Finite Group Databasebeta

← #37573060

Subgroups — #37573060

Order 24448.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 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#27133222C211yesmaximal, characteristic
2#27133138C211yesmaximal, characteristic, complemented
3C2×#20259236C212yesmaximal, complemented
4#27132430C212yesmaximal, complemented
5C2×#99×C191C211yesmaximal, characteristic, complemented
6C22×C22⋊D191C2211yescharacteristic
7#20259305—21no—
8#20259303C412yescomplemented
9#20259294—22no—
10#20259291—21no—
11#20259290C412yescomplemented
12D4×Dic191—24no—
13#20259281—21no—
14C2×C22⋊Dic191—21no—
15C2×C22⋊Dic191C2211yescharacteristic
16C2×C22⋊Dic191C2212yes—
17#20259236—24no—
18C22×D4×C191C2211yescharacteristic
19C2×C22⋊C4×C191C2212yes—
20#99×C191—22no—
21C23×D191C2×C411yescharacteristic, complemented
22C2×C22⋊D191—22no—
23C2×C22⋊D191—22no—
24C2×C22⋊D191—21no—
25C2×C22⋊D191—21no—
26C2×C22⋊D191—22no—
27C22×Dic191—22no—
28C22×Dic191—22no—
29C22×Dic191—21no—
30C22×Dic191C2×C411yescharacteristic, complemented
31C4⋊Dic191—22no—
32Dic191⋊C4—22no—
33C22⋊Dic191—22no—
34C22⋊Dic191—22no—
35C22⋊Dic191—22no—
36C22⋊Dic191—21no—
37C22⋊Dic191C2×C412yescomplemented
38C22⋊Dic191D414yescomplemented
39(C2×D191)⋊C4—22no—
40C4×Dic191—22no—
41C24×C191D411yescharacteristic
42C24×C191C2311yescharacteristic
43C2×D4×C191—22no—
44C2×D4×C191—22no—
45C2×D4×C191C2×C412yes—
46C2×D4×C191D412yes—
47C22×C4×C191—22no—
48C22×C4×C191D411yescharacteristic
49C22⋊C4×C191—22no—
50C22⋊C4×C191—22no—
51C22×D191—21no—
52C22×D191—22no—
53C22×D191—81no—
54C22⋊D191—41no—
55C22⋊D191—42no—
56C22⋊D191—41no—
57C22⋊D191—82no—
58C2×Dic191—22no—
59C2×Dic191—21no—
60C2×Dic191—22no—
61C2×Dic191—24no—
62C2×Dic191—82no—
63C2×Dic191—82no—
64C2×Dic191—81no—
65C23×C191—21no—
66C23×C191—21no—
67C23×C191—81no—
68C23×C191—81no—
69C23×C191C22⋊C412yes—
70C23×C191C22×C411yescharacteristic
71C23×C191C2×D412yes—
72C23×C191C2×D411yescharacteristic
73C23×C191C2×D411yescharacteristic
74C23×C191#4112yes—
75D4×C191—24no—
76D4×C191—24no—
77C2×C4×C191—21no—
78C2×C4×C191—21no—
79C2×C4×C191—42no—
80C2×C4×C191—82no—
81C2×C4×C191C22⋊C412yes—
82C22×C191—22no—
83C22×C191—42no—
84C22×C191—41no—
85C22×C191—41no—
86C22×C191—42no—
87C22×C191—41no—
88C22×C191—42no—
89C22×C191—42no—
90C22×C191—81no—
91C22×C191C2×C22⋊C411yescharacteristic
92C22×C191C4×D412yes—
93C22×C191#12011yescharacteristic
94C22×C191#12112yes—
95C22×C191#12311yescharacteristic
96C2×D191—42no—
97C2×D191—41no—
98C2×D191—82no—
99C4×C191—22no—
100C4×C191—82no—
101Dic191—44no—
102Dic191—42no—
103Dic191—41no—
104C2×C191—21no—
105C2×C191—21no—
106C2×C191—22no—
107C2×C191—22no—
108C2×C191—21no—
109C2×C191—42no—
110C2×C191#38611yescharacteristic
111D191—81no—
