Finite Group Databasebeta

← #36364577

Subgroups — #36364577

Order 22144.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#127×D173C211yesmaximal, characteristic, complemented
2#26012020C211yesmaximal, characteristic, complemented
3#26011881C211yesmaximal, characteristic, complemented
4#26011856C211yesmaximal, characteristic, complemented
5#26011855C211yesmaximal, characteristic, complemented
6#26011689C212yesmaximal, complemented
7#26011684C212yesmaximal, complemented
8#26011640C212yesmaximal, complemented
9#26011623C212yesmaximal, complemented
10#26011546C211yesmaximal, characteristic, complemented
11#496×C173C211yesmaximal, characteristic, complemented
12C2×D4×D173—21no—
13C2×D4×D173—21no—
14C2×D4×D173C2212yes—
15C2×#14583813C2212yescomplemented
16C2×#14583811C2212yescomplemented
17#19149676C2211yescharacteristic
18#19149675—21no—
19#19149675—21no—
20#19149675C2212yes—
21#19149672—21no—
22#19149671C2211yescharacteristic, complemented
23#19149661—21no—
24#19149655C2211yescharacteristic
25#19149650C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic, complemented
27D4×Dic173C2211yescharacteristic, complemented
28#19149639—21no—
29#19149635—21no—
30C42×D173C2211yescharacteristic
31#19149608C2212yescomplemented
32#19149601C2212yescomplemented
33#19149601C2212yescomplemented
34#19149600C2212yescomplemented
35#19149600C2212yescomplemented
36#19149591C2212yes—
37#19149590C2211yescharacteristic, complemented
38C2×SD16×C173C2212yescomplemented
39C2×D8×C173C2212yescomplemented
40#127×C173C2211yescharacteristic
41#124×C173C2211yescharacteristic, complemented
42#97×C173C2211yescharacteristic, complemented
43C23×D173—22no—
44C2×D692—21no—
45C2×D692—21no—
46C2×D692C2312yes—
47C2×C22⋊D173—21no—
48C2×C22⋊D173—22no—
49C2×C22⋊D173—22no—
50C2×C22⋊D173—22no—
51D4×D173—42no—
52D4×D173—42no—
53D4×D173—44no—
54D4×D173—44no—
55C22×Dic173—21no—
56C2×C4×D173D411yescharacteristic
57C2×C4×D173D411yescharacteristic
58C2×C4×D173C2311yescharacteristic
59#14583813—42no—
60#14583811—42no—
61C2×#13063436C2312yes—
62C4⋊Dic173—21no—
63C4⋊Dic173C2311yescharacteristic
64C4⋊Dic173C2311yescharacteristic
65#14583805—21no—
66#14583805—21no—
67#14583805—21no—
68C22⋊Dic173—21no—
69C22⋊Dic173—21no—
70#14583803—21no—
71C4×Dic173—21no—
72C4×Dic173D411yescharacteristic
73C4×Dic173D411yescharacteristic
74C4×Dic173C2311yescharacteristic
75C2×Q8×C173C2311yescharacteristic
76C2×D4×C173—21no—
77C2×D4×C173—21no—
78C2×D4×C173C2311yescharacteristic
79C2×D4×C173C2312yes—
80SD16×C173—42no—
81D8×C173—42no—
82C2×C8×C173C2312yes—
83C22⋊C4×C173—21no—
84C22⋊C4×C173—21no—
85C42×C173C2311yescharacteristic
86C22×D173—22no—
87C22×D173—44no—
88C22×D173—44no—
89C22×D173—44no—
90C22×D173C2×D411yescharacteristic
91D692—22no—
92D692—22no—
93D692—42no—
94D692—42no—
95C22⋊D173—44no—
96C22⋊D173—44no—
97C22⋊D173—42no—
98C22⋊D173—44no—
99C2×Dic173—21no—
100C2×Dic173—21no—
101C2×Dic173—42no—
102C2×Dic173C2×D411yescharacteristic
103C2×Dic173C2×D411yescharacteristic
104C2×Dic173C2×D411yescharacteristic
105C4×D173—22no—
106C4×D173—22no—
107C4×D173—22no—
108#13063436—22no—
109C23×C173—21no—
110C23×C173—22no—
111Q8×C173—22no—
112D4×C173—22no—
113D4×C173—22no—
114D4×C173—22no—
115D4×C173—42no—
116D4×C173—42no—
117C2×C4×C173—21no—
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173C2411yescharacteristic
121C8×C173—22no—
122C22×C173—41no—
123C22×C173—42no—
124C22×C173—42no—
125C22×C173—42no—
126C22×C173—41no—
127C22×C173—41no—
128C22×C173C22×D411yescharacteristic
129C2×D173—21no—
130C2×D173—21no—
131C2×D173—42no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—88no—
135C2×D173AGL(1,ℤ/8)12yes—
136C4×C173—21no—
137C4×C173—21no—
138C4×C173—41no—
139C4×C173H(5,𝔽2)12yes—
140Dic173—21no—
141Dic173—21no—
142Dic173—41no—
143Dic173—41no—
144Dic173AGL(1,ℤ/8)12yes—
145C2×C173—82no—
146C2×C173—81no—
147C2×C173#53511yescharacteristic
148C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
