Finite Group Databasebeta

← #36364030

Subgroups — #36364030

Order 22144.

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

normal subgroups only

This group has 572 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×#19149710C211yesmaximal, characteristic, complemented
2C2×#19149707C211yesmaximal, characteristic, complemented
3C2×#19149704C211yesmaximal, characteristic, complemented
4#26012037C211yesmaximal, characteristic, complemented
5C2×#19149668C211yesmaximal, characteristic, complemented
6#26011884C212yesmaximal, complemented
7#26011877C212yesmaximal, complemented
8#26011876C212yesmaximal, complemented
9#26011872C212yesmaximal, complemented
10M16×Dic173C211yesmaximal, characteristic, complemented
11C2×#137×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×D692C2211yescharacteristic
14#19149710—24no—
15#19149707—24no—
16#19149704—24no—
17C2×#14583816C2212yescomplemented
18C2×#14583815C2212yes—
19C2×#14583813C2212yes—
20C2×#14583813C2212yescomplemented
21C2×D1384C2212yes—
22C2×#14583811C2212yes—
23C2×#14583810C2211yescharacteristic
24#19149677C2211yescharacteristic, complemented
25#19149675C2211yescharacteristic, complemented
26#19149668—22no—
27#19149668C2212yes—
28#19149668C2212yes—
29#19149668C2211yescharacteristic, complemented
30#19149666C2211yescharacteristic, complemented
31#19149651—22no—
32C4×C22⋊D173—22no—
33C2×#14583803—22no—
34C2×C4×Dic173C2211yescharacteristic
35#19149589C2212yescomplemented
36C8×Dic173C2212yescomplemented
37C2×#41×C173C2211yescharacteristic
38C22×Q8×C173C2211yescharacteristic
39#137×C173—24no—
40C2×Q16×C173C2212yescomplemented
41C2×SD16×C173C2212yescomplemented
42C2×M16×C173C2211yescharacteristic
43C23×D173—21no—
44#14583826—22no—
45#14583826—22no—
46C2×Dic346C2311yescharacteristic
47C2×D692—22no—
48C2×D692—22no—
49C2×D692—21no—
50C2×D692C2312yes—
51C2×D692C2311yescharacteristic
52C2×C22⋊D173—22no—
53C2×C22⋊D173—21no—
54C22×Dic173D412yes—
55C2×C4×D173—21no—
56#14583816—24no—
57#14583815—24no—
58#14583813—24no—
59#14583813—24no—
60D1384—24no—
61#14583811—24no—
62#14583810D414yescomplemented
63C2×#13063436C2312yes—
64#14583805—22no—
65C22⋊Dic173—22no—
66#14583803—24no—
67#14583803—24no—
68#14583803—22no—
69C4×Dic173D412yes—
70C4×Dic173C2311yescharacteristic
71C4×Dic173C2311yescharacteristic
72#41×C173—22no—
73#41×C173—22no—
74C2×Q8×C173—21no—
75C2×Q8×C173—22no—
76C2×Q8×C173—22no—
77C2×Q8×C173C2312yes—
78C2×Q8×C173C2311yescharacteristic
79C2×D4×C173—21no—
80C2×D4×C173C2311yescharacteristic
81C22×C4×C173—21no—
82C22×C4×C173—21no—
83C22×C4×C173C2311yescharacteristic
84Q16×C173—24no—
85SD16×C173—24no—
86M16×C173D414yescomplemented
87C2×C8×C173C2312yes—
88C22×D173—21no—
89C22×D173—22no—
90C22×D173—44no—
91C22×D173—42no—
92Dic346—21no—
93Dic346—21no—
94D692—22no—
95D692—21no—
96D692—22no—
97D692—21no—
98D692—42no—
99C22⋊D173—44no—
100C22⋊D173—42no—
101C2×Dic173—21no—
102C2×Dic173—42no—
103C2×Dic173C2×D412yes—
104C2×Dic173C2×D412yes—
105C4×D173—42no—
106#13063436C2×D414yes—
107C23×C173—21no—
108C23×C173C2×D411yescharacteristic
109Q8×C173—21no—
110Q8×C173—22no—
111Q8×C173—22no—
112Q8×C173—21no—
113Q8×C173—42no—
114D4×C173—21no—
115D4×C173—21no—
116D4×C173—42no—
117C2×C4×C173—21no—
118C2×C4×C173—22no—
119C2×C4×C173—42no—
120C2×C4×C173—42no—
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2×D412yes—
123C2×C4×C173C2×D412yes—
124C2×C4×C173C2411yescharacteristic
125C8×C173C2×D414yes—
126C22×C173—22no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173#12712yes—
130C22×C173C22×D411yescharacteristic
131C2×D173—41no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—44no—
135C2×D173—84no—
136C4×C173—41no—
137C4×C173—42no—
138C4×C173#12712yes—
139C4×C173C22×D411yescharacteristic
140C4×C173C22×D411yescharacteristic
141Dic173—22no—
142Dic173—22no—
143Dic173—41no—
144C2×C173—22no—
145C2×C173—81no—
146C2×C173C2×#12711yescharacteristic
147C2×C173#57512yes—
148D173—81no—
