Finite Group Databasebeta

← #36364040

Subgroups — #36364040

Order 22144.

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

normal subgroups only

This group has 564 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×#19149709C211yesmaximal, characteristic, complemented
2C2×#19149704C211yesmaximal, characteristic, complemented
3C22×#14583815C211yesmaximal, characteristic, complemented
4#26012062C211yesmaximal, characteristic, complemented
5#26012055C211yesmaximal, characteristic, complemented
6#26011884C212yesmaximal, complemented
7#26011879C212yesmaximal, complemented
8#26011872C212yesmaximal, complemented
9#26011862C212yesmaximal, complemented
10#26011446C211yesmaximal, characteristic, complemented
11C2×#135×C173C211yesmaximal, characteristic, complemented
12C22×Dic346C2211yescharacteristic
13C22×D692C2211yescharacteristic
14#19149709—24no—
15#19149704—24no—
16C2×#14583816C2212yescomplemented
17C2×#14583815—24no—
18C2×#14583815C2212yes—
19C2×#14583815C2212yes—
20C2×#14583814C2212yescomplemented
21C2×#14583813C2212yes—
22C2×#14583811C2212yes—
23C2×#14583810C2211yescharacteristic
24#19149677C2212yescomplemented
25#19149675C2212yes—
26#19149668C2212yes—
27#19149666C2212yescomplemented
28#19149660—22no—
29#19149658—22no—
30#19149653—22no—
31#19149647—22no—
32#19149638C2211yescharacteristic
33#19149589C2212yescomplemented
34C8×Dic173C2212yescomplemented
35C2×#41×C173C2211yescharacteristic
36C2×#41×C173C2211yescharacteristic
37#135×C173—24no—
38C2×Q16×C173C2211yescharacteristic, complemented
39C2×SD16×C173C2211yescharacteristic, complemented
40C2×SD16×C173C2211yescharacteristic, complemented
41C2×D8×C173C2211yescharacteristic, complemented
42C22×C8×C173C2211yescharacteristic
43C23×D173—21no—
44C2×Dic346—21no—
45C2×Dic346—22no—
46C2×Dic346—22no—
47C2×Dic346C2312yes—
48C2×D692—22no—
49C2×D692—22no—
50C2×D692—21no—
51C2×D692C2312yes—
52C2×C22⋊D173—24no—
53C22×Dic173—21no—
54#14583816—24no—
55#14583815—24no—
56#14583815—24no—
57#14583814—24no—
58#14583813—24no—
59#14583811—24no—
60#14583810D414yescomplemented
61C2×#13063436C2312yes—
62C4⋊Dic173D412yes—
63#14583805—24no—
64C22⋊Dic173—24no—
65C22⋊Dic173D412yes—
66#14583803—24no—
67C4×Dic173C2312yes—
68#41×C173—22no—
69#41×C173—22no—
70#41×C173—22no—
71#41×C173—22no—
72C2×Q8×C173C2311yescharacteristic
73C2×Q8×C173C2311yescharacteristic
74C2×D4×C173—21no—
75C2×D4×C173—21no—
76C2×D4×C173C2311yescharacteristic
77C2×D4×C173C2311yescharacteristic
78C22×C4×C173—21no—
79C22×C4×C173—21no—
80C22×C4×C173C2311yescharacteristic
81Q16×C173—22no—
82SD16×C173—22no—
83SD16×C173—22no—
84D8×C173—22no—
85C2×C8×C173D414yescomplemented
86C2×C8×C173C2311yescharacteristic
87C2×C8×C173C2311yescharacteristic
88C22×D173—22no—
89C22×D173—42no—
90C22×D173—82no—
91Dic346—22no—
92Dic346—22no—
93Dic346—42no—
94D692—22no—
95D692—22no—
96D692—42no—
97C22⋊D173—84no—
98C2×Dic173—22no—
99C2×Dic173—42no—
100C2×Dic173C2×D414yes—
101#13063436C2×D414yes—
102C23×C173—21no—
103C23×C173—21no—
104C23×C173C2×D411yescharacteristic
105Q8×C173—21no—
106Q8×C173—21no—
107Q8×C173—21no—
108Q8×C173—21no—
109D4×C173—21no—
110D4×C173—21no—
111D4×C173—21no—
112D4×C173—21no—
113D4×C173—42no—
114D4×C173—42no—
115C2×C4×C173—21no—
116C2×C4×C173—21no—
117C2×C4×C173—42no—
118C2×C4×C173—42no—
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173C2×D412yes—
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2411yescharacteristic
123C8×C173C2×D412yes—
124C8×C173C2×D412yes—
125C22×C173—22no—
126C22×C173—41no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173—42no—
130C22×C173#12712yes—
131C22×C173C22×D411yescharacteristic
132C2×D173—42no—
133C2×D173—44no—
134C2×D173—84no—
135C4×C173—41no—
136C4×C173—41no—
137C4×C173#12712yes—
138C4×C173C22×D411yescharacteristic
139C4×C173C22×D411yescharacteristic
140Dic173—24no—
141Dic173—42no—
142C2×C173—22no—
143C2×C173—81no—
144C2×C173—81no—
145C2×C173C2×#12711yescharacteristic
146C2×C173#57712yes—