112C191#191311yescharacteristic, complemented, Sylow
113#1913—1911nomaximal, Sylow
114#534—1911no—
115#518—1911no—
116C2×#99—1912no—
117C2×#99D19111yescharacteristic, complemented
118#342—1912no—
119C22×D4—1911no—
120C22×D4C2×D19111yescharacteristic
121#123—3821no—
122#123—1912no—
123#121—3822no—
124#120—3821no—
125#120—1912no—
126C4×D4—3824no—
127#117—3821no—
128C2×C22⋊C4—3821no—
129C2×C22⋊C4—1912no—
130C2×C22⋊C4—1911no—
131C2×C22⋊C4C2×D19112yes—
132#99—22no—
133#99—3824no—
134C24—1911no—
135C24D76411yescharacteristic
136C24C22×D19111yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—3822no—
140C2×D4—3822no—
141C2×D4—3821no—
142C2×D4—3821no—
143C2×D4—3822no—
144C2×D4C4×D19112yes—
145C2×D4C22⋊D19112yes—
146C22×C4—22no—
147C22×C4—3822no—
148C22×C4—3822no—
149C22×C4—3821no—
150C22×C4—1911no—
151C22×C4D76411yescharacteristic
152C4⋊C4—3822no—
153C4⋊C4—3822no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—3822no—
157C22⋊C4—3822no—
158C22⋊C4—3822no—
159C22⋊C4—3821no—
160C22⋊C4—3822no—
161C22⋊C4—1912no—
162C22⋊C4—1914no—
163C42—3822no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—3822no—
169C23—3821no—
170C23—15281no—
171C23(C2×D191)⋊C412yes—
172C23C2×C4×D19111yescharacteristic
173C23D4×D19112yes—
174C23C2×C22⋊D19111yescharacteristic
175C23C2×D76411yescharacteristic
176C23#1569028612yes—
177D4—24no—
178D4—24no—
179D4—15282no—
180D4—7641no—
181D4—7642no—
182D4—7641no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—3822no—
188C2×C4—3821no—
189C2×C4—3822no—
190C2×C4—3824no—
191C2×C4—15282no—
192C2×C4—15282no—
193C2×C4—15281no—
194C2×C4(C2×D191)⋊C412yes—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—15282no—
205C22—7642no—
206C22—7641no—
207C22C2×(C2×D191)⋊C411yescharacteristic
208C22(C22⋊D191)⋊C412yes—
209C22#2025929211yescharacteristic
210C22#2025929412yes—
211C22#2025930611yescharacteristic
212C4—22no—
213C4—82no—
214C4—7644no—
215C4—7642no—
216C4—7641no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—15281no—
224C2#2713254011yescharacteristic
subgroupG/H
centerC2#27132540
commutatorC23×C191C22×C4
FrattiniC24C22×D191
FittingC2×#99×C191C2
socleC2×C191#386
prime radical#27133222C2
cosocleC24×C191C23
hypercenterC2×#99D191
hypocenterC191#1913
LCSG ▷ C23×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #27133138C2semidirect
3. C2×#20259236C2semidirect
4. #27132430C2semidirect
5. C2×#99×C191C2semidirect
8. #20259303C4semidirect
11. #20259290C4semidirect
21. C23×D191C2×C4semidirect
30. C22×Dic191C2×C4semidirect
37. C22⋊Dic191C2×C4semidirect
38. C22⋊Dic191D4semidirect
112. C191#1913semidirect
117. C2×#99D191semidirect

Maximal subgroups

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

subgroupindex
#271332222
#271331382
C2×#202592362
#271324302
C2×#99×C1912
#1913191