Finite Group Databasebeta

← #34001228

Subgroups — #34001228

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#22625068C211yesmaximal, characteristic, complemented
2#22624940C211yesmaximal, characteristic, complemented
3#22609076C211yesmaximal, characteristic
4#17989510—21no—
5#17989498—21no—
6#17988966C2211yescharacteristic
7#17988877—21no—
8#17988336C411yescharacteristic
9D4⋊#10596831—21no—
10#555×C73C411yescharacteristic
11#14556665—41no—
12#14556652—41no—
13C2×#11924018—41no—
14#14556591D411yescharacteristic
15#14556522—41no—
16C4⋊#10596831—21no—
17#14556512—41no—
18C4×#10596831D411yescharacteristic
19C4⋊Q8×C73C2×C411yescharacteristic
20#124×C73—41no—
21#121×C73—41no—
22C4×D4×C73—21no—
23#117×C73—41no—
24#11924037—81no—
25#11924031—81no—
26C2×Dic146—21no—
27#11924018—81no—
28#11924018—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—81no—
44D4×C73—41no—
45D4×C73—41no—
46D4×C73—161no—
47C2×C4×C73—21no—
48C2×C4×C73—41no—
49C2×C4×C73—41no—
50C2×C4×C73—81no—
51C2×C4×C73—81no—
52C2×C4×C73#9911yescharacteristic
53C2×Dic73—41no—
54Dic146—81no—
55#10596833—161no—
56#10596831—21no—
57#10596831—21no—
58#10596831—81no—
59C22×C73—81no—
60C22×C73—81no—
61C22×C73—81no—
62C22×C73AΣL(1,𝔽16)11yescharacteristic
63C4×C73—21no—
64C4×C73—21no—
65C4×C73—21no—
66C4×C73—81no—
67C4×C73—81no—
68Dic73—81no—
69#7521—731nomaximal, Sylow
70C2×C73—21no—
71C2×C73—161no—
72C2×C73#140211yescharacteristic
73#1634—731no—
74#1411—731no—
75#1358—731no—
76C73#752111yescharacteristic, complemented, Sylow
77#555#6440811yescharacteristic
78#494—731no—
79#476—1461no—
80#364—1461no—
81#356—1461no—
82#333—731no—
83D4⋊C8—1461no—
84C2×Q16—2921no—
85C4⋊Q8—731no—
86C4⋊Q8C2×#6440811yescharacteristic
87#124—41no—
88#121—41no—
89C4×D4—21no—
90#117—41no—
91M32—2921no—
92#106—2921no—
93C4⋊C8—1461no—
94#102—2921no—
95#101—2921no—
96C4×C8—731no—
97C2×Q8—21no—
98C2×Q8—1461no—
99C2×D4—21no—
100C2×D4—41no—
101C22×C4—41no—
102Q16—5841no—
103Q16—5841no—
104M16—5841no—
105C2×C8—2921no—
106C2×C8—1461no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—2921no—
110C22⋊C4—42no—
111C22⋊C4—41no—
112C42—41no—
113C42C22⋊#6440811yescharacteristic
114C16—5841no—
115C23—41no—
116Q8—81no—
117Q8—5841no—
118D4—41no—
119D4—41no—
120D4—161no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—81no—
125C2×C4—81no—
126C2×C4—2921no—
127C2×C4#1455664911yescharacteristic
128C8—5841no—
129C8—11681no—
130C8—1461no—
131C8—1461no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22#1798948711yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—81no—
140C4—81no—
141C4—5841no—
142C2—21no—
143C2—161no—
144C2#2262505011yescharacteristic
subgroupG/H
centerC2#22625050
commutatorC4⋊Q8×C73C2×C4
FrattiniC4⋊Q8C2×#64408
Fitting, prime radical#555×C73C4
socleC2×C73#1402
cosocle#17988966C22
hypercenter#555#64408
hypocenterC73#7521
LCSG ▷ C4⋊Q8×C73 ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4⋊Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22625068C2semidirect
2. #22624940C2semidirect
76. C73#7521semidirect

Maximal subgroups

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

subgroupindex
#226250682
#226249402
#226090762
#752173