Finite Group Databasebeta

← #34001000

Subgroups — #34001000

Order 18688.

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

normal subgroups only

This group has 174 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#22625020C211yesmaximal, characteristic
2#22624854C211yesmaximal, characteristic
3#22611483C211yesmaximal, characteristic
4#17989566—21no—
5C22⋊#11924031—21no—
6#17988615—21no—
7#17988603—21no—
8#17988555C2211yescharacteristic
9#17988344C411yescharacteristic, complemented
10#430×C73C411yescharacteristic, complemented
11#14556670—41no—
12C2×#11924031—41no—
13#14556650—41no—
14#14556649—22no—
15C2×#11924028D411yescharacteristic
16C22×#10596831D411yescharacteristic
17#14556550—21no—
18#14556520—42no—
19#14556517—42no—
20#14556503—41no—
21C2×M16×C73—21no—
22C2×M16×C73C2×C411yescharacteristic
23#108×C73—41no—
24C4⋊#64408—81no—
25C22⋊#64408—42no—
26C4×#64408—81no—
27#11924031—81no—
28#11924028—41no—
29#11924028—41no—
30C2×Dic146Q1611yescharacteristic
31C2×D292SD1611yescharacteristic
32C2×C22⋊D73—21no—
33C2×C4×D73—21no—
34C2×#10596831—21no—
35C2×#10596831—41no—
36C4⋊Dic73—41no—
37C22⋊Dic73—41no—
38C4×Dic73—22no—
39C22×C4×C73C22⋊C411yescharacteristic
40M16×C73—41no—
41M16×C73—41no—
42M16×C73—41no—
43C2×C8×C73—21no—
44C2×C8×C73—41no—
45C23×C73#10411yescharacteristic
46C22×D73—21no—
47C2×#64408—82no—
48C2×C4×C73—21no—
49C2×C4×C73—21no—
50C2×C4×C73#9911yescharacteristic
51C2×C4×C73#10311yescharacteristic
52C22⋊D73—81no—
53C2×Dic73—21no—
54C2×Dic73—42no—
55D292—41no—
56C4×D73—41no—
57C4×D73—41no—
58Dic146—41no—
59C8×C73—41no—
60C8×C73—81no—
61#10596831—41no—
62C22×C73—21no—
63C22×C73—41no—
64C22×C73#32811yescharacteristic
65C2×D73—41no—
66C2×D73—81no—
67#64408—162no—
68C4×C73—21no—
69C4×C73#35211yescharacteristic
70C4×C73#35311yescharacteristic
71Dic73—41no—
72Dic73—82no—
73#7438—731nomaximal, Sylow
74C2×C73—21no—
75C2×C73—41no—
76C2×C73#135211yescharacteristic
77D73—81no—
78#2098—731no—
79#1391—731no—
80#1315—731no—
81C73#743811yescharacteristic, complemented, Sylow
82#430#6440811yescharacteristic, complemented
83#424—1461no—
84#421—1461no—
85#410—1461no—
86#408—731no—
87C22⋊C16—1461no—
88H(3,ℤ/4)—731no—
89C2×#41—731no—
90C2×M16—21no—
91C2×M16C2×#6440811yescharacteristic
92C22×C8—731no—
93#117—2921no—
94#117—1461no—
95C2×C16—2921no—
96#108—41no—
97C4⋊C8—2922no—
98#104—2922no—
99#99—2921no—
100#99—1462no—
101C22⋊C8—2921no—
102#41—2921no—
103#41—2921no—
104C2×Q8—731no—
105C2×D4—731no—
106C2×D4—1461no—
107C22×C4—1461no—
108C22×C4C22⋊#6440811yescharacteristic
109M16—41no—
110M16—41no—
111M16—41no—
112C2×C8—21no—
113C2×C8—41no—
114C2×C8—2921no—
115C2×C8—1461no—
116C4⋊C4—2921no—
117C4⋊C4—5841no—
118C22⋊C4—2921no—
119C22⋊C4—2922no—
120C42—5841no—
121C42—1462no—
122C16—5841no—
123C23—1461no—
124C23#1455665611yescharacteristic
125Q8—2921no—
126D4—2921no—
127D4—5841no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—2921no—
131C2×C4—2921no—
132C2×C4—2922no—
133C2×C4—5842no—
134C2×C4—1461no—
135C2×C4#1455664911yescharacteristic
136C2×C4#1455665411yescharacteristic
137C8—41no—
138C8—81no—
139C8—2921no—
140C22—21no—
141C22—41no—
142C22—2921no—
143C22—5841no—
144C22#1798944111yescharacteristic
145C4—21no—
146C4—2921no—
147C4—5842no—
148C4—11682no—
149C4#1798948911yescharacteristic
150C4#1798949111yescharacteristic
151C2—21no—
152C2—41no—
153C2—5841no—
154C2#2262492811yescharacteristic
subgroupG/H
centerC4#17989491
commutatorC2×M16×C73C2×C4
FrattiniC2×M16C2×#64408
Fitting#430×C73C4
socleC2×C73#1352
prime radicalC2×#11924028D4
cosocle#17988555C22
hypercenter#430#64408
hypocenterC73#7438
LCSG ▷ C2×M16×C73 ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16 ◁ #430

Complements

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

NH ≅ G/Ntype
9. #17988344C4semidirect
10. #430×C73C4semidirect
81. C73#7438semidirect
82. #430#64408semidirect

Maximal subgroups

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

subgroupindex
#226250202
#226248542
#226114832
#743873