Finite Group Databasebeta

← #36323247

Subgroups — #36323247

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#24923992C211yesmaximal, characteristic, complemented
2#24923615C211yesmaximal, characteristic, complemented
3#24923461C211yesmaximal, characteristic
4#24923014C211yesmaximal, characteristic, complemented
5Q8⋊(C167⋊C8)C211yesmaximal, characteristic, complemented
6#24922996C211yesmaximal, characteristic, complemented
7#460×C167C211yesmaximal, characteristic, complemented
8C2×#14581935—21no—
9#19128832C2211yescharacteristic
10#19128827—22no—
11#19128820—21no—
12#19128808—21no—
13C4×Dic334—21no—
14C4×D668C2211yescharacteristic
15#19128790—21no—
16C4⋊(C167⋊C8)C2211yescharacteristic
17#19128753—21no—
18#19128753—22no—
19C4×C167⋊C8C2211yescharacteristic
20C2×SD16×C167—21no—
21#127×C167C2211yescharacteristic
22C4×Q8×C167C2211yescharacteristic
23C4⋊C8×C167C2211yescharacteristic
24#102×C167—21no—
25C2×Dic334—21no—
26C2×D668D411yescharacteristic
27C2×C4×D167—21no—
28#14581935—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×C167D411yescharacteristic
40C2×D4×C167—21no—
41C2×D4×C167—21no—
42C2×D4×C167—21no—
43SD16×C167—42no—
44C2×C8×C167—21no—
45C4⋊C4×C167—21no—
46C4⋊C4×C167D411yescharacteristic
47C42×C167—21no—
48C42×C167C2311yescharacteristic
49C22×D167—21no—
50Dic334—41no—
51Dic334—41no—
52D668—21no—
53D668—21no—
54C2×Dic167—21no—
55C2×Dic167—22no—
56C4×D167—41no—
57C4×D167—41no—
58C167⋊C8—41no—
59C167⋊C8—41no—
60C23×C167—41no—
61Q8×C167—21no—
62Q8×C167—21no—
63D4×C167—42no—
64D4×C167—41no—
65D4×C167—41no—
66D4×C167—41no—
67D4×C167—41no—
68C2×C4×C167—21no—
69C2×C4×C167—21no—
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—41no—
83C4×C167—41no—
84C4×C167#13511yescharacteristic
85C4×C167#13511yescharacteristic
86C4×C167AGL(1,ℤ/8)11yescharacteristic
87C4×C167AGL(1,ℤ/8)11yescharacteristic
88Dic167—41no—
89Dic167—42no—
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#486—1671no—
99#460D16711yescharacteristic, complemented
100#459—1671no—
101#332—1671no—
102Q8⋊C8—1671no—
103D4⋊C8—1671no—
104C2×SD16—21no—
105C2×D8—3341no—
106C4⋊Q8—1671no—
107#127C2×D16711yescharacteristic
108#126—3342no—
109#124—3341no—
110C22⋊Q8—3341no—
111C4×Q8—3341no—
112C4×Q8C2×D16711yescharacteristic
113C4×D4—1671no—
114#117—3341no—
115C4⋊C8—1671no—
116C4⋊C8C2×D16711yescharacteristic
117#102—21no—
118#102—3341no—
119#102—3342no—
120C4×C8—1671no—
121C2×Q8—3341no—
122C2×Q8C22⋊D16711yescharacteristic
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1671no—
127C22×C4—3341no—
128SD16—42no—
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—1672no—
138C4⋊C4C22⋊D16711yescharacteristic
139C22⋊C4—3342no—
140C22⋊C4—3342no—
141C22⋊C4—3341no—
142C42—21no—
143C42—3342no—
144C42C22×D16711yescharacteristic
145C23—41no—
146C23—3341no—
147Q8—21no—
148Q8—21no—
149Q8—6681no—
150Q8—6681no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3341no—
157D4—3341no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—3341no—
161C2×C4—3342no—
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—41no—
179C4—41no—
180C4—6681no—
181C4—6682no—
182C4#1912885211yescharacteristic
183C4#1912885311yescharacteristic
184C4#1912885811yescharacteristic
185C4#1912885911yescharacteristic
186C2—161no—
187C2—13361no—
188C2#2492341211yescharacteristic
189C2#2492341311yescharacteristic
190C2#2492344011yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#460×C167C2
socleC22×C167#120
prime radical#24923461C2
hypercenter#460D167
hypocenterC167#1628
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #24923992C2semidirect
2. #24923615C2semidirect
4. #24923014C2semidirect
5. Q8⋊(C167⋊C8)C2semidirect
6. #24922996C2semidirect
7. #460×C167C2semidirect
95. C167#1628semidirect
99. #460D167semidirect

Maximal subgroups

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

subgroupindex
#249239922
#249236152
#249234612
#249230142
Q8⋊(C167⋊C8)2
#249229962
#460×C1672
#1628167