Finite Group Databasebeta

← #31884823

Subgroups — #31884823

Order 18176.

200 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 43 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#22586270C212yesmaximal
2C71×#1820C211yesmaximal, characteristic
3#17985723—22no—
4C2×#117×C71C411yescharacteristic
5#381×C71C411yescharacteristic
6#381×C71C2211yescharacteristic
7#342×C71—22no—
8C22⋊(C71⋊C8)—42no—
9C23×C4×C71C2×C411yescharacteristic
10#117×C71—41no—
11#117×C71—41no—
12C2×C4⋊C4×C71—21no—
13C2×C22⋊C4×C71—21no—
14C2×C22⋊C4×C71—21no—
15C2×C22⋊C4×C71D412yes—
16C2×C42×C71C812yescomplemented
17#95×C71—41no—
18#95×C71—41no—
19C2×C71⋊C8—82no—
20C24×C71C22⋊C411yescharacteristic
21C22×C4×C71—21no—
22C22×C4×C71—21no—
23C22×C4×C71—22no—
24C22×C4×C71—22no—
25C22×C4×C71—42no—
26C22×C4×C71—41no—
27C22×C4×C71C2×C811yescharacteristic
28C22×C4×C71M1611yescharacteristic
29C4⋊C4×C71—42no—
30C22⋊C4×C71—41no—
31C22⋊C4×C71—41no—
32C22⋊C4×C71—42no—
33C22⋊C4×C71—42no—
34C42×C71—42no—
35C23×C71—22no—
36C23×C71—81no—
37C23×C71C22⋊C811yescharacteristic
38C23×C71#9911yescharacteristic
39C23×C71#10011yescharacteristic
40C2×C4×C71—21no—
41C2×C4×C71—21no—
42C2×C4×C71—42no—
43C2×C4×C71—42no—
44C2×C4×C71—41no—
45C2×C4×C71—41no—
46C2×C4×C71—42no—
47C2×C4×C71—42no—
48C2×C4×C71—82no—
49C2×C4×C71—82no—
50C2×C4×C71—81no—
51C2×C4×C71—82no—
52C71⋊C8—162no—
53C22×C71—21no—
54C22×C71—21no—
55C22×C71—41no—
56C22×C71—41no—
57C22×C71—41no—
58C22×C71—82no—
59C22×C71#32311yescharacteristic
60C22×C71#35212yes—
61C4×C71—41no—
62C4×C71—41no—
63C4×C71—82no—
64C4×C71—82no—
65C4×C71—161no—
66#7348—711nomaximal, Sylow
67C2×C71—21no—
68C2×C71—22no—
69C2×C71—41no—
70C2×C71—41no—
71C2×C71#131711yescharacteristic
72#1820D7111yescharacteristic
73#1317—712no—
74C71#734811yescharacteristic, complemented, Sylow
75C2×#117Dic7111yescharacteristic
76#381Dic7111yescharacteristic
77#381C2×D7111yescharacteristic
78#342—22no—
79#323—1422no—
80C23×C4C2×Dic7111yescharacteristic
81#117—41no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4C22⋊D7112yes—
87C2×C42C71⋊C812yescomplemented
88C22⋊C8—2842no—
89#95—41no—
90#95—41no—
91C24C22⋊Dic7111yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—42no—
97C22×C4—41no—
98C22×C4C2×C71⋊C811yescharacteristic
99C22×C4#1176584911yescharacteristic
100C2×C8—5682no—
101C4⋊C4—42no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C42—42no—
107C23—22no—
108C23—81no—
109C23C22⋊(C71⋊C8)11yescharacteristic
110C23#1455604311yescharacteristic
111C23#1455604511yescharacteristic
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—82no—
121C2×C4—82no—
122C2×C4—81no—
123C2×C4—82no—
124C8—11362no—
125C22—21no—
126C22—21no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—82no—
131C22#1798572311yescharacteristic
132C22#1798578912yes—
133C4—41no—
134C4—41no—
135C4—82no—
136C4—82no—
137C4—161no—
138C2—21no—
139C2—22no—
140C2—41no—
141C2—41no—
142C2#2258627011yescharacteristic
subgroupG/H
centerC2#22586270
commutatorC22×C4×C71C2×C8
Frattini#381C2×D71
FittingC71×#1820C2
socleC2×C71#1317
prime radicalC24×C71C22⋊C4
cosocle#381×C71C22
hypercenter#1820D71
hypocenterC71#7348
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1820

Complements

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

NH ≅ G/Ntype
16. C2×C42×C71C8semidirect
74. C71#7348semidirect
87. C2×C42C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225862702
C71×#18202
#734871