Finite Group Databasebeta

← #36364039

Subgroups — #36364039

Order 22144.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19149712C211yesmaximal, characteristic, complemented
2C2×#19149707C211yesmaximal, characteristic, complemented
3C2×#19149701C211yesmaximal, characteristic, complemented
4C22×#14583815C211yesmaximal, characteristic, complemented
5#26012067C211yesmaximal, characteristic, complemented
6#26012060C211yesmaximal, characteristic, complemented
7#26011885C212yesmaximal, complemented
8#26011880C212yesmaximal, complemented
9#26011875C212yesmaximal, complemented
10#26011864C212yesmaximal, complemented
11#405×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C2×#14583826C2211yescharacteristic
14C22×Dic346C2211yescharacteristic
15C22×D692C2211yescharacteristic
16#19149712—24no—
17#19149707—24no—
18#19149701—24no—
19C2×Dic692C2211yescharacteristic, complemented
20C2×Dic692C2212yescomplemented
21C2×#14583815—24no—
22C2×#14583815C2212yes—
23C2×#14583815C2212yes—
24C2×#14583815C2212yes—
25C2×#14583815C2211yescharacteristic, complemented
26C2×#14583815C2211yescharacteristic, complemented
27C2×#14583815C2212yescomplemented
28C2×D1384C2212yes—
29C2×D1384C2211yescharacteristic, complemented
30#19149679C2212yescomplemented
31#19149676C2212yes—
32#19149667C2212yes—
33#19149667C2212yescomplemented
34#19149663—22no—
35#19149659—22no—
36#19149654—22no—
37#19149649—22no—
38C2×M16×C173C2211yescharacteristic
39C22×C8×C173C2211yescharacteristic
40#117×C173C2211yescharacteristic
41#97×C173C2212yescomplemented
42C4×C8×C173C2212yescomplemented
43C23×D173—21no—
44#14583826—22no—
45#14583826—22no—
46#14583826—22no—
47#14583826—22no—
48C2×Dic346—21no—
49C2×Dic346—22no—
50C2×Dic346—22no—
51C2×Dic346C2311yescharacteristic
52C2×Dic346C2312yes—
53C2×Dic346C2311yescharacteristic
54C2×D692—22no—
55C2×D692—22no—
56C2×D692—21no—
57C2×D692—24no—
58C2×D692C2312yes—
59C2×D692C2311yescharacteristic
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C22×Dic173—21no—
64C2×C4×D173—21no—
65C2×C4×D173—21no—
66Dic692—22no—
67Dic692—24no—
68#14583815—24no—
69#14583815—24no—
70#14583815—24no—
71#14583815—22no—
72#14583815—22no—
73#14583815—24no—
74D1384—24no—
75D1384—22no—
76C4⋊Dic173—24no—
77#14583803—24no—
78#14583803—24no—
79C22×C4×C173C2311yescharacteristic
80M16×C173D414yescomplemented
81C2×C8×C173D414yescomplemented
82C2×C8×C173C2311yescharacteristic
83C2×C8×C173C2311yescharacteristic
84C2×C8×C173C2312yes—
85C4⋊C4×C173D412yes—
86C22⋊C4×C173D412yes—
87C42×C173C2312yes—
88C22×D173—21no—
89C22×D173—21no—
90C22×D173—22no—
91C22×D173—42no—
92C22×D173—82no—
93Dic346—22no—
94Dic346—21no—
95Dic346—22no—
96Dic346—21no—
97Dic346—21no—
98Dic346—21no—
99Dic346—42no—
100D692—21no—
101D692—22no—
102D692—21no—
103D692—21no—
104D692—22no—
105D692—21no—
106D692—42no—
107D692—84no—
108C22⋊D173—42no—
109C22⋊D173—42no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—22no—
113C2×Dic173—42no—
114C4×D173—42no—
115C4×D173—42no—
116C23×C173C2×D411yescharacteristic
117C2×C4×C173C2×D414yes—
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2×D412yes—
121C2×C4×C173C2411yescharacteristic
122C8×C173C2×D412yes—
123C8×C173C2×D414yes—
124C8×C173C2×D412yes—
125C22×C173—22no—
126C22×C173#12712yes—
127C22×C173C22×D411yescharacteristic
128C2×D173—41no—
129C2×D173—41no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—44no—
133C2×D173—42no—
134C2×D173—84no—
135C4×C173—24no—
136C4×C173#12712yes—
137C4×C173C22×D411yescharacteristic
138C4×C173C22×D411yescharacteristic
139Dic173—41no—
140Dic173—41no—
141Dic173—42no—
142C2×C173—22no—
143C2×C173C2×#12711yescharacteristic
144C2×C173#57712yes—
145D173—81no—
146D173—81no—
147D173—82no—
148C173#315411yescharacteristic, complemented, Sylow
149#3154—1731nomaximal, Sylow
150C2×#137—1731no—
151C2×AGL(1,ℤ/8)—1731no—
152C2×#135—1731no—
