Finite Group Databasebeta

← #36323270

Subgroups — #36323270

Order 21376.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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#24923976C211yesmaximal, characteristic
2#24923648C211yesmaximal, characteristic, complemented
3#24923561C211yesmaximal, characteristic, complemented
4#24923017C211yesmaximal, characteristic, complemented
5#24922999C211yesmaximal, characteristic, complemented
6D4⋊(C167⋊C8)C211yesmaximal, characteristic
7#476×C167C211yesmaximal, characteristic
8C2×#14581940—22no—
9#19128832C2211yescharacteristic
10#19128823—21no—
11#19128811—21no—
12#19128809—22no—
13C4×Dic334C2211yescharacteristic
14D4×Dic167—21no—
15C2×C4⋊Dic167—21no—
16#19128765—21no—
17C4⋊(C167⋊C8)C2211yescharacteristic
18#19128755—21no—
19C4×C167⋊C8C2211yescharacteristic
20C4⋊Q8×C167C2211yescharacteristic
21C4×D4×C167C2211yescharacteristic
22#106×C167—21no—
23C4⋊C8×C167C2211yescharacteristic
24#102×C167—21no—
25C2×Dic334D411yescharacteristic
26C2×Dic334D412yes—
27C22×Dic167—22no—
28#14581940—44no—
29C2×C167⋊C8—21no—
30C2×C167⋊C8—21no—
31C4⋊Dic167—22no—
32C4⋊Dic167—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167D411yescharacteristic
35Dic167⋊C4—22no—
36Dic167⋊C4—21no—
37C22⋊Dic167—22no—
38C4×Dic167—21no—
39C2×Q8×C167—21no—
40C2×D4×C167D411yescharacteristic
41C22×C4×C167—21no—
42C2×C8×C167—21no—
43C4⋊C4×C167—21no—
44C4⋊C4×C167—21no—
45C4⋊C4×C167D411yescharacteristic
46C22⋊C4×C167—21no—
47C42×C167C2311yescharacteristic
48Dic334—21no—
49Dic334—21no—
50Dic334—42no—
51C2×Dic167—21no—
52C2×Dic167—21no—
53C2×Dic167—22no—
54C2×Dic167—82no—
55C167⋊C8—22no—
56C167⋊C8—41no—
57C23×C167—21no—
58Q8×C167—42no—
59D4×C167—21no—
60D4×C167Q1612yescomplemented
61C2×C4×C167—21no—
62C2×C4×C167—41no—
63C2×C4×C167—41no—
64C2×C4×C167—41no—
65C2×C4×C167C2×D411yescharacteristic
66C2×C4×C167C2×D411yescharacteristic
67C2×C4×C167C2×D411yescharacteristic
68C8×C167—41no—
69C22×C167—22no—
70C22×C167—41no—
71C22×C167—41no—
72C22×C167#12011yescharacteristic
73C4×C167—21no—
74C4×C167—41no—
75C4×C167—81no—
76C4×C167C2×Q1611yescharacteristic
77C4×C167#13511yescharacteristic
78C4×C167#13711yescharacteristic
79C4×C167#13711yescharacteristic
80Dic167—41no—
81Dic167—41no—
82Dic167—42no—
83C2×C167—42no—
84C2×C167#45111yescharacteristic
85C2×C167#45211yescharacteristic
86C2×C167#45311yescharacteristic
87C167#163211yescharacteristic, complemented, Sylow
88#1632—1671nomaximal, Sylow
89#554—1671no—
90#494—1671no—
91#477—1671no—
92#476D16711yescharacteristic
93#333—1671no—
94Q8⋊C8—1671no—
95D4⋊C8—1671no—
96C2×Q16—3342no—
97C4⋊Q8—1671no—
98C4⋊Q8C2×D16711yescharacteristic
99#125—3341no—
100C22⋊Q8—3342no—
101C22⋊Q8—3341no—
102C4×Q8—1671no—
103C4×D4—3341no—
104C4×D4C2×D16711yescharacteristic
105C2×C4⋊C4—3341no—
106#107—3341no—
107#106—21no—
108C4⋊C8—1671no—
109C4⋊C8C2×D16711yescharacteristic
110#103—3341no—
111#102—21no—
112C4×C8—1671no—
113C2×Q8—21no—
114C2×Q8—1671no—
115C2×Q8—1672no—
116C2×D4C22⋊D16711yescharacteristic
117C22×C4—21no—
118C22×C4—3342no—
119Q16—6684no—
120C2×C8—21no—
121C2×C8—3341no—
122C2×C8—3341no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—3342no—
126C4⋊C4—3342no—
127C4⋊C4—3341no—
128C4⋊C4—3341no—
129C4⋊C4—3341no—
130C4⋊C4—1671no—
131C4⋊C4C22⋊D16711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3342no—
134C42—3341no—
135C42C22×D16711yescharacteristic
136C23—21no—
137Q8—42no—
138Q8—3341no—
139Q8—3341no—
140Q8—6682no—
141D4—21no—
142D4#1458194012yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—13362no—
148C2×C4—3341no—
149C2×C4—3341no—
150C2×C4—3342no—
151C2×C4D4×D16711yescharacteristic
152C2×C4D4×D16711yescharacteristic
153C2×C4C2×C22⋊D16711yescharacteristic
154C8—41no—
155C8—3342no—
156C8—6681no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1912880011yescharacteristic
161C4—21no—
162C4—41no—
163C4—81no—
164C4—6681no—
165C4—6681no—
166C4—6682no—
167C4C2×#1458194011yescharacteristic
168C4#1912885311yescharacteristic
169C4#1912886111yescharacteristic
170C4#1912886211yescharacteristic
171C2—42no—
172C2#2492342411yescharacteristic
173C2#2492342811yescharacteristic
174C2#2492343511yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#476×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenter#476D167
hypocenterC167#1632
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #24923648C2semidirect
3. #24923561C2semidirect
4. #24923017C2semidirect
5. #24922999C2semidirect
60. D4×C167Q16semidirect
87. C167#1632semidirect
142. D4#14581940semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249236482
#249235612
#249230172
#249229992
D4⋊(C167⋊C8)2
#476×C1672
#1632167