Finite Group Databasebeta

← #37508309

Subgroups — #37508309

Order 22912.

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

normal subgroups only

This group has 196 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#26039189C211yesmaximal, characteristic
2#26038845C212yesmaximal, complemented
3#26038180C212yesmaximal, complemented
4#26038179C211yesmaximal, characteristic
5#497×C179C211yesmaximal, characteristic
6#19153111—21no—
7#19153106—22no—
8#19153105C2212yes—
9#19153094—21no—
10#19153090—22no—
11#19153046—22no—
12#19153044—22no—
13#19153024C2212yes—
14C2×Q16×C179—21no—
15C2×SD16×C179—21no—
16C4⋊Q8×C179C2211yescharacteristic
17#124×C179C2211yescharacteristic
18#97×C179C2211yescharacteristic
19C2×Dic358—21no—
20C22×Dic179—21no—
21C2×C179⋊C8—22no—
22C4⋊Dic179D412yes—
23C4⋊Dic179D412yes—
24Dic179⋊C4—22no—
25Dic179⋊C4—22no—
26Dic179⋊C4—22no—
27C22⋊Dic179—22no—
28C22⋊Dic179—21no—
29C4×Dic179—21no—
30C2×Q8×C179—21no—
31C2×Q8×C179D411yescharacteristic
32C2×D4×C179D411yescharacteristic
33Q16×C179—42no—
34SD16×C179—42no—
35C2×C8×C179—21no—
36C4⋊C4×C179—22no—
37C22⋊C4×C179—22no—
38C42×C179C2311yescharacteristic
39Dic358—81no—
40C2×Dic179—22no—
41C2×Dic179—22no—
42C2×Dic179—81no—
43C179⋊C8—24no—
44C23×C179—21no—
45Q8×C179—41no—
46Q8×C179—41no—
47Q8×C179—41no—
48D4×C179—41no—
49C2×C4×C179—21no—
50C2×C4×C179—41no—
51C2×C4×C179C2×D412yes—
52C2×C4×C179C2×D411yescharacteristic
53C8×C179—22no—
54C22×C179—41no—
55C22×C179—42no—
56C22×C179#12011yescharacteristic
57C4×C179—21no—
58C4×C179—22no—
59C4×C179—41no—
60C4×C179—81no—
61Dic179—42no—
62Dic179—42no—
63C2×C179—81no—
64C2×C179#45311yescharacteristic
65C2×C179#45612yes—
66C179#166011yescharacteristic, complemented, Sylow
67#1660—1791nomaximal, Sylow
68#563—1791no—
69#501—1792no—
70#497D17911yescharacteristic
71#330—1792no—
72#329—1791no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—3581no—
76C4⋊Q8C2×D17911yescharacteristic
77#126—3582no—
78#125—1792no—
79#124C2×D17911yescharacteristic
80#123—3581no—
81C22⋊Q8—3582no—
82#107—3582no—
83#106—3582no—
84#97—1792no—
85#97C2×D17911yescharacteristic
86C2×Q8—21no—
87C2×Q8—3581no—
88C2×Q8C22⋊D17911yescharacteristic
89C2×D4C22⋊D17911yescharacteristic
90C22×C4—3581no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—3582no—
95C4⋊C4—22no—
96C4⋊C4—3582no—
97C4⋊C4—3582no—
98C4⋊C4—3582no—
99C4⋊C4—1792no—
100C4⋊C4—1792no—
101C22⋊C4—22no—
102C22⋊C4—3582no—
103C22⋊C4—3581no—
104C42—3581no—
105C42C22×D17911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—14321no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—3582no—
115C2×C4—3582no—
116C2×C4—14321no—
117C2×C4D4×D17912yes—
118C2×C4C2×C22⋊D17911yescharacteristic
119C8—22no—
120C8—3584no—
121C22—41no—
122C22—42no—
123C22#1915308111yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—7162no—
129C4—7162no—
130C2—81no—
131C2#2603860611yescharacteristic
132C2#2603862012yes—
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#497×C179C2
socleC22×C179#120
prime radicalC2×D4×C179D4
hypercenter#497D179
hypocenterC179#1660
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #26038845C2semidirect
3. #26038180C2semidirect
66. C179#1660semidirect

Maximal subgroups

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

subgroupindex
#260391892
#260388452
#260381802
#260381792
#497×C1792
#1660179