Finite Group Databasebeta

← #34001222

Subgroups — #34001222

Order 18688.

148 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 22 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#22625066C211yesmaximal, characteristic
2#22624941C211yesmaximal, characteristic
3#22609052C211yesmaximal, characteristic
4#17989510—21no—
5#17989498—21no—
6#17988959C2211yescharacteristic
7#17988882—21no—
8#17988331C411yescharacteristic
9Q8⋊#10596831—21no—
10#559×C73C411yescharacteristic
11#14556665—41no—
12#14556652—41no—
13C2×#11924016—41no—
14#14556591D411yescharacteristic
15#14556524—41no—
16C4⋊#10596831—21no—
17#14556515—41no—
18C4×#10596831D411yescharacteristic
19#127×C73C2×C411yescharacteristic
20#124×C73—41no—
21#123×C73—41no—
22C4×Q8×C73—21no—
23C4×D4×C73—41no—
24#11924037—81no—
25#11924031—81no—
26C2×Dic146—21no—
27#11924016—81no—
28#11924016—81no—
29C2×#10596831—21no—
30C2×#10596831—41no—
31C4⋊Dic73—41no—
32C2×Q8×C73—21no—
33C2×D4×C73—21no—
34C2×D4×C73—41no—
35C22×C4×C73—41no—
36C4⋊C4×C73—21no—
37C4⋊C4×C73—41no—
38C22⋊C4×C73—42no—
39C22⋊C4×C73—41no—
40C42×C73—41no—
41C42×C73C22⋊C411yescharacteristic
42C23×C73—41no—
43Q8×C73—41no—
44Q8×C73—41no—
45D4×C73—81no—
46D4×C73—81no—
47D4×C73—81no—
48C2×C4×C73—21no—
49C2×C4×C73—41no—
50C2×C4×C73—41no—
51C2×C4×C73—161no—
52C2×C4×C73#9911yescharacteristic
53C2×Dic73—41no—
54Dic146—81no—
55#10596833—161no—
56#10596831—21no—
57#10596831—21no—
58#10596831—81no—
59C22×C73—82no—
60C22×C73—81no—
61C22×C73AΣL(1,𝔽16)11yescharacteristic
62C4×C73—21no—
63C4×C73—21no—
64C4×C73—21no—
65C4×C73—81no—
66C4×C73—81no—
67Dic73—81no—
68#7518—731nomaximal, Sylow
69C2×C73—21no—
70C2×C73—161no—
71C2×C73#140211yescharacteristic
72#1630—731no—
73#1410—731no—
74#1359—731no—
75C73#751811yescharacteristic, complemented, Sylow
76#559#6440811yescharacteristic
77#492—731no—
78#477—1461no—
79#364—1461no—
80#356—1461no—
81#331—731no—
82Q8⋊C8—1461no—
83C2×SD16—2921no—
84C4⋊Q8—731no—
85#127C2×#6440811yescharacteristic
86#124—41no—
87#123—41no—
88C4×Q8—21no—
89C4×D4—41no—
90M32—2921no—
91#107—2921no—
92C4⋊C8—1461no—
93#103—2921no—
94#101—2921no—
95C4×C8—731no—
96C2×Q8—21no—
97C2×Q8—1461no—
98C2×D4—21no—
99C2×D4—41no—
100C22×C4—41no—
101SD16—5841no—
102SD16—5841no—
103M16—5841no—
104C2×C8—2921no—
105C2×C8—1461no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—2921no—
109C22⋊C4—42no—
110C22⋊C4—41no—
111C42—41no—
112C42C22⋊#6440811yescharacteristic
113C16—5841no—
114C23—41no—
115Q8—41no—
116Q8—41no—
117Q8—5841no—
118D4—81no—
119D4—81no—
120D4—81no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—161no—
125C2×C4—2921no—
126C2×C4#1455664911yescharacteristic
127C8—5841no—
128C8—11681no—
129C8—1461no—
130C8—1461no—
131C22—82no—
132C22—81no—
133C22#1798948711yescharacteristic
134C4—21no—
135C4—21no—
136C4—21no—
137C4—81no—
138C4—81no—
139C4—5841no—
140C2—21no—
141C2—161no—
142C2#2262505011yescharacteristic
subgroupG/H
centerC2#22625050
commutator, prime radical#127×C73C2×C4
Frattini#127C2×#64408
Fitting#559×C73C4
socleC2×C73#1402
cosocle#17988959C22
hypercenter#559#64408
hypocenterC73#7518
LCSG ▷ #127×C73 ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #127 ◁ #559

Complements

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

NH ≅ G/Ntype
75. C73#7518semidirect

Maximal subgroups

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

subgroupindex
#226250662
#226249412
#226090522
#751873