Finite Group Databasebeta

← #37532809

Subgroups — #37532809

Order 23168.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#26041934C211yesmaximal, characteristic, complemented
2#26041625C211yesmaximal, characteristic, complemented
3#26041471C211yesmaximal, characteristic, complemented
4#26041024C211yesmaximal, characteristic, complemented
5D4⋊#13089678C211yesmaximal, characteristic, complemented
6#26041006C211yesmaximal, characteristic, complemented
7#459×C181C211yesmaximal, characteristic, complemented
8C2×#15668449—21no—
9C2×#15668434—21no—
10#19153816C2211yescharacteristic
11#19153798—22no—
12#19153795—21no—
13#19153787—21no—
14C4×D724C2211yescharacteristic
15C4⋊C4×D181—21no—
16C4⋊#13089678C2211yescharacteristic
17#19153737—22no—
18#19153737—21no—
19C4×#13089678C2211yescharacteristic
20C2×D8×C181—21no—
21#127×C181C2211yescharacteristic
22C4×D4×C181C2211yescharacteristic
23C4⋊C8×C181C2211yescharacteristic
24#102×C181—21no—
25#15668449—42no—
26#15668449—42no—
27C2×Dic362—21no—
28C2×D724D411yescharacteristic
29C2×C22⋊D181—22no—
30C2×C4×D181—22no—
31C2×C4×D181—21no—
32#15668434—42no—
33C2×#13089678—21no—
34C2×#13089678—21no—
35C4⋊Dic181—21no—
36C4⋊Dic181D411yescharacteristic
37C4⋊Dic181D412yes—
38#15668428—22no—
39C22⋊Dic181—22no—
40#15668426—21no—
41C2×D4×C181—21no—
42C2×D4×C181—21no—
43C2×D4×C181—21no—
44C2×D4×C181D411yescharacteristic
45C22×C4×C181—21no—
46D8×C181—42no—
47C2×C8×C181—21no—
48C4⋊C4×C181D411yescharacteristic
49C22⋊C4×C181—21no—
50C42×C181C2311yescharacteristic
51C22×D181—21no—
52Dic362—41no—
53Dic362—41no—
54D724—21no—
55D724D812yescomplemented
56C22⋊D181—44no—
57C2×Dic181—21no—
58C2×Dic181—22no—
59C4×D181—44no—
60C4×D181—41no—
61C4×D181—41no—
62#13089678—41no—
63#13089678—41no—
64C23×C181—21no—
65C23×C181—41no—
66D4×C181—21no—
67D4×C181—21no—
68D4×C181—42no—
69D4×C181—41no—
70D4×C181—41no—
71D4×C181—41no—
72D4×C181—41no—
73C2×C4×C181—21no—
74C2×C4×C181—41no—
75C2×C4×C181—41no—
76C2×C4×C181C2×D411yescharacteristic
77C2×C4×C181C2×D411yescharacteristic
78C2×C4×C181C2×D411yescharacteristic
79C8×C181—41no—
80C22×C181—41no—
81C22×C181—41no—
82C22×C181—41no—
83C22×C181—81no—
84C22×C181—81no—
85C22×C181—81no—
86C22×C181#12011yescharacteristic
87C2×D181—22no—
88C2×D181—41no—
89C2×D181—41no—
90C4×C181—21no—
91C4×C181—41no—
92C4×C181C2×D811yescharacteristic
93C4×C181#13511yescharacteristic
94C4×C181AGL(1,ℤ/8)11yescharacteristic
95C4×C181AGL(1,ℤ/8)11yescharacteristic
96Dic181—42no—
97Dic181—41no—
98C2×C181—81no—
99C2×C181—161no—
100C2×C181#44711yescharacteristic
101C2×C181#44911yescharacteristic
102C2×C181#45411yescharacteristic
103D181—42no—
104C181#162511yescharacteristic, complemented, Sylow
105#1625—1811nomaximal, Sylow
106#547—1811no—
107#486—1811no—
108#459—1811no—
109#459D18111yescharacteristic, complemented
110#332—1811no—
111D4⋊C8—1811no—
112D4⋊C8—1811no—
113C2×#41—3621no—
114C2×D8—21no—
115C2×D8—3621no—
116C4⋊Q8—1811no—
117#127C2×D18111yescharacteristic
118#123—3622no—
119C22⋊Q8—3621no—
120#121—3621no—
121C4×D4—1811no—
122C4×D4C2×D18111yescharacteristic
123C2×C4⋊C4—3621no—
124C4⋊C8—1811no—
125C4⋊C8C2×D18111yescharacteristic
126#102—21no—
127#102—3621no—
128#102—3622no—
129C4×C8—1811no—
130#41—7242no—
131#41—7242no—
132C2×Q8—3621no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3622no—
137C2×D4—1811no—
138C2×D4C22⋊D18111yescharacteristic
139C22×C4—21no—
140C22×C4—3622no—
141C22×C4—3621no—
142D8—42no—
143D8—7242no—
144C2×C8—21no—
145C2×C8—3621no—
146C2×C8—3621no—
147C4⋊C4—3622no—
148C4⋊C4—3621no—
149C4⋊C4—1811no—
150C4⋊C4—1812no—
151C4⋊C4C22⋊D18111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3622no—
154C22⋊C4—3621no—
155C42C22×D18111yescharacteristic
156C23—21no—
157C23—41no—
158C23—3621no—
159Q8—7241no—
160Q8—7241no—
161D4—21no—
162D4—21no—
163D4—42no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—3621no—
169D4—1812no—
170D4—7244no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—3621no—
175C2×C4—3622no—
176C2×C4—7244no—
177C2×C4—7241no—
178C2×C4—7241no—
179C2×C4D4×D18111yescharacteristic
180C2×C4D4×D18111yescharacteristic
181C2×C4C2×C22⋊D18111yescharacteristic
182C8—41no—
183C8—7241no—
184C8—7241no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—3622no—
192C22—7241no—
193C22—7241no—
194C22#1915378411yescharacteristic
195C4—21no—
196C4—41no—
197C4—7242no—
198C4—7241no—
199C4D8×D18111yescharacteristic
200C4#1915383611yescharacteristic
201C4#1915384011yescharacteristic
202C4#1915384211yescharacteristic
203C2—81no—
204C2—161no—
205C2—7242no—
206C2#2604141111yescharacteristic
207C2#2604142211yescharacteristic
208C2#2604145011yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#459×C181C2
socleC22×C181#120
hypercenter#459D181
hypocenterC181#1625
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26041934C2semidirect
2. #26041625C2semidirect
3. #26041471C2semidirect
4. #26041024C2semidirect
5. D4⋊#13089678C2semidirect
6. #26041006C2semidirect
7. #459×C181C2semidirect
55. D724D8semidirect
104. C181#1625semidirect
109. #459D181semidirect

Maximal subgroups

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

subgroupindex
#260419342
#260416252
#260414712
#260410242
D4⋊#130896782
#260410062
#459×C1812
#1625181