Finite Group Databasebeta

← #36362843

Subgroups — #36362843

Order 22144.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19149725C211yesmaximal, characteristic
2C2×#19149709C211yesmaximal, characteristic, complemented
3C22×#14583815C211yesmaximal, characteristic, complemented
4#26012067C211yesmaximal, characteristic, complemented
5#26011656C214yesmaximal, complemented
6#26011643C214yesmaximal, complemented
7#26011495C211yesmaximal, characteristic, complemented
8#26011457C211yesmaximal, characteristic, complemented
9#416×C173C211yesmaximal, characteristic, complemented
10#19149725—24no—
11#19149725—24no—
12C2×#14583827C2212yes—
13C2×#14583826—21no—
14C2×#14583826C2211yescharacteristic
15C22×Dic346C2211yescharacteristic, complemented
16C22×D692—21no—
17C22×D692C2211yescharacteristic
18C2×D4×D173—22no—
19C2×D4×D173C2212yes—
20#19149709—24no—
21C2×#14583816C2212yescomplemented
22C2×#14583815—24no—
23C2×#14583815C2214yescomplemented
24C2×#14583814C2212yescomplemented
25C2×#14583810C2211yescharacteristic, complemented
26#19149679—21no—
27#19149667—21no—
28#19149663—22no—
29#19149659C2214yescomplemented
30#19149599C2214yescomplemented
31#19149591C2214yescomplemented
32C2×#41×C173C2211yescharacteristic
33C22×C8×C173C2211yescharacteristic, complemented
34#117×C173C2211yescharacteristic, complemented
35#103×C173C2212yescomplemented
36#102×C173C2212yescomplemented
37C23×D173—22no—
38C23×D173—21no—
39#14583827—24no—
40#14583827—24no—
41#14583826—28no—
42#14583826—22no—
43#14583826—22no—
44C2×Dic346—21no—
45C2×Dic346—22no—
46C2×Dic346—22no—
47C2×Dic346D411yescharacteristic
48C2×Dic346C2312yes—
49C2×D692—28no—
50C2×D692—22no—
51C2×D692—21no—
52C2×D692—22no—
53C2×D692D411yescharacteristic
54C2×D692D414yes—
55C2×D692C2312yes—
56C2×C22⋊D173—22no—
57C2×C22⋊D173D412yes—
58D4×D173—216no—
59D4×D173—24no—
60D4×D173—24no—
61C22×Dic173—21no—
62C2×C4×D173—22no—
63C2×C4×D173C2312yes—
64#14583816—24no—
65#14583815—28no—
66#14583814—24no—
67#14583810—22no—
68C2×#13063436C2312yes—
69C4⋊Dic173—24no—
70#14583803—24no—
71#41×C173—22no—
72#41×C173D414yes—
73C2×Q8×C173C2311yescharacteristic
74C2×D4×C173—21no—
75C2×D4×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173C2311yescharacteristic
78C2×C8×C173—22no—
79C2×C8×C173C2312yes—
80C4⋊C4×C173C2312yes—
81C22⋊C4×C173—21no—
82C42×C173—21no—
83C22×D173—22no—
84C22×D173—28no—
85C22×D173—24no—
86C22×D173—84no—
87C22×D173—82no—
88C22×D173C2×D412yes—
89Dic346—22no—
90Dic346—22no—
91Dic346—22no—
92Dic346—42no—
93D692—28no—
94D692—22no—
95D692—24no—
96D692—22no—
97D692C2×D414yes—
98C22⋊D173—28no—
99C22⋊D173—24no—
100C2×Dic173—22no—
101C2×Dic173—42no—
102C2×Dic173C2×D412yes—
103C4×D173—28no—
104C4×D173—22no—
105C4×D173—22no—
106#13063436—22no—
107C23×C173—21no—
108C23×C173C2×D411yescharacteristic
109Q8×C173—21no—
110Q8×C173C2×D412yes—
111D4×C173—24no—
112D4×C173—21no—
113D4×C173C2×D412yes—
114C2×C4×C173—21no—
115C2×C4×C173—24no—
116C2×C4×C173—22no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2411yescharacteristic
121C8×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—42no—
125C22×C173#12012yes—
126C22×C173C22×D411yescharacteristic
127C2×D173—22no—
128C2×D173—24no—
129C2×D173—24no—
130C2×D173—44no—
131C2×D173—84no—
132C2×D173—88no—
133C4×C173—22no—
134C4×C173—42no—
135C4×C173#12012yes—
136C4×C173C22×D411yescharacteristic
137C4×C173C22×D411yescharacteristic
138Dic173—22no—
139Dic173—42no—
140C2×C173—22no—
141C2×C173—42no—
142C2×C173C2×#12011yescharacteristic
143C2×C173#57712yes—
144D173—42no—
145D173—44no—
146C173#300911yescharacteristic, complemented, Sylow
147#3009—1731nomaximal, Sylow
148C2×H(5,𝔽2)—1731no—
149C2×#137—1731no—
