Finite Group Databasebeta

← #37549149

Subgroups — #37549149

Order 23168.

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#19153873C211yesmaximal, characteristic, complemented
2#26042222C211yesmaximal, characteristic, complemented
3#26042132C212yesmaximal, complemented
4#26042131C212yesmaximal, complemented
5#26041791C211yesmaximal, characteristic
6C22×#13089685—21no—
7C2×C22⋊#10624382—21no—
8#19153873—21no—
9#19153873C2212yes—
10(C2×#10624382)⋊C4—21no—
11(C2×#10624382)⋊C4C2212yes—
12C2×#15668449C411yescharacteristic, complemented
13C22×C4×D181C2211yescharacteristic
14#19153800—21no—
15#19153796C411yescharacteristic, complemented
16#19153796C2211yescharacteristic
17#19153791C411yescharacteristic, complemented
18#19153791C2211yescharacteristic
19#19153786—21no—
20C4×D724—21no—
21C4×D724—21no—
22#117×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24C2×#13089685—22no—
25C2×#13089685—42no—
26C22⋊#10624382—42no—
27C23×D181D411yescharacteristic
28#15668449—42no—
29C2×Dic362C2×C411yescharacteristic
30C2×D724—21no—
31C2×D724—21no—
32C2×D724C2×C411yescharacteristic
33C2×C22⋊D181—21no—
34C22×Dic181D411yescharacteristic
35C2×C4×D181—21no—
36C2×C4×D181—42no—
37C2×C4×D181C2×C411yescharacteristic
38C2×C4×D181D411yescharacteristic
39C2×C4×D181D411yescharacteristic
40C2×C4×D181C2311yescharacteristic
41C4⋊Dic181—21no—
42C4⋊Dic181—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47C22×C4×C181C2×C411yescharacteristic
48C4⋊C4×C181C2×C411yescharacteristic
49C4⋊C4×C181C2×C411yescharacteristic
50C22⋊C4×C181—21no—
51C42×C181—21no—
52C2×#10624382—42no—
53C2×#10624382—42no—
54C2×#10624382—42no—
55#13089685—42no—
56C22×D181—22no—
57C22×D181—22no—
58C22×D181—21no—
59C22×D181—41no—
60C22×D181C22⋊C411yescharacteristic
61C22×D181C2×D411yescharacteristic
62Dic362—41no—
63D724—41no—
64D724—42no—
65D724—42no—
66C22⋊D181—42no—
67C2×Dic181—22no—
68C2×Dic181—21no—
69C2×Dic181C22⋊C411yescharacteristic
70C2×Dic181C2×D411yescharacteristic
71C4×D181—41no—
72C4×D181—41no—
73C4×D181—41no—
74C4×D181—41no—
75C4×D181—42no—
76C23×C181C22⋊C411yescharacteristic
77C2×C4×C181—22no—
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181C22⋊C411yescharacteristic
81C2×C4×C181C22×C411yescharacteristic
82C22×C181—21no—
83C22×C181—21no—
84C22×C181#9912yes—
85C22×C181C2×C22⋊C411yescharacteristic
86C2×D181—21no—
87C2×D181—21no—
88C2×D181—21no—
89C2×D181—21no—
90C2×D181—21no—
91C2×D181—44no—
92C2×D181—41no—
93C2×D181—41no—
94C2×D181—41no—
95C2×D181#13512yes—
96#10624382—82no—
97C4×C181—21no—
98C4×C181—21no—
99C4×C181—41no—
100C4×C181—41no—
101Dic181—21no—
102Dic181—21no—
103Dic181—41no—
104C2×C181—22no—
105C2×C181C2×#9911yescharacteristic
106C2×C181#41611yescharacteristic
107C2×C181#42111yescharacteristic
108D181—22no—
109D181—41no—
110D181—81no—
111C181#151611yescharacteristic, complemented, Sylow
112#1516—1811nomaximal, Sylow
113#525—1811no—
114C2×C22⋊C8—1811no—
115#381—1811no—
116#328—1812no—
117#327—1812no—
118C2×#41—1811no—
119C23×C4—1811no—
120C22×C8—3621no—
121#123—3621no—
122C22⋊Q8—1811no—
123C22⋊Q8—1811no—
124#121—1811no—
125#121—1811no—
126#120—3621no—
127C4×D4—3621no—
128C4×D4—3621no—
129#117#1062438211yescharacteristic, complemented
130C2×C22⋊C4—3621no—
131C22⋊C8—3621no—
132C22⋊C8—1812no—
133#95—3621no—
134#95—1812no—
135C24—1811no—
136#41—7242no—
137C2×Q8—1811no—
138C2×D4—3621no—
139C2×D4—3621no—
140C2×D4—3621no—
141C2×D4—1811no—
142C22×C4—3621no—
143C22×C4—3622no—
144C22×C4—1811no—
145C22×C4—1811no—
146C22×C4—1811no—
147C22×C4—1811no—
148C22×C4—1811no—
149C22×C4—7242no—
150C22×C4C2×#1062438211yescharacteristic
151C2×C8—3622no—
152C2×C8—7242no—
153C4⋊C4—3621no—
154C4⋊C4—3621no—
155C4⋊C4C2×#1062438211yescharacteristic
156C4⋊C4C2×#1062438211yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3621no—
159C22⋊C4—3621no—
160C22⋊C4—3621no—
161C22⋊C4—3621no—
162C22⋊C4—7242no—
163C42—21no—
164C23—3622no—
165C23—3622no—
166C23—3621no—
167C23—1811no—
168C23—1811no—
169C23—7241no—
170C23C22⋊#1062438211yescharacteristic
171Q8—7241no—
172D4—7241no—
173D4—7242no—
174D4—7242no—
175D4—7242no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—3622no—
180C2×C4—3621no—
181C2×C4—1811no—
182C2×C4—1811no—
183C2×C4—7241no—
184C2×C4—7241no—
185C2×C4—7241no—
186C2×C4—7241no—
187C2×C4—7242no—
188C2×C4—7242no—
189C2×C4—7242no—
190C2×C4—7242no—
191C2×C4C22⋊#1062438211yescharacteristic
192C2×C4C22×#1062438211yescharacteristic
193C8—7242no—
194C22—21no—
195C22—21no—
196C22—3621no—
197C22—3621no—
198C22—3621no—
199C22—3621no—
200C22—3621no—
201C22—1812no—
202C22—7244no—
203C22—7241no—
204C22—7241no—
205C22—7241no—
206C22#1915387512yes—
207C22C2×C22⋊#1062438211yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—3621no—
213C4—3621no—
214C4—14482no—
215C4—7241no—
216C2—22no—
217C2—3622no—
218C2—14481no—
219C2—7241no—
220C2C2×#1915387511yescharacteristic
221C2#2604226911yescharacteristic
222C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#117×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041791C2
cosocleC2×C4×D181C23
hypercenter#117#10624382
hypocenterC181#1516
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#19153873C2semidirect
2. #26042222C2semidirect
3. #26042132C2semidirect
4. #26042131C2semidirect
12. C2×#15668449C4semidirect
15. #19153796C4semidirect
17. #19153791C4semidirect
22. #117×C181C4semidirect
111. C181#1516semidirect
129. #117#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538732
#260422222
#260421322
#260421312
#260417912
#1516181