Finite Group Databasebeta

← #36352512

Subgroups — #36352512

Order 22144.

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

normal subgroups only

This group has 320 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#26012081C211yesmaximal, characteristic, complemented
2#26011641C211yesmaximal, characteristic, complemented
3#26011627C211yesmaximal, characteristic, complemented
4#26011401C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8C22×D692—21no—
9C22×D692C2211yescharacteristic
10C2×#14583813—21no—
11C2×D1384—21no—
12#19149674—21no—
13#19149668—21no—
14#19149664—21no—
15#19149654—21no—
16#19149653C2211yescharacteristic
17#19149649—21no—
18#19149639—21no—
19C2×#14583803—21no—
20C2×#14583803C2211yescharacteristic
21#19149601—21no—
22#19149599—21no—
23C22⋊#13063436C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×D692—21no—
30C2×D692—21no—
31C2×D692—21no—
32C2×D692—21no—
33C2×D692—42no—
34C2×D692—42no—
35C2×D692—42no—
36C2×D692—42no—
37C2×D692D411yescharacteristic, complemented
38C2×C22⋊D173—21no—
39C22×Dic173D411yescharacteristic
40C2×C4×D173—21no—
41#14583813—42no—
42D1384—42no—
43C2×#13063436—21no—
44C4⋊Dic173D411yescharacteristic, complemented
45#14583805—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51#14583803—21no—
52#14583803—42no—
53C4×Dic173—21no—
54C2×Q8×C173D411yescharacteristic, complemented
55C22×C4×C173—21no—
56C22×C4×C173C2311yescharacteristic
57C2×C8×C173—21no—
58C4⋊C4×C173—21no—
59C4⋊C4×C173D411yescharacteristic, complemented
60C22⋊C4×C173—21no—
61C22×D173—22no—
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—21no—
66C22×D173—41no—
67C22×D173—41no—
68C22×D173—41no—
69C22×D173—41no—
70C22×D173—41no—
71C22×D173—41no—
72C22×D173—41no—
73C22×D173—41no—
74D692—22no—
75D692—22no—
76D692—41no—
77D692—41no—
78D692—41no—
79D692—48no—
80D692—81no—
81D692—81no—
82C22⋊D173—41no—
83C22⋊D173—41no—
84C2×Dic173—21no—
85C2×Dic173—21no—
86C2×Dic173—41no—
87C4×D173—81no—
88#13063436—41no—
89C23×C173C2×D411yescharacteristic
90Q8×C173—41no—
91C2×C4×C173—22no—
92C2×C4×C173—21no—
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#12011yescharacteristic
103C22×C173#13511yescharacteristic
104C22×C173AGL(1,ℤ/8)11yescharacteristic
105C2×D173—42no—
106C2×D173—42no—
107C2×D173—42no—
108C2×D173—41no—
109C2×D173—41no—
110C2×D173—41no—
111C2×D173—41no—
112C2×D173—42no—
113C2×D173—41no—
114C2×D173—41no—
115C2×D173—81no—
116C2×D173—81no—
117C2×D173—81no—
118C2×D173—81no—
119C2×D173—81no—
120C4×C173—21no—
121C4×C173—21no—
122C4×C173—42no—
123C4×C173—41no—
124C4×C173—41no—
125Dic173—41no—
126Dic173—41no—
127C2×C173—21no—
128C2×C173—21no—
129C2×C173#44911yescharacteristic
130C2×C173#45311yescharacteristic
131C2×C173B(4,𝔽2)11yescharacteristic
132D173—81no—
133D173—81no—
134D173—82no—
135C173#160011yescharacteristic, complemented, Sylow
136#1600—1731nomaximal, Sylow
137#538—1731no—
138#450—1731no—
139#447—1731no—
140#392—1731no—
141#328D17311yescharacteristic, complemented
142#327—1731no—
143#323—1731no—
144C22×D4—3461no—
145C22×D4—1731no—
146C2×SD16—3461no—
147C2×D8—3461no—
148#126—3461no—
149#124—3461no—
150#123—3461no—
151C22⋊Q8C2×D17311yescharacteristic
152#121—3461no—
153#121—1731no—
154#120—3461no—
155C4×D4—3461no—
156C2×C22⋊C4—3461no—
157C2×C22⋊C4—1731no—
158#102—3461no—
159#102—3461no—
160C22⋊C8—1731no—
161C22⋊C8C2×D17311yescharacteristic
162#95C2×D17311yescharacteristic
163C24—3461no—
164C24—1731no—
165C2×Q8D69211yescharacteristic, complemented
166C2×D4—3461no—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—1731no—
172C2×D4—6922no—
173C2×D4—6922no—
174C2×D4—6922no—
175C2×D4—6922no—
176C22×C4—21no—
177C22×C4—3461no—
178C22×C4—1731no—
179C22×C4C22×D17311yescharacteristic
180SD16—6922no—
181D8—6922no—
182C2×C8—21no—
183C2×C8—3461no—
184C4⋊C4—21no—
185C4⋊C4—3461no—
186C4⋊C4—1731no—
187C4⋊C4D69211yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—3461no—
190C22⋊C4—3461no—
191C22⋊C4—3461no—
192C22⋊C4—3461no—
193C22⋊C4—3461no—
194C22⋊C4—3461no—
195C22⋊C4—6922no—
196C42—3461no—
197C23—3462no—
198C23—3461no—
199C23—3461no—
200C23—3461no—
201C23—3461no—
202C23—6921no—
203C23—6921no—
204C23—6921no—
205C23—6921no—
206C23—6921no—
207C23—6921no—
208C23—6921no—
209C23—6921no—
210C23D4×D17311yescharacteristic
211Q8—41no—
212D4—3462no—
213D4—3462no—
214D4—13841no—
215D4—13841no—
216D4—6921no—
217D4—6921no—
218D4—6921no—
219D4—6928no—
220D4—6921no—
221D4—6921no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3461no—
229C2×C4—3461no—
230C2×C4—13841no—
231C2×C4—6921no—
232C2×C4D4×D17311yescharacteristic
233C2×C4C2×D69211yescharacteristic
234C8—41no—
235C8—6921no—
236C22—21no—
237C22—21no—
238C22—13841no—
239C22—13841no—
240C22—13841no—
241C22—13841no—
242C22—13841no—
243C22—6922no—
244C22—6922no—
245C22—6922no—
246C22—6921no—
247C22—6921no—
248C22—6922no—
249C22—6921no—
250C22—6921no—
251C22—6921no—
252C22—6921no—
253C22#1914964911yescharacteristic
254C22#1914970211yescharacteristic
255C22#1914970611yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—6921no—
262C4—6921no—
263C2—21no—
264C2—21no—
265C2—13842no—
266C2—13841no—
267C2—13841no—
268C2#2601163911yescharacteristic
269C2#2601166011yescharacteristic
270C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
hypercenter#328D173
hypocenterC173#1600
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #26012081C2semidirect
2. #26011641C2semidirect
3. #26011627C2semidirect
4. #26011401C2semidirect
5. #26011224C2semidirect
6. #26011214C2semidirect
7. #328×C173C2semidirect
37. C2×D692D4semidirect
44. C4⋊Dic173D4semidirect
54. C2×Q8×C173D4semidirect
59. C4⋊C4×C173D4semidirect
135. C173#1600semidirect
141. #328D173semidirect
165. C2×Q8D692semidirect
187. C4⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260120812
#260116412
#260116272
#260114012
#260112242
#260112142
#328×C1732
#1600173