Finite Group Databasebeta

← #36352513

Subgroups — #36352513

Order 22144.

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

normal subgroups only

This group has 308 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#26011630C211yesmaximal, characteristic, complemented
3#26011402C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#448×C173C211yesmaximal, characteristic, complemented
8C2×D4×D173—21no—
9C2×#14583813—21no—
10#19149664—21no—
11#19149661—21no—
12#19149653C2211yescharacteristic
13#19149651—21no—
14#19149649—21no—
15#19149647—21no—
16C2×#14583803—21no—
17C2×#14583803C2211yescharacteristic
18#19149603—21no—
19C22⋊#13063436C2211yescharacteristic, complemented
20#19149586—21no—
21#19149586C2211yescharacteristic, complemented
22C22×Q8×C173C2211yescharacteristic, complemented
23C2×SD16×C173—21no—
24#121×C173C2211yescharacteristic
25#103×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—
39#14583813—42no—
40C2×#13063436—21no—
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43C22⋊Dic173—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—42no—
49C2×Q8×C173—21no—
50C2×Q8×C173—21no—
51C2×Q8×C173—42no—
52C2×D4×C173—21no—
53C2×D4×C173D411yescharacteristic
54C22×C4×C173—21no—
55C22×C4×C173C2311yescharacteristic
56SD16×C173—42no—
57C2×C8×C173—21no—
58C4⋊C4×C173D411yescharacteristic
59C22⋊C4×C173—21no—
60C22×D173—21no—
61C22×D173—21no—
62C22×D173—21no—
63C22×D173—42no—
64C22×D173—41no—
65C22×D173—41no—
66C22×D173—81no—
67C22×D173—81no—
68C22×D173SD1612yescomplemented
69D692—41no—
70D692—42no—
71C22⋊D173—42no—
72C22⋊D173—41no—
73C22⋊D173—41no—
74C22⋊D173—81no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—41no—
78C2×Dic173—41no—
79C2×Dic173—41no—
80C2×Dic173—41no—
81C4×D173—42no—
82#13063436—41no—
83C23×C173—21no—
84C23×C173C2×D411yescharacteristic
85Q8×C173—22no—
86Q8×C173—22no—
87Q8×C173—41no—
88Q8×C173—41no—
89D4×C173—41no—
90D4×C173—41no—
91D4×C173—41no—
92C2×C4×C173—22no—
93C2×C4×C173—21no—
94C2×C4×C173—41no—
95C2×C4×C173—41no—
96C2×C4×C173—41no—
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173C2×D411yescharacteristic
99C8×C173—41no—
100C22×C173—21no—
101C22×C173—21no—
102C22×C173—41no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173#12011yescharacteristic
106C22×C173C2×SD1611yescharacteristic
107C22×C173AGL(1,ℤ/8)11yescharacteristic
108C2×D173—22no—
109C2×D173—41no—
110C2×D173—41no—
111C2×D173—42no—
112C2×D173—41no—
113C2×D173—41no—
114C2×D173—41no—
115C2×D173—81no—
116C2×D173—84no—
117C4×C173—21no—
118C4×C173—21no—
119C4×C173—41no—
120C4×C173—42no—
121Dic173—41no—
122Dic173—41no—
123Dic173—81no—
124C2×C173—21no—
125C2×C173—21no—
126C2×C173—81no—
127C2×C173#45011yescharacteristic
128C2×C173#45511yescharacteristic
129C2×C173B(4,𝔽2)11yescharacteristic
130D173—42no—
131D173—81no—
132C173#160111yescharacteristic, complemented, Sylow
133#1601—1731nomaximal, Sylow
134#534—1731no—
135#448—1731no—
136#448D17311yescharacteristic, complemented
137#393—1731no—
138#327—1731no—
139#327—1731no—
140#323—1731no—
141C22×Q8C2×D17311yescharacteristic, complemented
142C22×D4—3461no—
143C2×SD16—21no—
144C2×SD16—3461no—
145#123—3461no—
146#123—3461no—
147#121—3461no—
148#121—1731no—
149#121C2×D17311yescharacteristic
150#120—3461no—
151#120—3461no—
152C2×C22⋊C4—3461no—
153C2×C22⋊C4—1731no—
154#103—21no—
155#103—3461no—
156C22⋊C8—1731no—
157C22⋊C8C2×D17311yescharacteristic, complemented
158#95—3461no—
159#95—1731no—
160C24—3461no—
161C24—1731no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×D4—21no—
166C2×D4—3461no—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3461no—
170C2×D4—1731no—
171C2×D4—6922no—
172C2×D4—6922no—
173C2×D4C22⋊D17311yescharacteristic
174C22×C4—21no—
175C22×C4—3461no—
176C22×C4—3461no—
177C22×C4—1731no—
178C22×C4C22×D17311yescharacteristic
179SD16—42no—
180SD16—6922no—
181C2×C8—21no—
182C2×C8—3461no—
183C4⋊C4—3461no—
184C4⋊C4—1731no—
185C4⋊C4C22⋊D17311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3461no—
188C22⋊C4—3461no—
189C22⋊C4—3461no—
190C22⋊C4—3461no—
191C22⋊C4—3461no—
192C22⋊C4—6922no—
193C23—21no—
194C23—3461no—
195C23—3461no—
196C23—3461no—
197C23—13841no—
198C23—13841no—
199C23—1732no—
200C23—6921no—
201C23—6921no—
202C23—6922no—
203C23D4×D17311yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—13841no—
212D4—6921no—
213D4—6922no—
214D4—6922no—
215D4—6921no—
216D4—6921no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3461no—
223C2×C4—3461no—
224C2×C4—6922no—
225C2×C4—6921no—
226C2×C4—6921no—
227C2×C4—6921no—
228C2×C4—6921no—
229C2×C4D4×D17311yescharacteristic
230C2×C4C2×C22⋊D17311yescharacteristic
231C8—41no—
232C8—6921no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—3462no—
239C22—13841no—
240C22—13844no—
241C22—6922no—
242C22—6921no—
243C22—6921no—
244C22—6921no—
245C22—6921no—
246C22—6921no—
247C22#1914964711yescharacteristic
248C22SD16×D17311yescharacteristic
249C22#1914970611yescharacteristic
250C4—21no—
251C4—21no—
252C4—42no—
253C4—41no—
254C4—13841no—
255C4—6921no—
256C4—6921no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—13841no—
261C2—6922no—
262C2#2601164011yescharacteristic
263C2#2601166711yescharacteristic
264C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#448×C173C2
socleC22×C173#120
prime radical#26012049C2
hypercenter#448D173
hypocenterC173#1601
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #26011630C2semidirect
3. #26011402C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011214C2semidirect
7. #448×C173C2semidirect
19. C22⋊#13063436C22semidirect
21. #19149586C22semidirect
22. C22×Q8×C173C22semidirect
26. C22⋊C8×C173C22semidirect
68. C22×D173SD16semidirect
132. C173#1601semidirect
136. #448D173semidirect
141. C22×Q8C2×D173semidirect
157. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120492
#260116302
#260114022
#260112262
#260112252
#260112142
#448×C1732
#1601173