Finite Group Databasebeta

← #37574506

Subgroups — #37574506

Order 24448.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133298C211yesmaximal, characteristic, complemented
2#27132851C212yesmaximal, complemented
3#27132830C211yesmaximal, characteristic, complemented
4#27132449C212yesmaximal, complemented
5#353×C191C211yesmaximal, characteristic, complemented
6#20259367C2211yescharacteristic, complemented
7C2×D4×D191—22no—
8#20259348—21no—
9#20259293—22no—
10#20259292—22no—
11#20259290—21no—
12#20259290C2212yes—
13C4×D764—21no—
14#20259250—21no—
15#20259239C2211yescharacteristic, complemented
16#20259236—22no—
17#127×C191C2211yescharacteristic
18#99×C191C2212yescomplemented
19C23×D191—21no—
20C23×D191D412yescomplemented
21#15690287—21no—
22#15690286—41no—
23C2×D764—21no—
24C2×D764—42no—
25C2×C22⋊D191—22no—
26C2×C22⋊D191—22no—
27C2×C22⋊D191—22no—
28D4×D191—41no—
29D4×D191—48no—
30C2×C4×D191—22no—
31#15690275—41no—
32#15690272—41no—
33#15690271—21no—
34C4⋊Dic191—21no—
35C22⋊Dic191—21no—
36C22⋊Dic191D412yescomplemented
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—42no—
39C2×D4×C191—22no—
40C2×D4×C191D411yescharacteristic, complemented
41C2×D4×C191C2311yescharacteristic
42C22⋊C4×C191—22no—
43C42×C191D411yescharacteristic, complemented
44C22×D191—22no—
45C22×D191—22no—
46C22×D191—44no—
47C22×D191—44no—
48C22×D191—42no—
49C22×D191—81no—
50Dic382—21no—
51D764—24no—
52D764—21no—
53D764—84no—
54C22⋊D191—44no—
55C22⋊D191—44no—
56C22⋊D191—42no—
57C22⋊D191—41no—
58C2×Dic191—22no—
59C2×Dic191—22no—
60C4×D191—44no—
61C4×D191—41no—
62C191⋊C8—41no—
63C23×C191—21no—
64C23×C191C2×D412yes—
65D4×C191—41no—
66D4×C191—41no—
67D4×C191—44no—
68C2×C4×C191—21no—
69C2×C4×C191—42no—
70C2×C4×C191C2×D411yescharacteristic
71C22×C191—22no—
72C22×C191—42no—
73C22×C191—42no—
74C22×C191—41no—
75C22×C191#12011yescharacteristic
76C2×D191—24no—
77C2×D191—42no—
78C2×D191—42no—
79C2×D191—44no—
80C2×D191—41no—
81C2×D191—82no—
82C2×D191—84no—
83C2×D191—82no—
84C4×C191—22no—
85C4×C191—21no—
86C4×C191—82no—
87Dic191—42no—
88Dic191—41no—
89C2×C191—21no—
90C2×C191—42no—
91C2×C191—81no—
92C2×C191B(4,𝔽2)11yescharacteristic
93D191—44no—
94D191—81no—
95C191#219611yescharacteristic, complemented, Sylow
96#2196—1911nomaximal, Sylow
97D42—1911no—
98B(4,𝔽2)—1912no—
99#453—1911no—
100#353D19111yescharacteristic, complemented
101AΣL(1,𝔽16)—1912no—
102H(5,𝔽2)—1911no—
103C22×D4—3822no—
104AGL(1,ℤ/8)—3821no—
105#127C2×D19111yescharacteristic
106#121—3822no—
107#120—3822no—
108#120—3821no—
109#120—1912no—
110C4×D4—3821no—
111#104—3821no—
112#100—1911no—
113#99—3822no—
114#99C2×D19112yescomplemented
115C24—3821no—
116C24—1912no—
117#41—3821no—
118#41—7641no—
119C2×D4—22no—
120C2×D4—3821no—
121C2×D4—3822no—
122C2×D4—3822no—
123C2×D4—3822no—
124C2×D4—7642no—
125C2×D4—7641no—
126C2×D4—7648no—
127C2×D4D76411yescharacteristic, complemented
128C2×D4C22×D19111yescharacteristic
129C22×C4—3822no—
130SD16—7641no—
131D8—7641no—
132M16—3821no—
133C4⋊C4—3821no—
134C22⋊C4—22no—
135C22⋊C4—3821no—
136C22⋊C4—3822no—
137C22⋊C4—1912no—
138C22⋊C4—7642no—
139C42D76411yescharacteristic, complemented
140C23—21no—
141C23—3822no—
142C23—3822no—
143C23—15281no—
144C23—7642no—
145C23—7644no—
146C23—7644no—
147C23D4×D19112yes—
148Q8—3821no—
149D4—41no—
150D4—41no—
151D4—44no—
152D4—3824no—
153D4—3821no—
154D4—15284no—
155D4—7642no—
156D4—7644no—
157D4—7644no—
158D4—7641no—
159C2×C4—21no—
160C2×C4—42no—
161C2×C4—3822no—
162C2×C4—3822no—
163C2×C4—7644no—
164C2×C4—7641no—
165C2×C4C2×D76411yescharacteristic
166C8—7641no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—3824no—
172C22—15282no—
173C22—15282no—
174C22—15284no—
175C22—7642no—
176C22—7644no—
177C22—7641no—
178C22—7642no—
179C22#2025929211yescharacteristic
180C4—22no—
181C4—21no—
182C4—82no—
183C4—7642no—
184C4—7641no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—15281no—
189C2—7644no—
190C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#353×C191C2
socleC2×C191B(4,𝔽2)
hypercenter#353D191
hypocenterC191#2196
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #27133298C2semidirect
2. #27132851C2semidirect
3. #27132830C2semidirect
4. #27132449C2semidirect
5. #353×C191C2semidirect
6. #20259367C22semidirect
15. #20259239C22semidirect
18. #99×C191C22semidirect
20. C23×D191D4semidirect
36. C22⋊Dic191D4semidirect
40. C2×D4×C191D4semidirect
43. C42×C191D4semidirect
95. C191#2196semidirect
100. #353D191semidirect
114. #99C2×D191semidirect
127. C2×D4D764semidirect
139. C42D764semidirect

Maximal subgroups

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

subgroupindex
#271332982
#271328512
#271328302
#271324492
#353×C1912
#2196191