153C22×SD16—1731no—
154#536—1731no—
155#535—1731no—
156#497—1732no—
157#496—1732no—
158#495—1732no—
159#492—1732no—
160#405D17311yescharacteristic, complemented
161C2×#41—1731no—
162C2×#41—1731no—
163C22×Q8—1731no—
164C22×D4—1731no—
165#137—3464no—
166AGL(1,ℤ/8)—3464no—
167#135—3464no—
168C2×Q16—1731no—
169C2×Q16—1732no—
170C2×SD16—3464no—
171C2×SD16—1732no—
172C2×SD16—1732no—
173C2×SD16—1731no—
174C2×SD16—1731no—
175C2×SD16—1732no—
176C2×SD16—1732no—
177C2×D8—1731no—
178C2×D8—1732no—
179C2×M16C2×D17311yescharacteristic
180C22×C8C2×D17311yescharacteristic
181C4⋊Q8—1732no—
182#127—1732no—
183#124—1732no—
184#124—1732no—
185#123—3462no—
186C22⋊Q8—3462no—
187#121—3462no—
188#120—3462no—
189#117C2×D17311yescharacteristic
190#97C2×D17312yescomplemented
191C4×C8C2×D17312yescomplemented
192C24—3461no—
193#41—3462no—
194#41—3462no—
195#41—3462no—
196#41—3462no—
197C2×Q8—3461no—
198C2×Q8—3462no—
199C2×Q8—3462no—
200C2×Q8—1731no—
201C2×Q8—1732no—
202C2×Q8—1731no—
203C2×D4—3461no—
204C2×D4—3461no—
205C2×D4—3462no—
206C2×D4—3462no—
207C2×D4—3464no—
208C2×D4—3461no—
209C2×D4—1731no—
210C2×D4—1732no—
211C2×D4—1731no—
212C22×C4—3461no—
213C22×C4—3461no—
214C22×C4—3461no—
215C22×C4C22×D17311yescharacteristic
216Q16—3462no—
217Q16—3464no—
218SD16—3464no—
219SD16—3464no—
220SD16—3462no—
221SD16—3462no—
222SD16—3464no—
223SD16—3464no—
224D8—3462no—
225D8—3464no—
226M16D69214yescomplemented
227C2×C8D69214yescomplemented
228C2×C8C22×D17311yescharacteristic
229C2×C8C22×D17312yes—
230C2×C8C22×D17311yescharacteristic
231C4⋊C4—3464no—
232C4⋊C4D69212yes—
233C22⋊C4—3464no—
234C22⋊C4—3464no—
235C22⋊C4D69212yes—
236C42C22×D17312yes—
237C23—3461no—
238C23—3462no—
239C23—3461no—
240C23—13842no—
241C23—6922no—
242C23C2×D69211yescharacteristic
243Q8—3462no—
244Q8—3461no—
245Q8—3462no—
246Q8—3461no—
247Q8—3461no—
248Q8—3461no—
249Q8—6922no—
250D4—3461no—
251D4—3462no—
252D4—3461no—
253D4—3461no—
254D4—3462no—
255D4—3461no—
256D4—13844no—
257D4—6922no—
258D4—6922no—
259D4—6922no—
260C2×C4—3461no—
261C2×C4—3462no—
262C2×C4—3461no—
263C2×C4—6922no—
264C2×C4—6922no—
265C2×C4—6922no—
266C2×C4C2×D69214yes—
267C2×C4C2×D69211yescharacteristic
268C2×C4C2×D69212yes—
269C2×C4C2×D69212yes—
270C2×C4C23×D17311yescharacteristic
271C8C2×D69212yes—
272C8C2×D69212yes—
273C8C2×D69214yes—
274C22—22no—
275C22—13844no—
276C22—6921no—
277C22—6921no—
278C22—6922no—
279C22—6922no—
280C22—6924no—
281C22—6922no—
282C22#1914967612yes—
283C22C22×D69211yescharacteristic
284C4—24no—
285C4—6921no—
286C4—6921no—
287C4—6922no—
288C4#1914967612yes—
289C4C22×D69211yescharacteristic
290C4C22×D69211yescharacteristic
291C2—22no—
292C2—13842no—
293C2—13841no—
294C2—13841no—
295C2C2×#1914967611yescharacteristic
296C2#2601230312yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#405×C173C2
socleC22×C173C22×D4
hypercenter#405D173
hypocenterC173#3154
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×#19149707C2semidirect
3. C2×#19149701C2semidirect
4. C22×#14583815C2semidirect
5. #26012067C2semidirect
6. #26012060C2semidirect
7. #26011885C2semidirect
8. #26011880C2semidirect
9. #26011875C2semidirect
10. #26011864C2semidirect
11. #405×C173C2semidirect
19. C2×Dic692C22semidirect
20. C2×Dic692C22semidirect
25. C2×#14583815C22semidirect
26. C2×#14583815C22semidirect
27. C2×#14583815C22semidirect
29. C2×D1384C22semidirect
30. #19149679C22semidirect
33. #19149667C22semidirect
41. #97×C173C22semidirect
42. C4×C8×C173C22semidirect
80. M16×C173D4semidirect
81. C2×C8×C173D4semidirect
148. C173#3154semidirect
160. #405D173semidirect
190. #97C2×D173semidirect
191. C4×C8C2×D173semidirect
226. M16D692semidirect
227. C2×C8D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×#191497072
C2×#191497012
C22×#145838152
#260120672
#260120602
#260118852
#260118802
#260118752
#260118642
#405×C1732
#3154173