Finite Group Databasebeta

← #37573050

Subgroups — #37573050

Order 24448.

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

normal subgroups only

This group has 362 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#27133230C211yesmaximal, characteristic
2#27133138C211yesmaximal, characteristic, complemented
3#27132679C212yesmaximal, complemented
4#27132420C212yesmaximal, complemented
5C2×#100×C191C211yesmaximal, characteristic, complemented
6C2×#15690287C2211yescharacteristic
7#20259318C412yescomplemented
8#20259309C412yescomplemented
9#20259293—22no—
10#20259291—22no—
11D4×Dic191—24no—
12#20259281C2211yescharacteristic
13#20259281C2212yes—
14C2×C22⋊Dic191—22no—
15#20259248—24no—
16C22×D4×C191C2211yescharacteristic
17C2×M16×C191C2212yes—
18#100×C191—22no—
19#15690287—22no—
20#15690287—22no—
21C2×Dic382C2×C411yescharacteristic, complemented
22C2×D764C2×C411yescharacteristic, complemented
23C2×C22⋊D191—24no—
24C2×C22⋊D191—21no—
25C22×Dic191—24no—
26C2×C4×D191—21no—
27C4⋊Dic191—22no—
28C4⋊Dic191—21no—
29C22⋊Dic191—24no—
30C22⋊Dic191—24no—
31C22⋊Dic191—22no—
32C22⋊Dic191—21no—
33C4×Dic191C2×C412yescomplemented
34C4×Dic191D414yescomplemented
35C24×C191D412yes—
36C2×D4×C191—22no—
37C2×D4×C191—22no—
38C2×D4×C191C2×C412yes—
39C2×D4×C191D412yes—
40C22×C4×C191C2311yescharacteristic
41M16×C191—22no—
42M16×C191—22no—
43C2×C8×C191—22no—
44C22×D191—21no—
45Dic382—22no—
46D764—22no—
47C22⋊D191—41no—
48C22⋊D191—41no—
49C22⋊D191—84no—
50C2×Dic191—22no—
51C2×Dic191—21no—
52C2×Dic191—24no—
53C2×Dic191—84no—
54C4×D191—42no—
55C23×C191—22no—
56C23×C191—22no—
57C23×C191—82no—
58C23×C191C22⋊C414yes—
59C23×C191C2×D411yescharacteristic
60D4×C191—24no—
61D4×C191—24no—
62C2×C4×C191C22×C411yescharacteristic
63C2×C4×C191C2×D412yes—
64C2×C4×C191C2×D411yescharacteristic
65C2×C4×C191#4112yes—
66C8×C191—42no—
67C22×C191—24no—
68C22×C191—41no—
69C22×C191—44no—
70C22×C191—44no—
71C22×C191—82no—
72C22×C191C2×C22⋊C411yescharacteristic
73C22×C191#12011yescharacteristic
74C22×C191#12311yescharacteristic
75C2×D191—42no—
76C2×D191—41no—
77C4×C191C4×D412yes—
78C4×C191#12112yes—
79Dic191—44no—
80Dic191—42no—
81Dic191—41no—
82C2×C191—21no—
83C2×C191—21no—
84C2×C191—21no—
85C2×C191—44no—
86C2×C191#38611yescharacteristic
87D191—81no—
88C191#191111yescharacteristic, complemented, Sylow
89#1911—1911nomaximal, Sylow
90#535—1911no—
91#518—1911no—
92#421—1912no—
93C2×#100D19111yescharacteristic, complemented
94H(3,ℤ/4)—1912no—
95C2×#41—1911no—
96C22×D4C2×D19111yescharacteristic
97C2×M16C2×D19112yes—
98#127—1912no—
99#124—1912no—
100#121—3822no—
101#120—3822no—
102C4×D4—3824no—
103#117—1912no—
104#117—1911no—
105C2×C22⋊C4—3822no—
106#104—3824no—
107#100—22no—
108C24D76412yes—
109#41—3822no—
110#41—3822no—
111C2×Q8—1911no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—3824no—
115C2×D4—3821no—
116C2×D4—1911no—
117C2×D4C4×D19112yes—
118C2×D4C22⋊D19112yes—
119C22×C4—3824no—
120C22×C4—3821no—
121C22×C4C22×D19111yescharacteristic
122M16—22no—
123M16—22no—
124C2×C8—22no—
125C4⋊C4—3822no—
126C4⋊C4—3821no—
127C22⋊C4—3824no—
128C22⋊C4—3824no—
129C22⋊C4—3822no—
130C22⋊C4—3821no—
131C42—1912no—
132C42—1914no—
133C23—22no—
134C23—22no—
135C23—82no—
136C23—3821no—
137C23(C2×D191)⋊C414yes—
138C23C2×D76411yescharacteristic
139Q8—3822no—
140D4—24no—
141D4—24no—
142D4—3822no—
143D4—15284no—
144D4—7641no—
145D4—7641no—
146C2×C4—3822no—
147C2×C4—3821no—
148C2×C4—3824no—
149C2×C4—15284no—
150C2×C4—7642no—
151C2×C4C2×C4×D19111yescharacteristic
152C2×C4D4×D19112yes—
153C2×C4C2×C22⋊D19111yescharacteristic
154C2×C4#1569028612yes—
155C8—42no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—7642no—
162C22—7641no—
163C22C2×(C2×D191)⋊C411yescharacteristic
164C22#2025929211yescharacteristic
165C22#2025930611yescharacteristic
166C4—7644no—
167C4—7642no—
168C4—7641no—
169C4(C22⋊D191)⋊C412yes—
170C4#2025929412yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—15281no—
176C2#2713254011yescharacteristic
subgroupG/H
centerC2#27132540
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
FittingC2×#100×C191C2
socleC2×C191#386
prime radical#27133230C2
cosocleC22×C4×C191C23
hypercenterC2×#100D191
hypocenterC191#1911
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #27133138C2semidirect
3. #27132679C2semidirect
4. #27132420C2semidirect
5. C2×#100×C191C2semidirect
7. #20259318C4semidirect
8. #20259309C4semidirect
21. C2×Dic382C2×C4semidirect
22. C2×D764C2×C4semidirect
33. C4×Dic191C2×C4semidirect
34. C4×Dic191D4semidirect
88. C191#1911semidirect
93. C2×#100D191semidirect

Maximal subgroups

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

subgroupindex
#271332302
#271331382
#271326792
#271324202
C2×#100×C1912
#1911191