Finite Group Databasebeta

← #31808427

Subgroups — #31808427

Order 17792.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#22581138C212yesmaximal, complemented
3#22581125C211yesmaximal, characteristic
4#22580751C212yesmaximal, complemented
5#355×C139C211yesmaximal, characteristic, complemented
6#17984890C2211yescharacteristic
7#17984875—22no—
8#17984868—21no—
9#17984840C2212yes—
10#17984835—22no—
11#17984834—21no—
12#17984823—21no—
13#17984819—22no—
14#17984770—21no—
15#17984768—22no—
16#17984761C2211yescharacteristic
17#124×C139C2211yescharacteristic
18#101×C139C2212yes—
19#14555786—41no—
20#14555785—22no—
21#14555785—21no—
22C2×Dic278D412yes—
23Q8×D139—41no—
24C22×Dic139—21no—
25Dic556—42no—
26#14555774—42no—
27#14555773—41no—
28#14555770—41no—
29#14555769—21no—
30C4⋊Dic139—21no—
31C4⋊Dic139—21no—
32Dic139⋊C4—22no—
33Dic139⋊C4—22no—
34Dic139⋊C4—22no—
35C22⋊Dic139—22no—
36C22⋊Dic139—21no—
37C4×Dic139D412yes—
38C2×Q8×C139C2311yescharacteristic
39C2×D4×C139D411yescharacteristic
40M16×C139—22no—
41C22⋊C4×C139—22no—
42C42×C139D411yescharacteristic
43Dic278—21no—
44Dic278—22no—
45Dic278—42no—
46D556—21no—
47D556—22no—
48C22⋊D139—41no—
49C2×Dic139—22no—
50C2×Dic139—22no—
51C2×Dic139—81no—
52C4×D139—42no—
53C4×D139—41no—
54C139⋊C8—41no—
55C23×C139—21no—
56Q8×C139—41no—
57D4×C139—41no—
58C2×C4×C139—21no—
59C2×C4×C139C2×D412yes—
60C2×C4×C139C2×D411yescharacteristic
61C8×C139—42no—
62C22×C139—42no—
63C22×C139—41no—
64C22×C139#12011yescharacteristic
65C2×D139—41no—
66C4×C139—22no—
67C4×C139—21no—
68C4×C139—41no—
69Dic139—42no—
70Dic139—41no—
71Dic139—42no—
72C2×C139—21no—
73C2×C139—81no—
74C2×C139B(4,𝔽2)11yescharacteristic
75D139—81no—
76C139#220311yescharacteristic, complemented, Sylow
77#2203—1391nomaximal, Sylow
78#563—1391no—
79#456—1392no—
80#454—1391no—
81#356—1392no—
82#355D13911yescharacteristic, complemented
832−1+4—1391no—
84#137—2782no—
85AGL(1,ℤ/8)—2781no—
86C4⋊Q8—1392no—
87#126—2782no—
88#125—2781no—
89#124C2×D13911yescharacteristic
90#123—2781no—
91C22⋊Q8—2782no—
92#104—2782no—
93#104—2781no—
94#101—1391no—
95#101C2×D13912yes—
96#41—2782no—
97#41—2781no—
98#41—5561no—
99C2×Q8—5561no—
100C2×Q8—1392no—
101C2×Q8C22×D13911yescharacteristic
102C2×D4D55611yescharacteristic
103C22×C4—2781no—
104Q16—5562no—
105SD16—5562no—
106SD16—5561no—
107D8—5561no—
108M16—22no—
109M16—2781no—
110C4⋊C4—2782no—
111C4⋊C4—2781no—
112C4⋊C4—2781no—
113C4⋊C4—2782no—
114C4⋊C4—2782no—
115C22⋊C4—22no—
116C22⋊C4—2782no—
117C22⋊C4—2781no—
118C42—1392no—
119C42D55611yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—2781no—
123Q8—2782no—
124Q8—5562no—
125D4—41no—
126D4—2781no—
127D4—2782no—
128D4—5561no—
129C2×C4—21no—
130C2×C4—2782no—
131C2×C4—2782no—
132C2×C4—5562no—
133C2×C4—5561no—
134C2×C4—11121no—
135C2×C4D4×D13912yes—
136C2×C4C2×D55611yescharacteristic
137C8—42no—
138C8—5561no—
139C22—42no—
140C22—41no—
141C22—5561no—
142C22#1798481211yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—5562no—
147C4—5562no—
148C4—5561no—
149C2—21no—
150C2—81no—
151C2—11121no—
152C2#2258114111yescharacteristic
subgroupG/H
centerC2#22581141
commutator, cosocleC2×Q8×C139C23
FrattiniC2×Q8C22×D139
Fitting#355×C139C2
socleC2×C139B(4,𝔽2)
prime radical#22581125C2
hypercenter#355D139
hypocenterC139#2203
LCSG ▷ C2×Q8×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #22581703C2semidirect
2. #22581138C2semidirect
4. #22580751C2semidirect
5. #355×C139C2semidirect
76. C139#2203semidirect
82. #355D139semidirect

Maximal subgroups

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

subgroupindex
#225817032
#225811382
#225811252
#225807512
#355×C1392
#2203139