Finite Group Databasebeta

← #31805703

Subgroups — #31805703

Order 17792.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#22581703C211yesmaximal, characteristic, complemented
2#22581343C211yesmaximal, characteristic
3#22581318C212yesmaximal, complemented
4#22580694C212yesmaximal, complemented
5#329×C139C211yesmaximal, characteristic, complemented
6C2×#14555772—21no—
7C2×#14555770—21no—
8#17984840—21no—
9#17984839C2211yescharacteristic
10#17984835—22no—
11#17984834C2212yes—
12#17984823—21no—
13#17984819—22no—
14#17984764—22no—
15#17984764—22no—
16#17984753C2211yescharacteristic
17#124×C139C2211yescharacteristic
18#97×C139C2212yes—
19C2×Dic278—21no—
20C2×D556—21no—
21C2×D556—22no—
22C22×Dic139—21no—
23#14555772—42no—
24#14555770—42no—
25C2×C139⋊C8—21no—
26C4⋊Dic139D412yes—
27C4⋊Dic139D412yes—
28Dic139⋊C4—22no—
29Dic139⋊C4—22no—
30Dic139⋊C4—22no—
31C22⋊Dic139—22no—
32C22⋊Dic139—21no—
33C4×Dic139—21no—
34C2×Q8×C139D411yescharacteristic
35C2×D4×C139D411yescharacteristic
36C2×C8×C139—22no—
37C22⋊C4×C139—22no—
38C42×C139C2311yescharacteristic
39C22×D139—41no—
40Dic278—81no—
41D556—41no—
42D556—41no—
43D556—44no—
44C2×Dic139—22no—
45C2×Dic139—22no—
46C2×Dic139—81no—
47C139⋊C8—22no—
48C23×C139—21no—
49Q8×C139—41no—
50D4×C139—41no—
51C2×C4×C139—21no—
52C2×C4×C139C2×D412yes—
53C2×C4×C139C2×D411yescharacteristic
54C8×C139—42no—
55C22×C139—41no—
56C22×C139—42no—
57C22×C139#12011yescharacteristic
58C2×D139—81no—
59C2×D139—82no—
60C4×C139—21no—
61C4×C139—22no—
62C4×C139—41no—
63Dic139—42no—
64Dic139—42no—
65C2×C139—81no—
66C2×C139#45311yescharacteristic
67C2×C139#45412yes—
68D139—161no—
69C139#165711yescharacteristic, complemented, Sylow
70#1657—1391nomaximal, Sylow
71#563—1391no—
72#496—1391no—
73#490—1392no—
74#330—1392no—
75#329D13911yescharacteristic, complemented
76C2×SD16—2781no—
77C2×D8—2781no—
78C4⋊Q8—2781no—
79#127—1391no—
80#126—2782no—
81#125—1392no—
82#124C2×D13911yescharacteristic
83#123—2781no—
84C22⋊Q8—2782no—
85#102—2782no—
86#102—2782no—
87#97—1391no—
88#97C2×D13912yes—
89C2×Q8—2781no—
90C2×Q8D55611yescharacteristic
91C2×D4—2782no—
92C2×D4—2781no—
93C2×D4D55611yescharacteristic
94C22×C4—2781no—
95SD16—5562no—
96D8—5562no—
97C2×C8—22no—
98C2×C8—2781no—
99C4⋊C4—2782no—
100C4⋊C4—2782no—
101C4⋊C4—2782no—
102C4⋊C4—1392no—
103C4⋊C4—1392no—
104C22⋊C4—22no—
105C22⋊C4—2782no—
106C22⋊C4—2781no—
107C42—2781no—
108C42C22×D13911yescharacteristic
109C23—21no—
110C23—5561no—
111Q8—41no—
112Q8—11121no—
113D4—41no—
114D4—5561no—
115D4—5561no—
116D4—5564no—
117C2×C4—21no—
118C2×C4—2782no—
119C2×C4—2782no—
120C2×C4—11121no—
121C2×C4D4×D13912yes—
122C2×C4C2×D55611yescharacteristic
123C8—42no—
124C8—2782no—
125C22—41no—
126C22—42no—
127C22—11121no—
128C22—11122no—
129C22#1798481211yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—5562no—
134C4—5562no—
135C2—81no—
136C2—22241no—
137C2#2258112311yescharacteristic
138C2#2258112712yes—
subgroupG/H
centerC22#17984812
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#329×C139C2
socleC22×C139#120
prime radical#22581343C2
hypercenter#329D139
hypocenterC139#1657
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #22581703C2semidirect
3. #22581318C2semidirect
4. #22580694C2semidirect
5. #329×C139C2semidirect
69. C139#1657semidirect
75. #329D139semidirect

Maximal subgroups

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

subgroupindex
#225817032
#225813432
#225813182
#225806942
#329×C1392
#1657139