Finite Group Databasebeta

← #37573047

Subgroups — #37573047

Order 24448.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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#27133227C211yesmaximal, characteristic
2#27132669C212yesmaximal, complemented
3#27132637C211yesmaximal, characteristic, complemented
4#27132420C212yesmaximal, complemented
5#568×C191C211yesmaximal, characteristic, complemented
6C22×D764C2211yescharacteristic
7#20259318C412yescomplemented
8#20259311C412yescomplemented
9#20259296—22no—
10#20259290—22no—
11#20259281C2212yes—
12#20259281C2211yescharacteristic
13#20259244—24no—
14#20259237—22no—
15C2×#41×C191C2211yescharacteristic
16#131×C191—24no—
17C2×M16×C191—22no—
18C2×M16×C191C2212yes—
19C23×D191—21no—
20C2×D764—22no—
21C2×D764—21no—
22C2×D764—22no—
23C2×D764C2×C412yes—
24C2×C22⋊D191—24no—
25C4⋊Dic191—21no—
26C4⋊Dic191—22no—
27C22⋊Dic191—21no—
28C22⋊Dic191—22no—
29(C2×D191)⋊C4—24no—
30C4×Dic191—22no—
31C4×Dic191C2×C412yes—
32#41×C191—22no—
33#41×C191—22no—
34C2×Q8×C191C2×C411yescharacteristic
35C2×D4×C191C2×C411yescharacteristic
36C2×D4×C191D412yes—
37C22×C4×C191D412yes—
38C22×C4×C191C2311yescharacteristic
39M16×C191—24no—
40M16×C191—22no—
41M16×C191—22no—
42C2×C8×C191—24no—
43C2×C8×C191D414yescomplemented
44C22×D191—22no—
45C22×D191—42no—
46C22×D191—82no—
47D764—24no—
48D764—41no—
49D764—41no—
50C22⋊D191—84no—
51C2×Dic191—22no—
52C2×Dic191—42no—
53C23×C191C22⋊C412yes—
54C23×C191C2×D411yescharacteristic
55Q8×C191—22no—
56D4×C191—24no—
57D4×C191—22no—
58C2×C4×C191—22no—
59C2×C4×C191—22no—
60C2×C4×C191C22⋊C412yes—
61C2×C4×C191C22×C411yescharacteristic
62C2×C4×C191C2×D412yes—
63C2×C4×C191C2×D411yescharacteristic
64C2×C4×C191#4112yes—
65C8×C191—24no—
66C22×C191—22no—
67C22×C191—41no—
68C22×C191—42no—
69C22×C191C2×C22⋊C411yescharacteristic
70C22×C191#12112yes—
71C2×D191—44no—
72C2×D191—42no—
73C2×D191—84no—
74C4×C191—22no—
75C4×C191C4×D412yes—
76C4×C191#12011yescharacteristic
77C4×C191#12311yescharacteristic
78Dic191—42no—
79Dic191—82no—
80C2×C191—21no—
81C2×C191—22no—
82C2×C191—42no—
83C2×C191#38611yescharacteristic
84D191—82no—
85C191#191011yescharacteristic, complemented, Sylow
86#1910—1911nomaximal, Sylow
87#568D19111yescharacteristic, complemented
88#535—1911no—
89#418—1912no—
90#410—1911no—
91H(3,ℤ/4)—1912no—
92C2×#41C2×D19111yescharacteristic
93C22×D4—1911no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D19112yes—
97#127—1912no—
98#124—1912no—
99#121—3822no—
100#120—3822no—
101#117—1912no—
102#117—1911no—
103#102—3824no—
104#99—3822no—
105C24—3821no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D19111yescharacteristic
109C2×D4—3824no—
110C2×D4—3822no—
111C2×D4—3821no—
112C2×D4—3822no—
113C2×D4—1912no—
114C2×D4C4×D19111yescharacteristic
115C2×D4D76412yes—
116C22×C4C22⋊D19112yes—
117C22×C4C22×D19111yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D19114yescomplemented
123C4⋊C4—3821no—
124C4⋊C4—3822no—
125C22⋊C4—3821no—
126C22⋊C4—3824no—
127C22⋊C4—3822no—
128C42—3822no—
129C42—1912no—
130C23—3822no—
131C23—15282no—
132C23—7642no—
133C23(C2×D191)⋊C412yes—
134C23C2×D76411yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—3824no—
139D4—15284no—
140D4—7641no—
141D4—7641no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—3822no—
145C2×C4—7642no—
146C2×C4(C2×D191)⋊C412yes—
147C2×C4C2×C4×D19111yescharacteristic
148C2×C4C2×C22⋊D19112yes—
149C2×C4C2×C22⋊D19111yescharacteristic
150C2×C4#1569028712yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—15284no—
156C22—7644no—
157C22—7642no—
158C22C2×(C2×D191)⋊C411yescharacteristic
159C22#2025929612yes—
160C4—22no—
161C4—15282no—
162C4—7642no—
163C4C4×C22⋊D19112yes—
164C4#2025929111yescharacteristic
165C4#2025930511yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—15282no—
170C2#2713254211yescharacteristic
subgroupG/H
centerC2#27132542
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#568×C191C2
socleC2×C191#386
prime radical#27133227C2
cosocleC22×C4×C191C23
hypercenter#568D191
hypocenterC191#1910
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27132669C2semidirect
3. #27132637C2semidirect
4. #27132420C2semidirect
5. #568×C191C2semidirect
7. #20259318C4semidirect
8. #20259311C4semidirect
43. C2×C8×C191D4semidirect
85. C191#1910semidirect
87. #568D191semidirect
122. C2×C8C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
#271332272
#271326692
#271326372
#271324202
#568×C1912
#1910191