Finite Group Databasebeta

← #37510342

Subgroups — #37510342

Order 22912.

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

normal subgroups only

This group has 246 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#26038482C212yesmaximal
2#26038308C211yesmaximal, characteristic
3#26038208C211yesmaximal, characteristic
4#26038198C212yesmaximal
5#544×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7C2×C4⋊Dic179C2211yescharacteristic
8C2×Dic179⋊C4—22no—
9C2×C4×Dic179C2211yescharacteristic
10#19153046—22no—
11#19153044—22no—
12C4⋊Q8×C179—22no—
13#125×C179—22no—
14C22⋊Q8×C179—22no—
15#117×C179C2212yes—
16C2×C4⋊C4×C179C2211yescharacteristic
17C22×Dic179—21no—
18C22×Dic179—21no—
19#15668243—22no—
20#15668243Q814yescomplemented
21C2×C179⋊C8—22no—
22C4⋊Dic179—22no—
23Dic179⋊C4—44no—
24C4×Dic179—21no—
25C4×Dic179Q812yescomplemented
26C2×Q8×C179—22no—
27C22×C4×C179D412yes—
28C22×C4×C179C2311yescharacteristic
29C4⋊C4×C179—22no—
30C4⋊C4×C179—24no—
31C4⋊C4×C179—22no—
32C4⋊C4×C179D412yes—
33C4⋊C4×C179D412yes—
34C22⋊C4×C179—22no—
35C42×C179—22no—
36C2×Dic179—22no—
37C2×Dic179—22no—
38C2×Dic179—22no—
39C2×Dic179—42no—
40C2×Dic179—41no—
41C179⋊C8—24no—
42C23×C179C2×D411yescharacteristic
43Q8×C179—82no—
44C2×C4×C179—22no—
45C2×C4×C179—22no—
46C2×C4×C179—22no—
47C2×C4×C179—42no—
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—42no—
58C4×C179—42no—
59C4×C179—42no—
60C4×C179C22⋊Q812yes—
61C4×C179C4⋊Q812yes—
62Dic179—24no—
63Dic179—81no—
64C2×C179—21no—
65C2×C179—21no—
66C2×C179#39511yescharacteristic
67C2×C179#45311yescharacteristic
68C2×C179#45611yescharacteristic
69C179#206011yescharacteristic, complemented, Sylow
70#2060—1791nomaximal, Sylow
71#544D17911yescharacteristic
72#428—1792no—
73#384—1791no—
74#343—1791no—
75#339—1792no—
76C2×M16—1792no—
77C4⋊Q8—22no—
78#125—22no—
79C22⋊Q8—22no—
80#117C2×D17912yes—
81C2×C4⋊C4—3582no—
82C2×C4⋊C4—1791no—
83C2×C4⋊C4C2×D17911yescharacteristic
84C2×C42—1791no—
85#107—3582no—
86#106—3582no—
87C2×Q8—22no—
88C22×C4—3581no—
89C22×C4—3581no—
90C22×C4C22⋊D17912yes—
91C22×C4C22×D17911yescharacteristic
92M16—3582no—
93M16—1794no—
94C2×C8—3582no—
95C4⋊C4—22no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—3582no—
99C4⋊C4—7164no—
100C4⋊C4C22⋊D17912yes—
101C4⋊C4C22⋊D17912yes—
102C22⋊C4—22no—
103C42—22no—
104C42—3581no—
105C42—1792no—
106C23C2×C22⋊D17911yescharacteristic
107Q8—82no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—42no—
112C2×C4—3582no—
113C2×C4—3582no—
114C2×C4—3582no—
115C2×C4—7162no—
116C2×C4—7161no—
117C2×C4C2×C22⋊D17912yes—
118C2×C4Q8×D17912yes—
119C2×C4Q8×D17911yescharacteristic
120C2×C4#1566826011yescharacteristic
121C8—3584no—
122C22—21no—
123C22—21no—
124C22#1915308211yescharacteristic
125C22#1915308911yescharacteristic
126C22#1915311111yescharacteristic
127C4—42no—
128C4—42no—
129C4—42no—
130C4—3584no—
131C4—14321no—
132C4#1915308912yes—
133C4#1915311112yes—
134C2—21no—
135C2—21no—
136C2#2603835811yescharacteristic
137C2#2603860711yescharacteristic
138C2#2603862211yescharacteristic
subgroupG/H
centerC22#19153082
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#544×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#544D179
hypocenterC179#2060
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. #15668243Q8semidirect
25. C4×Dic179Q8semidirect
69. C179#2060semidirect

Maximal subgroups

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

subgroupindex
#260384822
#260383082
#260382082
#260381982
#544×C1792
#2060179