Finite Group Databasebeta

← #37574516

Subgroups — #37574516

Order 24448.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#27132852C212yesmaximal, complemented
2#27132830C211yesmaximal, characteristic
3#27132456C211yesmaximal, characteristic, complemented
4#27132453C212yesmaximal, complemented
5#559×C191C211yesmaximal, characteristic, complemented
6#20259367C2211yescharacteristic
7#20259348—21no—
8#20259305—22no—
9#20259250—21no—
10#20259239C2211yescharacteristic
11#20259237C2212yes—
12#20259236—22no—
13#127×C191C2211yescharacteristic
14#124×C191—22no—
15#123×C191—21no—
16#123×C191C2212yes—
17C4×Q8×C191—21no—
18C4×D4×C191—22no—
19#15690287—21no—
20#15690286—41no—
21C2×C22⋊D191—22no—
22D4×D191—41no—
23#15690275—41no—
24#15690272—41no—
25#15690271—21no—
26Dic191⋊C4—42no—
27C22⋊Dic191—22no—
28(C2×D191)⋊C4—42no—
29C2×Q8×C191—21no—
30C2×D4×C191—22no—
31C2×D4×C191D411yescharacteristic
32C2×D4×C191C2311yescharacteristic
33C22×C4×C191—21no—
34C22×C4×C191D412yes—
35C4⋊C4×C191—21no—
36C4⋊C4×C191—22no—
37C22⋊C4×C191—22no—
38C22⋊C4×C191—21no—
39C22⋊C4×C191—22no—
40C22⋊C4×C191D412yes—
41C42×C191—22no—
42C42×C191D411yescharacteristic
43C22×D191—22no—
44Dic382—21no—
45D764—21no—
46C22⋊D191—42no—
47C22⋊D191—41no—
48C2×Dic191—22no—
49C2×Dic191—42no—
50C4×D191—41no—
51C191⋊C8—41no—
52C23×C191—21no—
53C23×C191C2×D412yes—
54Q8×C191—42no—
55D4×C191—41no—
56D4×C191—41no—
57D4×C191—44no—
58C2×C4×C191—21no—
59C2×C4×C191—22no—
60C2×C4×C191—22no—
61C2×C4×C191—42no—
62C2×C4×C191—81no—
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—42no—
74C4×C191—42no—
75Dic191—41no—
76Dic191—82no—
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#559D19111yescharacteristic, complemented
85#458—1912no—
86#453—1911no—
87#353—1911no—
88#352—1912no—
89H(5,𝔽2)—1911no—
90AGL(1,ℤ/8)—3821no—
91#127C2×D19111yescharacteristic
92#124—22no—
93#123—21no—
94#123—3822no—
95#123C2×D19112yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—3821no—
99#100—1911no—
100#99—3822no—
101#99—1912no—
102#41—3821no—
103#41—7641no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—3822no—
107C2×D4—7641no—
108C2×D4C22⋊D19111yescharacteristic
109C2×D4C22×D19111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D19112yes—
112SD16—7641no—
113D8—7641no—
114M16—3821no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—7642no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—3822no—
122C22⋊C4—7642no—
123C22⋊C4C22⋊D19112yes—
124C42—22no—
125C42C22⋊D19111yescharacteristic
126C23—21no—
127C23—3822no—
128C23C2×C22⋊D19112yes—
129Q8—42no—
130Q8—3821no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3821no—
135D4—7642no—
136D4—7641no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—3822no—
143C2×C4—7641no—
144C2×C4—7642no—
145C2×C4C2×C22⋊D19111yescharacteristic
146C8—7641no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—15282no—
152C22—7641no—
153C22#2025929111yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—15282no—
159C4—7641no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—15281no—
164C2#2713285011yescharacteristic
subgroupG/H
centerC2#27132850
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#559×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132830C2
hypercenter#559D191
hypocenterC191#2198
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #27132852C2semidirect
3. #27132456C2semidirect
4. #27132453C2semidirect
5. #559×C191C2semidirect
82. C191#2198semidirect
84. #559D191semidirect

Maximal subgroups

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

subgroupindex
#271328522
#271328302
#271324562
#271324532
#559×C1912
#2198191