Finite Group Databasebeta

← #36352509

Subgroups — #36352509

Order 22144.

320 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 53 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#26011641C211yesmaximal, characteristic, complemented
2#26011624C211yesmaximal, characteristic, complemented
3#26011400C211yesmaximal, characteristic, complemented
4#26011228C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011215C211yesmaximal, characteristic, complemented
7#538×C173C211yesmaximal, characteristic, complemented
8C22×D692C2211yescharacteristic
9C22×C22⋊D173—21no—
10C2×#14583813—21no—
11C2×#14583811—21no—
12C2×#14583803—21no—
13#19149599—21no—
14#19149599—21no—
15C22⋊#13063436C2211yescharacteristic
16C22⋊#13063436C2211yescharacteristic
17#19149586C2211yescharacteristic
18#126×C173—21no—
19#124×C173—21no—
20#123×C173—21no—
21C22⋊Q8×C173C2211yescharacteristic
22#121×C173—21no—
23#121×C173C2211yescharacteristic
24#120×C173—21no—
25C4×D4×C173—21no—
26C2×C22⋊C4×C173C2211yescharacteristic
27C23×D173—21no—
28C2×D692—21no—
29C2×D692—21no—
30C2×D692—42no—
31C2×C22⋊D173—42no—
32C2×C22⋊D173—42no—
33C2×C22⋊D173—42no—
34C22×Dic173—21no—
35#14583813—42no—
36#14583811—42no—
37C2×#13063436—21no—
38C2×#13063436—21no—
39#14583803—42no—
40C24×C173D411yescharacteristic
41C2×Q8×C173D411yescharacteristic, complemented
42C2×D4×C173—21no—
43C2×D4×C173—21no—
44C2×D4×C173—21no—
45C2×D4×C173D411yescharacteristic, complemented
46C22×C4×C173—21no—
47C22×C4×C173D411yescharacteristic
48C22×C4×C173C2311yescharacteristic
49C4⋊C4×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173D411yescharacteristic, complemented
52C4⋊C4×C173D411yescharacteristic, complemented
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173—21no—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—21no—
58C22⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C42×C173—21no—
61C22×D173—22no—
62C22×D173—41no—
63C22×D173—41no—
64C22×D173—41no—
65C22×D173—41no—
66C22×D173—41no—
67C22×D173—41no—
68D692—22no—
69D692—22no—
70D692—41no—
71D692—41no—
72C22⋊D173—48no—
73C2×Dic173—22no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76#13063436—41no—
77#13063436—41no—
78C23×C173—21no—
79C23×C173—21no—
80C23×C173—21no—
81C23×C173—21no—
82C23×C173—41no—
83C23×C173—41no—
84C23×C173C2×D411yescharacteristic
85Q8×C173—41no—
86D4×C173—41no—
87D4×C173—41no—
88D4×C173—41no—
89D4×C173—81no—
90D4×C173—81no—
91C2×C4×C173—21no—
92C2×C4×C173—21no—
93C2×C4×C173—21no—
94C2×C4×C173—21no—
95C2×C4×C173—41no—
96C2×C4×C173—41no—
97C2×C4×C173—81no—
98C2×C4×C173C2×D411yescharacteristic
99C2×C4×C173C2×D411yescharacteristic
100C22×C173—21no—
101C22×C173—21no—
102C22×C173—42no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173—41no—
106C22×C173—41no—
107C22×C173—41no—
108C22×C173—41no—
109C22×C173—81no—
110C22×C173#12011yescharacteristic
111C22×C173#13511yescharacteristic
112C22×C173AGL(1,ℤ/8)11yescharacteristic
113C2×D173—42no—
114C2×D173—42no—
115C2×D173—42no—
116C2×D173—81no—
117C2×D173—81no—
118C2×D173—81no—
119C2×D173—81no—
120C4×C173—21no—
121C4×C173—21no—
122C4×C173—41no—
123C4×C173—41no—
124C4×C173—41no—
125C4×C173—41no—
126Dic173—42no—
127C2×C173—21no—
128C2×C173—21no—
129C2×C173—81no—
130C2×C173—81no—
131C2×C173#44911yescharacteristic
132C2×C173#45311yescharacteristic
133C2×C173B(4,𝔽2)11yescharacteristic
134D173—82no—
135C173#160011yescharacteristic, complemented, Sylow
136#1600—1731nomaximal, Sylow
137#538D17311yescharacteristic, complemented
138#450—1731no—
139#447—1731no—
140#392—1731no—
141#328—1731no—
142#327—1731no—
143#323—1731no—
144C22×D4—3461no—
145C22×D4—1731no—
146C2×SD16—3461no—
147C2×D8—3461no—
148#126—21no—
149#124—21no—
150#123—21no—
151C22⋊Q8C2×D17311yescharacteristic
152#121—21no—
153#121C2×D17311yescharacteristic
154#120—21no—
155C4×D4—21no—
156C2×C22⋊C4—3461no—
157C2×C22⋊C4C2×D17311yescharacteristic
158#102—3461no—
159#102—3461no—
160C22⋊C8—1731no—
161C22⋊C8—1731no—
162#95—1731no—
163C24—3461no—
164C24C22⋊D17311yescharacteristic
165C2×Q8C22⋊D17311yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—6922no—
172C2×D4—6922no—
173C2×D4—6922no—
174C2×D4—6922no—
175C2×D4C22⋊D17311yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—3461no—
178C22×C4C22⋊D17311yescharacteristic
179C22×C4C22×D17311yescharacteristic
180SD16—6922no—
181D8—6922no—
182C2×C8—3461no—
183C2×C8—3461no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C22⋊D17311yescharacteristic, complemented
187C4⋊C4C22⋊D17311yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—6922no—
196C42—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—3462no—
204C23—6921no—
205C23—6921no—
206C23—6921no—
207C23—6921no—
208C23—6921no—
209C23—6921no—
210C23C2×C22⋊D17311yescharacteristic
211Q8—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—3462no—
218D4—3462no—
219D4—6921no—
220D4—6921no—
221D4—6928no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—3462no—
230C2×C4—6921no—
231C2×C4—6921no—
232C2×C4C2×C22⋊D17311yescharacteristic
233C2×C4C2×C22⋊D17311yescharacteristic
234C8—6921no—
235C8—6921no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—13841no—
247C22—13841no—
248C22—13841no—
249C22—13841no—
250C22—6922no—
251C22—6922no—
252C22—6922no—
253C22#1914964811yescharacteristic
254C22#1914969911yescharacteristic
255C22#1914970411yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—6922no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—81no—
267C2—13842no—
268C2#2601163611yescharacteristic
269C2#2601165911yescharacteristic
270C2#2601167811yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#538×C173C2
socleC22×C173#120
hypercenter#538D173
hypocenterC173#1600
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #26011641C2semidirect
2. #26011624C2semidirect
3. #26011400C2semidirect
4. #26011228C2semidirect
5. #26011225C2semidirect
6. #26011215C2semidirect
7. #538×C173C2semidirect
41. C2×Q8×C173D4semidirect
45. C2×D4×C173D4semidirect
51. C4⋊C4×C173D4semidirect
52. C4⋊C4×C173D4semidirect
135. C173#1600semidirect
137. #538D173semidirect
165. C2×Q8C22⋊D173semidirect
175. C2×D4C22⋊D173semidirect
186. C4⋊C4C22⋊D173semidirect
187. C4⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260116412
#260116242
#260114002
#260112282
#260112252
#260112152
#538×C1732
#1600173