Finite Group Databasebeta

← #36323444

Subgroups — #36323444

Order 21376.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#24923664C211yesmaximal, characteristic, complemented
2#24923628C212yesmaximal, complemented
3#24923009C211yesmaximal, characteristic, complemented
4#24923009C212yesmaximal, complemented
5H(3,𝔽4)×C167C211yesmaximal, characteristic
6C2×#14581940—21no—
7C2×#14581938—21no—
8#19128832C2211yescharacteristic, complemented
9#19128756—22no—
10#19128753—22no—
11#19128743C2211yescharacteristic, complemented
12#19128743C2212yescomplemented
13#124×C167—21no—
14#124×C167—22no—
15#124×C167C2211yescharacteristic
16#124×C167C2212yes—
17#120×C167—21no—
18#120×C167—22no—
19C2×Dic334—21no—
20#14581940—42no—
21#14581938—42no—
22C2×C167⋊C8—21no—
23C2×C167⋊C8—22no—
24C4⋊Dic167—22no—
25C24×C167—21no—
26C2×Q8×C167D411yescharacteristic
27C2×Q8×C167D412yes—
28C2×D4×C167—21no—
29C2×D4×C167—22no—
30C2×D4×C167D411yescharacteristic
31C2×D4×C167D412yes—
32C22⋊C4×C167—22no—
33C22⋊C4×C167—21no—
34C22⋊C4×C167—22no—
35C22⋊C4×C167—22no—
36C22⋊C4×C167—22no—
37C42×C167—21no—
38C42×C167C2311yescharacteristic
39Dic334—41no—
40Dic334—41no—
41C2×Dic167—41no—
42C167⋊C8—22no—
43C167⋊C8—42no—
44C23×C167—21no—
45C23×C167—22no—
46C23×C167—81no—
47C23×C167—82no—
48Q8×C167—42no—
49Q8×C167—41no—
50D4×C167—42no—
51D4×C167—41no—
52D4×C167—81no—
53D4×C167—82no—
54C2×C4×C167—21no—
55C2×C4×C167—22no—
56C2×C4×C167C2×D412yes—
57C2×C4×C167C2×D411yescharacteristic
58C22×C167—41no—
59C22×C167—42no—
60C22×C167—42no—
61C22×C167—42no—
62C22×C167—42no—
63C22×C167—84no—
64C22×C167#12011yescharacteristic
65C4×C167—21no—
66C4×C167—22no—
67C4×C167—41no—
68C4×C167—42no—
69Dic167—81no—
70C2×C167—81no—
71C2×C167—82no—
72C2×C167#45311yescharacteristic
73C2×C167#45512yes—
74C167#165911yescharacteristic, complemented, Sylow
75#1659—1671nomaximal, Sylow
76H(3,𝔽4)D16711yescharacteristic
77#497—1671no—
78#489—1672no—
79#329—1671no—
80#329—1672no—
81C2×Q16—3341no—
82C2×SD16—3341no—
83C4⋊Q8—1671no—
84#124—22no—
85#124—21no—
86#124C2×D16712yes—
87#124C2×D16711yescharacteristic
88#120—22no—
89#120—21no—
90#103—3342no—
91#102—3342no—
92#97—1672no—
93#97—1671no—
94C24—21no—
95C2×Q8—3341no—
96C2×Q8C22⋊D16712yes—
97C2×Q8C22⋊D16711yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D16712yes—
101C2×D4C22⋊D16711yescharacteristic
102Q16—6682no—
103SD16—6682no—
104C2×C8—3341no—
105C2×C8—3342no—
106C4⋊C4—3342no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D16711yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—6681no—
121Q8—6681no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—6681no—
129C2×C4C2×C22⋊D16712yes—
130C2×C4C2×C22⋊D16711yescharacteristic
131C8—3342no—
132C8—6682no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1912880111yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—13361no—
145C2—81no—
146C2—82no—
147C2#2492343711yescharacteristic
148C2#2492344712yes—
subgroupG/H
centerC22#19128801
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting, prime radicalH(3,𝔽4)×C167C2
socleC22×C167#120
hypercenterH(3,𝔽4)D167
hypocenterC167#1659
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #24923664C2semidirect
2. #24923628C2semidirect
3. #24923009C2semidirect
4. #24923009C2semidirect
8. #19128832C22semidirect
11. #19128743C22semidirect
12. #19128743C22semidirect
74. C167#1659semidirect

Maximal subgroups

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

subgroupindex
#249236642
#249236282
#249230092
H(3,𝔽4)×C1672
#1659167