Finite Group Databasebeta

← #36364337

Subgroups — #36364337

Order 22144.

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

normal subgroups only

This group has 416 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
1C2×#19149659C211yesmaximal, characteristic
2C22⋊Q8×D173C211yesmaximal, characteristic
3#26011828C212yesmaximal, complemented
4#26011826C211yesmaximal, characteristic, complemented
5#26011733C212yesmaximal, complemented
6#26011731C212yesmaximal, complemented
7#26011649C211yesmaximal, characteristic, complemented
8#26011648C212yesmaximal, complemented
9#26011646C211yesmaximal, characteristic, complemented
10C22⋊C8×D173C211yesmaximal, characteristic
11#484×C173C211yesmaximal, characteristic, complemented
12C22×Dic346C2211yescharacteristic
13C2×Q8×D173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15C2×Dic692C2212yescomplemented
16C2×#14583816C2212yescomplemented
17C2×C8×D173C2212yes—
18#19149659—21no—
19#19149659—21no—
20#19149659C2212yes—
21#19149659C2211yescharacteristic
22#19149659C2212yes—
23#19149658C2211yescharacteristic
24#19149657—21no—
25#19149656—21no—
26#19149655C2211yescharacteristic
27C2×C4⋊Dic173—21no—
28C4⋊C4×D173—21no—
29C4⋊C4×D173C2211yescharacteristic
30C2×#14583803—21no—
31C22⋊C4×D173—21no—
32#19149612C2212yescomplemented
33#19149604C2212yescomplemented
34#19149602C2212yescomplemented
35#19149602C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38C22⋊Q8×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#107×C173C2212yescomplemented
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44C2×Dic346—21no—
45C2×Dic346—21no—
46C2×Dic346—42no—
47C2×Dic346C2311yescharacteristic
48C2×Dic346C2312yes—
49Q8×D173—42no—
50C22×Dic173—21no—
51C22×Dic173D411yescharacteristic
52C2×C4×D173—21no—
53C2×C4×D173—21no—
54C2×C4×D173—42no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59Dic692—42no—
60#14583816—42no—
61C2×#13063436C2312yes—
62C8×D173—42no—
63C4⋊Dic173—21no—
64C4⋊Dic173—22no—
65C4⋊Dic173—22no—
66C4⋊Dic173C2311yescharacteristic
67#14583805—21no—
68#14583805—21no—
69#14583805—21no—
70C22⋊Dic173—21no—
71#14583803—21no—
72#14583803—21no—
73#14583803—22no—
74#14583803—22no—
75C2×Q8×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173C2311yescharacteristic
78C2×C8×C173C2312yes—
79C4⋊C4×C173—21no—
80C4⋊C4×C173—21no—
81C4⋊C4×C173C2311yescharacteristic
82C4⋊C4×C173C2312yes—
83C22⋊C4×C173—21no—
84C22×D173—41no—
85C22×D173—41no—
86C22×D173Q1614yescomplemented
87C22×D173C2×D411yescharacteristic
88C22×D173C2×D411yescharacteristic
89Dic346—22no—
90Dic346—22no—
91Dic346—22no—
92Dic346—42no—
93Dic346—42no—
94C2×Dic173—21no—
95C2×Dic173—21no—
96C2×Dic173—22no—
97C2×Dic173—41no—
98C2×Dic173—42no—
99C2×Dic173C2×D411yescharacteristic
100C2×Dic173C2×D411yescharacteristic
101C4×D173—22no—
102C4×D173—22no—
103C4×D173—41no—
104C4×D173—41no—
105C4×D173—42no—
106C4×D173—42no—
107#13063436—22no—
108C23×C173C2×D411yescharacteristic
109Q8×C173—22no—
110C2×C4×C173—22no—
111C2×C4×C173—21no—
112C2×C4×C173—21no—
113C2×C4×C173—22no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2411yescharacteristic
117C8×C173—22no—
118C22×C173—21no—
119C22×C173—21no—
120C22×C173C2×Q1612yes—
121C22×C173C22×D411yescharacteristic
122C2×D173—21no—
123C2×D173—21no—
124C2×D173—21no—
125C2×D173—21no—
126C2×D173—44no—
127C2×D173C2×Q1612yes—
128C2×D173C2×Q1612yes—
129C4×C173—21no—
130C4×C173—41no—
131C4×C173—42no—
132C4×C173—41no—
133C4×C173H(5,𝔽2)12yes—
134Dic173—21no—
135Dic173—21no—
136Dic173—41no—
137Dic173—42no—
138Dic173—41no—
139C2×C173—22no—
140C2×C173#53411yescharacteristic
141C2×C173C22×Q1611yescharacteristic
142C2×C173#57511yescharacteristic
143D173—22no—
144D173—22no—
