Finite Group Databasebeta

← #37508304

Subgroups — #37508304

Order 22912.

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

normal subgroups only

This group has 228 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#26039181C211yesmaximal, characteristic
2#26038801C212yesmaximal, complemented
3#26038179C211yesmaximal, characteristic, complemented
4#26038178C212yesmaximal, complemented
5#497×C179C211yesmaximal, characteristic, complemented
6#19153102—21no—
7#19153101C2212yes—
8#19153099—22no—
9#19153083—22no—
10#19153081—21no—
11#19153036—22no—
12#19153033—22no—
13#19153024C2212yescomplemented
14C2×Q16×C179—21no—
15C2×SD16×C179—21no—
16C4⋊Q8×C179C2211yescharacteristic, complemented
17#124×C179C2211yescharacteristic
18#97×C179C2211yescharacteristic, complemented
19C23×D179—21no—
20C2×Dic358D412yes—
21C2×D716—21no—
22C2×D716D412yes—
23C2×C22⋊D179—22no—
24C2×C179⋊C8—22no—
25C22⋊Dic179—21no—
26(C2×D179)⋊C4—22no—
27(C2×D179)⋊C4—22no—
28(C2×D179)⋊C4—22no—
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
39C22×D179—22no—
40C22×D179—81no—
41C22×D179—82no—
42Dic358—42no—
43D716—42no—
44D716—81no—
45C22⋊D179—82no—
46C2×Dic179—22no—
47C179⋊C8—42no—
48C23×C179—21no—
49Q8×C179—41no—
50Q8×C179—41no—
51Q8×C179—41no—
52D4×C179—41no—
53C2×C4×C179—21no—
54C2×C4×C179—41no—
55C2×C4×C179C2×D412yes—
56C2×C4×C179C2×D411yescharacteristic
57C8×C179—22no—
58C22×C179—41no—
59C22×C179—42no—
60C22×C179#12011yescharacteristic
61C2×D179—42no—
62C2×D179—42no—
63C2×D179—42no—
64C2×D179—84no—
65C4×C179—21no—
66C4×C179—22no—
67C4×C179—41no—
68C4×C179—81no—
69Dic179—42no—
70C2×C179—81no—
71C2×C179#45311yescharacteristic
72C2×C179#45512yes—
73D179—82no—
74C179#165911yescharacteristic, complemented, Sylow
75#1659—1791nomaximal, Sylow
76H(3,𝔽4)—1791no—
77#497D17911yescharacteristic, complemented
78#489—1792no—
79#329—1791no—
80#329—1792no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D17911yescharacteristic, complemented
84#124—3582no—
85#124—3581no—
86#124—1792no—
87#124C2×D17911yescharacteristic
88#120—3582no—
89#120—3581no—
90#103—3582no—
91#102—3582no—
92#97—1792no—
93#97C2×D17911yescharacteristic, complemented
94C24—3581no—
95C2×Q8—21no—
96C2×Q8—1792no—
97C2×Q8C22⋊D17911yescharacteristic
98C2×D4—3581no—
99C2×D4—3582no—
100C2×D4—1792no—
101C2×D4C22⋊D17911yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—3582no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—3582no—
109C22⋊C4—3581no—
110C22⋊C4—3582no—
111C22⋊C4—3582no—
112C42—3581no—
113C42C22×D17911yescharacteristic
114C23—21no—
115C23—3582no—
116C23—14321no—
117C23—14322no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—7162no—
122D4—41no—
123D4—14321no—
124D4—14322no—
125D4—7162no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—3582no—
129C2×C4D4×D17912yes—
130C2×C4C2×C22⋊D17911yescharacteristic
131C8—22no—
132C8—7162no—
133C22—41no—
134C22—42no—
135C22—14324no—
136C22—7162no—
137C22—7162no—
138C22—7162no—
139C22#1915308111yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—7162no—
145C2—81no—
146C2—14322no—
147C2#2603860611yescharacteristic
148C2#2603861512yes—
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#497×C179C2
socleC22×C179#120
prime radical#26039181C2
hypercenter#497D179
hypocenterC179#1659
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. #26038801C2semidirect
3. #26038179C2semidirect
4. #26038178C2semidirect
5. #497×C179C2semidirect
13. #19153024C22semidirect
16. C4⋊Q8×C179C22semidirect
18. #97×C179C22semidirect
74. C179#1659semidirect
77. #497D179semidirect
83. C4⋊Q8C2×D179semidirect
93. #97C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260391812
#260388012
#260381792
#260381782
#497×C1792
#1659179