Finite Group Databasebeta

← #37508311

Subgroups — #37508311

Order 22912.

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

normal subgroups only

This group has 196 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#26038849C212yesmaximal, complemented
2#26038834C211yesmaximal, characteristic
3#26038181C212yesmaximal, complemented
4#26038179C211yesmaximal, characteristic
5#563×C179C211yesmaximal, characteristic
6C2×#15668249—21no—
7C2×#15668247—21no—
8#19153113C2211yescharacteristic
9#19153046—22no—
10#19153044—22no—
11#19153024C2211yescharacteristic
12#19153024C2212yes—
13C4⋊Q8×C179—21no—
14#126×C179—22no—
15#125×C179C2212yes—
16#124×C179C2211yescharacteristic
17#123×C179—21no—
18C22⋊Q8×C179—22no—
19C2×Dic358—21no—
20#15668249—42no—
21#15668247—42no—
22C2×C179⋊C8—22no—
23C2×C179⋊C8—21no—
24C4⋊Dic179—22no—
25C2×Q8×C179—21no—
26C2×Q8×C179D411yescharacteristic
27C2×D4×C179D411yescharacteristic
28C22×C4×C179—21no—
29C4⋊C4×C179—22no—
30C4⋊C4×C179—22no—
31C4⋊C4×C179—22no—
32C4⋊C4×C179D412yes—
33C4⋊C4×C179D412yes—
34C22⋊C4×C179—22no—
35C22⋊C4×C179—21no—
36C22⋊C4×C179—22no—
37C42×C179—21no—
38C42×C179C2311yescharacteristic
39Dic358—41no—
40Dic358—41no—
41C2×Dic179—41no—
42C179⋊C8—24no—
43C179⋊C8—22no—
44C23×C179—21no—
45Q8×C179—41no—
46Q8×C179—81no—
47D4×C179—41no—
48C2×C4×C179—22no—
49C2×C4×C179—21no—
50C2×C4×C179—22no—
51C2×C4×C179—81no—
52C2×C4×C179C2×D411yescharacteristic
53C2×C4×C179C2×D412yes—
54C22×C179—41no—
55C22×C179—42no—
56C22×C179#12011yescharacteristic
57C4×C179—21no—
58C4×C179—22no—
59C4×C179—42no—
60C4×C179—41no—
61C4×C179—42no—
62Dic179—81no—
63C2×C179—81no—
64C2×C179#45311yescharacteristic
65C2×C179#45612yes—
66C179#166011yescharacteristic, complemented, Sylow
67#1660—1791nomaximal, Sylow
68#563D17911yescharacteristic
69#501—1792no—
70#497—1791no—
71#330—1792no—
72#329—1791no—
73C2×Q16—3581no—
74C2×SD16—3581no—
75C4⋊Q8—21no—
76C4⋊Q8—1791no—
77#126—22no—
78#125C2×D17912yes—
79#124C2×D17911yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—3582no—
83#106—3582no—
84#97—1792no—
85#97—1791no—
86C2×Q8—21no—
87C2×Q8—3581no—
88C2×Q8C22⋊D17911yescharacteristic
89C2×D4C22⋊D17911yescharacteristic
90C22×C4—21no—
91Q16—7162no—
92SD16—7162no—
93C2×C8—3582no—
94C2×C8—3581no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—3582no—
99C4⋊C4C22⋊D17912yes—
100C4⋊C4C22⋊D17912yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D17911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—7161no—
110Q8—7161no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—7161no—
117C2×C4C2×C22⋊D17912yes—
118C2×C4C2×C22⋊D17911yescharacteristic
119C8—3584no—
120C8—3582no—
121C22—41no—
122C22—42no—
123C22#1915308211yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—14321no—
130C2—81no—
131C2#2603860711yescharacteristic
132C2#2603862212yes—
subgroupG/H
centerC22#19153082
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#563×C179C2
socleC22×C179#120
prime radicalC2×D4×C179D4
hypercenter#563D179
hypocenterC179#1660
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26038849C2semidirect
3. #26038181C2semidirect
66. C179#1660semidirect

Maximal subgroups

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

subgroupindex
#260388492
#260388342
#260381812
#260381792
#563×C1792
#1660179