Finite Group Databasebeta

← #36364563

Subgroups — #36364563

Order 22144.

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

normal subgroups only

This group has 400 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#26012043C211yesmaximal, characteristic, complemented
2C2×#19149666C211yesmaximal, characteristic
3#26011882C212yesmaximal, complemented
4#26011849C212yesmaximal, complemented
5#26011712C212yesmaximal, complemented
6#26011711C212yesmaximal, complemented
7#26011649C211yesmaximal, characteristic, complemented
8#26011646C211yesmaximal, characteristic, complemented
9#26011642C211yesmaximal, characteristic
10#26011545C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic
12C22×Dic346C2211yescharacteristic
13C2×Dic692C2212yescomplemented
14C2×#14583816C2212yescomplemented
15C2×#14583814C2212yes—
16#19149678C2211yescharacteristic, complemented
17#19149677C2211yescharacteristic, complemented
18#19149669—21no—
19#19149666—21no—
20#19149666—21no—
21#19149666C2212yes—
22#19149666C2212yes—
23#19149658C2211yescharacteristic
24#19149656—21no—
25Q8×Dic173C2211yescharacteristic, complemented
26#19149644—21no—
27#19149644C2211yescharacteristic, complemented
28#19149635—21no—
29C2×C22⋊Dic173—21no—
30C2×C22⋊Dic173—21no—
31C2×C4×Dic173C2211yescharacteristic
32#19149608C2212yescomplemented
33#19149604C2212yescomplemented
34#19149602C2212yescomplemented
35#19149600C2212yes—
36C22⋊#13063436C2211yescharacteristic
37#19149589C2212yescomplemented
38C22×D4×C173C2211yescharacteristic
39C2×SD16×C173C2212yes—
40C22⋊Q8×C173C2211yescharacteristic
41#102×C173C2212yes—
42C22⋊C8×C173C2211yescharacteristic
43C2×Dic346—21no—
44C2×Dic346—21no—
45C2×Dic346—42no—
46C2×Dic346C2312yes—
47C2×Dic346C2311yescharacteristic
48C22×Dic173—21no—
49C22×Dic173D411yescharacteristic
50C22×Dic173D411yescharacteristic
51Dic692—42no—
52#14583816—42no—
53#14583814—42no—
54C2×#13063436C2312yes—
55C4⋊Dic173—21no—
56C4⋊Dic173C2311yescharacteristic
57#14583805—21no—
58#14583805—21no—
59#14583805—21no—
60#14583805—21no—
61#14583805—21no—
62C22⋊Dic173—22no—
63C22⋊Dic173—22no—
64C22⋊Dic173—22no—
65C22⋊Dic173—22no—
66C22⋊Dic173—21no—
67C4×Dic173—21no—
68C4×Dic173—21no—
69C4×Dic173D411yescharacteristic
70C4×Dic173D411yescharacteristic
71C4×Dic173C2311yescharacteristic
72C4×Dic173C2311yescharacteristic
73C24×C173—21no—
74C2×Q8×C173C2311yescharacteristic
75C2×D4×C173—21no—
76C2×D4×C173—42no—
77C2×D4×C173C2312yes—
78C22×C4×C173C2311yescharacteristic
79SD16×C173—42no—
80C2×C8×C173C2312yes—
81C4⋊C4×C173—21no—
82C4⋊C4×C173C2311yescharacteristic
83C22⋊C4×C173—21no—
84Dic346—22no—
85Dic346—22no—
86Dic346—22no—
87Dic346—42no—
88Dic346—42no—
89C2×Dic173—22no—
90C2×Dic173—22no—
91C2×Dic173—21no—
92C2×Dic173—21no—
93C2×Dic173—41no—
94C2×Dic173—42no—
95C2×Dic173C2×D411yescharacteristic
96C2×Dic173C2×D411yescharacteristic
97C2×Dic173C2×D411yescharacteristic
98C2×Dic173C2×D411yescharacteristic
99#13063436—22no—
100C23×C173—22no—
101C23×C173—42no—
102C23×C173—42no—
103C23×C173—42no—
104C23×C173C2×D411yescharacteristic
105Q8×C173—22no—
106D4×C173—22no—
107D4×C173—22no—
108D4×C173—42no—
109C2×C4×C173—22no—
110C2×C4×C173—21no—
111C2×C4×C173—21no—
112C2×C4×C173C2×D411yescharacteristic
113C2×C4×C173C2411yescharacteristic
114C8×C173—22no—
115C22×C173—21no—
116C22×C173—21no—
117C22×C173—42no—
118C22×C173—42no—
119C22×C173—42no—
120C22×C173—84no—
121C22×C173#13712yes—
122C22×C173C22×D411yescharacteristic
123C4×C173—21no—
124C4×C173—41no—
125C4×C173—41no—
126C4×C173H(5,𝔽2)12yes—
127Dic173—21no—
128Dic173—21no—
129Dic173—21no—
130Dic173—21no—
