Finite Group Databasebeta

← #36352547

Subgroups — #36352547

Order 22144.

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

normal subgroups only

This group has 280 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#26012049C211yesmaximal, characteristic
2#26011804C211yesmaximal, characteristic, complemented
3#26011417C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#480×C173C211yesmaximal, characteristic, complemented
8C2×D4×D173—21no—
9#19149664—21no—
10#19149661—21no—
11#19149653C2211yescharacteristic
12#19149651—21no—
13#19149649—21no—
14#19149647—21no—
15C2×#14583803—21no—
16C2×#14583803C2211yescharacteristic
17#19149612—21no—
18#19149599—21no—
19C22⋊#13063436C2211yescharacteristic, complemented
20#19149586—21no—
21#19149586C2211yescharacteristic, complemented
22#121×C173C2211yescharacteristic
23C2×C4⋊C4×C173C2211yescharacteristic, complemented
24#107×C173—21no—
25#102×C173—21no—
26C22⋊C8×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×D692—21no—
30C2×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34D4×D173—42no—
35D4×D173—42no—
36C22×Dic173—21no—
37C22×Dic173D411yescharacteristic
38C2×C4×D173—21no—
39C2×#13063436—21no—
40C4⋊Dic173D411yescharacteristic
41#14583805—21no—
42C22⋊Dic173—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—42no—
48C2×D4×C173—21no—
49C2×D4×C173D411yescharacteristic
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C2×C8×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173—21no—
55C4⋊C4×C173D411yescharacteristic
56C22⋊C4×C173—21no—
57C22×D173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173—42no—
61C22×D173—41no—
62C22×D173—41no—
63C22×D173—81no—
64C22×D173—81no—
65C22×D173D812yes—
66D692—41no—
67D692—42no—
68C22⋊D173—42no—
69C22⋊D173—41no—
70C22⋊D173—41no—
71C22⋊D173—81no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76C2×Dic173—41no—
77C2×Dic173—41no—
78C4×D173—42no—
79#13063436—41no—
80C23×C173—21no—
81C23×C173C2×D411yescharacteristic
82D4×C173—41no—
83D4×C173—41no—
84D4×C173—41no—
85C2×C4×C173—21no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2×D411yescharacteristic
92C8×C173—41no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173#12011yescharacteristic
99C22×C173C2×D811yescharacteristic
100C22×C173AGL(1,ℤ/8)11yescharacteristic
101C2×D173—22no—
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—42no—
105C2×D173—41no—
106C2×D173—41no—
107C2×D173—41no—
108C2×D173—81no—
109C2×D173—84no—
110C4×C173—21no—
111C4×C173—21no—
112C4×C173—41no—
113C4×C173—81no—
114Dic173—41no—
115Dic173—41no—
116Dic173—81no—
117C2×C173—21no—
118C2×C173—21no—
119C2×C173—81no—
120C2×C173#44711yescharacteristic
121C2×C173#45511yescharacteristic
122C2×C173#45811yescharacteristic
123D173—42no—
124D173—81no—
125C173#160711yescharacteristic, complemented, Sylow
126#1607—1731nomaximal, Sylow
127#534—1731no—
128#480—1731no—
129#480D17311yescharacteristic, complemented
130#397—1731no—
131#327—1731no—
132#327—1731no—
133#323—1731no—
134C22×D4—3461no—
135#123—3461no—
136#123—3461no—
137#121—3461no—
138#121—1731no—
139#121C2×D17311yescharacteristic
140#120—3461no—
141#120—3461no—
142C2×C4⋊C4C2×D17311yescharacteristic, complemented
143C2×C22⋊C4—3461no—
144C2×C22⋊C4—1731no—
145#107—21no—
146#107—3461no—
147#102—21no—
148#102—3461no—
149C22⋊C8—1731no—
150C22⋊C8C2×D17311yescharacteristic, complemented
151#95—3461no—
152#95—1731no—
153C24—3461no—
154C24—1731no—
155C2×D4—21no—
156C2×D4—3461no—
157C2×D4—3461no—
158C2×D4—3461no—
159C2×D4—3461no—
160C2×D4—1731no—
161C2×D4—6922no—
162C2×D4—6922no—
163C2×D4C22⋊D17311yescharacteristic
164C22×C4—21no—
165C22×C4—3461no—
166C22×C4—3461no—
167C22×C4—1731no—
168C22×C4C22×D17311yescharacteristic
169C2×C8—21no—
170C2×C8—3461no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—3461no—
174C4⋊C4—1731no—
175C4⋊C4C22⋊D17311yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—3461no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—6922no—
183C23—21no—
184C23—3461no—
185C23—3461no—
186C23—3461no—
187C23—13841no—
188C23—13841no—
189C23—1732no—
190C23—6921no—
191C23—6921no—
192C23—6922no—
193C23D4×D17311yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—13841no—
198D4—6921no—
199D4—6922no—
200D4—6922no—
201D4—6921no—
202D4—6921no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—3461no—
209C2×C4—3461no—
210C2×C4—6922no—
211C2×C4—6921no—
212C2×C4—6921no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4D4×D17311yescharacteristic
216C2×C4C2×C22⋊D17311yescharacteristic
217C8—41no—
218C8—6921no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—3462no—
225C22—13841no—
226C22—13844no—
227C22—6922no—
228C22—6921no—
229C22—6921no—
230C22—6921no—
231C22—6921no—
232C22—6921no—
233C22#1914964711yescharacteristic
234C22D8×D17311yescharacteristic
235C22#1914970311yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—13841no—
241C4—6921no—
242C4—6921no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—13841no—
247C2—6922no—
248C2#2601162311yescharacteristic
249C2#2601166711yescharacteristic
250C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#480×C173C2
socleC22×C173#120
prime radical#26012049C2
hypercenter#480D173
hypocenterC173#1607
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #26011804C2semidirect
3. #26011417C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011214C2semidirect
7. #480×C173C2semidirect
19. C22⋊#13063436C22semidirect
21. #19149586C22semidirect
23. C2×C4⋊C4×C173C22semidirect
26. C22⋊C8×C173C22semidirect
125. C173#1607semidirect
129. #480D173semidirect
142. C2×C4⋊C4C2×D173semidirect
150. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120492
#260118042
#260114172
#260112262
#260112252
#260112142
#480×C1732
#1607173