Finite Group Databasebeta

← #37574535

Subgroups — #37574535

Order 24448.

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

normal subgroups only

This group has 226 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#27133413C211yesmaximal, characteristic, complemented
2#27132854C212yesmaximal
3#27132840C211yesmaximal, characteristic
4#27132452C212yesmaximal, complemented
5#354×C191C211yesmaximal, characteristic, complemented
6#20259367C2211yescharacteristic
7#20259353—21no—
8#20259320—21no—
9#20259315—22no—
10#20259314—21no—
11#20259306—22no—
12#20259303C2212yes—
13#20259299—22no—
14#20259250—21no—
15#20259239C2211yescharacteristic
16#20259236—22no—
17#124×C191C2211yescharacteristic
18#99×C191C2212yes—
19#15690287—21no—
20#15690286—41no—
21C2×Dic382—21no—
22C2×C22⋊D191—22no—
23D4×D191—41no—
24C22×Dic191D412yes—
25#15690277—41no—
26#15690274—41no—
27#15690271—21no—
28C4⋊Dic191—21no—
29C4⋊Dic191—21no—
30C4⋊Dic191—42no—
31Dic191⋊C4—22no—
32Dic191⋊C4—22no—
33Dic191⋊C4—22no—
34C22⋊Dic191—22no—
35C22⋊Dic191D412yes—
36(C2×D191)⋊C4—42no—
37C4×Dic191—21no—
38C2×Q8×C191D411yescharacteristic
39C2×D4×C191C2311yescharacteristic
40C22⋊C4×C191—22no—
41C22⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43C22×D191—22no—
44Dic382—21no—
45Dic382—81no—
46D764—21no—
47C22⋊D191—42no—
48C22⋊D191—41no—
49C2×Dic191—22no—
50C2×Dic191—22no—
51C2×Dic191—22no—
52C2×Dic191—42no—
53C4×D191—41no—
54C191⋊C8—41no—
55C23×C191C2×D412yes—
56Q8×C191—41no—
57D4×C191—41no—
58C2×C4×C191—21no—
59C2×C4×C191—21no—
60C2×C4×C191—42no—
61C2×C4×C191C2×D411yescharacteristic
62C22×C191—22no—
63C22×C191—42no—
64C22×C191#12011yescharacteristic
65C2×D191—41no—
66C2×D191—82no—
67C4×C191—21no—
68C4×C191—41no—
69C4×C191—41no—
70C4×C191—82no—
71Dic191—42no—
72Dic191—41no—
73Dic191—42no—
74C2×C191—21no—
75C2×C191—42no—
76C2×C191B(4,𝔽2)11yescharacteristic
77D191—81no—
78C191#220111yescharacteristic, complemented, Sylow
79#2201—1911nomaximal, Sylow
80#563—1911no—
81#458—1912no—
82#455—1911no—
83#354D19111yescharacteristic, complemented
84#352—1912no—
85H(5,𝔽2)—1911no—
86#137—3821no—
87C4⋊Q8—3821no—
88#126—3822no—
89#125—3821no—
90#124C2×D19111yescharacteristic
91#123—3822no—
92#123—1912no—
93C22⋊Q8—3822no—
94#104—3821no—
95#100—1911no—
96#99—3822no—
97#99C2×D19112yes—
98#41—3821no—
99#41—7641no—
100C2×Q8—3821no—
101C2×Q8D76411yescharacteristic
102C2×D4—3822no—
103C2×D4—7641no—
104C2×D4C22×D19111yescharacteristic
105C22×C4—1912no—
106Q16—7641no—
107SD16—7641no—
108M16—3821no—
109C4⋊C4—3822no—
110C4⋊C4—3822no—
111C4⋊C4—3821no—
112C4⋊C4—3821no—
113C4⋊C4—3822no—
114C4⋊C4—7642no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3822no—
118C22⋊C4—1912no—
119C22⋊C4—7642no—
120C42—3821no—
121C42D76411yescharacteristic
122C23—3822no—
123C23D4×D19112yes—
124Q8—41no—
125Q8—3821no—
126Q8—15281no—
127D4—41no—
128D4—3821no—
129D4—7642no—
130D4—7641no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—3822no—
135C2×C4—3822no—
136C2×C4—3822no—
137C2×C4—7641no—
138C2×C4—7642no—
139C2×C4C2×D76411yescharacteristic
140C8—7641no—
141C22—22no—
142C22—42no—
143C22—15282no—
144C22—7641no—
145C22#2025929211yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—7642no—
151C4—7641no—
152C4—7642no—
153C2—21no—
154C2—42no—
155C2—15281no—
156C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#354×C191C2
socleC2×C191B(4,𝔽2)
prime radical#20259367C22
hypercenter#354D191
hypocenterC191#2201
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #27133413C2semidirect
4. #27132452C2semidirect
5. #354×C191C2semidirect
78. C191#2201semidirect
83. #354D191semidirect

Maximal subgroups

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

subgroupindex
#271334132
#271328542
#271328402
#271324522
#354×C1912
#2201191