Finite Group Databasebeta

← #31805694

Subgroups — #31805694

Order 17792.

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

normal subgroups only

This group has 262 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#22581695C211yesmaximal, characteristic, complemented
2#22581343C211yesmaximal, characteristic, complemented
3#22581342C212yesmaximal, complemented
4#22580692C212yesmaximal, complemented
5#329×C139C211yesmaximal, characteristic, complemented
6C2×#14555774—22no—
7C2×#14555772—21no—
8C2×D1112—22no—
9C2×#14555770—21no—
10#17984839C2211yescharacteristic, complemented
11#17984831—21no—
12#17984830C2212yes—
13#17984828—22no—
14#17984812—22no—
15#17984810—21no—
16#17984753C2211yescharacteristic, complemented
17#124×C139C2211yescharacteristic
18#97×C139C2212yescomplemented
19C23×D139—21no—
20C2×Dic278D412yes—
21C2×D556—21no—
22C2×D556—21no—
23C2×D556—22no—
24C2×D556D412yes—
25C2×C22⋊D139—22no—
26#14555774—44no—
27#14555772—42no—
28D1112—44no—
29#14555770—42no—
30C2×C139⋊C8—21no—
31C22⋊Dic139—21no—
32(C2×D139)⋊C4—22no—
33(C2×D139)⋊C4—22no—
34(C2×D139)⋊C4—22no—
35C4×Dic139—21no—
36C2×Q8×C139D411yescharacteristic
37C2×D4×C139D411yescharacteristic
38C2×C8×C139—22no—
39C22⋊C4×C139—22no—
40C42×C139C2311yescharacteristic
41C22×D139—22no—
42C22×D139—41no—
43C22×D139—82no—
44C22×D139—81no—
45Dic278—42no—
46D556—42no—
47D556—41no—
48D556—42no—
49D556—41no—
50D556—42no—
51D556—81no—
52C22⋊D139—82no—
53C2×Dic139—22no—
54C139⋊C8—22no—
55C23×C139—21no—
56Q8×C139—41no—
57D4×C139—41no—
58C2×C4×C139—21no—
59C2×C4×C139C2×D412yes—
60C2×C4×C139C2×D411yescharacteristic
61C8×C139—24no—
62C22×C139—41no—
63C22×C139—42no—
64C22×C139#12011yescharacteristic
65C2×D139—42no—
66C2×D139—42no—
67C2×D139—42no—
68C2×D139—84no—
69C2×D139—82no—
70C2×D139—81no—
71C4×C139—22no—
72C4×C139—21no—
73C4×C139—41no—
74Dic139—42no—
75C2×C139—81no—
76C2×C139#45312yes—
77C2×C139#45311yescharacteristic
78D139—82no—
79D139—161no—
80C139#165511yescharacteristic, complemented, Sylow
81#1655—1391nomaximal, Sylow
82H(3,𝔽4)—1391no—
83#496—1391no—
84#496—1392no—
85#329—1392no—
86#329D13911yescharacteristic, complemented
87C2×SD16—2782no—
88C2×SD16—2781no—
89C2×D8—2782no—
90C2×D8—2781no—
91#127—1391no—
92#124—2782no—
93#124—2781no—
94#124—1392no—
95#124C2×D13911yescharacteristic
96#120—2782no—
97#120—2781no—
98#97—1391no—
99#97C2×D13912yescomplemented
100C24—2781no—
101C2×Q8—1392no—
102C2×Q8D55611yescharacteristic
103C2×D4—2781no—
104C2×D4—2782no—
105C2×D4—2782no—
106C2×D4—2781no—
107C2×D4—1392no—
108C2×D4D55611yescharacteristic
109SD16—5564no—
110SD16—5562no—
111D8—5564no—
112D8—5562no—
113C2×C8—22no—
114C2×C8—2781no—
115C22⋊C4—22no—
116C22⋊C4—2782no—
117C22⋊C4—2781no—
118C22⋊C4—2782no—
119C22⋊C4—2782no—
120C42—2781no—
121C42C22×D13911yescharacteristic
122C23—21no—
123C23—2782no—
124C23—5561no—
125C23—11122no—
126C23—11121no—
127Q8—41no—
128Q8—5562no—
129D4—41no—
130D4—5562no—
131D4—5562no—
132D4—5561no—
133D4—5562no—
134D4—5561no—
135D4—11121no—
136D4—11122no—
137C2×C4—21no—
138C2×C4—2782no—
139C2×C4D4×D13912yes—
140C2×C4C2×D55611yescharacteristic
141C8—24no—
142C8—2782no—
143C22—42no—
144C22—41no—
145C22—5562no—
146C22—5562no—
147C22—5562no—
148C22—11124no—
149C22—11122no—
150C22—11121no—
151C22#1798481211yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—5562no—
156C2—81no—
157C2—11122no—
158C2—22241no—
159C2#2258112212yes—
160C2#2258112311yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#329×C139C2
socleC22×C139#120
hypercenter#329D139
hypocenterC139#1655
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. #22581695C2semidirect
2. #22581343C2semidirect
3. #22581342C2semidirect
4. #22580692C2semidirect
5. #329×C139C2semidirect
10. #17984839C22semidirect
16. #17984753C22semidirect
18. #97×C139C22semidirect
80. C139#1655semidirect
86. #329D139semidirect
99. #97C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225816952
#225813432
#225813422
#225806922
#329×C1392
#1655139