Finite Group Databasebeta

← #36323365

Subgroups — #36323365

Order 21376.

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

normal subgroups only

This group has 212 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#24923984C211yesmaximal, characteristic
2#24923621C211yesmaximal, characteristic, complemented
3#24923471C211yesmaximal, characteristic, complemented
4#24923017C211yesmaximal, characteristic, complemented
5D4⋊(C167⋊C8)C211yesmaximal, characteristic, complemented
6#24922996C211yesmaximal, characteristic, complemented
7#474×C167C211yesmaximal, characteristic, complemented
8C2×#14581937—21no—
9#19128832C2211yescharacteristic
10#19128818—22no—
11#19128806—21no—
12#19128803—21no—
13C4×D668C2211yescharacteristic
14#19128791—21no—
15#19128790—21no—
16C4⋊(C167⋊C8)C2211yescharacteristic
17#19128756—21no—
18#19128755—22no—
19C4×C167⋊C8C2211yescharacteristic
20C4⋊Q8×C167C2211yescharacteristic
21C4×D4×C167C2211yescharacteristic
22#107×C167—21no—
23C4⋊C8×C167C2211yescharacteristic
24#102×C167—21no—
25C2×Dic334D412yes—
26C2×D668D411yescharacteristic
27C2×C22⋊D167—22no—
28C2×C4×D167—21no—
29#14581937—42no—
30C2×C167⋊C8—21no—
31C2×C167⋊C8—21no—
32C4⋊Dic167—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167D411yescharacteristic
35C22⋊Dic167—22no—
36(C2×D167)⋊C4—22no—
37(C2×D167)⋊C4—21no—
38C4×Dic167—22no—
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
48C22×D167—21no—
49Dic334—42no—
50D668—21no—
51D668—21no—
52C22⋊D167—82no—
53C2×Dic167—21no—
54C2×Dic167—22no—
55C4×D167—41no—
56C4×D167—41no—
57C167⋊C8—41no—
58C167⋊C8—41no—
59C23×C167—21no—
60Q8×C167—41no—
61Q8×C167—41no—
62D4×C167—21no—
63D4×C167—21no—
64C2×C4×C167—21no—
65C2×C4×C167—41no—
66C2×C4×C167—41no—
67C2×C4×C167—41no—
68C2×C4×C167C2×D411yescharacteristic
69C2×C4×C167C2×D411yescharacteristic
70C2×C4×C167C2×D411yescharacteristic
71C8×C167—41no—
72C22×C167—41no—
73C22×C167—41no—
74C22×C167—41no—
75C22×C167#12011yescharacteristic
76C2×D167—41no—
77C2×D167—41no—
78C2×D167—41no—
79C4×C167—21no—
80C4×C167—41no—
81C4×C167—81no—
82C4×C167#13511yescharacteristic
83C4×C167#13511yescharacteristic
84C4×C167AGL(1,ℤ/8)11yescharacteristic
85C4×C167#13711yescharacteristic
86Dic167—41no—
87Dic167—42no—
88C2×C167—81no—
89C2×C167#44911yescharacteristic
90C2×C167#45211yescharacteristic
91C2×C167#45511yescharacteristic
92D167—81no—
93C167#164611yescharacteristic, complemented, Sylow
94#1646—1671nomaximal, Sylow
95#555—1671no—
96#487—1671no—
97#474D16711yescharacteristic, complemented
98#461—1671no—
99#333—1671no—
100D4⋊C8—1671no—
101D4⋊C8—1671no—
102C2×SD16—3341no—
103C4⋊Q8—1671no—
104C4⋊Q8C2×D16711yescharacteristic
105#124—3342no—
106#121—3341no—
107#121—3341no—
108C4×D4—1671no—
109C4×D4C2×D16711yescharacteristic
110#117—3341no—
111#117—3341no—
112#107—21no—
113C4⋊C8—1671no—
114C4⋊C8C2×D16711yescharacteristic
115#103—3342no—
116#103—3341no—
117#102—21no—
118C4×C8—1671no—
119C2×Q8—21no—
120C2×Q8—1672no—
121C2×D4—3342no—
122C2×D4—1671no—
123C2×D4C22⋊D16711yescharacteristic
124C22×C4—21no—
125C22×C4—3341no—
126SD16—6682no—
127C2×C8—21no—
128C2×C8—3341no—
129C2×C8—3341no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3341no—
133C4⋊C4—3341no—
134C4⋊C4—1671no—
135C4⋊C4C22⋊D16711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3342no—
138C22⋊C4—3342no—
139C22⋊C4—3341no—
140C42—3342no—
141C42C22×D16711yescharacteristic
142C23—21no—
143C23—3341no—
144Q8—41no—
145Q8—41no—
146Q8—6682no—
147D4—21no—
148D4—21no—
149D4—13362no—
150D4—3341no—
151D4—3341no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3341no—
157C2×C4—3342no—
158C2×C4—6681no—
159C2×C4—6681no—
160C2×C4D4×D16711yescharacteristic
161C2×C4D4×D16711yescharacteristic
162C2×C4C2×C22⋊D16711yescharacteristic
163C8—41no—
164C8—6681no—
165C8—6681no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—6681no—
170C22—6681no—
171C22—6681no—
172C22#1912880011yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—6681no—
177C4—6682no—
178C4#1912885111yescharacteristic
179C4#1912885211yescharacteristic
180C4#1912885611yescharacteristic
181C4#1912886311yescharacteristic
182C2—81no—
183C2—13361no—
184C2#2492341111yescharacteristic
185C2#2492342911yescharacteristic
186C2#2492344511yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#474×C167C2
socleC22×C167#120
prime radical#24923984C2
hypercenter#474D167
hypocenterC167#1646
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #24923621C2semidirect
3. #24923471C2semidirect
4. #24923017C2semidirect
5. D4⋊(C167⋊C8)C2semidirect
6. #24922996C2semidirect
7. #474×C167C2semidirect
93. C167#1646semidirect
97. #474D167semidirect

Maximal subgroups

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

subgroupindex
#249239842
#249236212
#249234712
#249230172
D4⋊(C167⋊C8)2
#249229962
#474×C1672
#1646167