Finite Group Databasebeta

← #37574533

Subgroups — #37574533

Order 24448.

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

normal subgroups only

This group has 226 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
2#27132840C211yesmaximal, characteristic
3#27132458C211yesmaximal, characteristic, complemented
4#27132453C212yesmaximal, complemented
5#563×C191C211yesmaximal, characteristic, complemented
6#20259367C2211yescharacteristic
7#20259353—21no—
8#20259305—22no—
9#20259250—21no—
10#20259239C2211yescharacteristic
11#20259237C2212yes—
12#20259236—22no—
13C4⋊Q8×C191—21no—
14#126×C191—22no—
15#125×C191—21no—
16#124×C191C2211yescharacteristic
17#123×C191C2212yes—
18C22⋊Q8×C191—22no—
19#15690287—21no—
20#15690286—41no—
21C2×C22⋊D191—22no—
22D4×D191—41no—
23#15690277—41no—
24#15690274—41no—
25#15690271—21no—
26Dic191⋊C4—42no—
27C22⋊Dic191—22no—
28(C2×D191)⋊C4—42no—
29C2×Q8×C191—21no—
30C2×Q8×C191D411yescharacteristic
31C2×D4×C191C2311yescharacteristic
32C22×C4×C191D412yes—
33C4⋊C4×C191—21no—
34C4⋊C4×C191—22no—
35C4⋊C4×C191—22no—
36C4⋊C4×C191—21no—
37C4⋊C4×C191—22no—
38C22⋊C4×C191—22no—
39C22⋊C4×C191—22no—
40C22⋊C4×C191D412yes—
41C42×C191—21no—
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×C191C2×D412yes—
53Q8×C191—41no—
54Q8×C191—81no—
55D4×C191—41no—
56C2×C4×C191—21no—
57C2×C4×C191—22no—
58C2×C4×C191—21no—
59C2×C4×C191—22no—
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—42no—
69C4×C191—41no—
70C4×C191—42no—
71C4×C191—41no—
72Dic191—41no—
73Dic191—82no—
74C2×C191—21no—
75C2×C191—42no—
76C2×C191B(4,𝔽2)11yescharacteristic
77D191—81no—
78C191#220111yescharacteristic, complemented, Sylow
79#2201—1911nomaximal, Sylow
80#563D19111yescharacteristic, complemented
81#458—1912no—
82#455—1911no—
83#354—1911no—
84#352—1912no—
85H(5,𝔽2)—1911no—
86#137—3821no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D19111yescharacteristic
91#123—3822no—
92#123C2×D19112yes—
93C22⋊Q8—22no—
94#104—3821no—
95#100—1911no—
96#99—3822no—
97#99—1912no—
98#41—3821no—
99#41—7641no—
100C2×Q8—21no—
101C2×Q8C22⋊D19111yescharacteristic
102C2×D4—3822no—
103C2×D4—7641no—
104C2×D4C22×D19111yescharacteristic
105C22×C4C22⋊D19112yes—
106Q16—7641no—
107SD16—7641no—
108M16—3821no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—7642no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3822no—
118C22⋊C4—7642no—
119C22⋊C4C22⋊D19112yes—
120C42—21no—
121C42C22⋊D19111yescharacteristic
122C23—3822no—
123C23C2×C22⋊D19112yes—
124Q8—41no—
125Q8—81no—
126Q8—3821no—
127D4—41no—
128D4—3821no—
129D4—7642no—
130D4—7641no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3822no—
137C2×C4—7641no—
138C2×C4—7642no—
139C2×C4C2×C22⋊D19111yescharacteristic
140C8—7641no—
141C22—22no—
142C22—42no—
143C22—15282no—
144C22—7641no—
145C22#2025929111yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—15282no—
152C4—7641no—
153C2—21no—
154C2—42no—
155C2—15281no—
156C2#2713285011yescharacteristic
subgroupG/H
centerC2#27132850
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#563×C191C2
socleC2×C191B(4,𝔽2)
prime radical#20259367C22
hypercenter#563D191
hypocenterC191#2201
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #27132458C2semidirect
4. #27132453C2semidirect
5. #563×C191C2semidirect
78. C191#2201semidirect
80. #563D191semidirect

Maximal subgroups

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

subgroupindex
#271328522
#271328402
#271324582
#271324532
#563×C1912
#2201191