Finite Group Databasebeta

← #37532807

Subgroups — #37532807

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#26041932C211yesmaximal, characteristic, complemented
2#26041471C211yesmaximal, characteristic, complemented
3#26041471C211yesmaximal, characteristic, complemented
4#26041025C211yesmaximal, characteristic, complemented
5#26041008C211yesmaximal, characteristic, complemented
6#26041008C211yesmaximal, characteristic, complemented
7#486×C181C211yesmaximal, characteristic, complemented
8C2×#15668450—21no—
9C2×#15668434—21no—
10C2×#15668434—21no—
11#19153801—22no—
12#19153792—21no—
13#19153791—21no—
14C4×D724C2211yescharacteristic
15C4×D724C2211yescharacteristic
16C4⋊C4×D181—21no—
17C4⋊#13089678C2211yescharacteristic
18C4⋊#13089678C2211yescharacteristic
19#19153737—21no—
20#19153737—21no—
21C4⋊Q8×C181C2211yescharacteristic
22#127×C181C2211yescharacteristic
23#102×C181—22no—
24C4×C8×C181C2211yescharacteristic
25#15668450—42no—
26#15668450—42no—
27C2×D724—22no—
28C2×D724D411yescharacteristic
29C2×D724D411yescharacteristic
30C2×C4×D181—22no—
31C2×C4×D181—21no—
32C2×C4×D181—21no—
33#15668434—42no—
34#15668434—42no—
35C2×#13089678—21no—
36C2×#13089678—21no—
37C4⋊Dic181D411yescharacteristic
38C4⋊Dic181D411yescharacteristic
39#15668428—22no—
40#15668426—22no—
41#15668426—21no—
42#15668426—21no—
43C2×Q8×C181—21no—
44C2×D4×C181—21no—
45C2×D4×C181—21no—
46C2×D4×C181—21no—
47C2×C8×C181—21no—
48C4⋊C4×C181—21no—
49C4⋊C4×C181D412yes—
50C42×C181C2311yescharacteristic
51C22×D181—21no—
52C22×D181—21no—
53D724—21no—
54D724—21no—
55D724—21no—
56D724—44no—
57D724D812yescomplemented
58C2×Dic181—21no—
59C2×Dic181—21no—
60C4×D181—44no—
61C4×D181—41no—
62C4×D181—41no—
63C4×D181—41no—
64C4×D181—41no—
65#13089678—41no—
66#13089678—41no—
67C23×C181—41no—
68Q8×C181—41no—
69Q8×C181—41no—
70D4×C181—42no—
71D4×C181—41no—
72D4×C181—41no—
73D4×C181—41no—
74D4×C181—41no—
75C2×C4×C181—22no—
76C2×C4×C181C2×D411yescharacteristic
77C2×C4×C181C2×D411yescharacteristic
78C2×C4×C181C2×D411yescharacteristic
79C8×C181—41no—
80C22×C181—81no—
81C22×C181—81no—
82C22×C181—81no—
83C22×C181#12011yescharacteristic
84C2×D181—22no—
85C2×D181—41no—
86C2×D181—41no—
87C2×D181—41no—
88C2×D181—41no—
89C2×D181—41no—
90C4×C181—21no—
91C4×C181—42no—
92C4×C181C2×D811yescharacteristic
93C4×C181#13511yescharacteristic
94C4×C181AGL(1,ℤ/8)11yescharacteristic
95C4×C181AGL(1,ℤ/8)11yescharacteristic
96Dic181—41no—
97Dic181—41no—
98C2×C181—161no—
99C2×C181#44711yescharacteristic
100C2×C181#44911yescharacteristic
101C2×C181#45411yescharacteristic
102D181—42no—
103D181—81no—
104C181#162511yescharacteristic, complemented, Sylow
105#1625—1811nomaximal, Sylow
106#547—1811no—
107#486D18111yescharacteristic, complemented
108#459—1811no—
109#459—1811no—
110#332—1811no—
111D4⋊C8—1811no—
112D4⋊C8—1811no—
113C2×#41—3621no—
114C2×D8—3621no—
115C2×D8—3621no—
116C4⋊Q8C2×D18111yescharacteristic
117#127C2×D18111yescharacteristic
118#123—3622no—
119C22⋊Q8—3621no—
120#121—3621no—
121C4×D4—1811no—
122C4×D4—1811no—
123C2×C4⋊C4—3621no—
124C4⋊C8—1811no—
125C4⋊C8—1811no—
126#102—22no—
127#102—3621no—
128#102—3621no—
129C4×C8C2×D18111yescharacteristic
130#41—7242no—
131#41—7242no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3622no—
137C2×D4—1811no—
138C2×D4—1811no—
139C22×C4—3622no—
140C22×C4—3621no—
141C22×C4—3621no—
142D8—7242no—
143D8—7242no—
144C2×C8—21no—
145C2×C8—3621no—
146C2×C8—3621no—
147C4⋊C4—21no—
148C4⋊C4—3622no—
149C4⋊C4—1811no—
150C4⋊C4—1811no—
151C4⋊C4C22⋊D18112yes—
152C22⋊C4—3622no—
153C22⋊C4—3621no—
154C22⋊C4—3621no—
155C42C22×D18111yescharacteristic
156C23—41no—
157C23—3621no—
158C23—3621no—
159Q8—41no—
160Q8—41no—
161D4—42no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—3621no—
167D4—3621no—
168D4—3621no—
169D4—1812no—
170D4—7244no—
171C2×C4—22no—
172C2×C4—3621no—
173C2×C4—3621no—
174C2×C4—7244no—
175C2×C4—7241no—
176C2×C4—7241no—
177C2×C4—7241no—
178C2×C4—7241no—
179C2×C4D4×D18111yescharacteristic
180C2×C4D4×D18111yescharacteristic
181C2×C4C2×C22⋊D18111yescharacteristic
182C8—41no—
183C8—7241no—
184C8—7241no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—3622no—
189C22—7241no—
190C22—7241no—
191C22—7241no—
192C22—7241no—
193C22—7241no—
194C22#1915378411yescharacteristic
195C4—21no—
196C4—42no—
197C4—7241no—
198C4—7241no—
199C4D8×D18111yescharacteristic
200C4#1915383711yescharacteristic
201C4#1915384011yescharacteristic
202C4#1915384311yescharacteristic
203C2—161no—
204C2—14481no—
205C2—7242no—
206C2#2604141111yescharacteristic
207C2#2604142311yescharacteristic
208C2#2604145111yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#486×C181C2
socleC22×C181#120
hypercenter#486D181
hypocenterC181#1625
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #26041932C2semidirect
2. #26041471C2semidirect
3. #26041471C2semidirect
4. #26041025C2semidirect
5. #26041008C2semidirect
6. #26041008C2semidirect
7. #486×C181C2semidirect
57. D724D8semidirect
104. C181#1625semidirect
107. #486D181semidirect

Maximal subgroups

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

subgroupindex
#260419322
#260414712
#260410252
#260410082
#486×C1812
#1625181