Finite Group Databasebeta

← #34000932

Subgroups — #34000932

Order 18688.

166 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 32 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#22625019C211yesmaximal, characteristic
2#22624866C211yesmaximal, characteristic
3#22611377C211yesmaximal, characteristic
4#17989566—21no—
5#17989483—21no—
6#17988586—21no—
7#17988553C2211yescharacteristic
8#17988356—21no—
9#17988352C411yescharacteristic, complemented
10C2×#106×C73C411yescharacteristic, complemented
11#14556670—41no—
12#14556665—41no—
13#14556650—41no—
14#14556649—22no—
15C2×#11924028D411yescharacteristic
16C2×#11924012—21no—
17C2×#11924012D411yescharacteristic
18#14556514—41no—
19#14556511—41no—
20#14556503—41no—
21C22×C8×C73C2×C411yescharacteristic
22C2×C4⋊C4×C73—21no—
23#106×C73—41no—
24C4⋊#64408—81no—
25C22⋊#64408—42no—
26C4×#64408—81no—
27#11924031—81no—
28#11924028—41no—
29#11924028—41no—
30C2×Dic146SD1611yescharacteristic
31C2×D292D811yescharacteristic
32C2×C22⋊D73—21no—
33C2×C4×D73—21no—
34#11924012—41no—
35#11924012—41no—
36#11924012—41no—
37C2×#10596831—21no—
38C2×#10596831—41no—
39C22×C4×C73—41no—
40C22×C4×C73C22⋊C411yescharacteristic
41C2×C8×C73—21no—
42C2×C8×C73—41no—
43C4⋊C4×C73—41no—
44C4⋊C4×C73—41no—
45C23×C73#10411yescharacteristic
46C22×D73—21no—
47C2×#64408—82no—
48C2×C4×C73—21no—
49C2×C4×C73—21no—
50C2×C4×C73—81no—
51C2×C4×C73—161no—
52C2×C4×C73#9911yescharacteristic
53C2×C4×C73#10211yescharacteristic
54C22⋊D73—81no—
55C2×Dic73—21no—
56D292—41no—
57C4×D73—41no—
58C4×D73—41no—
59Dic146—41no—
60C8×C73—41no—
61#10596831—41no—
62#10596831—81no—
63C22×C73—21no—
64C22×C73—41no—
65C22×C73#32711yescharacteristic
66C2×D73—41no—
67C2×D73—81no—
68#64408—162no—
69C4×C73—21no—
70C4×C73—161no—
71C4×C73#35211yescharacteristic
72C4×C73#35411yescharacteristic
73Dic73—41no—
74#7410—731nomaximal, Sylow
75C2×C73—21no—
76C2×C73—41no—
77C2×C73#134411yescharacteristic
78D73—81no—
79#2079—731no—
80#1391—731no—
81#1321—731no—
82C73#741011yescharacteristic, complemented, Sylow
83C2×#106#6440811yescharacteristic, complemented
84#417—1461no—
85#410—1461no—
86#408—731no—
87#349—1461no—
88#343—1461no—
89#341—731no—
90C2×#41—731no—
91C2×M16—731no—
92C2×M16—1461no—
93C22×C8C2×#6440811yescharacteristic
94#117—2921no—
95C2×C4⋊C4—21no—
96M32—2921no—
97#106—41no—
98#103—2921no—
99#102—2921no—
100#99—2921no—
101#99—1462no—
102C22⋊C8—2921no—
103#41—2921no—
104#41—2921no—
105C2×Q8—731no—
106C2×D4—731no—
107C2×D4—1461no—
108C22×C4—41no—
109C22×C4—1461no—
110C22×C4C22⋊#6440811yescharacteristic
111M16—2921no—
112M16—2921no—
113M16—2921no—
114C2×C8—21no—
115C2×C8—41no—
116C2×C8—2921no—
117C2×C8—1461no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—5841no—
121C22⋊C4—2922no—
122C42—5841no—
123C16—5841no—
124C23—1461no—
125C23#1455665511yescharacteristic
126Q8—2921no—
127D4—2921no—
128D4—5841no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—81no—
132C2×C4—161no—
133C2×C4—2921no—
134C2×C4—2921no—
135C2×C4—5842no—
136C2×C4—1461no—
137C2×C4#1455664911yescharacteristic
138C2×C4#1455665311yescharacteristic
139C8—41no—
140C8—2921no—
141C8—5841no—
142C22—21no—
143C22—41no—
144C22—2921no—
145C22—5841no—
146C22#1798943811yescharacteristic
147C4—21no—
148C4—161no—
149C4—2921no—
150C4—11682no—
151C4#1798948911yescharacteristic
152C4#1798949311yescharacteristic
153C2—21no—
154C2—41no—
155C2—5841no—
156C2#2262491111yescharacteristic
subgroupG/H
centerC2#22624911
commutatorC22×C8×C73C2×C4
FrattiniC22×C8C2×#64408
FittingC2×#106×C73C4
socleC2×C73#1344
prime radicalC2×#11924028D4
cosocle#17988553C22
hypercenterC2×#106#64408
hypocenterC73#7410
LCSG ▷ C22×C8×C73 ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
9. #17988352C4semidirect
10. C2×#106×C73C4semidirect
82. C73#7410semidirect
83. C2×#106#64408semidirect

Maximal subgroups

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

subgroupindex
#226250192
#226248662
#226113772
#741073