Finite Group Databasebeta

← #36362799

Subgroups — #36362799

Order 22144.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×Q8×D173C211yesmaximal, characteristic, complemented
2C2×#19149712C211yesmaximal, characteristic, complemented
3C2×#19149710C211yesmaximal, characteristic, complemented
4#26012003C211yesmaximal, characteristic
5#26011648C214yesmaximal, complemented
6#26011631C214yesmaximal, complemented
7#26011499C211yesmaximal, characteristic, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#419×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C22×Dic346—21no—
12C22×Dic346C2211yescharacteristic
13C2×Q8×D173—24no—
14C2×Q8×D173—24no—
15C2×Q8×D173—22no—
16C2×Q8×D173C2214yes—
17C22×C4×D173—21no—
18C22×C4×D173C2211yescharacteristic
19#19149712—24no—
20#19149710—24no—
21C2×Dic692C2212yescomplemented
22C2×#14583816C2212yescomplemented
23C2×#14583815C2212yescomplemented
24C2×#14583813C2212yescomplemented
25C2×#14583810C2211yescharacteristic
26#19149663—21no—
27#19149659C2212yes—
28#19149654C2212yes—
29#19149649—21no—
30C4×D692—22no—
31#19149602C2214yescomplemented
32#19149591C2214yes—
33C22×Q8×C173C2211yescharacteristic
34C2×M16×C173C2211yescharacteristic
35#117×C173C2211yescharacteristic
36#103×C173C2214yescomplemented
37C23×D173D411yescharacteristic
38#14583826—22no—
39#14583826—22no—
40C2×Dic346—28no—
41C2×Dic346—24no—
42C2×Dic346—21no—
43C2×Dic346—22no—
44C2×Dic346D414yescomplemented
45C2×Dic346C2311yescharacteristic
46C2×Dic346C2312yes—
47Q8×D173—216no—
48Q8×D173—28no—
49Q8×D173—28no—
50C2×D692—22no—
51C2×D692C2311yescharacteristic
52C2×C22⋊D173—21no—
53C22×Dic173—21no—
54C22×Dic173D411yescharacteristic
55C2×C4×D173—28no—
56C2×C4×D173—24no—
57C2×C4×D173—22no—
58C2×C4×D173—22no—
59C2×C4×D173—21no—
60C2×C4×D173D412yes—
61C2×C4×D173C2312yes—
62Dic692—24no—
63#14583816—24no—
64#14583815—24no—
65#14583813—24no—
66#14583810—22no—
67C2×#13063436C2312yes—
68C4⋊Dic173—22no—
69#14583803—22no—
70#14583803—22no—
71C2×Q8×C173—22no—
72C2×Q8×C173—21no—
73C2×Q8×C173D414yescomplemented
74C2×Q8×C173C2312yes—
75C22×C4×C173—21no—
76C22×C4×C173C2311yescharacteristic
77M16×C173—22no—
78C2×C8×C173C2312yes—
79C4⋊C4×C173C2312yes—
80C22⋊C4×C173—21no—
81C42×C173—21no—
82C22×D173—22no—
83C22×D173—24no—
84C22×D173—21no—
85C22×D173C2×D412yes—
86Dic346—216no—
87Dic346—24no—
88Dic346—22no—
89Dic346—21no—
90Dic346—21no—
91Dic346C2×D414yes—
92D692—21no—
93D692—21no—
94D692—44no—
95C22⋊D173—42no—
96C2×Dic173—24no—
97C2×Dic173—22no—
98C2×Dic173—22no—
99C2×Dic173—21no—
100C2×Dic173C2×D412yes—
101C4×D173—216no—
102C4×D173—24no—
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—42no—
106#13063436—22no—
107C23×C173C2×D411yescharacteristic
108Q8×C173—24no—
109Q8×C173—22no—
110Q8×C173C2×D414yes—
111C2×C4×C173—24no—
112C2×C4×C173—22no—
113C2×C4×C173—22no—
114C2×C4×C173C2×D411yescharacteristic
115C2×C4×C173C2×D412yes—
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C22×C173—22no—
120C22×C173#12012yes—
121C22×C173C22×D411yescharacteristic
122C2×D173—28no—
123C2×D173—22no—
124C2×D173—22no—
125C2×D173—41no—
126C2×D173—42no—
127C2×D173#13714yescomplemented
128C4×C173—24no—
129C4×C173—42no—
130C4×C173#12012yes—
131C4×C173C22×D411yescharacteristic
132C4×C173C22×D411yescharacteristic
133Dic173—24no—
134Dic173—22no—
135Dic173—41no—
136C2×C173—22no—
137C2×C173C2×#12011yescharacteristic
138C2×C173C2×#13712yes—
139D173—24no—
140D173—81no—
141C173#300411yescharacteristic, complemented, Sylow
142#3004—1731nomaximal, Sylow
