Finite Group Databasebeta

← #37508293

Subgroups — #37508293

Order 22912.

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

normal subgroups only

This group has 230 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#26038829C212yesmaximal, complemented
2#26038805C211yesmaximal, characteristic, complemented
3#26038181C211yesmaximal, characteristic, complemented
4#26038179C212yesmaximal, complemented
5#562×C179C211yesmaximal, characteristic, complemented
6C2×#15668246—22no—
7C2×#15668244—22no—
8#19153110C2211yescharacteristic
9#19153033—22no—
10#19153024C2211yescharacteristic
11#19153024C2212yes—
12#125×C179C2211yescharacteristic
13#124×C179C2212yes—
14#123×C179—22no—
15C22⋊Q8×C179—22no—
16#121×C179—22no—
17C2×D716—21no—
18C2×D716—22no—
19#15668246—44no—
20#15668244—44no—
21C2×C179⋊C8—22no—
22C2×C179⋊C8—21no—
23C2×Q8×C179D412yes—
24C2×D4×C179—22no—
25C2×D4×C179D412yes—
26C22×C4×C179—22no—
27C4⋊C4×C179—22no—
28C4⋊C4×C179—22no—
29C4⋊C4×C179D412yes—
30C22⋊C4×C179—22no—
31C22⋊C4×C179—22no—
32C22⋊C4×C179—22no—
33C42×C179C2311yescharacteristic
34C22×D179—41no—
35D716—42no—
36D716—42no—
37D716—42no—
38C179⋊C8—24no—
39C179⋊C8—41no—
40C23×C179—22no—
41Q8×C179—42no—
42D4×C179—42no—
43D4×C179—82no—
44C2×C4×C179—22no—
45C2×C4×C179—22no—
46C2×C4×C179—82no—
47C2×C4×C179C2×D412yes—
48C2×C4×C179C2×D411yescharacteristic
49C22×C179—42no—
50C22×C179—42no—
51C22×C179—42no—
52C22×C179#12011yescharacteristic
53C2×D179—82no—
54C2×D179—81no—
55C4×C179—22no—
56C4×C179—21no—
57C4×C179—42no—
58C4×C179—42no—
59C2×C179—82no—
60C2×C179#45312yes—
61C2×C179#45411yescharacteristic
62D179—161no—
63C179#165611yescharacteristic, complemented, Sylow
64#1656—1791nomaximal, Sylow
65#562D17911yescharacteristic, complemented
66#496—1792no—
67#490—1791no—
68#330—1791no—
69#329—1792no—
70C2×SD16—3582no—
71C2×D8—3582no—
72#127—1791no—
73#125C2×D17911yescharacteristic
74#124C2×D17912yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—3582no—
79#97—1791no—
80#97—1792no—
81C2×Q8C22⋊D17912yes—
82C2×D4—22no—
83C2×D4—3581no—
84C2×D4—3582no—
85C2×D4C22⋊D17912yes—
86C22×C4—22no—
87SD16—7164no—
88D8—7164no—
89C2×C8—3582no—
90C2×C8—3581no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D17912yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D17911yescharacteristic
98C23—22no—
99C23—7161no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—7162no—
104D4—7162no—
105D4—7162no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D17911yescharacteristic
110C2×C4C2×C22⋊D17912yes—
111C8—3584no—
112C8—7161no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—14322no—
117C22—14321no—
118C22#1915308211yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—28641no—
125C2#2603860712yes—
126C2#2603861211yescharacteristic
subgroupG/H
centerC22#19153082
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#562×C179C2
socleC22×C179#120
hypercenter#562D179
hypocenterC179#1656
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. #26038829C2semidirect
2. #26038805C2semidirect
3. #26038181C2semidirect
4. #26038179C2semidirect
5. #562×C179C2semidirect
63. C179#1656semidirect
65. #562D179semidirect

Maximal subgroups

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

subgroupindex
#260388292
#260388052
#260381812
#260381792
#562×C1792
#1656179