Finite Group Databasebeta

← #37574518

Subgroups — #37574518

Order 24448.

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

normal subgroups only

This group has 246 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#27133399C211yesmaximal, characteristic, complemented
2#27132854C212yesmaximal, complemented
3#27132830C211yesmaximal, characteristic
4#27132452C212yesmaximal, complemented
5#353×C191C211yesmaximal, characteristic, complemented
6#20259367C2211yescharacteristic
7#20259348—21no—
8#20259309—22no—
9#20259306—22no—
10#20259303—21no—
11#20259303C2212yes—
12C4×Dic382—21no—
13D4×Dic191—22no—
14#20259250—21no—
15#20259239C2211yescharacteristic
16#20259236—22no—
17#127×C191C2211yescharacteristic
18#99×C191C2212yes—
19#15690287—21no—
20#15690286—41no—
21C2×Dic382—21no—
22C2×C22⋊D191—22no—
23D4×D191—41no—
24C22×Dic191—21no—
25C22×Dic191D412yes—
26#15690275—41no—
27#15690272—41no—
28#15690271—21no—
29C4⋊Dic191—21no—
30C4⋊Dic191—42no—
31Dic191⋊C4—22no—
32C22⋊Dic191—22no—
33C22⋊Dic191—21no—
34C22⋊Dic191—22no—
35C22⋊Dic191D412yes—
36(C2×D191)⋊C4—42no—
37C4×Dic191—22no—
38C2×D4×C191—22no—
39C2×D4×C191D411yescharacteristic
40C2×D4×C191C2311yescharacteristic
41C22⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43C22×D191—22no—
44Dic382—21no—
45Dic382—42no—
46D764—21no—
47C22⋊D191—42no—
48C22⋊D191—41no—
49C2×Dic191—22no—
50C2×Dic191—22no—
51C2×Dic191—22no—
52C2×Dic191—42no—
53C2×Dic191—81no—
54C4×D191—41no—
55C191⋊C8—41no—
56C23×C191—21no—
57C23×C191C2×D412yes—
58D4×C191—41no—
59D4×C191—41no—
60D4×C191—44no—
61C2×C4×C191—21no—
62C2×C4×C191—42no—
63C2×C4×C191C2×D411yescharacteristic
64C22×C191—22no—
65C22×C191—42no—
66C22×C191—42no—
67C22×C191—41no—
68C22×C191#12011yescharacteristic
69C2×D191—41no—
70C2×D191—82no—
71C4×C191—22no—
72C4×C191—21no—
73C4×C191—82no—
74Dic191—42no—
75Dic191—41no—
76Dic191—42no—
77C2×C191—21no—
78C2×C191—42no—
79C2×C191—81no—
80C2×C191B(4,𝔽2)11yescharacteristic
81D191—81no—
82C191#219811yescharacteristic, complemented, Sylow
83#2198—1911nomaximal, Sylow
84#559—1911no—
85#458—1912no—
86#453—1911no—
87#353D19111yescharacteristic, complemented
88#352—1912no—
89H(5,𝔽2)—1911no—
90AGL(1,ℤ/8)—3821no—
91#127C2×D19111yescharacteristic
92#124—3822no—
93#123—3822no—
94#123—3821no—
95#123—1912no—
96C4×Q8—3821no—
97C4×D4—3822no—
98#104—3821no—
99#100—1911no—
100#99—3822no—
101#99C2×D19112yes—
102#41—3821no—
103#41—7641no—
104C2×Q8—3821no—
105C2×D4—22no—
106C2×D4—3822no—
107C2×D4—7641no—
108C2×D4D76411yescharacteristic
109C2×D4C22×D19111yescharacteristic
110C22×C4—3821no—
111C22×C4—1912no—
112SD16—7641no—
113D8—7641no—
114M16—3821no—
115C4⋊C4—3821no—
116C4⋊C4—3822no—
117C4⋊C4—7642no—
118C22⋊C4—22no—
119C22⋊C4—3822no—
120C22⋊C4—3821no—
121C22⋊C4—3822no—
122C22⋊C4—1912no—
123C22⋊C4—7642no—
124C42—3822no—
125C42D76411yescharacteristic
126C23—21no—
127C23—3822no—
128C23D4×D19112yes—
129Q8—3821no—
130Q8—7642no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3821no—
135D4—7642no—
136D4—7641no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3822no—
140C2×C4—3822no—
141C2×C4—3822no—
142C2×C4—15281no—
143C2×C4—7641no—
144C2×C4—7642no—
145C2×C4C2×D76411yescharacteristic
146C8—7641no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—15282no—
152C22—7641no—
153C22#2025929211yescharacteristic
154C4—22no—
155C4—21no—
156C4—82no—
157C4—7642no—
158C4—7641no—
159C4—7642no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—15281no—
164C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#353×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132830C2
hypercenter#353D191
hypocenterC191#2198
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. #27133399C2semidirect
2. #27132854C2semidirect
4. #27132452C2semidirect
5. #353×C191C2semidirect
82. C191#2198semidirect
87. #353D191semidirect

Maximal subgroups

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

subgroupindex
#271333992
#271328542
#271328302
#271324522
#353×C1912
#2198191