Finite Group Databasebeta

← #37571818

Subgroups — #37571818

Order 24448.

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

normal subgroups only

This group has 206 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#27133053C211yesmaximal, characteristic
2#27133029C212yesmaximal, complemented
3#27132405C212yesmaximal, complemented
4#27132403C211yesmaximal, characteristic, complemented
5#563×C191C211yesmaximal, characteristic, complemented
6C2×#15690274—21no—
7C2×#15690272—21no—
8#20259319C2211yescharacteristic
9#20259242—22no—
10#20259242—22no—
11#20259233C2212yes—
12#20259233C2211yescharacteristic
13C4⋊Q8×C191—21no—
14#126×C191—22no—
15#125×C191C2212yes—
16#124×C191C2211yescharacteristic
17#123×C191—21no—
18C22⋊Q8×C191—22no—
19C2×D764—21no—
20C2×D764—22no—
21#15690274—42no—
22#15690272—42no—
23C2×C191⋊C8—21no—
24C2×C191⋊C8—22no—
25C2×Q8×C191—21no—
26C2×Q8×C191D411yescharacteristic
27C2×D4×C191D411yescharacteristic
28C22×C4×C191—21no—
29C4⋊C4×C191—22no—
30C4⋊C4×C191—22no—
31C4⋊C4×C191—22no—
32C4⋊C4×C191D412yes—
33C4⋊C4×C191D412yes—
34C22⋊C4×C191—22no—
35C22⋊C4×C191—21no—
36C22⋊C4×C191—22no—
37C42×C191—21no—
38C42×C191C2311yescharacteristic
39C22×D191—41no—
40D764—41no—
41D764—41no—
42D764—44no—
43C191⋊C8—22no—
44C191⋊C8—42no—
45C23×C191—21no—
46Q8×C191—41no—
47Q8×C191—81no—
48D4×C191—41no—
49C2×C4×C191—22no—
50C2×C4×C191—21no—
51C2×C4×C191—22no—
52C2×C4×C191—81no—
53C2×C4×C191C2×D412yes—
54C2×C4×C191C2×D411yescharacteristic
55C22×C191—41no—
56C22×C191—42no—
57C22×C191#12011yescharacteristic
58C2×D191—81no—
59C2×D191—82no—
60C4×C191—21no—
61C4×C191—22no—
62C4×C191—42no—
63C4×C191—41no—
64C4×C191—42no—
65C2×C191—81no—
66C2×C191#45311yescharacteristic
67C2×C191#45412yes—
68D191—161no—
69C191#165711yescharacteristic, complemented, Sylow
70#1657—1911nomaximal, Sylow
71#563D19111yescharacteristic, complemented
72#496—1911no—
73#490—1912no—
74#330—1912no—
75#329—1911no—
76C2×SD16—3821no—
77C2×D8—3821no—
78C4⋊Q8—21no—
79#127—1911no—
80#126—22no—
81#125C2×D19112yes—
82#124C2×D19111yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—3822no—
86#102—3822no—
87#97—1912no—
88#97—1911no—
89C2×Q8—21no—
90C2×Q8C22⋊D19111yescharacteristic
91C2×D4—3822no—
92C2×D4—3821no—
93C2×D4C22⋊D19111yescharacteristic
94C22×C4—21no—
95SD16—7642no—
96D8—7642no—
97C2×C8—3821no—
98C2×C8—3822no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D19112yes—
103C4⋊C4C22⋊D19112yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D19111yescharacteristic
109C23—21no—
110C23—7641no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—7641no—
115D4—7641no—
116D4—7644no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D19112yes—
122C2×C4C2×C22⋊D19111yescharacteristic
123C8—3822no—
124C8—7642no—
125C22—41no—
126C22—42no—
127C22—15281no—
128C22—15282no—
129C22#2025929111yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—30561no—
137C2#2713283111yescharacteristic
138C2#2713283612yes—
subgroupG/H
centerC22#20259291
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#563×C191C2
socleC22×C191#120
prime radical#27133053C2
hypercenter#563D191
hypocenterC191#1657
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #27133029C2semidirect
3. #27132405C2semidirect
4. #27132403C2semidirect
5. #563×C191C2semidirect
69. C191#1657semidirect
71. #563D191semidirect

Maximal subgroups

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

subgroupindex
#271330532
#271330292
#271324052
#271324032
#563×C1912
#1657191