149D173—82no—
150C173#315311yescharacteristic, complemented, Sylow
151#3153—1731nomaximal, Sylow
152C2×#137—1731no—
153C2×#137D17311yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—1731no—
155C2×AGL(1,ℤ/8)—1731no—
156#532—1731no—
157C2×#124—1731no—
158#497—1732no—
159#496—1732no—
160#495—1732no—
161#495—1732no—
162C4×M16—1731no—
163C2×#41—1731no—
164C2×#41C2×D17311yescharacteristic
165C22×Q8C2×D17311yescharacteristic
166C22×D4—1731no—
167#137—24no—
168#137—3464no—
169AGL(1,ℤ/8)—3464no—
170AGL(1,ℤ/8)—3464no—
171C2×Q16—1732no—
172C2×Q16C2×D17312yescomplemented
173C2×SD16—1732no—
174C2×SD16—1732no—
175C2×SD16—1732no—
176C2×SD16C2×D17312yescomplemented
177C2×D8—1732no—
178C2×D8—1732no—
179C2×M16—1731no—
180C2×M16C2×D17311yescharacteristic
181C4⋊Q8—1731no—
182#127—1731no—
183#124—3462no—
184#124—1732no—
185#124—1732no—
186#124—1731no—
187#124—1731no—
188#121—3462no—
189C4×D4—3462no—
190C2×C22⋊C4—3462no—
191C2×C42—1731no—
192#97—1732no—
193C4×C8—1732no—
194C24—3461no—
195#41—22no—
196#41—22no—
197#41—3462no—
198#41—3462no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—1731no—
203C2×Q8C22×D17312yes—
204C2×Q8C22×D17311yescharacteristic
205C2×D4—21no—
206C2×D4—3461no—
207C2×D4—3462no—
208C2×D4—3462no—
209C2×D4—3461no—
210C2×D4—3462no—
211C2×D4—1732no—
212C2×D4—1731no—
213C2×D4C22×D17311yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—3461no—
217C22×C4—1732no—
218C22×C4C22×D17311yescharacteristic
219Q16—24no—
220Q16—3464no—
221SD16—24no—
222SD16—3464no—
223SD16—3464no—
224SD16—3464no—
225D8—3464no—
226D8—3464no—
227M16—1734no—
228M16C22⋊D17314yescomplemented
229C2×C8—1732no—
230C2×C8C22×D17312yes—
231C4⋊C4—3462no—
232C22⋊C4—3464no—
233C22⋊C4—3464no—
234C22⋊C4—3462no—
235C22⋊C4—3462no—
236C42—1732no—
237C42—1731no—
238C42—1731no—
239C23—21no—
240C23—3462no—
241C23—3461no—
242C23—6924no—
243C23—6922no—
244C23D4×D17311yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—3461no—
251Q8—3461no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—3462no—
256D4—3461no—
257D4—3461no—
258D4—3462no—
259D4—6924no—
260D4—6922no—
261D4—6922no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—3461no—
267C2×C4—1732no—
268C2×C4—1732no—
269C2×C4—6922no—
270C2×C4—6922no—
271C2×C4D4×D17311yescharacteristic
272C2×C4D4×D17312yes—
273C2×C4C2×C22⋊D17312yes—
274C2×C4C23×D17311yescharacteristic
275C8—1734no—
276C8C2×C22⋊D17314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—13844no—
281C22—6921no—
282C22—6922no—
283C22—6922no—
284C22—6924no—
285C22#1914967512yes—
286C22C2×D4×D17311yescharacteristic
287C4—41no—
288C4—42no—
289C4—3462no—
290C4—3462no—
291C4—6921no—
292C4#1914967512yes—
293C4C2×D4×D17311yescharacteristic
294C4C22×C22⋊D17311yescharacteristic
295C2—22no—
296C2—81no—
297C2—13842no—
298C2—13841no—
299C2C2×#1914967511yescharacteristic
300C2#2601229012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
hypercenterC2×#137D173
hypocenterC173#3153
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19149710C2semidirect
2. C2×#19149707C2semidirect
3. C2×#19149704C2semidirect
4. #26012037C2semidirect
5. C2×#19149668C2semidirect
6. #26011884C2semidirect
7. #26011877C2semidirect
8. #26011876C2semidirect
9. #26011872C2semidirect
10. M16×Dic173C2semidirect
11. C2×#137×C173C2semidirect
17. C2×#14583816C22semidirect
20. C2×#14583813C22semidirect
24. #19149677C22semidirect
25. #19149675C22semidirect
29. #19149668C22semidirect
30. #19149666C22semidirect
35. #19149589C22semidirect
36. C8×Dic173C22semidirect
40. C2×Q16×C173C22semidirect
41. C2×SD16×C173C22semidirect
62. #14583810D4semidirect
86. M16×C173D4semidirect
150. C173#3153semidirect
153. C2×#137D173semidirect
172. C2×Q16C2×D173semidirect
176. C2×SD16C2×D173semidirect
228. M16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
C2×#191497072
C2×#191497042
#260120372
C2×#191496682
#260118842
#260118772
#260118762
#260118722
M16×Dic1732
C2×#137×C1732
#3153173