Finite Group Databasebeta

← #37576158

Subgroups — #37576158

Order 24448.

1348 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C23×D764C211yesmaximal, characteristic, complemented
2C2×#20259318C212yesmaximal, complemented
3C2×#20259290C214yesmaximal, complemented
4#27132572C214yesmaximal, complemented
5#27132564C212yesmaximal, complemented
6C4×C22⋊Dic191C211yesmaximal, characteristic, complemented
7C2×#121×C191C211yesmaximal, characteristic, complemented
8C24×D191C2212yes—
9C22×D764—28no—
10C22×D764—24no—
11C22×D764C2212yes—
12C22×D764C2212yes—
13C22×C22⋊D191C2218yescomplemented
14#20259318—28no—
15#20259290—216no—
16C2×C22⋊Dic191C2212yes—
17C2×(C2×D191)⋊C4C2218yescomplemented
18C2×C4×Dic191C2212yescomplemented
19C191⋊#95C2212yescomplemented
20C22×D4×C191C2212yes—
21C22×D4×C191C2211yescharacteristic, complemented
22C23×C4×C191C2211yescharacteristic
23#121×C191—24no—
24C2×C4⋊C4×C191C2211yescharacteristic, complemented
25C2×C22⋊C4×C191C2212yescomplemented
26C23×D191—28no—
27C23×D191—24no—
28C23×D191—28no—
29C23×D191—44no—
30C23×D191C2314yes—
31C2×D764—232no—
32C2×D764—28no—
33C2×D764—24no—
34C2×D764—24no—
35C2×D764—24no—
36C2×D764D418yescomplemented
37C2×C22⋊D191—216no—
38C2×C22⋊D191—216no—
39C2×C22⋊D191—48no—
40C22×Dic191C2314yes—
41C22⋊Dic191D418yescomplemented
42(C2×D191)⋊C4—216no—
43C4×Dic191—24no—
44C24×C191C2312yes—
45C24×C191C2311yescharacteristic
46C2×D4×C191—24no—
47C2×D4×C191—28no—
48C2×D4×C191—22no—
49C2×D4×C191—24no—
50C22×C4×C191—24no—
51C22×C4×C191D414yescomplemented
52C22×C4×C191C2312yes—
53C22×C4×C191C2311yescharacteristic
54C22×C4×C191C2311yescharacteristic
55C4⋊C4×C191—22no—
56C22⋊C4×C191—24no—
57C22×D191—216no—
58C22×D191—28no—
59C22×D191—24no—
60C22×D191—24no—
61C22×D191—28no—
62C22×D191—432no—
63C22×D191—48no—
64C22×D191—48no—
65C22×D191C2×D418yes—
66D764—216no—
67D764—216no—
68C22⋊D191—432no—
69C2×Dic191—24no—
70C2×Dic191—24no—
71C2×Dic191C2×D418yescomplemented
72C23×C191—22no—
73C23×C191—22no—
74C23×C191—22no—
75C23×C191—24no—
76C23×C191—24no—
77C23×C191—24no—
78C23×C191C2×D414yes—
79C23×C191C2×D412yes—
80C23×C191C2411yescharacteristic
81D4×C191—48no—
82D4×C191—44no—
83C2×C4×C191—22no—
84C2×C4×C191—28no—
85C2×C4×C191—22no—
86C2×C4×C191—24no—
87C2×C4×C191C2×D414yes—
88C2×C4×C191C2×D412yes—
89C2×C4×C191C2×D412yes—
90C22×C191—22no—
91C22×C191—22no—
92C22×C191—28no—
93C22×C191—42no—
94C22×C191—42no—
95C22×C191—42no—
96C22×C191—48no—
97C22×C191#12714yescomplemented
98C22×C191C22×D412yes—
99C22×C191C22×D412yes—
100C22×C191C22×D411yescharacteristic
101C22×C191H(5,𝔽2)12yes—
102C2×D191—28no—
103C2×D191—216no—
104C2×D191—432no—
105C2×D191—44no—
106C2×D191—44no—
107C2×D191—48no—
108C4×C191—22no—
109C4×C191—42no—
110C4×C191#12014yescomplemented
111Dic191—28no—
112C2×C191—24no—
113C2×C191—82no—
114C2×C191C2×#12011yescharacteristic
115C2×C191C2×#12711yescharacteristic
116C2×C191#53511yescharacteristic
117C2×C191D4214yes—
118D191—48no—
119C191#243311yescharacteristic, complemented, Sylow
120#2433—1911nomaximal, Sylow
121C23×D4—1911no—
122C2×#127—1912no—
123C2×#121D19111yescharacteristic, complemented
124C2×#120—1914no—
125#392—1914no—
126#390—1912no—
127C4×C22⋊C4—1911no—
128C25—1912no—
129C22×D4—3828no—
130C22×D4—3824no—
