Finite Group Databasebeta

← #37541983

Subgroups — #37541983

Order 23168.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041220C212yesmaximal, complemented
2#26041212C212yesmaximal, complemented
3#26041209C212yesmaximal, complemented
4#26041202C212yesmaximal
5#26041199C212yesmaximal, complemented
6#26041191C211yesmaximal, characteristic
7#26041140C211yesmaximal, characteristic
8#26041137C212yesmaximal
9#399×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C23×Dic181C2211yescharacteristic
12C2×C4⋊Dic181C2212yes—
13C2×#15668428C2212yes—
14C2×#15668428C2212yes—
15C2×#15668428C2212yes—
16C2×#15668428C2212yes—
17C2×C22⋊Dic181C2212yes—
18C2×C22⋊Dic181C2211yescharacteristic
19#19153723C2212yes—
20#19153723C2212yes—
21#19153723C2212yes—
22#19153723C2212yes—
23#19153723C2211yescharacteristic
24#19153723C2212yes—
25#19153723C2212yes—
26C2×C4⋊C4×C181C2212yes—
27C2×C4⋊C4×C181C2211yescharacteristic
28C2×C22⋊C4×C181C2212yes—
29C2×C22⋊C4×C181C2211yescharacteristic
30#95×C181C2211yescharacteristic
31C2×Dic362—41no—
32C2×Dic362—42no—
33C22×Dic181—41no—
34C22×Dic181—41no—
35C22×Dic181Q814yescomplemented
36C22×Dic181C2312yes—
37C22×Dic181C2312yes—
38C22×Dic181C2312yes—
39C22×Dic181C2312yes—
40C4⋊Dic181—42no—
41#15668428—42no—
42#15668428—42no—
43#15668428—42no—
44#15668428—42no—
45C22⋊Dic181—42no—
46C22⋊Dic181—41no—
47C24×C181C2311yescharacteristic
48C22×C4×C181C2311yescharacteristic
49C22×C4×C181C2312yes—
50C22×C4×C181C2311yescharacteristic
51C22×C4×C181C2312yes—
52C4⋊C4×C181—41no—
53C4⋊C4×C181—42no—
54C22⋊C4×C181—42no—
55C22⋊C4×C181—41no—
56Dic362—44no—
57C2×Dic181—22no—
58C2×Dic181—22no—
59C2×Dic181—22no—
60C2×Dic181—22no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—22no—
65C2×Dic181—22no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—44no—
69C2×Dic181C2×Q814yes—
70C23×C181—22no—
71C23×C181—21no—
72C23×C181—21no—
73C23×C181—22no—
74C23×C181C2×Q812yes—
75C23×C181C2411yescharacteristic
76C2×C4×C181—22no—
77C2×C4×C181—21no—
78C2×C4×C181—22no—
79C2×C4×C181—21no—
80C2×C4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181—22no—
86C22×C181—22no—
87C22×C181—24no—
88C22×C181—41no—
89C22×C181—42no—
90C22×C181—41no—
91C22×C181C22×Q811yescharacteristic
92C22×C181H(5,𝔽2)11yescharacteristic
93C22×C181H(5,𝔽2)12yes—
94C22×C181H(5,𝔽2)11yescharacteristic
95C22×C1812−1+412yes—
96C4×C181—42no—
97C4×C181—41no—
98C4×C181—42no—
99C4×C181—41no—
100Dic181—24no—
101Dic181—42no—
102Dic181—42no—
103Dic181—42no—
104C2×C181—42no—
105C2×C181#54311yescharacteristic
106C2×C181#54412yes—
107C2×C181#56011yescharacteristic
108C2×C181#56212yes—
109C2×C181#56311yescharacteristic
110C181#284111yescharacteristic, complemented, Sylow
111#2841—1811nomaximal, Sylow
112#400—1812no—
113#399D18111yescharacteristic
114#398—1812no—
115#398—1812no—
116#396—1812no—
117#395—1812no—
118#393—1811no—
119#381—1811no—
120#380—1812no—
121C22×Q8—1811no—
122C23×C4—1811no—
123C2×C4⋊C4—1812no—
124C2×C4⋊C4—1812no—
125C2×C4⋊C4—1812no—
126C2×C4⋊C4—1812no—
127C2×C4⋊C4—1812no—
128C2×C4⋊C4C2×D18112yes—
129C2×C4⋊C4C2×D18111yescharacteristic
130C2×C22⋊C4—1812no—
131C2×C22⋊C4—1811no—
132C2×C22⋊C4C2×D18112yes—
133C2×C22⋊C4C2×D18111yescharacteristic
134#95—1812no—
135#95—1812no—
136#95—1812no—
137#95—1812no—
138#95—1811no—
139#95—1812no—
140#95—1812no—
141#95C2×D18111yescharacteristic
142C24C22×D18111yescharacteristic
143C2×Q8—7241no—
144C2×Q8—7242no—
145C22×C4—1812no—
146C22×C4—1812no—
147C22×C4—1812no—
148C22×C4—1812no—
149C22×C4—1814no—
150C22×C4—7241no—
151C22×C4—7241no—
152C22×C4C22×D18111yescharacteristic
153C22×C4C22×D18111yescharacteristic
154C22×C4C22×D18112yes—
155C22×C4C22×D18112yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—7242no—
159C4⋊C4—7242no—
160C4⋊C4—7242no—
161C4⋊C4—7242no—
162C4⋊C4—7242no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—7242no—
166C22⋊C4—7241no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D18112yes—
172C23C23×D18111yescharacteristic
173Q8—7244no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—3622no—
185C2×C4—3622no—
186C2×C4—3622no—
187C2×C4—3622no—
188C2×C4—3622no—
189C2×C4—3622no—
190C2×C4—3622no—
191C2×C4—3622no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—1814no—
196C2×C4—7244no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D18111yescharacteristic
203C22#1915386111yescharacteristic
204C22#1915386112yes—
205C22#1915386211yescharacteristic
206C22#1915386312yes—
207C4—41no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—3624no—
212C4—7242no—
213C4—7242no—
214C4—7242no—
215C2—42no—
216C2#2604190411yescharacteristic
217C2#2604190812yes—
218C2#2604201611yescharacteristic
219C2#2604202612yes—
220C2#2604203211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#399×C181C2
prime radicalC24×C181C23
hypercenter#399D181
hypocenterC181#2841
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #26041220C2semidirect
2. #26041212C2semidirect
3. #26041209C2semidirect
5. #26041199C2semidirect
35. C22×Dic181Q8semidirect
110. C181#2841semidirect

Maximal subgroups

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

subgroupindex
#260412202
#260412122
#260412092
#260412022
#260411992
#260411912
#260411402
#260411372
#399×C1812
#2841181