Finite Group Databasebeta

← #37510351

Subgroups — #37510351

Order 22912.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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#26038481C212yesmaximal
3#26038304C211yesmaximal, characteristic
4#26038200C212yesmaximal
5#343×C179C211yesmaximal, characteristic
6#19153112—22no—
7#19153104—22no—
8#19153092—22no—
9#19153072C2212yes—
10C2×C4⋊Dic179C2211yescharacteristic
11C2×C4×Dic179C2211yescharacteristic
12#19153047—22no—
13#19153045—22no—
14C179⋊#95—22no—
15C2×M16×C179C2212yes—
16C2×C4⋊C4×C179C2211yescharacteristic
17C2×Dic358—22no—
18C22×Dic179—21no—
19C22×Dic179—21no—
20C4⋊Dic179—22no—
21C4⋊Dic179—22no—
22C4⋊Dic179D412yes—
23C4⋊Dic179D412yes—
24Dic179⋊C4—24no—
25C22⋊Dic179—22no—
26C4×Dic179—22no—
27C4×Dic179—21no—
28C4×Dic179Q812yescomplemented
29C22×C4×C179D412yes—
30C22×C4×C179C2311yescharacteristic
31M16×C179—22no—
32M16×C179Q814yescomplemented
33C2×C8×C179—22no—
34C4⋊C4×C179—22no—
35Dic358—82no—
36C2×Dic179—22no—
37C2×Dic179—22no—
38C2×Dic179—22no—
39C2×Dic179—41no—
40C2×Dic179—41no—
41C2×Dic179—42no—
42C2×Dic179—41no—
43C23×C179C2×D411yescharacteristic
44C2×C4×C179—22no—
45C2×C4×C179—42no—
46C2×C4×C179C2×D412yes—
47C2×C4×C179C2×Q811yescharacteristic
48C2×C4×C179C2×Q812yes—
49C2×C4×C179#4111yescharacteristic
50C8×C179—24no—
51C22×C179—21no—
52C22×C179—21no—
53C22×C179#12011yescharacteristic
54C22×C179C22⋊Q811yescharacteristic
55C22×C179C4⋊Q811yescharacteristic
56C4×C179—42no—
57C4×C179C22⋊Q812yes—
58C4×C179C4⋊Q812yes—
59Dic179—42no—
60Dic179—42no—
61Dic179—42no—
62Dic179—81no—
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#382—1791no—
73#343D17911yescharacteristic
74#339—1792no—
75C2×M16C2×D17912yes—
76C4⋊Q8—3582no—
77#125—3582no—
78C22⋊Q8—3582no—
79#117—1792no—
80C2×C4⋊C4—1791no—
81C2×C4⋊C4C2×D17911yescharacteristic
82C2×C42—1791no—
83#107—3582no—
84#106—3582no—
85#95—3582no—
86C2×Q8—3582no—
87C22×C4—3581no—
88C22×C4—3581no—
89C22×C4D71612yes—
90C22×C4C22×D17911yescharacteristic
91M16—22no—
92M16Dic35814yescomplemented
93C2×C8—22no—
94C4⋊C4—22no—
95C4⋊C4—3584no—
96C4⋊C4—3582no—
97C4⋊C4—3582no—
98C4⋊C4—1792no—
99C4⋊C4—1792no—
100C22⋊C4—3582no—
101C42—3582no—
102C42—3581no—
103C42—1792no—
104C23C2×D71611yescharacteristic
105Q8—14322no—
106C2×C4—22no—
107C2×C4—42no—
108C2×C4—3582no—
109C2×C4—3582no—
110C2×C4—3582no—
111C2×C4—7162no—
112C2×C4—7161no—
113C2×C4—7161no—
114C2×C4—7161no—
115C2×C4D4×D17912yes—
116C2×C4Q8×D17911yescharacteristic
117C2×C4C2×Dic35812yes—
118C2×C4#1566825811yescharacteristic
119C8—24no—
120C22—21no—
121C22—21no—
122C22#1915308311yescharacteristic
123C22#1915309311yescharacteristic
124C22#1915311311yescharacteristic
125C4—42no—
126C4—14321no—
127C4—7162no—
128C4—7162no—
129C4—7162no—
130C4#1915309012yes—
131C4#1915311212yes—
132C2—21no—
133C2—21no—
134C2#2603836011yescharacteristic
135C2#2603861411yescharacteristic
136C2#2603861911yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#343×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#343D179
hypocenterC179#2061
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
28. C4×Dic179Q8semidirect
32. M16×C179Q8semidirect
68. C179#2061semidirect
92. M16Dic358semidirect

Maximal subgroups

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

subgroupindex
#260390702
#260384812
#260383042
#260382002
#343×C1792
#2061179