Finite Group Databasebeta

← #37508323

Subgroups — #37508323

Order 22912.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#26038809C211yesmaximal, characteristic, complemented
2#26038798C212yesmaximal, complemented
3#26038181C211yesmaximal, characteristic, complemented
4#26038179C212yesmaximal, complemented
5#562×C179C211yesmaximal, characteristic
6#19153113C2211yescharacteristic
7#19153037—22no—
8#19153036—22no—
9#19153034—22no—
10#19153024C2212yes—
11#19153024C2211yescharacteristic
12#125×C179C2211yescharacteristic
13#124×C179C2212yes—
14#123×C179—22no—
15C22⋊Q8×C179—22no—
16#121×C179—22no—
17C2×Dic358—21no—
18C2×C179⋊C8—22no—
19C2×C179⋊C8—21no—
20C4⋊Dic179—22no—
21C2×Q8×C179D412yes—
22C2×D4×C179—22no—
23C2×D4×C179D412yes—
24C22×C4×C179—22no—
25C4⋊C4×C179—22no—
26C4⋊C4×C179—22no—
27C4⋊C4×C179D412yes—
28C22⋊C4×C179—22no—
29C22⋊C4×C179—22no—
30C22⋊C4×C179—22no—
31C42×C179C2311yescharacteristic
32Dic358—42no—
33C2×Dic179—41no—
34C179⋊C8—41no—
35C179⋊C8—42no—
36C23×C179—22no—
37Q8×C179—42no—
38D4×C179—42no—
39D4×C179—82no—
40C2×C4×C179—22no—
41C2×C4×C179—22no—
42C2×C4×C179—82no—
43C2×C4×C179C2×D412yes—
44C2×C4×C179C2×D411yescharacteristic
45C22×C179—42no—
46C22×C179—42no—
47C22×C179—42no—
48C22×C179#12011yescharacteristic
49C4×C179—21no—
50C4×C179—22no—
51C4×C179—42no—
52C4×C179—42no—
53Dic179—81no—
54C2×C179—82no—
55C2×C179#45411yescharacteristic
56C2×C179#45512yes—
57C179#166211yescharacteristic, complemented, Sylow
58#1662—1791nomaximal, Sylow
59#562D17911yescharacteristic
60#491—1791no—
61#489—1792no—
62#330—1791no—
63#329—1792no—
64C4⋊Q8—1791no—
65#125C2×D17911yescharacteristic
66#124C2×D17912yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—3582no—
71#103—3582no—
72#102—3582no—
73#97—1791no—
74#97—1792no—
75C2×Q8—3581no—
76C2×Q8C22⋊D17912yes—
77C2×D4—22no—
78C2×D4C22⋊D17912yes—
79C22×C4—22no—
80C2×C8—3582no—
81C2×C8—3581no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—3582no—
85C4⋊C4C22⋊D17912yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D17911yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—7162no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—7161no—
99C2×C4C2×C22⋊D17911yescharacteristic
100C2×C4C2×C22⋊D17912yes—
101C8—7161no—
102C8—7162no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1915308211yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—14321no—
112C2—82no—
113C2#2603861211yescharacteristic
114C2#2603861712yes—
subgroupG/H
centerC22#19153082
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting, prime radical#562×C179C2
socleC22×C179#120
hypercenter#562D179
hypocenterC179#1662
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #26038809C2semidirect
2. #26038798C2semidirect
3. #26038181C2semidirect
4. #26038179C2semidirect
57. C179#1662semidirect

Maximal subgroups

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

subgroupindex
#260388092
#260387982
#260381812
#260381792
#562×C1792
#1662179