131Dic173—42no—
132Dic173—41no—
133Dic173—41no—
134C2×C173—22no—
135C2×C173—82no—
136C2×C173#53511yescharacteristic
137C2×C173C2×#13711yescharacteristic
138C2×C173#57511yescharacteristic
139C173#321511yescharacteristic, complemented, Sylow
140#3215—1731nomaximal, Sylow
141#533—1731no—
142C2×#124—1731no—
143#497—1732no—
144#489—1732no—
145#464—1732no—
146#464—1732no—
147#451—1731no—
148#451—1731no—
149#450—1731no—
150#450D17311yescharacteristic
151#432—1731no—
152C22×Q8—1731no—
153C22×D4C2×D17311yescharacteristic
154C2×Q16—1732no—
155C2×Q16—1732no—
156C2×SD16—1732no—
157C2×SD16C2×D17312yes—
158C4⋊Q8—1731no—
159C4⋊Q8—1731no—
160#125—3461no—
161#124—3461no—
162#124—3461no—
163#124—1732no—
164#124—1732no—
165C22⋊Q8—3461no—
166C22⋊Q8—1731no—
167C22⋊Q8C2×D17311yescharacteristic
168C4×Q8—3461no—
169C4×Q8—1731no—
170C4×Q8—1731no—
171#117—3461no—
172C2×C22⋊C4—3461no—
173C2×C22⋊C4—3461no—
174C2×C42—1731no—
175C4⋊C8—1732no—
176#103—1732no—
177#103—1732no—
178#102—1732no—
179#102C2×D17312yes—
180C22⋊C8—1731no—
181C22⋊C8C2×D17311yescharacteristic
182#97—1732no—
183C24—21no—
184C2×Q8—3461no—
185C2×Q8—3461no—
186C2×Q8—1732no—
187C2×Q8—1731no—
188C2×Q8—6922no—
189C2×Q8C22×D17311yescharacteristic
190C2×D4—21no—
191C2×D4—42no—
192C2×D4C22×D17312yes—
193C22×C4—3461no—
194C22×C4—1731no—
195C22×C4—1731no—
196C22×C4C22×D17311yescharacteristic
197Q16—6922no—
198Q16—6922no—
199SD16—42no—
200SD16—6922no—
201C2×C8—1732no—
202C2×C8C22×D17312yes—
203C4⋊C4—21no—
204C4⋊C4—3461no—
205C4⋊C4—3461no—
206C4⋊C4—3461no—
207C4⋊C4—3461no—
208C4⋊C4—3461no—
209C4⋊C4—3461no—
210C4⋊C4—1731no—
211C4⋊C4C22×D17311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—3461no—
214C22⋊C4—3462no—
215C22⋊C4—3462no—
216C22⋊C4—3462no—
217C22⋊C4—3462no—
218C42—3461no—
219C42—3461no—
220C42—1731no—
221C42—1731no—
222C42—1731no—
223C42—1731no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23D4×D17311yescharacteristic
229Q8—22no—
230Q8—3462no—
231Q8—3462no—
232Q8—3462no—
233Q8—6922no—
234Q8—6922no—
235D4—22no—
236D4—22no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—3462no—
242C2×C4—3461no—
243C2×C4—3461no—
244C2×C4—3462no—
245C2×C4—1731no—
246C2×C4—1731no—
247C2×C4—1731no—
248C2×C4—1731no—
249C2×C4—6921no—
250C2×C4—6922no—
251C2×C4D4×D17311yescharacteristic
252C2×C4C23×D17311yescharacteristic
253C8—22no—
254C8—3462no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—84no—
261C22#1914970812yes—
262C22C2×D4×D17311yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—3461no—
267C4—3461no—
268C4—3461no—
269C4—3461no—
270C4—6921no—
271C4—6922no—
272C4—6921no—
273C4#1914972412yes—
274C2—22no—
275C2—82no—
276C2#2601205611yescharacteristic
277C2C2×#1914970811yescharacteristic
278C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalC22×D4×C173C22
hypercenter#450D173
hypocenterC173#3215
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012043C2semidirect
3. #26011882C2semidirect
4. #26011849C2semidirect
5. #26011712C2semidirect
6. #26011711C2semidirect
7. #26011649C2semidirect
8. #26011646C2semidirect
10. #26011545C2semidirect
13. C2×Dic692C22semidirect
14. C2×#14583816C22semidirect
16. #19149678C22semidirect
17. #19149677C22semidirect
25. Q8×Dic173C22semidirect
27. #19149644C22semidirect
32. #19149608C22semidirect
33. #19149604C22semidirect
34. #19149602C22semidirect
37. #19149589C22semidirect
139. C173#3215semidirect

Maximal subgroups

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

subgroupindex
#260120432
C2×#191496662
#260118822
#260118492
#260117122
#260117112
#260116492
#260116462
#260116422
#260115452
#450×C1732
#3215173