Finite Group Databasebeta

← #31885303

Subgroups — #31885303

Order 18176.

164 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 25 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#22588004C211yesmaximal, characteristic
2#22586507C211yesmaximal, characteristic, complemented
3C71×#1409C211yesmaximal, characteristic, complemented
4#17986376C411yescharacteristic, complemented
5#17986375—21no—
6#17986347—21no—
7#17985798—21no—
8#17985720C411yescharacteristic, complemented
9#492×C71C2211yescharacteristic
10#353×C71—21no—
11C2×D568—41no—
12C2×#11765850—42no—
13#14556125—21no—
14#14556048—42no—
15#14556047—41no—
16C4×C71⋊C8—21no—
17C2×SD16×C71—41no—
18C4⋊Q8×C71D411yescharacteristic
19#127×C71D411yescharacteristic
20#99×C71—41no—
21C4×C8×C71C2×C411yescharacteristic
22C2×D284—41no—
23C2×D284—41no—
24C2×D284—41no—
25D568—82no—
26#11765850—84no—
27#11765849—81no—
28C2×C71⋊C8—22no—
29C2×Q8×C71—21no—
30C2×D4×C71—21no—
31C2×D4×C71—41no—
32SD16×C71—82no—
33C2×C8×C71—21no—
34C4⋊C4×C71—41no—
35C22⋊C4×C71—81no—
36C42×C71C22⋊C411yescharacteristic
37C23×C71—41no—
38C22×D71—81no—
39Q8×C71—81no—
40D4×C71—81no—
41D4×C71—82no—
42C2×C4×C71—21no—
43C2×C4×C71—41no—
44C2×C4×C71—161no—
45C2×C4×C71#9911yescharacteristic
46D284—82no—
47D284—82no—
48D284—82no—
49C8×C71—22no—
50C71⋊C8—42no—
51C71⋊C8—161no—
52C22×C71—81no—
53C22×C71—161no—
54C22×C71#35311yescharacteristic
55C2×D71—161no—
56C2×D71—161no—
57C2×D71—161no—
58C4×C71—22no—
59C4×C71—21no—
60C4×C71—81no—
61C4×C71—321no—
62#7527—711nomaximal, Sylow
63C2×C71—21no—
64C2×C71—161no—
65C2×C71#140811yescharacteristic
66D71—321no—
67#1721—711no—
68#1410—711no—
69#1409D7111yescharacteristic, complemented
70C71#752711yescharacteristic, complemented, Sylow
71#493—711no—
72#493—1421no—
73#492C2×D7111yescharacteristic
74#486—1421no—
75#356—1421no—
76#353—21no—
77#322—711no—
78C2×SD16—41no—
79C2×D8—2841no—
80C2×D8—2842no—
81C4⋊Q8D28411yescharacteristic
82#127—1421no—
83#127C22⋊D7111yescharacteristic
84#102—2842no—
85#101—2841no—
86#99—41no—
87C4×C8—1421no—
88C4×C8C4×D7111yescharacteristic
89C2×Q8—21no—
90C2×D4—21no—
91C2×D4—41no—
92C2×D4—2841no—
93C2×D4—2841no—
94C2×D4—2841no—
95SD16—82no—
96D8—5682no—
97D8—5684no—
98M16—5681no—
99C2×C8—21no—
100C2×C8—1422no—
101C4⋊C4—41no—
102C22⋊C4—81no—
103C42(C2×D71)⋊C411yescharacteristic
104C23—41no—
105C23—5681no—
106Q8—81no—
107D4—81no—
108D4—82no—
109D4—5682no—
110D4—5682no—
111D4—5682no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—161no—
115C2×C4#1455604211yescharacteristic
116C8—22no—
117C8—2842no—
118C8—11361no—
119C22—81no—
120C22—161no—
121C22—11361no—
122C22—11361no—
123C22—11361no—
124C22#1798579111yescharacteristic
125C4—22no—
126C4—21no—
127C4—81no—
128C4—321no—
129C2—21no—
130C2—161no—
131C2—22721no—
132C2#2258650211yescharacteristic
subgroupG/H
centerC2#22586502
commutatorC4×C8×C71C2×C4
Frattini#492C2×D71
FittingC71×#1409C2
socleC2×C71#1408
prime radical#22588004C2
cosocle#492×C71C22
hypercenter#1409D71
hypocenterC71#7527
LCSG ▷ C4×C8×C71 ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1409

Complements

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

NH ≅ G/Ntype
2. #22586507C2semidirect
3. C71×#1409C2semidirect
4. #17986376C4semidirect
8. #17985720C4semidirect
69. #1409D71semidirect
70. C71#7527semidirect

Maximal subgroups

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

subgroupindex
#225880042
#225865072
C71×#14092
#752771