150C22×SD16—1731no—
151#536—1731no—
152#452—1734no—
153#450—1734no—
154#418—1731no—
155#416D17311yescharacteristic, complemented
156#408—1731no—
157H(5,𝔽2)—3464no—
158H(5,𝔽2)—3464no—
159C2×#41—3461no—
160C2×#41—1732no—
161C2×#41—1731no—
162C2×#41C2×D17311yescharacteristic
163C22×Q8—1731no—
164C22×D4—3461no—
165C22×D4—3462no—
166C22×D4—1732no—
167C22×D4—1731no—
168#137—3464no—
169C2×Q16—1732no—
170C2×SD16—3464no—
171C2×SD16—1732no—
172C2×SD16—1734no—
173C2×M16—1731no—
174C22×C8C2×D17311yescharacteristic, complemented
175C4⋊Q8—3461no—
176#124—3461no—
177#123—3462no—
178C22⋊Q8—1734no—
179#117C2×D17311yescharacteristic, complemented
180#103C2×D17312yescomplemented
181#102—1734no—
182#102C2×D17312yescomplemented
183C22⋊C8—1734no—
184C24—3462no—
185C24—3461no—
186#41—22no—
187#41—3468no—
188#41—3464no—
189#41—3464no—
190#41—3462no—
191#41—3462no—
192#41D69214yes—
193C2×Q8—3461no—
194C2×Q8—3462no—
195C2×Q8—3462no—
196C2×Q8—1732no—
197C2×Q8—1731no—
198C2×Q8C22×D17311yescharacteristic
199C2×D4—21no—
200C2×D4—3468no—
201C2×D4—3462no—
202C2×D4—34616no—
203C2×D4—3464no—
204C2×D4—3464no—
205C2×D4—3462no—
206C2×D4—3461no—
207C2×D4—3462no—
208C2×D4—1732no—
209C2×D4—1731no—
210C2×D4—1734no—
211C2×D4—1732no—
212C2×D4C22×D17311yescharacteristic
213C22×C4—21no—
214C22×C4—3462no—
215C22×C4—3461no—
216C22×C4—1732no—
217C22×C4C22×D17311yescharacteristic
218Q16—3464no—
219SD16—3468no—
220SD16—3464no—
221M16—3462no—
222C2×C8—22no—
223C2×C8—1732no—
224C2×C8C22×D17312yes—
225C4⋊C4—3464no—
226C4⋊C4C22×D17312yes—
227C22⋊C4—21no—
228C22⋊C4—3464no—
229C42—21no—
230C23—21no—
231C23—3468no—
232C23—3464no—
233C23—3462no—
234C23—13842no—
235C23—13844no—
236C23—1732no—
237C23D4×D17311yescharacteristic
238Q8—21no—
239Q8—3462no—
240Q8—3462no—
241Q8—3462no—
242Q8—6922no—
243Q8C2×D69212yes—
244D4—24no—
245D4—21no—
246D4—3468no—
247D4—3468no—
248D4—3464no—
249D4—3462no—
250D4—3464no—
251D4—3462no—
252D4—1734no—
253D4C2×D69212yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—3468no—
258C2×C4—3462no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—1732no—
262C2×C4—6922no—
263C2×C4D4×D17311yescharacteristic
264C2×C4D4×D17312yes—
265C2×C4C2×D69212yes—
266C2×C4C23×D17311yescharacteristic
267C8—22no—
268C8—3462no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—3462no—
273C22—3464no—
274C22—3464no—
275C22—13844no—
276C22—13848no—
277C22—6924no—
278C22#1914964912yes—
279C22C2×D4×D17311yescharacteristic
280C4—22no—
281C4—42no—
282C4—3462no—
283C4—6922no—
284C4#1914964912yes—
285C4C2×D4×D17311yescharacteristic
286C4C22×D69211yescharacteristic
287C2—22no—
288C2—42no—
289C2—6922no—
290C2—6924no—
291C2C2×#1914964911yescharacteristic
292C2#2601229812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149725C2
hypercenter#416D173
hypocenterC173#3009
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#19149709C2semidirect
3. C22×#14583815C2semidirect
4. #26012067C2semidirect
5. #26011656C2semidirect
6. #26011643C2semidirect
7. #26011495C2semidirect
8. #26011457C2semidirect
9. #416×C173C2semidirect
15. C22×Dic346C22semidirect
21. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583814C22semidirect
25. C2×#14583810C22semidirect
29. #19149659C22semidirect
30. #19149599C22semidirect
31. #19149591C22semidirect
33. C22×C8×C173C22semidirect
34. #117×C173C22semidirect
35. #103×C173C22semidirect
36. #102×C173C22semidirect
146. C173#3009semidirect
155. #416D173semidirect
174. C22×C8C2×D173semidirect
179. #117C2×D173semidirect
180. #103C2×D173semidirect
182. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497092
C22×#145838152
#260120672
#260116562
#260116432
#260114952
#260114572
#416×C1732
#3009173