Finite Group Databasebeta

← #36363506

Subgroups — #36363506

Order 22144.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D173C211yesmaximal, characteristic
2C2×#19149637C211yesmaximal, characteristic
3#26011835C212yesmaximal, complemented
4#26011831C212yesmaximal, complemented
5#26011827C211yesmaximal, characteristic, complemented
6#26011824C211yesmaximal, characteristic, complemented
7#26011821C211yesmaximal, characteristic, complemented
8#26011530C212yesmaximal, complemented
9#26011498C212yesmaximal, complemented
10#26011453C211yesmaximal, characteristic
11#483×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic
13C22×C4×D173C2211yescharacteristic
14C2×#14583809C2212yes—
15#19149659C2211yescharacteristic
16#19149658C2211yescharacteristic
17#19149657—21no—
18#19149656—21no—
19#19149655C2211yescharacteristic
20#19149637—21no—
21#19149637—21no—
22#19149637C2212yes—
23#19149637C2212yes—
24C2×C4⋊Dic173C2211yescharacteristic
25C4⋊C4×D173—21no—
26C4⋊C4×D173C2211yescharacteristic
27C2×#14583803—21no—
28C22⋊C4×D173—21no—
29C2×C4×Dic173—21no—
30#19149613C2212yescomplemented
31#19149612C2212yescomplemented
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149603C2212yescomplemented
35#19149602C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38C22⋊Q8×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#106×C173C2212yescomplemented
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44C2×Dic346—21no—
45C2×Dic346C2311yescharacteristic
46Q8×D173—24no—
47C22×Dic173—21no—
48C22×Dic173D411yescharacteristic
49C2×C4×D173—24no—
50C2×C4×D173—21no—
51C2×C4×D173—21no—
52C2×C4×D173D411yescharacteristic
53C2×C4×D173D411yescharacteristic
54C2×C4×D173C2311yescharacteristic
55C2×C4×D173C2311yescharacteristic
56#14583809—24no—
57C2×#13063436C2312yes—
58C4⋊Dic173—21no—
59C4⋊Dic173—21no—
60C4⋊Dic173C2312yes—
61C4⋊Dic173C2311yescharacteristic
62#14583805—21no—
63#14583805—21no—
64#14583805—21no—
65C22⋊Dic173—21no—
66#14583803—21no—
67#14583803—21no—
68#14583803—22no—
69#14583803—22no—
70C4×Dic173—22no—
71C4×Dic173—22no—
72C2×Q8×C173C2311yescharacteristic
73C22×C4×C173—21no—
74C22×C4×C173C2311yescharacteristic
75C2×C8×C173C2312yes—
76C4⋊C4×C173—21no—
77C4⋊C4×C173—21no—
78C4⋊C4×C173C2312yes—
79C4⋊C4×C173C2311yescharacteristic
80C22⋊C4×C173—21no—
81C22×D173—22no—
82C22×D173—22no—
83C22×D173—41no—
84C22×D173C2×D411yescharacteristic
85C22×D173C2×D411yescharacteristic
86Dic346—22no—
87Dic346—42no—
88C2×Dic173—22no—
89C2×Dic173—21no—
90C2×Dic173—21no—
91C2×Dic173—22no—
92C2×Dic173—42no—
93C2×Dic173C2×D411yescharacteristic
94C2×Dic173C2×D411yescharacteristic
95C4×D173—22no—
96C4×D173—22no—
97C4×D173—22no—
98C4×D173—42no—
99C4×D173—42no—
100C4×D173#4114yes—
101#13063436—22no—
102C23×C173C2×D411yescharacteristic
103Q8×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—21no—
106C2×C4×C173—21no—
107C2×C4×C173—22no—
108C2×C4×C173—42no—
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2411yescharacteristic
111C8×C173—22no—
112C22×C173—21no—
113C22×C173—21no—
114C22×C173#13712yes—
115C22×C173C22×D411yescharacteristic
116C2×D173—21no—
117C2×D173—21no—
118C2×D173—21no—
119C2×D173—21no—
120C2×D173—21no—
121C2×D173—44no—
122C2×D173#12312yes—
123C4×C173—21no—
124C4×C173—41no—
125C4×C173—42no—
126C4×C173—41no—
127C4×C173C2×#4112yes—
128Dic173—21no—
129Dic173—41no—
130Dic173—41no—
131Dic173—42no—
132Dic173#12312yes—
133C2×C173—22no—
134C2×C173C2×#12311yescharacteristic
135C2×C173C2×#13711yescharacteristic
136C2×C173#57511yescharacteristic
