Finite Group Databasebeta

← #34001218

Subgroups — #34001218

Order 18688.

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

normal subgroups only

This group has 162 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#22625062C211yesmaximal, characteristic
2#22624939C211yesmaximal, characteristic
3#22608990C211yesmaximal, characteristic
4#17989510—21no—
5#17989492—21no—
6#17988963C2211yescharacteristic
7#17988787—21no—
8#17988331C411yescharacteristic, complemented
9Q8⋊#10596831—21no—
10#559×C73C411yescharacteristic, complemented
11#14556665—41no—
12#14556650—41no—
13C2×#11924015—41no—
14C2×#11924013—41no—
15#14556588D411yescharacteristic
16C4⋊#10596831—21no—
17#14556511—41no—
18C4×#10596831D411yescharacteristic
19#127×C73C2×C411yescharacteristic
20#124×C73—41no—
21#123×C73—41no—
22C4×Q8×C73—21no—
23C4×D4×C73—41no—
24C22⋊#64408—81no—
25#11924031—81no—
26C2×D292—21no—
27C2×D292—41no—
28#11924015—82no—
29#11924013—81no—
30#11924013—81no—
31C2×#10596831—21no—
32C2×#10596831—41no—
33C2×Q8×C73—21no—
34C2×D4×C73—21no—
35C2×D4×C73—41no—
36C22×C4×C73—41no—
37C4⋊C4×C73—21no—
38C4⋊C4×C73—41no—
39C22⋊C4×C73—42no—
40C22⋊C4×C73—41no—
41C42×C73—41no—
42C42×C73C22⋊C411yescharacteristic
43C23×C73—41no—
44C22×D73—41no—
45C2×#64408—161no—
46Q8×C73—41no—
47Q8×C73—41no—
48D4×C73—81no—
49D4×C73—81no—
50D4×C73—81no—
51C2×C4×C73—21no—
52C2×C4×C73—41no—
53C2×C4×C73—41no—
54C2×C4×C73—161no—
55C2×C4×C73#9911yescharacteristic
56D292—81no—
57D292—81no—
58D292—81no—
59#10596831—21no—
60#10596831—21no—
61#10596831—81no—
62C22×C73—82no—
63C22×C73—81no—
64C22×C73AΣL(1,𝔽16)11yescharacteristic
65C2×D73—81no—
66C2×D73—161no—
67#64408—321no—
68C4×C73—21no—
69C4×C73—21no—
70C4×C73—21no—
71C4×C73—81no—
72C4×C73—81no—
73#7516—731nomaximal, Sylow
74C2×C73—21no—
75C2×C73—161no—
76C2×C73#140211yescharacteristic
77D73—161no—
78#1620—731no—
79#1408—731no—
80#1357—731no—
81C73#751611yescharacteristic, complemented, Sylow
82#559#6440811yescharacteristic, complemented
83#493—731no—
84#460—1461no—
85#364—1461no—
86#353—1461no—
87#331—731no—
88Q8⋊C8—1461no—
89C2×SD16—2921no—
90C2×D8—2921no—
91#127—731no—
92#127C2×#6440811yescharacteristic
93#124—41no—
94#123—41no—
95C4×Q8—21no—
96C4×D4—41no—
97M32—2921no—
98C4⋊C8—1461no—
99#102—2921no—
100#99—2921no—
101C4×C8—731no—
102C2×Q8—21no—
103C2×D4—21no—
104C2×D4—41no—
105C2×D4—2921no—
106C2×D4—1461no—
107C22×C4—41no—
108SD16—5842no—
109D8—5841no—
110D8—5841no—
111C2×C8—2921no—
112C2×C8—1461no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—5841no—
118C42—41no—
119C42C22⋊#6440811yescharacteristic
120C16—5841no—
121C23—41no—
122C23—2921no—
123Q8—41no—
124Q8—41no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—5841no—
129D4—5841no—
130D4—5841no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—161no—
135C2×C4—11681no—
136C2×C4#1455664911yescharacteristic
137C8—5841no—
138C8—1461no—
139C8—1461no—
140C22—82no—
141C22—81no—
142C22—5841no—
143C22—11681no—
144C22#1798948711yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—81no—
149C4—81no—
150C4—23361no—
151C2—21no—
152C2—161no—
153C2—11681no—
154C2#2262505011yescharacteristic
subgroupG/H
centerC2#22625050
commutator#127×C73C2×C4
Frattini#127C2×#64408
Fitting#559×C73C4
socleC2×C73#1402
prime radical, cosocle#17988963C22
hypercenter#559#64408
hypocenterC73#7516
LCSG ▷ #127×C73 ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #127 ◁ #559

Complements

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

NH ≅ G/Ntype
8. #17988331C4semidirect
10. #559×C73C4semidirect
81. C73#7516semidirect
82. #559#64408semidirect

Maximal subgroups

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

subgroupindex
#226250622
#226249392
#226089902
#751673