150D173—22no—
151D173—82no—
152C173#321711yescharacteristic, complemented, Sylow
153#3217—1731nomaximal, Sylow
154C2×#127—1731no—
155#529—1731no—
156#496—1731no—
157#496D17311yescharacteristic, complemented
158#490—1731no—
159#490—1731no—
160#460—1732no—
161#459—1732no—
162#450—1732no—
163#447—1732no—
164#432—1731no—
165C22×D4—3461no—
166C22×D4—3461no—
167C22×D4—1732no—
168C2×SD16—1732no—
169C2×SD16C2×D17312yescomplemented
170C2×D8—1732no—
171C2×D8C2×D17312yescomplemented
172#127—3461no—
173#127—3461no—
174#127—1732no—
175#127—1731no—
176#127C2×D17311yescharacteristic
177#126—3461no—
178#125—1731no—
179#124C2×D17311yescharacteristic, complemented
180#123—3461no—
181C22⋊Q8—1731no—
182#121—1731no—
183C4×Q8—1731no—
184C4×D4—3461no—
185C4×D4—1731no—
186#117—3461no—
187C2×C42—1731no—
188C4⋊C8—1732no—
189#102—1732no—
190#102—1732no—
191#102—1732no—
192#102—1732no—
193C22⋊C8—1732no—
194#97—1731no—
195#97C2×D17311yescharacteristic, complemented
196C24—3462no—
197C2×Q8C22×D17311yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—3461no—
201C2×D4—3462no—
202C2×D4—3462no—
203C2×D4—3462no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4—1732no—
207C2×D4—6922no—
208C2×D4—6922no—
209C2×D4—6924no—
210C2×D4—6924no—
211C2×D4C22×D17312yes—
212C2×D4C22×D17311yescharacteristic
213C22×C4—3461no—
214C22×C4—1731no—
215C22×C4—1731no—
216C22×C4—1731no—
217SD16—42no—
218SD16—6922no—
219D8—42no—
220D8—6922no—
221C2×C8—1732no—
222C2×C8C22×D17312yes—
223C4⋊C4—3461no—
224C4⋊C4—3461no—
225C4⋊C4—3461no—
226C4⋊C4—3461no—
227C4⋊C4—1731no—
228C4⋊C4—1731no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—3461no—
232C22⋊C4—3461no—
233C22⋊C4—3461no—
234C42—3461no—
235C42—1731no—
236C42—1731no—
237C42—1731no—
238C42C22×D17311yescharacteristic
239C23—21no—
240C23—22no—
241C23—3462no—
242C23—1731no—
243C23—6924no—
244C23—6924no—
245C23—6924no—
246Q8—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—3462no—
253D4—3462no—
254D4—6922no—
255D4—6922no—
256D4—6924no—
257D4—6924no—
258D4—6924no—
259D4—6922no—
260C2×C4—21no—
261C2×C4—3462no—
262C2×C4—3462no—
263C2×C4—3462no—
264C2×C4—3461no—
265C2×C4—3461no—
266C2×C4—1731no—
267C2×C4—1731no—
268C2×C4—1731no—
269C2×C4—6922no—
270C2×C4D4×D17311yescharacteristic
271C2×C4D4×D17311yescharacteristic
272C2×C4C23×D17311yescharacteristic
273C8—22no—
274C8—3462no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—3461no—
282C22—3461no—
283C22—13848no—
284C22—1732no—
285C22—6922no—
286C22—6922no—
287C22—6922no—
288C22C2×D4×D17311yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—3461no—
293C4—3461no—
294C4—1732no—
295C4—6921no—
296C4—6921no—
297C4#1914972412yes—
298C2—81no—
299C2—82no—
300C2—3462no—
301C2—13842no—
302C2#2601205411yescharacteristic
303C2AGL(1,ℤ/8)×D17311yescharacteristic
304C2AGL(1,ℤ/8)×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3217
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #127×D173C2semidirect
2. #26012020C2semidirect
3. #26011881C2semidirect
4. #26011856C2semidirect
5. #26011855C2semidirect
6. #26011689C2semidirect
7. #26011684C2semidirect
8. #26011640C2semidirect
9. #26011623C2semidirect
10. #26011546C2semidirect
11. #496×C173C2semidirect
15. C2×#14583813C22semidirect
16. C2×#14583811C22semidirect
22. #19149671C22semidirect
26. Q8×Dic173C22semidirect
27. D4×Dic173C22semidirect
31. #19149608C22semidirect
32. #19149601C22semidirect
33. #19149601C22semidirect
34. #19149600C22semidirect
35. #19149600C22semidirect
37. #19149590C22semidirect
38. C2×SD16×C173C22semidirect
39. C2×D8×C173C22semidirect
41. #124×C173C22semidirect
42. #97×C173C22semidirect
152. C173#3217semidirect
157. #496D173semidirect
169. C2×SD16C2×D173semidirect
171. C2×D8C2×D173semidirect
179. #124C2×D173semidirect
195. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#127×D1732
#260120202
#260118812
#260118562
#260118552
#260116892
#260116842
#260116402
#260116232
#260115462
#496×C1732
#3217173