Finite Group Databasebeta

← #31805702

Subgroups — #31805702

Order 17792.

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#22581343C211yesmaximal, characteristic
2#22581319C212yesmaximal, complemented
3#22580695C212yesmaximal, complemented
4#22580693C211yesmaximal, characteristic, complemented
5#563×C139C211yesmaximal, characteristic, complemented
6C2×#14555772—21no—
7C2×#14555770—21no—
8#17984839C2211yescharacteristic
9#17984762—22no—
10#17984762—22no—
11#17984753C2212yes—
12#17984753C2211yescharacteristic
13C4⋊Q8×C139—21no—
14#126×C139—22no—
15#125×C139C2212yes—
16#124×C139C2211yescharacteristic
17#123×C139—21no—
18C22⋊Q8×C139—22no—
19C2×D556—21no—
20C2×D556—22no—
21#14555772—42no—
22#14555770—42no—
23C2×C139⋊C8—21no—
24C2×C139⋊C8—22no—
25C2×Q8×C139—21no—
26C2×Q8×C139D411yescharacteristic
27C2×D4×C139D411yescharacteristic
28C22×C4×C139—21no—
29C4⋊C4×C139—22no—
30C4⋊C4×C139—22no—
31C4⋊C4×C139—22no—
32C4⋊C4×C139D412yes—
33C4⋊C4×C139D412yes—
34C22⋊C4×C139—22no—
35C22⋊C4×C139—21no—
36C22⋊C4×C139—22no—
37C42×C139—21no—
38C42×C139C2311yescharacteristic
39C22×D139—41no—
40D556—41no—
41D556—41no—
42D556—44no—
43C139⋊C8—22no—
44C139⋊C8—42no—
45C23×C139—21no—
46Q8×C139—41no—
47Q8×C139—81no—
48D4×C139—41no—
49C2×C4×C139—22no—
50C2×C4×C139—21no—
51C2×C4×C139—22no—
52C2×C4×C139—81no—
53C2×C4×C139C2×D412yes—
54C2×C4×C139C2×D411yescharacteristic
55C22×C139—41no—
56C22×C139—42no—
57C22×C139#12011yescharacteristic
58C2×D139—81no—
59C2×D139—82no—
60C4×C139—21no—
61C4×C139—22no—
62C4×C139—42no—
63C4×C139—41no—
64C4×C139—42no—
65C2×C139—81no—
66C2×C139#45311yescharacteristic
67C2×C139#45412yes—
68D139—161no—
69C139#165711yescharacteristic, complemented, Sylow
70#1657—1391nomaximal, Sylow
71#563D13911yescharacteristic, complemented
72#496—1391no—
73#490—1392no—
74#330—1392no—
75#329—1391no—
76C2×SD16—2781no—
77C2×D8—2781no—
78C4⋊Q8—21no—
79#127—1391no—
80#126—22no—
81#125C2×D13912yes—
82#124C2×D13911yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—2782no—
86#102—2782no—
87#97—1392no—
88#97—1391no—
89C2×Q8—21no—
90C2×Q8C22⋊D13911yescharacteristic
91C2×D4—2782no—
92C2×D4—2781no—
93C2×D4C22⋊D13911yescharacteristic
94C22×C4—21no—
95SD16—5562no—
96D8—5562no—
97C2×C8—2781no—
98C2×C8—2782no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D13912yes—
103C4⋊C4C22⋊D13912yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D13911yescharacteristic
109C23—21no—
110C23—5561no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—5561no—
115D4—5561no—
116D4—5564no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D13912yes—
122C2×C4C2×C22⋊D13911yescharacteristic
123C8—2782no—
124C8—5562no—
125C22—41no—
126C22—42no—
127C22—11121no—
128C22—11122no—
129C22#1798481111yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—22241no—
137C2#2258112111yescharacteristic
138C2#2258112612yes—
subgroupG/H
centerC22#17984811
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#563×C139C2
socleC22×C139#120
prime radical#22581343C2
hypercenter#563D139
hypocenterC139#1657
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #22581319C2semidirect
3. #22580695C2semidirect
4. #22580693C2semidirect
5. #563×C139C2semidirect
69. C139#1657semidirect
71. #563D139semidirect

Maximal subgroups

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

subgroupindex
#225813432
#225813192
#225806952
#225806932
#563×C1392
#1657139