147D173—82no—
148C173#315411yescharacteristic, complemented, Sylow
149#3154—1731nomaximal, Sylow
150C2×#137—1731no—
151C2×AGL(1,ℤ/8)—1731no—
152C2×#135D17311yescharacteristic, complemented
153C22×SD16—1731no—
154#536—1731no—
155#535—1731no—
156#497—1732no—
157#496—1732no—
158#495—1732no—
159#492—1732no—
160#405—1731no—
161C2×#41C2×D17311yescharacteristic
162C2×#41C2×D17311yescharacteristic
163C22×Q8—1731no—
164C22×D4—1731no—
165#137—3464no—
166AGL(1,ℤ/8)—3464no—
167#135—24no—
168C2×Q16—1732no—
169C2×Q16C2×D17311yescharacteristic, complemented
170C2×SD16—3464no—
171C2×SD16—1732no—
172C2×SD16—1732no—
173C2×SD16—1732no—
174C2×SD16—1732no—
175C2×SD16C2×D17311yescharacteristic, complemented
176C2×SD16C2×D17311yescharacteristic, complemented
177C2×D8—1732no—
178C2×D8C2×D17311yescharacteristic, complemented
179C2×M16—1731no—
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#117—1731no—
190#97—1732no—
191C4×C8—1732no—
192C24—3461no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—3461no—
198C2×Q8—3462no—
199C2×Q8—3462no—
200C2×Q8—1732no—
201C2×Q8C22×D17311yescharacteristic
202C2×Q8C22×D17311yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—3464no—
206C2×D4—3462no—
207C2×D4—3461no—
208C2×D4—3462no—
209C2×D4—1732no—
210C2×D4C22×D17311yescharacteristic
211C2×D4C22×D17311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—3461no—
215C22×C4C22×D17311yescharacteristic
216Q16—22no—
217Q16—3464no—
218SD16—22no—
219SD16—22no—
220SD16—3464no—
221SD16—3464no—
222SD16—3464no—
223SD16—3464no—
224D8—22no—
225D8—3464no—
226M16—1734no—
227C2×C8—1732no—
228C2×C8C22⋊D17314yescomplemented
229C2×C8C22×D17311yescharacteristic
230C2×C8C22×D17311yescharacteristic
231C4⋊C4—3464no—
232C4⋊C4—1732no—
233C22⋊C4—3464no—
234C22⋊C4—3464no—
235C22⋊C4—1732no—
236C42—1732no—
237C23—21no—
238C23—21no—
239C23—3462no—
240C23—13842no—
241C23—6922no—
242C23D4×D17311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—3462no—
248Q8—3462no—
249Q8—6922no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—3462no—
257D4—3462no—
258D4—13844no—
259D4—6922no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3462no—
265C2×C4—1734no—
266C2×C4—6922no—
267C2×C4D4×D17311yescharacteristic
268C2×C4D4×D17312yes—
269C2×C4C2×C22⋊D17312yes—
270C2×C4C23×D17311yescharacteristic
271C8—1734no—
272C8C2×C22⋊D17312yes—
273C8C2×C22⋊D17312yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—13844no—
280C22—6922no—
281C22—6924no—
282C22#1914967512yes—
283C22C2×D4×D17311yescharacteristic
284C4—41no—
285C4—41no—
286C4—3464no—
287C4—6922no—
288C4#1914967512yes—
289C4C2×D4×D17311yescharacteristic
290C4C22×C22⋊D17311yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—13842no—
295C2C2×#1914967511yescharacteristic
296C2#2601229812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#135×C173C2
socleC22×C173C22×D4
hypercenterC2×#135D173
hypocenterC173#3154
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#19149709C2semidirect
2. C2×#19149704C2semidirect
3. C22×#14583815C2semidirect
4. #26012062C2semidirect
5. #26012055C2semidirect
6. #26011884C2semidirect
7. #26011879C2semidirect
8. #26011872C2semidirect
9. #26011862C2semidirect
10. #26011446C2semidirect
11. C2×#135×C173C2semidirect
16. C2×#14583816C22semidirect
20. C2×#14583814C22semidirect
24. #19149677C22semidirect
27. #19149666C22semidirect
33. #19149589C22semidirect
34. C8×Dic173C22semidirect
38. C2×Q16×C173C22semidirect
39. C2×SD16×C173C22semidirect
40. C2×SD16×C173C22semidirect
41. C2×D8×C173C22semidirect
60. #14583810D4semidirect
85. C2×C8×C173D4semidirect
148. C173#3154semidirect
152. C2×#135D173semidirect
169. C2×Q16C2×D173semidirect
175. C2×SD16C2×D173semidirect
176. C2×SD16C2×D173semidirect
178. C2×D8C2×D173semidirect
228. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497092
C2×#191497042
C22×#145838152
#260120622
#260120552
#260118842
#260118792
#260118722
#260118622
#260114462
C2×#135×C1732
#3154173