Finite Group Databasebeta

← #36323142

Subgroups — #36323142

Order 21376.

254 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 51 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#24923846C211yesmaximal, characteristic
2#24923583C211yesmaximal, characteristic, complemented
3#24923580C211yesmaximal, characteristic, complemented
4#24923201C211yesmaximal, characteristic, complemented
5#24923002C211yesmaximal, characteristic, complemented
6#24922994C211yesmaximal, characteristic, complemented
7#327×C167C211yesmaximal, characteristic, complemented
8C2×#14581949—21no—
9#19128810—21no—
10#19128809—21no—
11#19128806C2211yescharacteristic
12#19128805—21no—
13#19128804—21no—
14#19128804—21no—
15C2×C4⋊Dic167—21no—
16C2×C4⋊Dic167C2211yescharacteristic
17C2×Dic167⋊C4C2211yescharacteristic
18#19128766—21no—
19#19128765—21no—
20#19128754—21no—
21#19128753—21no—
22C22⋊(C167⋊C8)C2211yescharacteristic
23C167⋊#95—21no—
24#121×C167C2211yescharacteristic
25C22⋊C8×C167C2211yescharacteristic
26#95×C167C2211yescharacteristic
27#14581949—42no—
28#14581949—42no—
29C2×Dic334—21no—
30C2×D668D411yescharacteristic
31C2×C22⋊D167—21no—
32C2×C22⋊D167—21no—
33C2×C22⋊D167—21no—
34C22×Dic167—21no—
35C22×Dic167—21no—
36C22×Dic167D411yescharacteristic
37C22×Dic167D411yescharacteristic
38C2×C4×D167—21no—
39C2×C167⋊C8—21no—
40C4⋊Dic167—21no—
41C4⋊Dic167—21no—
42C4⋊Dic167—42no—
43C4⋊Dic167D411yescharacteristic
44Dic167⋊C4—21no—
45Dic167⋊C4—21no—
46Dic167⋊C4—21no—
47C22⋊Dic167—21no—
48(C2×D167)⋊C4—21no—
49(C2×D167)⋊C4—21no—
50C2×D4×C167—21no—
51C2×D4×C167D411yescharacteristic
52C22×C4×C167—21no—
53C22×C4×C167C2311yescharacteristic
54C2×C8×C167—21no—
55C4⋊C4×C167D411yescharacteristic
56C22⋊C4×C167—21no—
57C22×D167—21no—
58Dic334—42no—
59D668—41no—
60C22⋊D167—42no—
61C22⋊D167—41no—
62C22⋊D167—41no—
63C22⋊D167—81no—
64C2×Dic167—21no—
65C2×Dic167—21no—
66C2×Dic167—21no—
67C2×Dic167—21no—
68C2×Dic167—42no—
69C2×Dic167—41no—
70C2×Dic167—41no—
71C2×Dic167—41no—
72C2×Dic167—41no—
73C2×Dic167D812yes—
74C4×D167—42no—
75C167⋊C8—41no—
76C23×C167—21no—
77C23×C167C2×D411yescharacteristic
78D4×C167—41no—
79D4×C167—41no—
80D4×C167—41no—
81C2×C4×C167—22no—
82C2×C4×C167—21no—
83C2×C4×C167—41no—
84C2×C4×C167—41no—
85C2×C4×C167—41no—
86C2×C4×C167C2×D411yescharacteristic
87C2×C4×C167C2×D411yescharacteristic
88C8×C167—41no—
89C22×C167—21no—
90C22×C167—21no—
91C22×C167—41no—
92C22×C167—41no—
93C22×C167—41no—
94C22×C167#12011yescharacteristic
95C22×C167C2×D811yescharacteristic
96C22×C167AGL(1,ℤ/8)11yescharacteristic
97C2×D167—41no—
98C2×D167—41no—
99C2×D167—41no—
100C4×C167—21no—
101C4×C167—21no—
102C4×C167—42no—
103C4×C167—41no—
104Dic167—22no—
105Dic167—41no—
106Dic167—41no—
107Dic167—81no—
108C2×C167—21no—
109C2×C167—21no—
110C2×C167—81no—
111C2×C167#44711yescharacteristic
112C2×C167#45611yescharacteristic
113C2×C167#45811yescharacteristic
114D167—81no—
115C167#161011yescharacteristic, complemented, Sylow
116#1610—1671nomaximal, Sylow
117#537—1671no—
118#480—1671no—
119#480—1671no—
120#398—1671no—
121#327—1671no—
122#327D16711yescharacteristic, complemented
123#324—1671no—
124C2×#41—3341no—
125C22⋊Q8—3341no—
126C22⋊Q8—3341no—
127#121—3341no—
128#121—3341no—
129#121—3341no—
130#121—1671no—
131#121C2×D16711yescharacteristic
132C2×C4⋊C4—3341no—
133C2×C4⋊C4—1671no—
134C2×C4⋊C4—1671no—
135#107—3341no—
136#107—3341no—
137#102—3341no—
138#102—3341no—
139C22⋊C8—1671no—
140C22⋊C8C2×D16711yescharacteristic
141#95—3341no—
142#95C2×D16711yescharacteristic
143#41—6682no—
144#41—6682no—
145C2×Q8—3341no—
146C2×D4—21no—
147C2×D4—3341no—
148C2×D4—3341no—
149C2×D4—3341no—
150C2×D4—1671no—
151C2×D4D66811yescharacteristic
152C22×C4—21no—
153C22×C4—3341no—
154C22×C4—3341no—
155C22×C4—3341no—
156C22×C4—1671no—
157C22×C4—1671no—
158C22×C4C22×D16711yescharacteristic
159C2×C8—21no—
160C2×C8—3341no—
161C4⋊C4—3341no—
162C4⋊C4—3341no—
163C4⋊C4—3341no—
164C4⋊C4—3341no—
165C4⋊C4—3341no—
166C4⋊C4—6682no—
167C4⋊C4—1671no—
168C4⋊C4D66811yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—3341no—
171C22⋊C4—3341no—
172C22⋊C4—3341no—
173C23—21no—
174C23—3341no—
175C23D4×D16711yescharacteristic
176Q8—6682no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—13361no—
181D4—6681no—
182D4—6682no—
183D4—6681no—
184D4—6681no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3341no—
191C2×C4—3341no—
192C2×C4—3341no—
193C2×C4—3341no—
194C2×C4—6682no—
195C2×C4—6681no—
196C2×C4—6681no—
197C2×C4—6681no—
198C2×C4—6682no—
199C2×C4—6681no—
200C2×C4—1672no—
201C2×C4D4×D16711yescharacteristic
202C2×C4C2×D66811yescharacteristic
203C8—41no—
204C8—6681no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—6681no—
211C22—6681no—
212C22—6681no—
213C22#1912880211yescharacteristic
214C22D8×D16711yescharacteristic
215C22#1912885911yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—13361no—
221C4—3342no—
222C4—6681no—
223C4—6681no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—13361no—
228C2#2492340411yescharacteristic
229C2#2492345311yescharacteristic
230C2#2492346011yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#327×C167C2
socleC22×C167#120
prime radical#24923846C2
hypercenter#327D167
hypocenterC167#1610
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. #24923583C2semidirect
3. #24923580C2semidirect
4. #24923201C2semidirect
5. #24923002C2semidirect
6. #24922994C2semidirect
7. #327×C167C2semidirect
115. C167#1610semidirect
122. #327D167semidirect

Maximal subgroups

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

subgroupindex
#249238462
#249235832
#249235802
#249232012
#249230022
#249229942
#327×C1672
#1610167