Finite Group Databasebeta

← #37571819

Subgroups — #37571819

Order 24448.

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

normal subgroups only

This group has 206 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#27133413C211yesmaximal, characteristic, complemented
2#27133053C211yesmaximal, characteristic
3#27133028C212yesmaximal, complemented
4#27132404C212yesmaximal, complemented
5#329×C191C211yesmaximal, characteristic, complemented
6C2×#15690274—21no—
7C2×#15690272—21no—
8#20259320—21no—
9#20259319C2211yescharacteristic
10#20259315—22no—
11#20259314C2212yes—
12#20259303—21no—
13#20259299—22no—
14#20259244—22no—
15#20259244—22no—
16#20259233C2211yescharacteristic
17#124×C191C2211yescharacteristic
18#97×C191C2212yes—
19C2×Dic382—21no—
20C2×D764—21no—
21C2×D764—22no—
22C22×Dic191—21no—
23#15690274—42no—
24#15690272—42no—
25C2×C191⋊C8—21no—
26C4⋊Dic191D412yes—
27C4⋊Dic191D412yes—
28Dic191⋊C4—22no—
29Dic191⋊C4—22no—
30Dic191⋊C4—22no—
31C22⋊Dic191—22no—
32C22⋊Dic191—21no—
33C4×Dic191—21no—
34C2×Q8×C191D411yescharacteristic
35C2×D4×C191D411yescharacteristic
36C2×C8×C191—22no—
37C22⋊C4×C191—22no—
38C42×C191C2311yescharacteristic
39C22×D191—41no—
40Dic382—81no—
41D764—41no—
42D764—41no—
43D764—44no—
44C2×Dic191—22no—
45C2×Dic191—22no—
46C2×Dic191—81no—
47C191⋊C8—22no—
48C23×C191—21no—
49Q8×C191—41no—
50D4×C191—41no—
51C2×C4×C191—21no—
52C2×C4×C191C2×D412yes—
53C2×C4×C191C2×D411yescharacteristic
54C8×C191—42no—
55C22×C191—41no—
56C22×C191—42no—
57C22×C191#12011yescharacteristic
58C2×D191—81no—
59C2×D191—82no—
60C4×C191—21no—
61C4×C191—22no—
62C4×C191—41no—
63Dic191—42no—
64Dic191—42no—
65C2×C191—81no—
66C2×C191#45311yescharacteristic
67C2×C191#45412yes—
68D191—161no—
69C191#165711yescharacteristic, complemented, Sylow
70#1657—1911nomaximal, Sylow
71#563—1911no—
72#496—1911no—
73#490—1912no—
74#330—1912no—
75#329D19111yescharacteristic, complemented
76C2×SD16—3821no—
77C2×D8—3821no—
78C4⋊Q8—3821no—
79#127—1911no—
80#126—3822no—
81#125—1912no—
82#124C2×D19111yescharacteristic
83#123—3821no—
84C22⋊Q8—3822no—
85#102—3822no—
86#102—3822no—
87#97—1911no—
88#97C2×D19112yes—
89C2×Q8—3821no—
90C2×Q8D76411yescharacteristic
91C2×D4—3822no—
92C2×D4—3821no—
93C2×D4D76411yescharacteristic
94C22×C4—3821no—
95SD16—7642no—
96D8—7642no—
97C2×C8—22no—
98C2×C8—3821no—
99C4⋊C4—3822no—
100C4⋊C4—3822no—
101C4⋊C4—3822no—
102C4⋊C4—1912no—
103C4⋊C4—1912no—
104C22⋊C4—22no—
105C22⋊C4—3822no—
106C22⋊C4—3821no—
107C42—3821no—
108C42C22×D19111yescharacteristic
109C23—21no—
110C23—7641no—
111Q8—41no—
112Q8—15281no—
113D4—41no—
114D4—7641no—
115D4—7641no—
116D4—7644no—
117C2×C4—21no—
118C2×C4—3822no—
119C2×C4—3822no—
120C2×C4—15281no—
121C2×C4D4×D19112yes—
122C2×C4C2×D76411yescharacteristic
123C8—42no—
124C8—3822no—
125C22—41no—
126C22—42no—
127C22—15281no—
128C22—15282no—
129C22#2025929211yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—7642no—
134C4—7642no—
135C2—81no—
136C2—30561no—
137C2#2713283311yescharacteristic
138C2#2713283712yes—
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#329×C191C2
socleC22×C191#120
prime radical#27133053C2
hypercenter#329D191
hypocenterC191#1657
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #27133413C2semidirect
3. #27133028C2semidirect
4. #27132404C2semidirect
5. #329×C191C2semidirect
69. C191#1657semidirect
75. #329D191semidirect

Maximal subgroups

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

subgroupindex
#271334132
#271330532
#271330282
#271324042
#329×C1912
#1657191