145C173#318911yescharacteristic, complemented, Sylow
146#3189—1731nomaximal, Sylow
147C2×C22⋊Q8—1731no—
148C2×C22⋊Q8—1731no—
149#484—1731no—
150#484—1732no—
151#484D17311yescharacteristic, complemented
152#467—1732no—
153#467—1732no—
154#451—1731no—
155#451—1732no—
156#451—1731no—
157C2×C22⋊C8—1731no—
158C22×Q8—1731no—
159C22×Q8—1731no—
160C23×C4—1731no—
161C2×Q16—1732no—
162C2×Q16—1732no—
163C22×C8—1732no—
164C22⋊Q8—3461no—
165C22⋊Q8—3461no—
166C22⋊Q8—3461no—
167C22⋊Q8—3461no—
168C22⋊Q8—1731no—
169C22⋊Q8—1731no—
170C22⋊Q8—1732no—
171C22⋊Q8—1731no—
172C22⋊Q8—1732no—
173C22⋊Q8C2×D17311yescharacteristic
174C2×C4⋊C4—3461no—
175C2×C4⋊C4—3461no—
176C2×C4⋊C4—1731no—
177C2×C4⋊C4C2×D17311yescharacteristic
178C2×C22⋊C4—3461no—
179C2×C22⋊C4—3461no—
180#107—1732no—
181#107C2×D17312yescomplemented
182#103—1732no—
183#103—1732no—
184#103—1732no—
185#103C2×D17312yescomplemented
186C22⋊C8—1731no—
187C22⋊C8—1732no—
188C22⋊C8C2×D17311yescharacteristic
189C24—1731no—
190C2×Q8—3461no—
191C2×Q8—3461no—
192C2×Q8—1731no—
193C2×Q8—1732no—
194C2×Q8—6922no—
195C2×Q8—6922no—
196C2×Q8C22×D17311yescharacteristic
197C22×C4—21no—
198C22×C4—3461no—
199C22×C4—3461no—
200C22×C4—3461no—
201C22×C4—1731no—
202C22×C4—1731no—
203C22×C4—1731no—
204C22×C4—1731no—
205C22×C4—1731no—
206C22×C4—6922no—
207C22×C4C22×D17311yescharacteristic
208Q16—6922no—
209Q16—6922no—
210C2×C8—1732no—
211C2×C8—6922no—
212C2×C8C22×D17312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—3461no—
216C4⋊C4—3461no—
217C4⋊C4—3462no—
218C4⋊C4—3462no—
219C4⋊C4—3461no—
220C4⋊C4—3461no—
221C4⋊C4—1731no—
222C4⋊C4C22×D17311yescharacteristic
223C4⋊C4C22×D17312yes—
224C22⋊C4—21no—
225C22⋊C4—3461no—
226C22⋊C4—3461no—
227C22⋊C4—3462no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C23—1731no—
231C23—1731no—
232C23—1734no—
233C23—6921no—
234C23—6921no—
235C23D4×D17311yescharacteristic
236Q8—22no—
237Q8—3462no—
238Q8—3462no—
239Q8—3462no—
240Q8—6922no—
241Q8—6922no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—3462no—
248C2×C4—3462no—
249C2×C4—3461no—
250C2×C4—3461no—
251C2×C4—3462no—
252C2×C4—1731no—
253C2×C4—1731no—
254C2×C4—6921no—
255C2×C4—6921no—
256C2×C4—6922no—
257C2×C4—6922no—
258C2×C4—6921no—
259C2×C4—6922no—
260C2×C4D4×D17311yescharacteristic
261C2×C4C23×D17311yescharacteristic
262C8—22no—
263C8—3462no—
264C22—21no—
265C22—21no—
266C22—3461no—
267C22—3461no—
268C22—3461no—
269C22—3461no—
270C22—1732no—
271C22—1732no—
272C22—6924no—
273C22Q16×D17312yes—
274C22C2×D4×D17311yescharacteristic
275C4—21no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—3461no—
280C4—3461no—
281C4—6921no—
282C4—6921no—
283C4—6922no—
284C4#1914972512yes—
285C2—22no—
286C2—3462no—
287C2—3462no—
288C2#2601205211yescharacteristic
289C2C2×Q16×D17311yescharacteristic
290C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#484×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#484D173
hypocenterC173#3189
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #26011828C2semidirect
4. #26011826C2semidirect
5. #26011733C2semidirect
6. #26011731C2semidirect
7. #26011649C2semidirect
8. #26011648C2semidirect
9. #26011646C2semidirect
11. #484×C173C2semidirect
15. C2×Dic692C22semidirect
16. C2×#14583816C22semidirect
32. #19149612C22semidirect
33. #19149604C22semidirect
34. #19149602C22semidirect
35. #19149602C22semidirect
40. #107×C173C22semidirect
41. #103×C173C22semidirect
86. C22×D173Q16semidirect
145. C173#3189semidirect
151. #484D173semidirect
181. #107C2×D173semidirect
185. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496592
C22⋊Q8×D1732
#260118282
#260118262
#260117332
#260117312
#260116492
#260116482
#260116462
C22⋊C8×D1732
#484×C1732
#3189173