Finite Group Databasebeta

← #37549151

Subgroups — #37549151

Order 23168.

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

normal subgroups only

This group has 264 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#26042298C211yesmaximal, characteristic, complemented
2#26042223C211yesmaximal, characteristic, complemented
3#26042131C212yesmaximal, complemented
4#26042130C212yesmaximal, complemented
5#26041826C211yesmaximal, characteristic
6C2×C4⋊#10624382—21no—
7C4⋊#13089685—21no—
8#19153873C2212yes—
9C4×#13089685—21no—
10(C2×#10624382)⋊C4—21no—
11(C2×#10624382)⋊C4C2212yes—
12C2×#15668449—21no—
13#19153815C411yescharacteristic, complemented
14#19153812C411yescharacteristic, complemented
15#19153802—21no—
16#19153791C2211yescharacteristic
17#19153789—21no—
18#19153787C2211yescharacteristic
19C4×D724C411yescharacteristic, complemented
20C4×C22⋊D181—21no—
21C42×D181C2211yescharacteristic
22C4×D4×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24C2×#13089685—22no—
25C4⋊#10624382—42no—
26#15668449—42no—
27#15668449—42no—
28C2×Dic362—21no—
29C2×D724—21no—
30C2×D724C2×C411yescharacteristic
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34C2×C4×D181—21no—
35C2×C4×D181D411yescharacteristic
36C2×C4×D181D411yescharacteristic
37C2×C4×D181C2311yescharacteristic
38C4⋊Dic181C2×C411yescharacteristic
39#15668428—21no—
40C22⋊Dic181—21no—
41#15668426—21no—
42#15668426—21no—
43C4×Dic181C2×C411yescharacteristic
44C4×Dic181D411yescharacteristic
45C4×Dic181D411yescharacteristic
46C2×D4×C181C2×C411yescharacteristic
47C22×C4×C181—21no—
48C4⋊C4×C181C2×C411yescharacteristic
49C22⋊C4×C181—21no—
50C42×C181C2×C411yescharacteristic
51C2×#10624382—42no—
52C2×#10624382—42no—
53C2×#10624382—42no—
54#13089685—42no—
55C22×D181—21no—
56C22×D181C2×D411yescharacteristic
57Dic362—42no—
58D724—41no—
59D724—41no—
60D724—41no—
61C22⋊D181—42no—
62C22⋊D181—41no—
63C22⋊D181—41no—
64C22⋊D181—42no—
65C2×Dic181—21no—
66C2×Dic181C22⋊C411yescharacteristic
67C2×Dic181C22⋊C411yescharacteristic
68C2×Dic181C2×D411yescharacteristic
69C4×D181—41no—
70C4×D181—41no—
71C4×D181—42no—
72C4×D181D812yes—
73C4×D181SD1612yes—
74C23×C181—21no—
75D4×C181—41no—
76C2×C4×C181—21no—
77C2×C4×C181—42no—
78C2×C4×C181C22⋊C411yescharacteristic
79C2×C4×C181C22⋊C411yescharacteristic
80C2×C4×C181C22×C411yescharacteristic
81C22×C181—41no—
82C22×C181—41no—
83C22×C181—41no—
84C22×C181C2×C22⋊C411yescharacteristic
85C2×D181—21no—
86C2×D181—21no—
87C2×D181—41no—
88C2×D181—41no—
89C2×D181—41no—
90C2×D181C2×D811yescharacteristic
91C2×D181C2×SD1611yescharacteristic
92#10624382—82no—
93C4×C181—21no—
94C4×C181—21no—
95C4×C181—41no—
96C4×C181#10212yes—
97Dic181—21no—
98Dic181—21no—
99Dic181—41no—
100Dic181#10212yes—
101C2×C181—81no—
102C2×C181#41011yescharacteristic
103C2×C181C2×#10211yescharacteristic
104C2×C181#42111yescharacteristic
105D181—21no—
106D181—21no—
107D181—81no—
108C181#151711yescharacteristic, complemented, Sylow
109#1517—1811nomaximal, Sylow
110#532—1811no—
111#431—1811no—
112#382—1811no—
113#327—1812no—
114#327—1812no—
115C2×#41—3621no—
116C4⋊Q8—1811no—
117#127—1811no—
118#124—3621no—
119#121—3621no—
120#121—1811no—
121#121—1811no—
122C4×D4—3621no—
123C4×D4—1811no—
124C4×D4#1062438211yescharacteristic, complemented
125C2×C4⋊C4—3621no—
126C2×C42—1811no—
127C4⋊C8—3621no—
128C22⋊C8—1812no—
129C4×C8—3621no—
130#95—3621no—
131#95—1812no—
132#41—7242no—
133#41—7242no—
134C2×Q8—3621no—
135C2×D4—3621no—
136C2×D4—3621no—
137C2×D4—3621no—
138C2×D4—3621no—
139C2×D4—1811no—
140C2×D4C2×#1062438211yescharacteristic
141C22×C4—21no—
142C22×C4—3621no—
143C22×C4—3622no—
144C22×C4—1811no—
145C22×C4—1811no—
146C22×C4—1811no—
147C2×C8—3622no—
148C4⋊C4—3621no—
149C4⋊C4—1811no—
150C4⋊C4—7242no—
151C4⋊C4C2×#1062438211yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3621no—
154C22⋊C4—3621no—
155C22⋊C4—3621no—
156C42—1811no—
157C42—1811no—
158C42—1811no—
159C42C2×#1062438211yescharacteristic
160C23—21no—
161C23—3621no—
162C23—1811no—
163Q8—7242no—
164D4—41no—
165D4—7241no—
166D4—7241no—
167D4—7241no—
168D4—7242no—
169D4—7242no—
170D4—7241no—
171D4—7241no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—3621no—
175C2×C4—1811no—
176C2×C4—1811no—
177C2×C4—1811no—
178C2×C4—1812no—
179C2×C4—1812no—
180C2×C4—7241no—
181C2×C4—7241no—
182C2×C4—7242no—
183C2×C4—7242no—
184C2×C4—7242no—
185C2×C4—7242no—
186C2×C4C22⋊#1062438211yescharacteristic
187C2×C4C22⋊#1062438211yescharacteristic
188C2×C4C22×#1062438211yescharacteristic
189C8—7242no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—3621no—
194C22—3621no—
195C22—1811no—
196C22—1811no—
197C22—7241no—
198C22—7241no—
199C22—7241no—
200C22C2×C22⋊#1062438211yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—3621no—
205C4—3621no—
206C4—14482no—
207C4—1812no—
208C4—7241no—
209C4#1915387812yes—
210C2—81no—
211C2—3621no—
212C2—3621no—
213C2—14481no—
214C2#2604225811yescharacteristic
215C2C2×#1915387811yescharacteristic
216C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4×D4×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041826C2
cosocleC2×C4×D181C23
hypercenterC4×D4#10624382
hypocenterC181#1517
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26042298C2semidirect
2. #26042223C2semidirect
3. #26042131C2semidirect
4. #26042130C2semidirect
13. #19153815C4semidirect
14. #19153812C4semidirect
19. C4×D724C4semidirect
22. C4×D4×C181C4semidirect
108. C181#1517semidirect
124. C4×D4#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422982
#260422232
#260421312
#260421302
#260418262
#1517181