Finite Group Databasebeta

← #36323393

Subgroups — #36323393

Order 21376.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#24923992C211yesmaximal, characteristic
2#24923672C211yesmaximal, characteristic, complemented
3#24923471C211yesmaximal, characteristic
4#24923014C211yesmaximal, characteristic, complemented
5Q8⋊(C167⋊C8)C211yesmaximal, characteristic, complemented
6#24922996C211yesmaximal, characteristic
7#475×C167C211yesmaximal, characteristic, complemented
8C2×#14581937—21no—
9#19128832C2211yescharacteristic
10#19128827—22no—
11#19128820—21no—
12#19128808—21no—
13C4×Dic334—21no—
14C4×D668C2211yescharacteristic
15#19128790—21no—
16#19128765—22no—
17C4⋊(C167⋊C8)C2211yescharacteristic
18#19128756—21no—
19C4×C167⋊C8C2211yescharacteristic
20C4⋊Q8×C167C2211yescharacteristic
21C4×Q8×C167C2211yescharacteristic
22#106×C167—21no—
23C4⋊C8×C167C2211yescharacteristic
24#103×C167—21no—
25C2×Dic334—21no—
26C2×D668D411yescharacteristic
27C2×C4×D167—21no—
28#14581937—42no—
29C2×C167⋊C8—21no—
30C2×C167⋊C8—21no—
31C4⋊Dic167—21no—
32C4⋊Dic167D412yes—
33C4⋊Dic167D411yescharacteristic
34Dic167⋊C4—22no—
35(C2×D167)⋊C4—22no—
36(C2×D167)⋊C4—22no—
37(C2×D167)⋊C4—21no—
38C4×Dic167—22no—
39C2×Q8×C167—21no—
40C2×Q8×C167D411yescharacteristic
41C2×C8×C167—21no—
42C4⋊C4×C167—21no—
43C4⋊C4×C167—21no—
44C4⋊C4×C167—21no—
45C4⋊C4×C167D411yescharacteristic
46C42×C167—21no—
47C42×C167C2311yescharacteristic
48C22×D167—21no—
49Dic334—41no—
50Dic334—41no—
51D668—21no—
52D668—21no—
53C2×Dic167—21no—
54C2×Dic167—22no—
55C4×D167—41no—
56C4×D167—41no—
57C167⋊C8—22no—
58C167⋊C8—41no—
59Q8×C167—21no—
60Q8×C167—21no—
61Q8×C167—41no—
62Q8×C167—41no—
63C2×C4×C167—21no—
64C2×C4×C167—21no—
65C2×C4×C167—41no—
66C2×C4×C167C2×D411yescharacteristic
67C2×C4×C167C2×D411yescharacteristic
68C2×C4×C167C2×D411yescharacteristic
69C8×C167—41no—
70C22×C167#12011yescharacteristic
71C2×D167—41no—
72C2×D167—41no—
73C2×D167—41no—
74C4×C167—21no—
75C4×C167—41no—
76C4×C167—41no—
77C4×C167—81no—
78C4×C167#13511yescharacteristic
79C4×C167#13511yescharacteristic
80C4×C167AGL(1,ℤ/8)11yescharacteristic
81C4×C167#13711yescharacteristic
82Dic167—42no—
83Dic167—41no—
84C2×C167#44911yescharacteristic
85C2×C167#45211yescharacteristic
86C2×C167#45611yescharacteristic
87D167—81no—
88C167#165011yescharacteristic, complemented, Sylow
89#1650—1671nomaximal, Sylow
90#556—1671no—
91#500—1671no—
92#475D16711yescharacteristic, complemented
93#461—1671no—
94#332—1671no—
95Q8⋊C8—1671no—
96D4⋊C8—1671no—
97C2×SD16—3341no—
98C4⋊Q8—1671no—
99C4⋊Q8C2×D16711yescharacteristic
100#126—3342no—
101#124—3341no—
102C22⋊Q8—3341no—
103C4×Q8—3341no—
104C4×Q8C2×D16711yescharacteristic
105C4×D4—1671no—
106#117—3341no—
107#107—3342no—
108#106—21no—
109C4⋊C8—1671no—
110C4⋊C8C2×D16711yescharacteristic
111#103—21no—
112#103—3341no—
113C4×C8—1671no—
114C2×Q8—21no—
115C2×Q8—3341no—
116C2×Q8C22⋊D16711yescharacteristic
117C2×D4—1671no—
118C22×C4—3341no—
119SD16—6682no—
120C2×C8—21no—
121C2×C8—3341no—
122C2×C8—3341no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3342no—
127C4⋊C4—3341no—
128C4⋊C4—1671no—
129C4⋊C4—1672no—
130C4⋊C4C22⋊D16711yescharacteristic
131C22⋊C4—3342no—
132C22⋊C4—3342no—
133C22⋊C4—3341no—
134C42—21no—
135C42—3342no—
136C42C22×D16711yescharacteristic
137C23—3341no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—6681no—
143Q8—6681no—
144D4—3341no—
145D4—3341no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3341no—
150C2×C4—3342no—
151C2×C4—6681no—
152C2×C4—6681no—
153C2×C4D4×D16711yescharacteristic
154C2×C4D4×D16711yescharacteristic
155C2×C4C2×C22⋊D16711yescharacteristic
156C8—41no—
157C8—3342no—
158C8—6681no—
159C22—6681no—
160C22—6681no—
161C22—6681no—
162C22#1912880011yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—6682no—
168C4—6681no—
169C4#1912885211yescharacteristic
170C4#1912885311yescharacteristic
171C4#1912885911yescharacteristic
172C4#1912886311yescharacteristic
173C2—13361no—
174C2#2492341311yescharacteristic
175C2#2492342911yescharacteristic
176C2#2492345011yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#475×C167C2
socleC22×C167#120
prime radicalC2×D668D4
hypercenter#475D167
hypocenterC167#1650
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #24923672C2semidirect
4. #24923014C2semidirect
5. Q8⋊(C167⋊C8)C2semidirect
7. #475×C167C2semidirect
88. C167#1650semidirect
92. #475D167semidirect

Maximal subgroups

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

subgroupindex
#249239922
#249236722
#249234712
#249230142
Q8⋊(C167⋊C8)2
#249229962
#475×C1672
#1650167