137D173—22no—
138D173—41no—
139C173#309511yescharacteristic, complemented, Sylow
140#3095—1731nomaximal, Sylow
141C2×C22⋊Q8—1731no—
142C2×#117—1731no—
143#485—1732no—
144#485—1732no—
145#484—1731no—
146#484—1731no—
147#483—1731no—
148#483D17311yescharacteristic, complemented
149#428—1732no—
150#419—1732no—
151#407—1731no—
152C22×Q8—1731no—
153C23×C4—1731no—
154C2×M16—1732no—
155C22⋊Q8—3461no—
156C22⋊Q8—3461no—
157C22⋊Q8—1731no—
158C22⋊Q8—1731no—
159C22⋊Q8—1731no—
160C22⋊Q8C2×D17311yescharacteristic
161#117—3461no—
162#117—3461no—
163#117—1732no—
164#117—1732no—
165C2×C4⋊C4—3461no—
166C2×C4⋊C4—1731no—
167C2×C4⋊C4—1731no—
168C2×C4⋊C4C2×D17311yescharacteristic
169C2×C22⋊C4—3461no—
170C2×C22⋊C4—3461no—
171C2×C42—3461no—
172#107—1732no—
173#107—1732no—
174#106—1732no—
175#106C2×D17312yescomplemented
176#103—1732no—
177#103—1732no—
178#103—1732no—
179#103C2×D17312yescomplemented
180C22⋊C8—1731no—
181C22⋊C8—1732no—
182C22⋊C8C2×D17311yescharacteristic
183C24—1731no—
184C2×Q8—3461no—
185C2×Q8—3464no—
186C2×Q8—1731no—
187C2×Q8C22×D17311yescharacteristic
188C22×C4—21no—
189C22×C4—3464no—
190C22×C4—3461no—
191C22×C4—3461no—
192C22×C4—3461no—
193C22×C4—1731no—
194C22×C4—1731no—
195C22×C4—1731no—
196C22×C4—1731no—
197C22×C4—1731no—
198C22×C4C22×D17311yescharacteristic
199M16—3464no—
200C2×C8—1732no—
201C2×C8C22×D17312yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—3461no—
205C4⋊C4—3461no—
206C4⋊C4—3461no—
207C4⋊C4—3461no—
208C4⋊C4—3461no—
209C4⋊C4—1732no—
210C4⋊C4—1731no—
211C4⋊C4C22×D17312yes—
212C4⋊C4C22×D17311yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—3462no—
215C22⋊C4—3462no—
216C22⋊C4—3461no—
217C22⋊C4—3461no—
218C22⋊C4—3461no—
219C42—3462no—
220C42—3462no—
221C23—3462no—
222C23—3462no—
223C23—1731no—
224C23—1731no—
225C23—6921no—
226C23D4×D17311yescharacteristic
227Q8—22no—
228Q8—3462no—
229Q8—6922no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—3462no—
236C2×C4—3462no—
237C2×C4—3462no—
238C2×C4—3462no—
239C2×C4—3462no—
240C2×C4—3461no—
241C2×C4—3461no—
242C2×C4—1731no—
243C2×C4—1731no—
244C2×C4—1734no—
245C2×C4—6922no—
246C2×C4—6922no—
247C2×C4—6922no—
248C2×C4D4×D17311yescharacteristic
249C2×C4C23×D17311yescharacteristic
250C8—22no—
251C8—3462no—
252C22—21no—
253C22—21no—
254C22—3461no—
255C22—3461no—
256C22—3461no—
257C22—3461no—
258C22—3461no—
259C22—1732no—
260C22—6924no—
261C22#1914970812yes—
262C22C2×D4×D17311yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—3461no—
268C4—1732no—
269C4—6921no—
270C4—6922no—
271C4—6921no—
272C4#41×D17312yes—
273C2—22no—
274C2—3462no—
275C2—6921no—
276C2#123×D17311yescharacteristic
277C2C2×#1914970811yescharacteristic
278C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#483×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#483D173
hypocenterC173#3095
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #26011835C2semidirect
4. #26011831C2semidirect
5. #26011827C2semidirect
6. #26011824C2semidirect
7. #26011821C2semidirect
8. #26011530C2semidirect
9. #26011498C2semidirect
11. #483×C173C2semidirect
30. #19149613C22semidirect
31. #19149612C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149603C22semidirect
35. #19149602C22semidirect
40. #106×C173C22semidirect
41. #103×C173C22semidirect
139. C173#3095semidirect
148. #483D173semidirect
175. #106C2×D173semidirect
179. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
C2×#191496372
#260118352
#260118312
#260118272
#260118242
#260118212
#260115302
#260114982
#260114532
#483×C1732
#3095173