Finite Group Databasebeta

← #34000442

Subgroups — #34000442

Order 18688.

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 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#22626303C211yesmaximal, characteristic
2#22625861C211yesmaximal, characteristic
3#22608233C211yesmaximal, characteristic
4#17989510—21no—
5#17989484—21no—
6#17989480C412yescomplemented
7#17989469—21no—
8#17989458—21no—
9#17989456C411yescharacteristic
10#17989455C411yescharacteristic
11C4×#11924011—21no—
12C4×#11924011C2211yescharacteristic
13#17988308—21no—
14#17988308C411yescharacteristic, complemented
15C82×C73C411yescharacteristic, complemented
16D73⋊M16—21no—
17#14556670—21no—
18#14556665—22no—
19C2×#11924011—22no—
20C2×#11924011C2×C411yescharacteristic
21C2×#11924011D411yescharacteristic
22C42×D73D411yescharacteristic
23#14556501—22no—
24#14556501C2×C411yescharacteristic
25#14556501Q811yescharacteristic
26C4×#10596831—21no—
27C4×#10596831D411yescharacteristic
28C4×C8×C73—21no—
29C4×C8×C73C2×C411yescharacteristic
30#11924038—41no—
31#11924037—41no—
32#11924036—41no—
33C4⋊#64408—41no—
34C22⋊#64408—41no—
35C4×#64408—41no—
36#11924031—42no—
37C2×C4×D73—21no—
38C2×C4×D73C22⋊C411yescharacteristic
39#11924011—41no—
40#11924011—41no—
41#11924011—44no—
42C2×#10596831—22no—
43C2×#10596831C22⋊C411yescharacteristic
44C2×#10596831C4⋊C411yescharacteristic
45C4×Dic73—21no—
46C4×Dic73C4⋊C411yescharacteristic
47C2×C8×C73—22no—
48C2×C8×C73C4211yescharacteristic
49C2×C8×C73C4⋊C411yescharacteristic
50C42×C73C22⋊C411yescharacteristic
51C22×D73#10411yescharacteristic
52C2×#64408—81no—
53C2×C4×C73—21no—
54C2×C4×C73#9511yescharacteristic
55C2×Dic73—21no—
56C2×Dic73#10411yescharacteristic
57C4×D73—21no—
58C4×D73—21no—
59C4×D73—42no—
60#10596833—81no—
61C8×C73—24no—
62C8×C73—21no—
63C8×C73—21no—
64#10596831—21no—
65#10596831—21no—
66#10596831—42no—
67C22×C73#33911yescharacteristic
68C2×D73—21no—
69C2×D73—41no—
70#64408—161no—
71C4×C73—22no—
72C4×C73H(3,ℤ/4)11yescharacteristic
73C4×C73#34411yescharacteristic
74Dic73—21no—
75Dic73—41no—
76#7225—731nomaximal, Sylow
77C2×C73—21no—
78C2×C73#128811yescharacteristic
79D73—41no—
80#2165—731no—
81#1841—731no—
82#1455—731no—
83C73#722511yescharacteristic, complemented, Sylow
84C4×M16—731no—
85C4×M16—1461no—
86#364—1461no—
87#349—1461no—
88#347—732no—
89#344—1461no—
90#339—1461no—
91#338—731no—
92H(3,ℤ/4)—731no—
93#322—731no—
94#322—1461no—
95C82#6440811yescharacteristic, complemented
96C2×M16—731no—
97C2×M16—731no—
98C2×M16—1462no—
99C2×M16—1461no—
100#117—1461no—
101C2×C42—731no—
102M32—1462no—
103#97—731no—
104#97—731no—
105#97—1462no—
106C4×C8—21no—
107C4×C8—731no—
108C4×C8—1461no—
109C4×C8C2×#6440811yescharacteristic
110C22×C4—731no—
111C22×C4—1461no—
112M16—2924no—
113M16—2921no—
114M16—2921no—
115M16—2921no—
116M16—2921no—
117C2×C8—22no—
118C2×C8—2921no—
119C2×C8—731no—
120C2×C8—731no—
121C2×C8—1462no—
122C2×C8C4×#6440811yescharacteristic
123C2×C8C4⋊#6440811yescharacteristic
124C4⋊C4—2921no—
125C22⋊C4—2921no—
126C42—2921no—
127C42—731no—
128C42—1461no—
129C42C22⋊#6440811yescharacteristic
130C16—2922no—
131C23—731no—
132C2×C4—21no—
133C2×C4—2922no—
134C2×C4—5841no—
135C2×C4—731no—
136C2×C4—1461no—
137C2×C4—1461no—
138C2×C4—1461no—
139C2×C4(C2×#64408)⋊C411yescharacteristic
140C8—24no—
141C8—21no—
142C8—21no—
143C8—2922no—
144C8—5841no—
145C8—1461no—
146C8—1461no—
147C22—2921no—
148C22—1461no—
149C22#1798945811yescharacteristic
150C4—22no—
151C4—2921no—
152C4—11681no—
153C4—1461no—
154C4#1798945511yescharacteristic
155C4#1798946911yescharacteristic
156C2—21no—
157C2—2921no—
158C2#2262479111yescharacteristic
subgroupG/H
centerC4#17989455
commutatorC2×C8×C73C42
FrattiniC4×C8C2×#64408
FittingC82×C73C4
socleC2×C73#1288
prime radicalC22×D73#104
cosocleC4×#11924011C22
hypercenterC82#64408
hypocenterC73#7225
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C8 ◁ C82

Complements

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

NH ≅ G/Ntype
6. #17989480C4semidirect
14. #17988308C4semidirect
15. C82×C73C4semidirect
83. C73#7225semidirect
95. C82#64408semidirect

Maximal subgroups

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

subgroupindex
#226263032
#226258612
#226082332
#722573