented
48C2×C8×C181—24no—
49C2×C8×C181—22no—
50C22×D181—21no—
51Dic362—22no—
52D724—22no—
53C22⋊D181—44no—
54C22⋊D181—42no—
55C2×Dic181—21no—
56C2×Dic181—21no—
57C2×Dic181—21no—
58C2×Dic181—42no—
59C2×Dic181—42no—
60C4×D181—41no—
61C4×D181—41no—
62C23×C181C22⋊C412yes—
63C23×C181C2×D411yescharacteristic
64Q8×C181—22no—
65D4×C181—24no—
66D4×C181—22no—
67C2×C4×C181—22no—
68C2×C4×C181—22no—
69C2×C4×C181C22⋊C412yes—
70C2×C4×C181C22×C411yescharacteristic
71C2×C4×C181C2×D412yes—
72C2×C4×C181C2×D411yescharacteristic
73C2×C4×C181#4112yes—
74C8×C181—24no—
75C22×C181—22no—
76C22×C181—41no—
77C22×C181—42no—
78C22×C181C2×C22⋊C411yescharacteristic
79C22×C181#12112yes—
80C2×D181—42no—
81C2×D181—41no—
82C4×C181—22no—
83C4×C181C4×D412yes—
84C4×C181#12011yescharacteristic
85C4×C181#12311yescharacteristic
86Dic181—22no—
87Dic181—41no—
88Dic181—41no—
89Dic181—82no—
90C2×C181—21no—
91C2×C181—22no—
92C2×C181—42no—
93C2×C181#38611yescharacteristic
94D181—81no—
95C181#188411yescharacteristic, complemented, Sylow
96#1884—1811nomaximal, Sylow
97#568D18111yescharacteristic, complemented
98#532—1811no—
99#421—1812no—
100#410—1811no—
101#339—1812no—
102C2×#41—1811no—
103C2×#41C2×D18111yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D18112yes—
107C4⋊Q8—1811no—
108#127—1811no—
109#124—1811no—
110#124—1811no—
111#121—3622no—
112C4×D4—3622no—
113#117—1812no—
114C2×C42—1811no—
115#104—3624no—
116#99—3622no—
117#41—22no—
118#41—22no—
119#41—3622no—
120#41—3622no—
121C2×Q8—1811no—
122C2×Q8C4×D18111yescharacteristic
123C2×D4—3622no—
124C2×D4—3621no—
125C2×D4—1811no—
126C2×D4C4×D18111yescharacteristic
127C2×D4D72412yes—
128C22×C4—3621no—
129C22×C4—3621no—
130C22×C4C22⋊D18112yes—
131C22×C4C22×D18111yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D18114yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—3622no—
138C4⋊C4—3622no—
139C22⋊C4—3622no—
140C22⋊C4—3622no—
141C22⋊C4—3622no—
142C42—3621no—
143C42—3622no—
144C42—1811no—
145C42—1811no—
146C23—3621no—
147C23#1566842612yes—
148C23C2×D72411yescharacteristic
149Q8—22no—
150Q8—3622no—
151D4—24no—
152D4—22no—
153D4—3622no—
154D4—7244no—
155D4—7242no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—3621no—
159C2×C4—3621no—
160C2×C4—3621no—
161C2×C4—7242no—
162C2×C4—7242no—
163C2×C4—7241no—
164C2×C4—7241no—
165C2×C4#1566842612yes—
166C2×C4C2×C4×D18111yescharacteristic
167C2×C4C2×C22⋊D18112yes—
168C2×C4C2×C22⋊D18111yescharacteristic
169C2×C4#1566844912yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—7242no—
175C22—7241no—
176C22C2×#1566842611yescharacteristic
177C22#1915379012yes—
178C4—22no—
179C4—3622no—
180C4—14482no—
181C4—7241no—
182C4—7241no—
183C4C4×C22⋊D18112yes—
184C4#1915378511yescharacteristic
185C4#1915379911yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—14481no—
190C2#2604115811yescharacteristic
subgroupG/H
centerC4#19153785
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#568×C181C2
socleC2×C181#386
prime radical#26041825C2
cosocleC22×C4×C181C23
hypercenter#568D181
hypocenterC181#1884
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26041295C2semidirect
3. #26041253C2semidirect
4. #26041040C2semidirect
5. #568×C181C2semidirect
7. #19153814C4semidirect
8. #19153812C4semidirect
9. #19153805C4semidirect
10. #19153803C4semidirect
47. M16×C181D4semidirect
95. C181#1884semidirect
97. #568D181semidirect
134. M16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
#260418252
#260412952
#260412532
#260410402
#568×C1812
#1884181