Finite Group Databasebeta

← #31808417

Subgroups — #31808417

Order 17792.

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

normal subgroups only

This group has 226 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#22581142C212yesmaximal
2#22581130C211yesmaximal, characteristic
3#22580748C211yesmaximal, characteristic, complemented
4#22580743C212yesmaximal, complemented
5#563×C139C211yesmaximal, characteristic, complemented
6#17984887C2211yescharacteristic
7#17984873—21no—
8#17984825—22no—
9#17984770—21no—
10#17984759C2211yescharacteristic
11#17984757C2212yes—
12#17984756—22no—
13C4⋊Q8×C139—21no—
14#126×C139—22no—
15#125×C139—21no—
16#124×C139C2211yescharacteristic
17#123×C139C2212yes—
18C22⋊Q8×C139—22no—
19#14555785—21no—
20#14555784—41no—
21C2×C22⋊D139—22no—
22D4×D139—41no—
23#14555775—41no—
24#14555772—41no—
25#14555769—21no—
26Dic139⋊C4—42no—
27C22⋊Dic139—22no—
28(C2×D139)⋊C4—42no—
29C2×Q8×C139—21no—
30C2×Q8×C139D411yescharacteristic
31C2×D4×C139C2311yescharacteristic
32C22×C4×C139D412yes—
33C4⋊C4×C139—21no—
34C4⋊C4×C139—22no—
35C4⋊C4×C139—22no—
36C4⋊C4×C139—21no—
37C4⋊C4×C139—22no—
38C22⋊C4×C139—22no—
39C22⋊C4×C139—22no—
40C22⋊C4×C139D412yes—
41C42×C139—21no—
42C42×C139D411yescharacteristic
43C22×D139—22no—
44Dic278—21no—
45D556—21no—
46C22⋊D139—42no—
47C22⋊D139—41no—
48C2×Dic139—22no—
49C2×Dic139—42no—
50C4×D139—41no—
51C139⋊C8—41no—
52C23×C139C2×D412yes—
53Q8×C139—41no—
54Q8×C139—81no—
55D4×C139—41no—
56C2×C4×C139—21no—
57C2×C4×C139—22no—
58C2×C4×C139—21no—
59C2×C4×C139—22no—
60C2×C4×C139—42no—
61C2×C4×C139C2×D411yescharacteristic
62C22×C139—22no—
63C22×C139—42no—
64C22×C139#12011yescharacteristic
65C2×D139—41no—
66C2×D139—82no—
67C4×C139—21no—
68C4×C139—42no—
69C4×C139—41no—
70C4×C139—42no—
71C4×C139—41no—
72Dic139—41no—
73Dic139—82no—
74C2×C139—21no—
75C2×C139—42no—
76C2×C139B(4,𝔽2)11yescharacteristic
77D139—81no—
78C139#220111yescharacteristic, complemented, Sylow
79#2201—1391nomaximal, Sylow
80#563D13911yescharacteristic, complemented
81#458—1392no—
82#455—1391no—
83#354—1391no—
84#352—1392no—
85H(5,𝔽2)—1391no—
86#137—2781no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D13911yescharacteristic
91#123—2782no—
92#123C2×D13912yes—
93C22⋊Q8—22no—
94#104—2781no—
95#100—1391no—
96#99—2782no—
97#99—1392no—
98#41—2781no—
99#41—5561no—
100C2×Q8—21no—
101C2×Q8C22⋊D13911yescharacteristic
102C2×D4—2782no—
103C2×D4—5561no—
104C2×D4C22×D13911yescharacteristic
105C22×C4C22⋊D13912yes—
106Q16—5561no—
107SD16—5561no—
108M16—2781no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—5562no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—2782no—
118C22⋊C4—5562no—
119C22⋊C4C22⋊D13912yes—
120C42—21no—
121C42C22⋊D13911yescharacteristic
122C23—2782no—
123C23C2×C22⋊D13912yes—
124Q8—41no—
125Q8—81no—
126Q8—2781no—
127D4—41no—
128D4—2781no—
129D4—5562no—
130D4—5561no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—2782no—
137C2×C4—5561no—
138C2×C4—5562no—
139C2×C4C2×C22⋊D13911yescharacteristic
140C8—5561no—
141C22—22no—
142C22—42no—
143C22—5561no—
144C22—11122no—
145C22#1798481111yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—5561no—
152C4—11122no—
153C2—21no—
154C2—42no—
155C2—11121no—
156C2#2258114011yescharacteristic
subgroupG/H
centerC2#22581140
commutator, cosocleC2×D4×C139C23
FrattiniC2×D4C22×D139
Fitting#563×C139C2
socleC2×C139B(4,𝔽2)
prime radical#17984887C22
hypercenter#563D139
hypocenterC139#2201
LCSG ▷ C2×D4×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #22580748C2semidirect
4. #22580743C2semidirect
5. #563×C139C2semidirect
78. C139#2201semidirect
80. #563D139semidirect

Maximal subgroups

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

subgroupindex
#225811422
#225811302
#225807482
#225807432
#563×C1392
#2201139