Finite Group Databasebeta

← #37549767

Subgroups — #37549767

Order 23168.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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#26042346C211yesmaximal, characteristic, complemented
2#26042277C212yesmaximal, complemented
3D181⋊#100C211yesmaximal, characteristic, complemented
4#26042146C212yesmaximal, complemented
5#26041842C211yesmaximal, characteristic
6D181⋊M16C2212yes—
7D4×#10624382—24no—
8#19153895C2212yes—
9C2×C22⋊#10624382—22no—
10#19153880—24no—
11#19153876—22no—
12C2×#15668448C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14#19153812C411yescharacteristic, complemented
15#19153804C411yescharacteristic, complemented
16#19153803C411yescharacteristic, complemented
17#19153788—22no—
18#19153784—22no—
19#19153773C2211yescharacteristic
20#127×C181C411yescharacteristic, complemented
21C22×#10624382—24no—
22#15668459—22no—
23#15668458—22no—
24D181⋊C8—22no—
25C4⋊#10624382—22no—
26C22⋊#10624382—24no—
27C22⋊#10624382—22no—
28C4×#10624382D414yescomplemented
29C23×D181D412yes—
30#15668448—22no—
31#15668448—22no—
32C2×Dic362C2×C411yescharacteristic, complemented
33C2×D724C2×C411yescharacteristic
34C2×C22⋊D181—21no—
35C2×C22⋊D181—22no—
36C2×C22⋊D181D412yes—
37D4×D181—22no—
38D4×D181—22no—
39C22×Dic181—21no—
40C2×C4×D181C2311yescharacteristic
41#15668428—21no—
42C22⋊Dic181—22no—
43#15668426—22no—
44#15668426—21no—
45C4×Dic181C2×C411yescharacteristic, complemented
46C2×D4×C181—22no—
47C2×D4×C181C2×C411yescharacteristic
48C2×D4×C181C2×C411yescharacteristic, complemented
49C42×C181C2×C411yescharacteristic, complemented
50C2×#10624382—24no—
51C2×#10624382—84no—
52#13089685—42no—
53C22×D181—22no—
54C22×D181—22no—
55C22×D181—82no—
56C22×D181C22⋊C412yes—
57C22×D181C2×D411yescharacteristic
58Dic362—22no—
59D724—22no—
60C22⋊D181—24no—
61C22⋊D181—41no—
62C22⋊D181—41no—
63C22⋊D181—82no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—42no—
67C2×Dic181C2×D411yescharacteristic
68C4×D181C2×D412yes—
69C4×D181#4112yes—
70C23×C181—21no—
71C23×C181C22⋊C412yes—
72D4×C181—22no—
73D4×C181—22no—
74D4×C181—82no—
75C2×C4×C181—21no—
76C2×C4×C181C22×C411yescharacteristic
77C22×C181—22no—
78C22×C181—42no—
79C22×C181—42no—
80C22×C181—41no—
81C22×C181C2×C22⋊C411yescharacteristic
82C2×D181—22no—
83C2×D181—41no—
84C2×D181—44no—
85C2×D181—42no—
86C2×D181—82no—
87C2×D181#12011yescharacteristic
88C2×D181#12311yescharacteristic
89#10624382—44no—
90C4×C181—41no—
91C4×C181C4×D412yes—
92Dic181—41no—
93Dic181—41no—
94Dic181#12112yes—
95C2×C181—21no—
96C2×C181—42no—
97C2×C181—81no—
98C2×C181#38611yescharacteristic
99D181—21no—
100D181—21no—
101D181—42no—
102C181#191111yescharacteristic, complemented, Sylow
103#1911—1811nomaximal, Sylow
104#535—1811no—
105#518—1811no—
106#421—1812no—
107C2×#100—1811no—
108H(3,ℤ/4)—1812no—
109C2×#41—1811no—
110C22×D4—1811no—
111C2×M16—1812no—
112#127—1811no—
113#127#1062438211yescharacteristic, complemented
114#124—1811no—
115#124—1811no—
116#121—3622no—
117#120—3622no—
118C4×D4—3624no—
119#117—1812no—
120#117—1811no—
121C2×C22⋊C4—3622no—
122#104—3624no—
123#100—3622no—
124C24—1812no—
125#41—3622no—
126#41—3622no—
127C2×Q8—1811no—
128C2×D4—22no—
129C2×D4—3622no—
130C2×D4—3621no—
131C2×D4—3622no—
132C2×D4—3622no—
133C2×D4—1812no—
134C2×D4—1811no—
135C2×D4C2×#1062438211yescharacteristic
136C2×D4C2×#1062438211yescharacteristic, complemented
137C22×C4—3624no—
138C22×C4—3621no—
139C22×C4—1811no—
140M16—3622no—
141M16—3622no—
142C2×C8—3622no—
143C4⋊C4—3621no—
144C4⋊C4—3622no—
145C22⋊C4—3621no—
146C22⋊C4—3622no—
147C22⋊C4—3624no—
148C22⋊C4—3622no—
149C22⋊C4—3622no—
150C42—1814no—
151C42—1811no—
152C42C2×#1062438211yescharacteristic, complemented
153C23—21no—
154C23—3622no—
155C23—3622no—
156C23—14482no—
157C23—1812no—
158C23—1811no—
159C23C22⋊#1062438212yes—
160Q8—3622no—
161D4—22no—
162D4—22no—
163D4—82no—
164D4—3624no—
165D4—3622no—
166D4—14482no—
167D4—7241no—
168D4—7241no—
169C2×C4—21no—
170C2×C4—3624no—
171C2×C4—3621no—
172C2×C4—3621no—
173C2×C4—14484no—
174C2×C4—1811no—
175C2×C4—1812no—
176C2×C4—1812no—
177C2×C4—7242no—
178C2×C4C22×#1062438211yescharacteristic
179C8—7242no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—3622no—
185C22—14482no—
186C22—1811no—
187C22—1811no—
188C22—7241no—
189C22—7242no—
190C22—7244no—
191C22C2×C22⋊#1062438211yescharacteristic
192C4—41no—
193C4—1812no—
194C4—7241no—
195C4—7244no—
196C4—7241no—
197C4D4×#1062438212yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—3621no—
202C2—3621no—
203C2—7242no—
204C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#127×C181C4
socleC2×C181#386
prime radical#26041842C2
cosocleC2×C4×D181C23
hypercenter#127#10624382
hypocenterC181#1911
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26042346C2semidirect
2. #26042277C2semidirect
3. D181⋊#100C2semidirect
4. #26042146C2semidirect
14. #19153812C4semidirect
15. #19153804C4semidirect
16. #19153803C4semidirect
20. #127×C181C4semidirect
28. C4×#10624382D4semidirect
32. C2×Dic362C2×C4semidirect
45. C4×Dic181C2×C4semidirect
48. C2×D4×C181C2×C4semidirect
49. C42×C181C2×C4semidirect
102. C181#1911semidirect
113. #127#10624382semidirect
136. C2×D4C2×#10624382semidirect
152. C42C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423462
#260422772
D181⋊#1002
#260421462
#260418422
#1911181