131C22×D4—1918no—
132C22×D4—1912no—
133C22×D4—1912no—
134C22×D4C2×D19112yes—
135C22×D4C2×D19111yescharacteristic, complemented
136C23×C4C2×D19111yescharacteristic
137#127—3828no—
138#121—24no—
139#120—38216no—
140C2×C4⋊C4C2×D19111yescharacteristic, complemented
141C2×C22⋊C4—1918no—
142C2×C22⋊C4—1912no—
143C2×C22⋊C4C2×D19112yescomplemented
144C2×C42—1912no—
145#95—1912no—
146C24—3828no—
147C24—3824no—
148C24—3828no—
149C24—1914no—
150C24—7644no—
151C24C22×D19112yes—
152C24C22×D19111yescharacteristic
153C2×D4—28no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—38232no—
158C2×D4—3828no—
159C2×D4—38216no—
160C2×D4—38216no—
161C2×D4—3824no—
162C2×D4—3824no—
163C2×D4—3824no—
164C2×D4—1918no—
165C2×D4—7648no—
166C22×C4—24no—
167C22×C4—1914no—
168C22×C4C22⋊D19114yescomplemented
169C22×C4C22×D19112yes—
170C22×C4C22×D19111yescharacteristic
171C22×C4C22×D19111yescharacteristic
172C4⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—38216no—
175C22⋊C4—1918no—
176C42—3824no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—38216no—
184C23—3828no—
185C23—3824no—
186C23—3824no—
187C23—3828no—
188C23—1918no—
189C23—76432no—
190C23—7648no—
191C23—7648no—
192C23D4×D19114yes—
193C23C2×C22⋊D19112yes—
194C23C23×D19111yescharacteristic
195D4—48no—
196D4—44no—
197D4—38216no—
198D4—38216no—
199D4—76432no—
200C2×C4—22no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—3824no—
205C2×C4—3824no—
206C2×C4—1918no—
207C2×C4D4×D19114yes—
208C2×C4C2×C22⋊D19112yes—
209C2×C4C2×C22⋊D19112yes—
210C22—22no—
211C22—28no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—48no—
216C22—42no—
217C22—38216no—
218C22—3828no—
219C22—76432no—
220C22—7644no—
221C22—7644no—
222C22—7648no—
223C22#2025931814yescomplemented
224C22C2×D4×D19112yes—
225C22C2×D4×D19112yes—
226C22C22×C22⋊D19111yescharacteristic
227C22#2025936812yes—
228C4—22no—
229C4—42no—
230C4—3828no—
231C4#2025929014yescomplemented
232C2—24no—
233C2—82no—
234C2—7648no—
235C2C2×#2025929011yescharacteristic
236C2C2×#2025931811yescharacteristic
237C2#2713322711yescharacteristic
238C2#2713329714yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC2×#121×C191C2
hypercenterC2×#121D191
hypocenterC191#2433
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×D764C2semidirect
2. C2×#20259318C2semidirect
3. C2×#20259290C2semidirect
4. #27132572C2semidirect
5. #27132564C2semidirect
6. C4×C22⋊Dic191C2semidirect
7. C2×#121×C191C2semidirect
13. C22×C22⋊D191C22semidirect
17. C2×(C2×D191)⋊C4C22semidirect
18. C2×C4×Dic191C22semidirect
19. C191⋊#95C22semidirect
21. C22×D4×C191C22semidirect
24. C2×C4⋊C4×C191C22semidirect
25. C2×C22⋊C4×C191C22semidirect
36. C2×D764D4semidirect
41. C22⋊Dic191D4semidirect
51. C22×C4×C191D4semidirect
71. C2×Dic191C2×D4semidirect
97. C22×C191#127semidirect
110. C4×C191#120semidirect
119. C191#2433semidirect
123. C2×#121D191semidirect
135. C22×D4C2×D191semidirect
140. C2×C4⋊C4C2×D191semidirect
143. C2×C22⋊C4C2×D191semidirect
168. C22×C4C22⋊D191semidirect
223. C22#20259318semidirect
231. C4#20259290semidirect

Maximal subgroups

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

subgroupindex
C23×D7642
C2×#202593182
C2×#202592902
#271325722
#271325642
C4×C22⋊Dic1912
C2×#121×C1912
#2433191