Finite Group Databasebeta

← #37571813

Subgroups — #37571813

Order 24448.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#27133053C212yesmaximal, complemented
2#27133029C211yesmaximal, characteristic, complemented
3#27132405C211yesmaximal, characteristic, complemented
4#27132403C212yesmaximal, complemented
5#562×C191C211yesmaximal, characteristic, complemented
6C2×#15690274—22no—
7C2×#15690272—22no—
8#20259319C2211yescharacteristic
9#20259242—22no—
10#20259233C2211yescharacteristic
11#20259233C2212yes—
12#125×C191C2211yescharacteristic
13#124×C191C2212yes—
14#123×C191—22no—
15C22⋊Q8×C191—22no—
16#121×C191—22no—
17C2×D764—22no—
18C2×D764—21no—
19#15690274—44no—
20#15690272—44no—
21C2×C191⋊C8—22no—
22C2×C191⋊C8—21no—
23C2×Q8×C191D412yes—
24C2×D4×C191—22no—
25C2×D4×C191D412yes—
26C22×C4×C191—22no—
27C4⋊C4×C191—22no—
28C4⋊C4×C191—22no—
29C4⋊C4×C191D412yes—
30C22⋊C4×C191—22no—
31C22⋊C4×C191—22no—
32C22⋊C4×C191—22no—
33C42×C191C2311yescharacteristic
34C22×D191—41no—
35D764—42no—
36D764—42no—
37D764—42no—
38C191⋊C8—24no—
39C191⋊C8—41no—
40C23×C191—22no—
41Q8×C191—42no—
42D4×C191—42no—
43D4×C191—82no—
44C2×C4×C191—22no—
45C2×C4×C191—22no—
46C2×C4×C191—82no—
47C2×C4×C191C2×D411yescharacteristic
48C2×C4×C191C2×D412yes—
49C22×C191—42no—
50C22×C191—42no—
51C22×C191—42no—
52C22×C191#12011yescharacteristic
53C2×D191—82no—
54C2×D191—81no—
55C4×C191—22no—
56C4×C191—21no—
57C4×C191—42no—
58C4×C191—42no—
59C2×C191—82no—
60C2×C191#45312yes—
61C2×C191#45411yescharacteristic
62D191—161no—
63C191#165611yescharacteristic, complemented, Sylow
64#1656—1911nomaximal, Sylow
65#562D19111yescharacteristic, complemented
66#496—1912no—
67#490—1911no—
68#330—1911no—
69#329—1912no—
70C2×SD16—3822no—
71C2×D8—3822no—
72#127—1911no—
73#125C2×D19111yescharacteristic
74#124C2×D19112yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—3822no—
79#97—1911no—
80#97—1912no—
81C2×Q8C22⋊D19112yes—
82C2×D4—22no—
83C2×D4—3821no—
84C2×D4—3822no—
85C2×D4C22⋊D19112yes—
86C22×C4—22no—
87SD16—7644no—
88D8—7644no—
89C2×C8—3822no—
90C2×C8—3821no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D19112yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D19111yescharacteristic
98C23—22no—
99C23—7641no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—7642no—
104D4—7642no—
105D4—7642no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D19111yescharacteristic
110C2×C4C2×C22⋊D19112yes—
111C8—3824no—
112C8—7641no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—15282no—
117C22—15281no—
118C22#2025929111yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—30561no—
125C2#2713283112yes—
126C2#2713283611yescharacteristic
subgroupG/H
centerC22#20259291
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#562×C191C2
socleC22×C191#120
hypercenter#562D191
hypocenterC191#1656
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #27133053C2semidirect
2. #27133029C2semidirect
3. #27132405C2semidirect
4. #27132403C2semidirect
5. #562×C191C2semidirect
63. C191#1656semidirect
65. #562D191semidirect

Maximal subgroups

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

subgroupindex
#271330532
#271330292
#271324052
#271324032
#562×C1912
#1656191