Finite Group Databasebeta

← #37510345

Subgroups — #37510345

Order 22912.

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

normal subgroups only

This group has 246 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#384×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7#19153112—22no—
8#19153104—22no—
9#19153092—22no—
10#19153072C2212yes—
11#19153046—22no—
12#19153044—22no—
13C2×C4⋊C4×C179—22no—
14C2×C4⋊C4×C179C2211yescharacteristic
15C2×C4⋊C4×C179C2211yescharacteristic
16C2×C42×C179C2211yescharacteristic
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—
33C4⋊C4×C179—44no—
34C42×C179—21no—
35C42×C179Q812yescomplemented
36Dic358—82no—
37C2×Dic179—22no—
38C2×Dic179—22no—
39C179⋊C8—24no—
40C23×C179C2×D411yescharacteristic
41C2×C4×C179—22no—
42C2×C4×C179—22no—
43C2×C4×C179—22no—
44C2×C4×C179—22no—
45C2×C4×C179—42no—
46C2×C4×C179—42no—
47C2×C4×C179—41no—
48C2×C4×C179C2×D412yes—
49C2×C4×C179C2×Q812yes—
50C2×C4×C179C2×Q811yescharacteristic
51C2×C4×C179#4111yescharacteristic
52C22×C179—21no—
53C22×C179—21no—
54C22×C179#12011yescharacteristic
55C22×C179C22⋊Q811yescharacteristic
56C22×C179C4⋊Q811yescharacteristic
57C4×C179—24no—
58C4×C179—42no—
59C4×C179—81no—
60C4×C179C22⋊Q812yes—
61C4×C179C4⋊Q812yes—
62Dic179—42no—
63Dic179—42no—
64C2×C179—21no—
65C2×C179—21no—
66C2×C179#39511yescharacteristic
67C2×C179#45311yescharacteristic
68C2×C179#45611yescharacteristic
69C179#206011yescharacteristic, complemented, Sylow
70#2060—1791nomaximal, Sylow
71#544—1791no—
72#428—1792no—
73#384D17911yescharacteristic
74#343—1791no—
75#339—1792no—
76C2×M16—1792no—
77C4⋊Q8—3582no—
78#125—3582no—
79C22⋊Q8—3582no—
80#117—1792no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4C2×D17911yescharacteristic
83C2×C4⋊C4C2×D17911yescharacteristic
84C2×C42C2×D17911yescharacteristic
85#107—3582no—
86#106—3582no—
87C2×Q8—3582no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C22⋊D17912yes—
91C22×C4C22×D17911yescharacteristic
92M16—3582no—
93M16—1794no—
94C2×C8—3582no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—44no—
98C4⋊C4—3584no—
99C4⋊C4—3582no—
100C4⋊C4—1792no—
101C4⋊C4—1792no—
102C22⋊C4—3582no—
103C42—21no—
104C42—3582no—
105C42Dic35812yescomplemented
106C23C2×C22⋊D17911yescharacteristic
107Q8—14322no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—3582no—
116C2×C4—3582no—
117C2×C4D4×D17912yes—
118C2×C4Q8×D17912yes—
119C2×C4C2×Dic35811yescharacteristic
120C2×C4#1566825911yescharacteristic
121C8—3584no—
122C22—21no—
123C22—21no—
124C22#1915308111yescharacteristic
125C22#1915309211yescharacteristic
126C22#1915311111yescharacteristic
127C4—24no—
128C4—42no—
129C4—81no—
130C4—7162no—
131C4—7162no—
132C4#1915309112yes—
133C4#1915311212yes—
134C2—21no—
135C2—21no—
136C2#2603835911yescharacteristic
137C2#2603860611yescharacteristic
138C2#2603862111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#384×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#384D179
hypocenterC179#2060
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
19. #15668243Q8semidirect
35. C42×C179Q8semidirect
69. C179#2060semidirect
105. 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
#384×C1792
#2060179