Finite Group Databasebeta

← #37510350

Subgroups — #37510350

Order 22912.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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#26039070C211yesmaximal, characteristic
2#26038479C212yesmaximal
3#26038208C211yesmaximal, characteristic
4#26038199C212yesmaximal
5#382×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7#19153112—22no—
8#19153104—22no—
9#19153092—22no—
10#19153072C2212yes—
11#19153046—22no—
12#19153044—22no—
13C2×C4⋊C4×C179C2211yescharacteristic
14C2×C4⋊C4×C179C2211yescharacteristic
15C2×C42×C179C2211yescharacteristic
16#95×C179—22no—
17C2×Dic358—22no—
18#15668243—22no—
19#15668243Q814yescomplemented
20C2×C179⋊C8—22no—
21C4⋊Dic179—22no—
22C4⋊Dic179D412yes—
23C4⋊Dic179D412yes—
24Dic179⋊C4—24no—
25C22⋊Dic179—22no—
26C4×Dic179—22no—
27C22×C4×C179—21no—
28C22×C4×C179—21no—
29C22×C4×C179D412yes—
30C22×C4×C179C2311yescharacteristic
31C4⋊C4×C179—22no—
32C4⋊C4×C179—22no—
33C42×C179—21no—
34C42×C179Q812yescomplemented
35Dic358—82no—
36C2×Dic179—22no—
37C2×Dic179—22no—
38C179⋊C8—24no—
39C23×C179C2×D411yescharacteristic
40C2×C4×C179—22no—
41C2×C4×C179—22no—
42C2×C4×C179—41no—
43C2×C4×C179—41no—
44C2×C4×C179—42no—
45C2×C4×C179—42no—
46C2×C4×C179—41no—
47C2×C4×C179C2×D412yes—
48C2×C4×C179C2×Q812yes—
49C2×C4×C179C2×Q811yescharacteristic
50C2×C4×C179#4111yescharacteristic
51C22×C179—21no—
52C22×C179—21no—
53C22×C179#12011yescharacteristic
54C22×C179C22⋊Q811yescharacteristic
55C22×C179C4⋊Q811yescharacteristic
56C4×C179—42no—
57C4×C179—42no—
58C4×C179—81no—
59C4×C179C22⋊Q812yes—
60C4×C179C4⋊Q812yes—
61Dic179—42no—
62Dic179—42no—
63C2×C179—21no—
64C2×C179—21no—
65C2×C179#39511yescharacteristic
66C2×C179#45411yescharacteristic
67C2×C179#45511yescharacteristic
68C179#206111yescharacteristic, complemented, Sylow
69#2061—1791nomaximal, Sylow
70#544—1791no—
71#428—1792no—
72#382D17911yescharacteristic
73#343—1791no—
74#339—1792no—
75C2×M16—1792no—
76C4⋊Q8—3582no—
77#125—3582no—
78C22⋊Q8—3582no—
79#117—1792no—
80C2×C4⋊C4C2×D17911yescharacteristic
81C2×C4⋊C4C2×D17911yescharacteristic
82C2×C42C2×D17911yescharacteristic
83#107—3582no—
84#106—3582no—
85#95—22no—
86C2×Q8—3582no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4C22⋊D17912yes—
90C22×C4C22×D17911yescharacteristic
91M16—3582no—
92M16—1794no—
93C2×C8—3582no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—3584no—
97C4⋊C4—3582no—
98C4⋊C4—1792no—
99C4⋊C4—1792no—
100C22⋊C4—3582no—
101C42—21no—
102C42—3582no—
103C42Dic35812yescomplemented
104C23C2×C22⋊D17911yescharacteristic
105Q8—14322no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—3582no—
114C2×C4—3582no—
115C2×C4D4×D17912yes—
116C2×C4Q8×D17912yes—
117C2×C4C2×Dic35811yescharacteristic
118C2×C4#1566825911yescharacteristic
119C8—3584no—
120C22—21no—
121C22—21no—
122C22#1915308111yescharacteristic
123C22#1915309211yescharacteristic
124C22#1915311111yescharacteristic
125C4—42no—
126C4—42no—
127C4—81no—
128C4—7162no—
129C4—7162no—
130C4#1915309112yes—
131C4#1915311212yes—
132C2—21no—
133C2—21no—
134C2#2603835911yescharacteristic
135C2#2603861111yescharacteristic
136C2#2603861611yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#382×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#382D179
hypocenterC179#2061
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
19. #15668243Q8semidirect
34. C42×C179Q8semidirect
68. C179#2061semidirect
103. C42Dic358semidirect

Maximal subgroups

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

subgroupindex
#260390702
#260384792
#260382082
#260381992
#382×C1792
#2061179