Finite Group Databasebeta

← #37508298

Subgroups — #37508298

Order 22912.

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

normal subgroups only

This group has 206 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#26038829C211yesmaximal, characteristic
2#26038805C212yesmaximal, complemented
3#26038181C212yesmaximal, complemented
4#26038179C211yesmaximal, characteristic, complemented
5#563×C179C211yesmaximal, characteristic, complemented
6C2×#15668246—21no—
7C2×#15668244—21no—
8#19153110C2211yescharacteristic
9#19153033—22no—
10#19153033—22no—
11#19153024C2212yes—
12#19153024C2211yescharacteristic
13C4⋊Q8×C179—21no—
14#126×C179—22no—
15#125×C179C2212yes—
16#124×C179C2211yescharacteristic
17#123×C179—21no—
18C22⋊Q8×C179—22no—
19C2×D716—22no—
20C2×D716—21no—
21#15668246—42no—
22#15668244—42no—
23C2×C179⋊C8—21no—
24C2×C179⋊C8—22no—
25C2×Q8×C179—21no—
26C2×Q8×C179D411yescharacteristic
27C2×D4×C179D411yescharacteristic
28C22×C4×C179—21no—
29C4⋊C4×C179—22no—
30C4⋊C4×C179—22no—
31C4⋊C4×C179—22no—
32C4⋊C4×C179D412yes—
33C4⋊C4×C179D412yes—
34C22⋊C4×C179—22no—
35C22⋊C4×C179—21no—
36C22⋊C4×C179—22no—
37C42×C179—21no—
38C42×C179C2311yescharacteristic
39C22×D179—41no—
40D716—41no—
41D716—41no—
42D716—44no—
43C179⋊C8—22no—
44C179⋊C8—42no—
45C23×C179—21no—
46Q8×C179—41no—
47Q8×C179—81no—
48D4×C179—41no—
49C2×C4×C179—22no—
50C2×C4×C179—21no—
51C2×C4×C179—22no—
52C2×C4×C179—81no—
53C2×C4×C179C2×D411yescharacteristic
54C2×C4×C179C2×D412yes—
55C22×C179—41no—
56C22×C179—42no—
57C22×C179#12011yescharacteristic
58C2×D179—81no—
59C2×D179—82no—
60C4×C179—21no—
61C4×C179—22no—
62C4×C179—42no—
63C4×C179—41no—
64C4×C179—42no—
65C2×C179—81no—
66C2×C179#45311yescharacteristic
67C2×C179#45412yes—
68D179—161no—
69C179#165711yescharacteristic, complemented, Sylow
70#1657—1791nomaximal, Sylow
71#563D17911yescharacteristic, complemented
72#496—1791no—
73#490—1792no—
74#330—1792no—
75#329—1791no—
76C2×SD16—3581no—
77C2×D8—3581no—
78C4⋊Q8—21no—
79#127—1791no—
80#126—22no—
81#125C2×D17912yes—
82#124C2×D17911yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—3582no—
86#102—3582no—
87#97—1792no—
88#97—1791no—
89C2×Q8—21no—
90C2×Q8C22⋊D17911yescharacteristic
91C2×D4—3582no—
92C2×D4—3581no—
93C2×D4C22⋊D17911yescharacteristic
94C22×C4—21no—
95SD16—7162no—
96D8—7162no—
97C2×C8—3581no—
98C2×C8—3582no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D17912yes—
103C4⋊C4C22⋊D17912yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D17911yescharacteristic
109C23—21no—
110C23—7161no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—7161no—
115D4—7161no—
116D4—7164no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D17912yes—
122C2×C4C2×C22⋊D17911yescharacteristic
123C8—3582no—
124C8—7162no—
125C22—41no—
126C22—42no—
127C22—14321no—
128C22—14322no—
129C22#1915308211yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—28641no—
137C2#2603860711yescharacteristic
138C2#2603861212yes—
subgroupG/H
centerC22#19153082
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#563×C179C2
socleC22×C179#120
prime radical#26038829C2
hypercenter#563D179
hypocenterC179#1657
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #26038805C2semidirect
3. #26038181C2semidirect
4. #26038179C2semidirect
5. #563×C179C2semidirect
69. C179#1657semidirect
71. #563D179semidirect

Maximal subgroups

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

subgroupindex
#260388292
#260388052
#260381812
#260381792
#563×C1792
#1657179