Finite Group Databasebeta

← #36323245

Subgroups — #36323245

Order 21376.

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

normal subgroups only

This group has 216 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#24923993C211yesmaximal, characteristic, complemented
2#24923466C211yesmaximal, characteristic, complemented
3#24923461C211yesmaximal, characteristic
4#24923015C211yesmaximal, characteristic, complemented
5#24923001C211yesmaximal, characteristic, complemented
6#24922998C211yesmaximal, characteristic, complemented
7#486×C167C211yesmaximal, characteristic, complemented
8C2×#14581938—21no—
9C2×#14581935—21no—
10#19128827—22no—
11#19128821—21no—
12#19128812—21no—
13Q8×Dic167—21no—
14C4×Dic334C2211yescharacteristic
15C4×D668C2211yescharacteristic
16#19128790—21no—
17C4⋊(C167⋊C8)C2211yescharacteristic
18C4⋊(C167⋊C8)C2211yescharacteristic
19#19128753—21no—
20#19128753—21no—
21C4⋊Q8×C167C2211yescharacteristic
22#127×C167C2211yescharacteristic
23#102×C167—22no—
24C4×C8×C167C2211yescharacteristic
25C2×Dic334D411yescharacteristic
26C2×D668D411yescharacteristic
27C2×C4×D167—21no—
28#14581938—42no—
29#14581935—42no—
30C2×C167⋊C8—21no—
31C2×C167⋊C8—21no—
32C4⋊Dic167—22no—
33C4⋊Dic167D411yescharacteristic
34C4⋊Dic167D411yescharacteristic
35Dic167⋊C4—21no—
36(C2×D167)⋊C4—22no—
37(C2×D167)⋊C4—22no—
38(C2×D167)⋊C4—21no—
39C4×Dic167—22no—
40C4×Dic167—21no—
41C2×Q8×C167—21no—
42C2×D4×C167—21no—
43C2×D4×C167—21no—
44C2×D4×C167—21no—
45C2×C8×C167—21no—
46C4⋊C4×C167—21no—
47C4⋊C4×C167D412yes—
48C42×C167C2311yescharacteristic
49C22×D167—21no—
50Dic334—21no—
51Dic334—21no—
52D668—21no—
53D668—21no—
54C2×Dic167—21no—
55C2×Dic167—21no—
56C2×Dic167—21no—
57C4×D167—41no—
58C4×D167—41no—
59C167⋊C8—41no—
60C167⋊C8—41no—
61C23×C167—41no—
62Q8×C167—41no—
63Q8×C167—41no—
64D4×C167—42no—
65D4×C167—41no—
66D4×C167—41no—
67D4×C167—41no—
68D4×C167—41no—
69C2×C4×C167—22no—
70C2×C4×C167C2×D411yescharacteristic
71C2×C4×C167C2×D411yescharacteristic
72C2×C4×C167C2×D411yescharacteristic
73C8×C167—41no—
74C22×C167—81no—
75C22×C167—81no—
76C22×C167—81no—
77C22×C167#12011yescharacteristic
78C2×D167—41no—
79C2×D167—41no—
80C2×D167—41no—
81C4×C167—21no—
82C4×C167—42no—
83C4×C167#13511yescharacteristic
84C4×C167#13511yescharacteristic
85C4×C167AGL(1,ℤ/8)11yescharacteristic
86C4×C167AGL(1,ℤ/8)11yescharacteristic
87Dic167—41no—
88Dic167—41no—
89Dic167—41no—
90C2×C167—161no—
91C2×C167#44911yescharacteristic
92C2×C167#44911yescharacteristic
93C2×C167#45411yescharacteristic
94D167—81no—
95C167#162811yescharacteristic, complemented, Sylow
96#1628—1671nomaximal, Sylow
97#556—1671no—
98#486D16711yescharacteristic, complemented
99#460—1671no—
100#459—1671no—
101#332—1671no—
102Q8⋊C8—1671no—
103D4⋊C8—1671no—
104C2×SD16—3341no—
105C2×D8—3341no—
106C4⋊Q8C2×D16711yescharacteristic
107#127C2×D16711yescharacteristic
108#126—3342no—
109#124—3341no—
110C22⋊Q8—3341no—
111C4×Q8—3341no—
112C4×Q8—1671no—
113C4×D4—1671no—
114#117—3341no—
115C4⋊C8—1671no—
116C4⋊C8—1671no—
117#102—22no—
118#102—3341no—
119#102—3341no—
120C4×C8C2×D16711yescharacteristic
121C2×Q8—21no—
122C2×Q8—1671no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1671no—
127C22×C4—3341no—
128SD16—6682no—
129D8—6682no—
130C2×C8—21no—
131C2×C8—3341no—
132C2×C8—3341no—
133C4⋊C4—21no—
134C4⋊C4—3342no—
135C4⋊C4—3341no—
136C4⋊C4—1671no—
137C4⋊C4—1671no—
138C4⋊C4C22⋊D16712yes—
139C22⋊C4—3342no—
140C22⋊C4—3342no—
141C22⋊C4—3341no—
142C42—3342no—
143C42—3341no—
144C42C22×D16711yescharacteristic
145C23—41no—
146C23—3341no—
147Q8—41no—
148Q8—41no—
149Q8—3341no—
150Q8—3341no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3341no—
157D4—3341no—
158C2×C4—22no—
159C2×C4—3341no—
160C2×C4—3341no—
161C2×C4—3341no—
162C2×C4—6681no—
163C2×C4—6681no—
164C2×C4D4×D16711yescharacteristic
165C2×C4D4×D16711yescharacteristic
166C2×C4C2×C22⋊D16711yescharacteristic
167C8—41no—
168C8—6681no—
169C8—6681no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—6681no—
174C22—6681no—
175C22—6681no—
176C22#1912880011yescharacteristic
177C4—21no—
178C4—42no—
179C4—6681no—
180C4—6681no—
181C4—6681no—
182C4#1912885111yescharacteristic
183C4#1912885311yescharacteristic
184C4#1912885611yescharacteristic
185C4#1912885911yescharacteristic
186C2—161no—
187C2—13361no—
188C2#2492341111yescharacteristic
189C2#2492341311yescharacteristic
190C2#2492344111yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#486×C167C2
socleC22×C167#120
prime radical#24923461C2
hypercenter#486D167
hypocenterC167#1628
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #24923993C2semidirect
2. #24923466C2semidirect
4. #24923015C2semidirect
5. #24923001C2semidirect
6. #24922998C2semidirect
7. #486×C167C2semidirect
95. C167#1628semidirect
98. #486D167semidirect

Maximal subgroups

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

subgroupindex
#249239932
#249234662
#249234612
#249230152
#249230012
#249229982
#486×C1672
#1628167