143C23×Q8—1731no—
144C2×#137—1731no—
145C2×#137—1731no—
146#525—1731no—
147#451—1734no—
148#448—1734no—
149#419—1731no—
150#419D17311yescharacteristic, complemented
151#407—1731no—
152C2×#41—1731no—
153C22×Q8—3461no—
154C22×Q8—3464no—
155C22×Q8—3464no—
156C22×Q8—3462no—
157C22×Q8—1734no—
158C22×Q8—1731no—
159C22×Q8C2×D17311yescharacteristic
160C23×C4—3461no—
161C23×C4—1731no—
162#137—3464no—
163#137—3464no—
164C2×Q16—1732no—
165C2×Q16—1732no—
166C2×SD16—1732no—
167C2×SD16—1732no—
168C2×M16—1731no—
169C2×M16C2×D17311yescharacteristic
170#123—3461no—
171C22⋊Q8—1732no—
172#121—1732no—
173#120—3461no—
174C4×D4—3462no—
175#117C2×D17311yescharacteristic
176#103—1734no—
177#103C2×D17314yescomplemented
178C22⋊C8—1734no—
179C24—1731no—
180#41—3462no—
181#41—3462no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—3468no—
185C2×Q8—3464no—
186C2×Q8—34616no—
187C2×Q8—3468no—
188C2×Q8—3468no—
189C2×Q8—3461no—
190C2×Q8—3462no—
191C2×Q8—1734no—
192C2×Q8—1731no—
193C2×Q8—1732no—
194C2×Q8D69214yescomplemented
195C2×Q8C22×D17312yes—
196C2×D4—3462no—
197C2×D4—3461no—
198C2×D4—1731no—
199C22×C4—21no—
200C22×C4—3468no—
201C22×C4—3464no—
202C22×C4—3461no—
203C22×C4—3462no—
204C22×C4—3462no—
205C22×C4—3461no—
206C22×C4—1731no—
207C22×C4—1732no—
208C22×C4—1732no—
209C22×C4C22×D17311yescharacteristic
210Q16—3464no—
211Q16—3464no—
212SD16—3464no—
213SD16—3464no—
214M16—22no—
215M16—3462no—
216C2×C8—1732no—
217C2×C8C22×D17312yes—
218C4⋊C4—3462no—
219C4⋊C4C22×D17312yes—
220C22⋊C4—21no—
221C22⋊C4—3462no—
222C22⋊C4—3462no—
223C42—21no—
224C23—3464no—
225C23—3462no—
226C23—3461no—
227C23—1732no—
228C23D4×D17311yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—34616no—
232Q8—3464no—
233Q8—3462no—
234Q8—3461no—
235Q8—3461no—
236Q8—1734no—
237Q8C2×D69214yes—
238D4—3461no—
239D4—3461no—
240D4—6924no—
241D4—6922no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—34616no—
246C2×C4—3464no—
247C2×C4—3462no—
248C2×C4—3462no—
249C2×C4—3464no—
250C2×C4—3462no—
251C2×C4—3462no—
252C2×C4—3461no—
253C2×C4—1732no—
254C2×C4—6922no—
255C2×C4D4×D17311yescharacteristic
256C2×C4D4×D17312yes—
257C2×C4C2×D69212yes—
258C2×C4C23×D17311yescharacteristic
259C8—22no—
260C8—3462no—
261C22—22no—
262C22—3468no—
263C22—3462no—
264C22—3462no—
265C22—1734no—
266C22—6921no—
267C22—6922no—
268C22#1914964912yes—
269C22C2×D4×D17311yescharacteristic
270C4—24no—
271C4—42no—
272C4—3464no—
273C4—3462no—
274C4—6921no—
275C4#1914964912yes—
276C4C2×D4×D17311yescharacteristic
277C4C22×D69211yescharacteristic
278C2—22no—
279C2—3464no—
280C2—13841no—
281C2C2×#1914964911yescharacteristic
282C2#137×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#419×C173C2
socleC22×C173C22×D4
prime radical#26012003C2
hypercenter#419D173
hypocenterC173#3004
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D173C2semidirect
2. C2×#19149712C2semidirect
3. C2×#19149710C2semidirect
5. #26011648C2semidirect
6. #26011631C2semidirect
7. #26011499C2semidirect
8. #26011454C2semidirect
9. #419×C173C2semidirect
21. C2×Dic692C22semidirect
22. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583813C22semidirect
31. #19149602C22semidirect
36. #103×C173C22semidirect
44. C2×Dic346D4semidirect
73. C2×Q8×C173D4semidirect
127. C2×D173#137semidirect
141. C173#3004semidirect
150. #419D173semidirect
177. #103C2×D173semidirect
194. C2×Q8D692semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1732
C2×#191497122
C2×#191497102
#260120032
#260116482
#260116312
#260114992
#260114542
#419×C1732
#3004173