Finite Group Databasebeta

← #37510347

Subgroups — #37510347

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#26038482C212yesmaximal
2#26038304C211yesmaximal, characteristic
3#26038208C211yesmaximal, characteristic
4#26038198C212yesmaximal
5#544×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7C2×C4⋊Dic179C2211yescharacteristic
8C2×C4×Dic179C2211yescharacteristic
9#19153046—22no—
10#19153044—22no—
11C179⋊#95—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—
23C4×Dic179—21no—
24C4×Dic179Q812yescomplemented
25C2×Q8×C179—22no—
26C22×C4×C179D412yes—
27C22×C4×C179C2311yescharacteristic
28C4⋊C4×C179—22no—
29C4⋊C4×C179—24no—
30C4⋊C4×C179—22no—
31C4⋊C4×C179D412yes—
32C4⋊C4×C179D412yes—
33C22⋊C4×C179—22no—
34C42×C179—22no—
35C2×Dic179—22no—
36C2×Dic179—41no—
37C2×Dic179—41no—
38C2×Dic179—42no—
39C2×Dic179—41no—
40C179⋊C8—24no—
41C23×C179C2×D411yescharacteristic
42Q8×C179—82no—
43C2×C4×C179—22no—
44C2×C4×C179—22no—
45C2×C4×C179—22no—
46C2×C4×C179—42no—
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—42no—
59C4×C179C22⋊Q812yes—
60C4×C179C4⋊Q812yes—
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#544D17911yescharacteristic
71#428—1792no—
72#382—1791no—
73#343—1791no—
74#339—1792no—
75C2×M16—1792no—
76C4⋊Q8—22no—
77#125—22no—
78C22⋊Q8—22no—
79#117C2×D17912yes—
80C2×C4⋊C4—1791no—
81C2×C4⋊C4C2×D17911yescharacteristic
82C2×C42—1791no—
83#107—3582no—
84#106—3582no—
85#95—3582no—
86C2×Q8—22no—
87C22×C4—3581no—
88C22×C4—3581no—
89C22×C4C22⋊D17912yes—
90C22×C4C22×D17911yescharacteristic
91M16—3582no—
92M16—1794no—
93C2×C8—3582no—
94C4⋊C4—22no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—3582no—
98C4⋊C4C22⋊D17912yes—
99C4⋊C4C22⋊D17912yes—
100C22⋊C4—22no—
101C42—22no—
102C42—3581no—
103C42—1792no—
104C23C2×C22⋊D17911yescharacteristic
105Q8—82no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—3582no—
111C2×C4—7162no—
112C2×C4—7161no—
113C2×C4—7161no—
114C2×C4—7161no—
115C2×C4C2×C22⋊D17912yes—
116C2×C4Q8×D17912yes—
117C2×C4Q8×D17911yescharacteristic
118C2×C4#1566826011yescharacteristic
119C8—3584no—
120C22—21no—
121C22—21no—
122C22#1915308211yescharacteristic
123C22#1915308911yescharacteristic
124C22#1915311111yescharacteristic
125C4—42no—
126C4—42no—
127C4—42no—
128C4—14321no—
129C4—7162no—
130C4#1915308912yes—
131C4#1915311112yes—
132C2—21no—
133C2—21no—
134C2#2603835811yescharacteristic
135C2#2603861211yescharacteristic
136C2#2603861711yescharacteristic
subgroupG/H
centerC22#19153082
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#544×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#544D179
hypocenterC179#2061
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
24. C4×Dic179Q8semidirect
68. C179#2061semidirect

Maximal subgroups

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

subgroupindex
#260384822
#260383042
#260382082
#260381982
#544×C1792
#2061179