Finite Group Databasebeta

← #36323136

Subgroups — #36323136

Order 21376.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#24923864C211yesmaximal, characteristic
2#24923600C211yesmaximal, characteristic, complemented
3#24923580C211yesmaximal, characteristic
4#24923201C211yesmaximal, characteristic, complemented
5#24923005C211yesmaximal, characteristic, complemented
6#24922994C211yesmaximal, characteristic, complemented
7#327×C167C211yesmaximal, characteristic
8#19128825—21no—
9#19128822—21no—
10#19128813—21no—
11#19128813—21no—
12#19128811C2211yescharacteristic
13#19128809—21no—
14D4×Dic167—21no—
15C2×C4⋊Dic167—21no—
16C2×C4⋊Dic167C2211yescharacteristic
17C2×Dic167⋊C4C2211yescharacteristic
18#19128765—21no—
19#19128764—21no—
20#19128757—21no—
21#19128753—21no—
22C22⋊(C167⋊C8)C2211yescharacteristic
23C167⋊#95—21no—
24#121×C167C2211yescharacteristic
25C22⋊C8×C167C2211yescharacteristic
26#95×C167C2211yescharacteristic
27C2×Dic334D411yescharacteristic
28C22×Dic167—21no—
29C22×Dic167—21no—
30C22×Dic167D411yescharacteristic
31C22×Dic167D411yescharacteristic
32C2×C167⋊C8—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167—21no—
35C4⋊Dic167—21no—
36C4⋊Dic167—42no—
37C4⋊Dic167D411yescharacteristic
38Dic167⋊C4—21no—
39Dic167⋊C4—21no—
40Dic167⋊C4—21no—
41Dic167⋊C4—21no—
42Dic167⋊C4—21no—
43C22⋊Dic167—21no—
44C22⋊Dic167—21no—
45C22⋊Dic167—21no—
46C22⋊Dic167—21no—
47C4×Dic167—21no—
48C2×D4×C167—21no—
49C2×D4×C167D411yescharacteristic
50C22×C4×C167—21no—
51C22×C4×C167C2311yescharacteristic
52C2×C8×C167—21no—
53C4⋊C4×C167D411yescharacteristic
54C22⋊C4×C167—21no—
55Dic334—41no—
56C2×Dic167—21no—
57C2×Dic167—21no—
58C2×Dic167—21no—
59C2×Dic167—21no—
60C2×Dic167—21no—
61C2×Dic167—21no—
62C2×Dic167—41no—
63C2×Dic167—41no—
64C2×Dic167—41no—
65C2×Dic167—41no—
66C2×Dic167—81no—
67C167⋊C8—41no—
68C23×C167—21no—
69C23×C167C2×D411yescharacteristic
70D4×C167—41no—
71D4×C167—41no—
72D4×C167—41no—
73C2×C4×C167—22no—
74C2×C4×C167—21no—
75C2×C4×C167—41no—
76C2×C4×C167—41no—
77C2×C4×C167—41no—
78C2×C4×C167C2×D411yescharacteristic
79C2×C4×C167C2×D411yescharacteristic
80C8×C167—41no—
81C22×C167—21no—
82C22×C167—21no—
83C22×C167—41no—
84C22×C167—41no—
85C22×C167—41no—
86C22×C167#12011yescharacteristic
87C22×C167#13511yescharacteristic
88C22×C167AGL(1,ℤ/8)11yescharacteristic
89C4×C167—21no—
90C4×C167—21no—
91C4×C167—42no—
92C4×C167—41no—
93Dic167—41no—
94Dic167—41no—
95Dic167—41no—
96Dic167—41no—
97Dic167—81no—
98C2×C167—21no—
99C2×C167—21no—
100C2×C167—81no—
101C2×C167#44911yescharacteristic
102C2×C167#45611yescharacteristic
103C2×C167#45811yescharacteristic
104C167#160911yescharacteristic, complemented, Sylow
105#1609—1671nomaximal, Sylow
106#539—1671no—
107#483—1671no—
108#480—1671no—
109#398—1671no—
110#328—1671no—
111#327D16711yescharacteristic
112#324—1671no—
113#126—3341no—
114#125—3341no—
115#123—3341no—
116#123—3341no—
117C22⋊Q8—3341no—
118C22⋊Q8—1671no—
119#121C2×D16711yescharacteristic
120C4×D4—3341no—
121C2×C4⋊C4—3341no—
122C2×C4⋊C4—1671no—
123C2×C4⋊C4—1671no—
124#107—3341no—
125#106—3341no—
126#103—3341no—
127#102—3341no—
128C22⋊C8—1671no—
129C22⋊C8C2×D16711yescharacteristic
130#95—3341no—
131#95C2×D16711yescharacteristic
132C2×Q8—1671no—
133C2×D4—21no—
134C2×D4D66811yescharacteristic
135C22×C4—21no—
136C22×C4—3341no—
137C22×C4—3341no—
138C22×C4—1671no—
139C22×C4—1671no—
140C22×C4C22×D16711yescharacteristic
141C2×C8—21no—
142C2×C8—3341no—
143C4⋊C4—3341no—
144C4⋊C4—3341no—
145C4⋊C4—3341no—
146C4⋊C4—3341no—
147C4⋊C4—3341no—
148C4⋊C4—3341no—
149C4⋊C4—3341no—
150C4⋊C4—3341no—
151C4⋊C4—6682no—
152C4⋊C4—1671no—
153C4⋊C4D66811yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—3341no—
156C22⋊C4—3341no—
157C22⋊C4—3341no—
158C22⋊C4—3341no—
159C42—3341no—
160C23—21no—
161C23D4×D16711yescharacteristic
162Q8—6681no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—13361no—
172C2×C4—3341no—
173C2×C4—3341no—
174C2×C4—3341no—
175C2×C4—3341no—
176C2×C4—3341no—
177C2×C4—3341no—
178C2×C4—6681no—
179C2×C4—6681no—
180C2×C4—6681no—
181C2×C4—6681no—
182C2×C4D4×D16711yescharacteristic
183C2×C4C2×D66811yescharacteristic
184C8—41no—
185C8—6681no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#1912880211yescharacteristic
192C22#1912885311yescharacteristic
193C22#1912885611yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—13361no—
199C4—6681no—
200C4—6681no—
201C4—6681no—
202C4—6681no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#2492341511yescharacteristic
207C2#2492345311yescharacteristic
208C2#2492346011yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#327×C167C2
socleC22×C167#120
prime radical#121×C167C22
hypercenter#327D167
hypocenterC167#1609
LCSG ▷ C22×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #24923600C2semidirect
4. #24923201C2semidirect
5. #24923005C2semidirect
6. #24922994C2semidirect
104. C167#1609semidirect

Maximal subgroups

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

subgroupindex
#249238642
#249236002
#249235802
#249232012
#249230052
#249229942
#327×C1672
#1609167