Finite Group Databasebeta

← #37533029

Subgroups — #37533029

Order 23168.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#26042021C211yesmaximal, characteristic
2#26041673C211yesmaximal, characteristic, complemented
3#26041639C211yesmaximal, characteristic, complemented
4#26041638C211yesmaximal, characteristic, complemented
5#26041018C211yesmaximal, characteristic, complemented
6#26041018C211yesmaximal, characteristic, complemented
7#329×C181C211yesmaximal, characteristic, complemented
8C2×Dic724—21no—
9C2×#15668438—21no—
10#19153816C2211yescharacteristic, complemented
11#19153805—21no—
12#19153804C2211yescharacteristic
13#19153804C2211yescharacteristic
14#19153802—21no—
15#19153802—21no—
16#19153786—21no—
17#19153786—21no—
18#19153784—21no—
19#19153741—21no—
20#19153740—21no—
21#19153738—21no—
22#19153737—21no—
23#19153727C2211yescharacteristic, complemented
24#124×C181C2211yescharacteristic
25#97×C181C2211yescharacteristic, complemented
26#97×C181C2211yescharacteristic, complemented
27C23×D181—21no—
28C2×Dic362—21no—
29C2×Dic362D411yescharacteristic
30C2×Dic362D411yescharacteristic
31C2×D724—21no—
32C2×D724D411yescharacteristic
33C2×D724D411yescharacteristic
34C2×C22⋊D181—21no—
35C2×C22⋊D181—21no—
36Dic724—42no—
37#15668438—42no—
38C2×#13089678—21no—
39C4⋊Dic181—21no—
40C4⋊Dic181—21no—
41C22⋊Dic181—21no—
42#15668426—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—21no—
48C4×Dic181—21no—
49C2×Q8×C181D411yescharacteristic
50C2×D4×C181D411yescharacteristic
51C2×C8×C181—21no—
52C2×C8×C181—21no—
53C22⋊C4×C181—21no—
54C22⋊C4×C181—21no—
55C42×C181C2311yescharacteristic
56C22×D181—21no—
57C22×D181—21no—
58C22×D181—81no—
59C22×D181—81no—
60C22×D181—81no—
61Dic362—41no—
62Dic362—41no—
63Dic362—41no—
64Dic362—41no—
65D724—41no—
66D724—41no—
67D724—81no—
68C22⋊D181—81no—
69C22⋊D181—81no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—41no—
73#13089678—41no—
74C23×C181—21no—
75Q8×C181—41no—
76D4×C181—41no—
77C2×C4×C181—21no—
78C2×C4×C181C2×D411yescharacteristic
79C2×C4×C181C2×D411yescharacteristic
80C2×C4×C181C2×D411yescharacteristic
81C8×C181—22no—
82C8×C181—41no—
83C22×C181—41no—
84C22×C181—41no—
85C22×C181—41no—
86C22×C181#12011yescharacteristic
87C2×D181—41no—
88C2×D181—41no—
89C2×D181—41no—
90C2×D181—41no—
91C2×D181—41no—
92C2×D181—41no—
93C2×D181—84no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181—21no—
97C4×C181—41no—
98Dic181—41no—
99Dic181—41no—
100Dic181—81no—
101C2×C181—81no—
102C2×C181#45311yescharacteristic
103C2×C181#45511yescharacteristic
104C2×C181#45511yescharacteristic
105D181—81no—
106D181—81no—
107C181#165911yescharacteristic, complemented, Sylow
108#1659—1811nomaximal, Sylow
109H(3,𝔽4)—1811no—
110#497—1811no—
111#489—1811no—
112#489—1811no—
113#329—1811no—
114#329—1811no—
115#329D18111yescharacteristic, complemented
116C2×Q16—3621no—
117C2×SD16—3621no—
118C4⋊Q8—1811no—
119#124—3621no—
120#124—3621no—
121#124—3621no—
122#124—1811no—
123#124—1811no—
124#124C2×D18111yescharacteristic
125#120—3621no—
126#120—3621no—
127#120—3621no—
128#103—3621no—
129#103—3621no—
130#102—3621no—
131#102—3621no—
132#97—1811no—
133#97C2×D18111yescharacteristic, complemented
134#97C2×D18111yescharacteristic, complemented
135C24—3621no—
136C2×Q8—3621no—
137C2×Q8—1811no—
138C2×Q8—1811no—
139C2×Q8D72411yescharacteristic
140C2×D4—3621no—
141C2×D4—3621no—
142C2×D4—3621no—
143C2×D4—1811no—
144C2×D4—1811no—
145C2×D4D72411yescharacteristic
146Q16—7242no—
147SD16—7242no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—3621no—
151C4⋊C4—3621no—
152C4⋊C4—3621no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—3621no—
156C22⋊C4—3621no—
157C22⋊C4—3621no—
158C22⋊C4—3621no—
159C22⋊C4—3621no—
160C22⋊C4—3621no—
161C22⋊C4—3621no—
162C42—3621no—
163C42C22×D18111yescharacteristic
164C23—21no—
165C23—3621no—
166C23—3621no—
167C23—14481no—
168C23—14481no—
169C23—14481no—
170Q8—41no—
171Q8—7241no—
172Q8—7241no—
173Q8—7241no—
174Q8—7241no—
175D4—41no—
176D4—14481no—
177D4—14481no—
178D4—14481no—
179D4—7241no—
180D4—7241no—
181C2×C4—21no—
182C2×C4—3621no—
183C2×C4—3621no—
184C2×C4—7241no—
185C2×C4D4×D18111yescharacteristic
186C2×C4D4×D18111yescharacteristic
187C2×C4C2×D72411yescharacteristic
188C8—22no—
189C8—41no—
190C8—7241no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—14484no—
195C22—7241no—
196C22—7241no—
197C22—7241no—
198C22—7241no—
199C22—7241no—
200C22—7241no—
201C22#1915378611yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—14481no—
207C4—7241no—
208C4—7241no—
209C2—81no—
210C2—14481no—
211C2—14481no—
212C2#2604144811yescharacteristic
213C2#2604145811yescharacteristic
214C2#2604145911yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#329×C181C2
socleC22×C181#120
prime radical#26042021C2
hypercenter#329D181
hypocenterC181#1659
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26041673C2semidirect
3. #26041639C2semidirect
4. #26041638C2semidirect
5. #26041018C2semidirect
6. #26041018C2semidirect
7. #329×C181C2semidirect
10. #19153816C22semidirect
23. #19153727C22semidirect
25. #97×C181C22semidirect
26. #97×C181C22semidirect
107. C181#1659semidirect
115. #329D181semidirect
133. #97C2×D181semidirect
134. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420212
#260416732
#260416392
#260416382
#260410182
#329×C1812
#1659181