Finite Group Databasebeta

← #31808425

Subgroups — #31808425

Order 17792.

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

normal subgroups only

This group has 216 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#22581135C212yesmaximal, complemented
2#22581125C211yesmaximal, characteristic
3#22580752C212yesmaximal, complemented
4#22580750C211yesmaximal, characteristic, complemented
5#563×C139C211yesmaximal, characteristic, complemented
6#17984890C2211yescharacteristic
7#17984873—22no—
8#17984868—21no—
9#17984770—21no—
10#17984770—22no—
11#17984761C2212yes—
12#17984761C2211yescharacteristic
13C4⋊Q8×C139C2212yes—
14#126×C139—22no—
15#125×C139—21no—
16#124×C139C2211yescharacteristic
17#123×C139—21no—
18C22⋊Q8×C139—22no—
19#14555786—41no—
20#14555785—21no—
21#14555785—22no—
22Q8×D139—41no—
23#14555775—42no—
24#14555773—41no—
25#14555772—42no—
26#14555770—41no—
27#14555769—21no—
28#14555769—22no—
29C2×Q8×C139D412yes—
30C2×Q8×C139C2311yescharacteristic
31C2×D4×C139D411yescharacteristic
32C22×C4×C139—21no—
33C4⋊C4×C139—21no—
34C4⋊C4×C139—22no—
35C4⋊C4×C139—21no—
36C4⋊C4×C139—22no—
37C4⋊C4×C139—22no—
38C22⋊C4×C139—22no—
39C22⋊C4×C139—21no—
40C22⋊C4×C139—22no—
41C42×C139D411yescharacteristic
42C42×C139D412yes—
43Dic278—21no—
44Dic278—22no—
45D556—21no—
46D556—22no—
47C22⋊D139—41no—
48C4×D139—41no—
49C4×D139—42no—
50C139⋊C8—41no—
51C139⋊C8—42no—
52C23×C139—21no—
53Q8×C139—41no—
54Q8×C139—42no—
55D4×C139—41no—
56C2×C4×C139—21no—
57C2×C4×C139—22no—
58C2×C4×C139—22no—
59C2×C4×C139—81no—
60C2×C4×C139C2×D412yes—
61C2×C4×C139C2×D411yescharacteristic
62C22×C139—42no—
63C22×C139—41no—
64C22×C139#12011yescharacteristic
65C2×D139—41no—
66C4×C139—21no—
67C4×C139—22no—
68C4×C139—41no—
69C4×C139—42no—
70C4×C139—42no—
71Dic139—41no—
72C2×C139—21no—
73C2×C139—81no—
74C2×C139B(4,𝔽2)11yescharacteristic
75D139—81no—
76C139#220311yescharacteristic, complemented, Sylow
77#2203—1391nomaximal, Sylow
78#563D13911yescharacteristic, complemented
79#456—1392no—
80#454—1391no—
81#356—1392no—
82#355—1391no—
832−1+4—1391no—
84#137—2782no—
85AGL(1,ℤ/8)—2781no—
86C4⋊Q8C2×D13912yes—
87#126—22no—
88#125—21no—
89#124C2×D13911yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—2781no—
93#104—2782no—
94#101—1392no—
95#101—1391no—
96#41—2781no—
97#41—2782no—
98#41—5561no—
99C2×Q8—5561no—
100C2×Q8C22⋊D13912yes—
101C2×Q8C22×D13911yescharacteristic
102C2×D4C22⋊D13911yescharacteristic
103C22×C4—21no—
104Q16—5562no—
105SD16—5561no—
106SD16—5562no—
107D8—5561no—
108M16—2781no—
109M16—2782no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D13911yescharacteristic
119C42C22⋊D13912yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—2781no—
124Q8—2782no—
125D4—41no—
126D4—2781no—
127D4—2782no—
128D4—5561no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—5561no—
134C2×C4—5562no—
135C2×C4C2×C22⋊D13912yes—
136C2×C4C2×C22⋊D13911yescharacteristic
137C8—5561no—
138C8—5562no—
139C22—42no—
140C22—41no—
141C22—5561no—
142C22#1798481111yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—5561no—
149C2—21no—
150C2—81no—
151C2—11121no—
152C2#2258114011yescharacteristic
subgroupG/H
centerC2#22581140
commutator, cosocleC2×Q8×C139C23
FrattiniC2×Q8C22×D139
Fitting#563×C139C2
socleC2×C139B(4,𝔽2)
prime radical#22581125C2
hypercenter#563D139
hypocenterC139#2203
LCSG ▷ C2×Q8×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #22581135C2semidirect
3. #22580752C2semidirect
4. #22580750C2semidirect
5. #563×C139C2semidirect
76. C139#2203semidirect
78. #563D139semidirect

Maximal subgroups

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

subgroupindex
#225811352
#225811252
#225807522
#225807502
#563×C1392
#2203139