Finite Group Databasebeta

← #34001241

Subgroups — #34001241

Order 18688.

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

normal subgroups only

This group has 138 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#22625074C211yesmaximal, characteristic, complemented
2#22624938C211yesmaximal, characteristic, complemented
3#22609254C211yesmaximal, characteristic
4#17989498—21no—
5#17989479—21no—
6#17988957C2211yescharacteristic
7#17988946—21no—
8#17988329C411yescharacteristic
9#17988327—21no—
10#562×C73C411yescharacteristic
11#14556665—41no—
12#14556652—41no—
13#14556591D411yescharacteristic
14#14556515—41no—
15#14556514—41no—
16#14556512—41no—
17#14556502—21no—
18#14556502D411yescharacteristic
19#125×C73C2×C411yescharacteristic
20#124×C73—21no—
21#123×C73—41no—
22C22⋊Q8×C73—41no—
23#121×C73—41no—
24#11924037—81no—
25#11924031—42no—
26C2×Dic146—21no—
27C2×#10596831—21no—
28C2×#10596831—41no—
29C4⋊Dic73—41no—
30C2×Q8×C73—21no—
31C2×D4×C73—21no—
32C2×D4×C73—41no—
33C22×C4×C73—41no—
34C4⋊C4×C73—21no—
35C4⋊C4×C73—41no—
36C4⋊C4×C73—41no—
37C22⋊C4×C73—41no—
38C22⋊C4×C73—41no—
39C22⋊C4×C73—41no—
40C42×C73C22⋊C411yescharacteristic
41C23×C73—41no—
42Q8×C73—81no—
43D4×C73—81no—
44D4×C73—161no—
45C2×C4×C73—21no—
46C2×C4×C73—41no—
47C2×C4×C73—41no—
48C2×C4×C73—161no—
49C2×C4×C73#9911yescharacteristic
50C2×Dic73—41no—
51Dic146—81no—
52#10596833—161no—
53#10596831—41no—
54#10596831—81no—
55C22×C73—81no—
56C22×C73—81no—
57C22×C73—81no—
58C22×C73AΣL(1,𝔽16)11yescharacteristic
59C4×C73—21no—
60C4×C73—41no—
61C4×C73—81no—
62C4×C73—81no—
63Dic73—81no—
64#7525—731nomaximal, Sylow
65C2×C73—21no—
66C2×C73—161no—
67C2×C73#140411yescharacteristic
68#1662—731no—
69#1413—731no—
70#1356—731no—
71C73#752511yescharacteristic, complemented, Sylow
72#562#6440811yescharacteristic
73#491—731no—
74#489—1461no—
75#356—1461no—
76#347—1461no—
77#330—731no—
78#329—1461no—
79C4⋊Q8—731no—
80#125C2×#6440811yescharacteristic
81#124—21no—
82#123—41no—
83C22⋊Q8—41no—
84#121—41no—
85M32—2921no—
86#103—2921no—
87#103—2921no—
88#102—2921no—
89#101—2921no—
90#97—731no—
91#97—1461no—
92C2×Q8—21no—
93C2×Q8—1461no—
94C2×D4—21no—
95C2×D4—41no—
96C22×C4—41no—
97M16—5841no—
98C2×C8—2921no—
99C2×C8—1461no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—2921no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42C22⋊#6440811yescharacteristic
108C16—2922no—
109C23—41no—
110Q8—81no—
111Q8—5841no—
112D4—81no—
113D4—161no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—161no—
118C2×C4—2921no—
119C2×C4#1455664911yescharacteristic
120C8—2921no—
121C8—5841no—
122C8—11681no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22#1798948711yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—5841no—
132C2—21no—
133C2—161no—
134C2#2262505411yescharacteristic
subgroupG/H
centerC2#22625054
commutator#125×C73C2×C4
Frattini#125C2×#64408
Fitting, prime radical#562×C73C4
socleC2×C73#1404
cosocle#17988957C22
hypercenter#562#64408
hypocenterC73#7525
LCSG ▷ #125×C73 ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #125 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #22625074C2semidirect
2. #22624938C2semidirect
71. C73#7525semidirect

Maximal subgroups

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

subgroupindex
#226250742
#226249382
#226092542
#752573