Finite Group Databasebeta

← #36323342

Subgroups — #36323342

Order 21376.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#24923980C211yesmaximal, characteristic
2#24923551C211yesmaximal, characteristic, complemented
3#24923471C211yesmaximal, characteristic
4#24923018C211yesmaximal, characteristic, complemented
5#24923001C211yesmaximal, characteristic, complemented
6#24922998C211yesmaximal, characteristic
7#487×C167C211yesmaximal, characteristic, complemented
8C2×#14581937—21no—
9#19128823—21no—
10#19128812—21no—
11#19128808—22no—
12C4×Dic334C2211yescharacteristic
13C4×D668C2211yescharacteristic
14#19128794—21no—
15C4⋊C4×D167—21no—
16#19128763—21no—
17C4⋊(C167⋊C8)C2211yescharacteristic
18C4⋊(C167⋊C8)C2211yescharacteristic
19#19128756—21no—
20#19128755—21no—
21C4⋊Q8×C167C2211yescharacteristic
22C4⋊Q8×C167C2211yescharacteristic
23#103×C167—22no—
24C4×C8×C167C2211yescharacteristic
25C2×Dic334D411yescharacteristic
26C2×D668D411yescharacteristic
27C2×C4×D167—22no—
28C2×C4×D167—21no—
29#14581937—42no—
30C2×C167⋊C8—21no—
31C2×C167⋊C8—21no—
32C4⋊Dic167—22no—
33C4⋊Dic167D411yescharacteristic
34C4⋊Dic167D411yescharacteristic
35Dic167⋊C4—22no—
36Dic167⋊C4—21no—
37(C2×D167)⋊C4—22no—
38(C2×D167)⋊C4—21no—
39C4×Dic167—21no—
40C2×Q8×C167—21no—
41C2×Q8×C167D412yes—
42C2×C8×C167—21no—
43C4⋊C4×C167—21no—
44C4⋊C4×C167—21no—
45C4⋊C4×C167—21no—
46C4⋊C4×C167—21no—
47C42×C167C2311yescharacteristic
48C22×D167—21no—
49Dic334—21no—
50Dic334—21no—
51D668—21no—
52D668SD1612yes—
53C2×Dic167—21no—
54C2×Dic167—21no—
55C2×Dic167—21no—
56C4×D167—41no—
57C4×D167—41no—
58C4×D167—82no—
59C167⋊C8—41no—
60C167⋊C8—41no—
61Q8×C167—42no—
62Q8×C167—41no—
63Q8×C167—41no—
64C2×C4×C167—22no—
65C2×C4×C167—41no—
66C2×C4×C167C2×D411yescharacteristic
67C2×C4×C167C2×D411yescharacteristic
68C2×C4×C167C2×D411yescharacteristic
69C8×C167—41no—
70C22×C167#12011yescharacteristic
71C2×D167—22no—
72C2×D167—41no—
73C2×D167—41no—
74C4×C167—21no—
75C4×C167—42no—
76C4×C167—81no—
77C4×C167C2×SD1611yescharacteristic
78C4×C167#13511yescharacteristic
79C4×C167#13711yescharacteristic
80C4×C167#13711yescharacteristic
81Dic167—41no—
82Dic167—41no—
83Dic167—41no—
84C2×C167#44811yescharacteristic
85C2×C167#45211yescharacteristic
86C2×C167#45511yescharacteristic
87D167—42no—
88C167#164311yescharacteristic, complemented, Sylow
89#1643—1671nomaximal, Sylow
90#554—1671no—
91#487D16711yescharacteristic, complemented
92#475—1671no—
93#461—1671no—
94#333—1671no—
95Q8⋊C8—1671no—
96D4⋊C8—1671no—
97C2×SD16—3341no—
98C4⋊Q8C2×D16711yescharacteristic
99C4⋊Q8C2×D16711yescharacteristic
100#125—3341no—
101C22⋊Q8—3341no—
102C22⋊Q8—3342no—
103C4×Q8—1671no—
104C4×D4—3341no—
105C4×D4—1671no—
106C2×C4⋊C4—3341no—
107#106—3341no—
108C4⋊C8—1671no—
109C4⋊C8—1671no—
110#103—22no—
111#103—3341no—
112#103—3341no—
113C4×C8C2×D16711yescharacteristic
114C2×Q8—21no—
115C2×Q8—1671no—
116C2×Q8C22⋊D16712yes—
117C2×D4—1671no—
118C22×C4—3342no—
119C22×C4—3341no—
120SD16—6682no—
121C2×C8—21no—
122C2×C8—3341no—
123C2×C8—3341no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—3342no—
129C4⋊C4—3342no—
130C4⋊C4—3341no—
131C4⋊C4—1671no—
132C4⋊C4—1671no—
133C22⋊C4—3342no—
134C22⋊C4—3341no—
135C42—3341no—
136C42C22×D16711yescharacteristic
137C23—3341no—
138Q8—42no—
139Q8—41no—
140Q8—41no—
141Q8—3341no—
142Q8—3341no—
143D4—3341no—
144D4—1672no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—13362no—
148C2×C4—3341no—
149C2×C4—3341no—
150C2×C4—3341no—
151C2×C4—6681no—
152C2×C4—6681no—
153C2×C4D4×D16711yescharacteristic
154C2×C4D4×D16711yescharacteristic
155C2×C4C2×C22⋊D16711yescharacteristic
156C8—41no—
157C8—6681no—
158C8—6681no—
159C22—3342no—
160C22—6681no—
161C22—6681no—
162C22#1912880011yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—6681no—
167C4—6681no—
168C4—6681no—
169C4SD16×D16711yescharacteristic
170C4#1912885511yescharacteristic
171C4#1912886111yescharacteristic
172C4#1912886411yescharacteristic
173C2—6682no—
174C2#2492340611yescharacteristic
175C2#2492343011yescharacteristic
176C2#2492344611yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#487×C167C2
socleC22×C167#120
prime radicalC2×D668D4
hypercenter#487D167
hypocenterC167#1643
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #24923551C2semidirect
4. #24923018C2semidirect
5. #24923001C2semidirect
7. #487×C167C2semidirect
88. C167#1643semidirect
91. #487D167semidirect

Maximal subgroups

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

subgroupindex
#249239802
#249235512
#249234712
#249230182
#249230012
#249229982
#487×C1672
#1643167