Finite Group Databasebeta

← C4⋊C4×C22⋊D173 #36356462

Subgroups — C4⋊C4×C22⋊D173 #36356462

Order 22144.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D173C212yesmaximal
2C2×C4×C22⋊D173C211yesmaximal, characteristic
3C2×C4⋊C4×D173C211yesmaximal, characteristic, complemented
4#26011386C212yesmaximal, complemented
5#26011364C212yesmaximal, complemented
6#26011362C212yesmaximal, complemented
7#26011360C211yesmaximal, characteristic, complemented
8#26011351C211yesmaximal, characteristic, complemented
9#26011348C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic173C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C173C211yesmaximal, characteristic, complemented
12C22×C22⋊D173C2211yescharacteristic
13C22×C4×D173C2212yes—
14C22×C4×D173C2211yescharacteristic
15C4×C22⋊D173—28no—
16C4×C22⋊D173C418yescomplemented
17C2×C4⋊Dic173C2211yescharacteristic
18C2×C4⋊Dic173C2211yescharacteristic
19C2×#14583805C2212yes—
20C2×#14583805C2212yes—
21C2×#14583805C2212yes—
22C2×#14583805C2211yescharacteristic
23C4⋊C4×D173—24no—
24C2×C22⋊Dic173C2212yes—
25C2×C22⋊Dic173C2211yescharacteristic
26C2×#14583803C2212yes—
27C2×#14583803C2211yescharacteristic
28C2×C4×Dic173C2212yes—
29C2×C4×Dic173C2211yescharacteristic
30#19149586C2212yes—
31#19149586C2212yes—
32#19149586C2212yes—
33C23×C4×C173C2212yes—
34C23×C4×C173C2211yescharacteristic
35C2×C4⋊C4×C173—24no—
36C2×C4⋊C4×C173C2211yescharacteristic
37C2×C4⋊C4×C173C2212yes—
38C2×C4⋊C4×C173C2211yescharacteristic
39C23×D173C2311yescharacteristic
40C2×C22⋊D173C2×C414yes—
41C2×C22⋊D173D414yes—
42C2×C22⋊D173Q814yes—
43C22×Dic173C2312yes—
44C22×Dic173C2312yes—
45C22×Dic173C2311yescharacteristic
46C22×Dic173C2311yescharacteristic
47C22×Dic173C2311yescharacteristic
48C2×C4×D173—24no—
49C2×C4×D173—22no—
50C2×C4×D173—24no—
51C2×C4×D173—22no—
52C2×C4×D173—42no—
53C2×C4×D173C2×C414yescomplemented
54C4⋊Dic173—22no—
55C4⋊Dic173—22no—
56#14583805—24no—
57#14583805—24no—
58#14583805—42no—
59#14583805C2×C414yescomplemented
60C22⋊Dic173—24no—
61C22⋊Dic173C2×C414yescomplemented
62#14583803—24no—
63#14583803C2×C414yescomplemented
64C4×Dic173—24no—
65C4×Dic173C2×C414yescomplemented
66C24×C173C2311yescharacteristic
67C22×C4×C173—22no—
68C22×C4×C173—22no—
69C22×C4×C173—24no—
70C22×C4×C173—24no—
71C22×C4×C173—42no—
72C22×C4×C173C2×C414yescomplemented
73C22×C4×C173C2312yes—
74C22×C4×C173C2311yescharacteristic
75C22×C4×C173C2312yes—
76C22×C4×C173C2311yescharacteristic
77C4⋊C4×C173—24no—
78C4⋊C4×C173—22no—
79C4⋊C4×C173D414yescomplemented
80C22×D173—21no—
81C22×D173—21no—
82C22×D173—21no—
83C22×D173—21no—
84C22×D173C22×C412yes—
85C22×D173C2×D412yes—
86C22×D173C2×Q812yes—
87C22⋊D173C4⋊C4116yesdirect product factor
88C2×Dic173—21no—
89C2×Dic173—21no—
90C2×Dic173—21no—
91C2×Dic173—21no—
92C2×Dic173—22no—
93C2×Dic173—22no—
94C2×Dic173—22no—
95C2×Dic173—22no—
96C2×Dic173—22no—
97C2×Dic173—22no—
98C2×Dic173C22×C412yes—
99C2×Dic173C22×C412yes—
100C2×Dic173C22×C412yes—
101C2×Dic173C2×D412yes—
102C2×Dic173C2×Q812yes—
103C4×D173—28no—
104C4×D173—48no—
105C23×C173—21no—
106C23×C173—21no—
107C23×C173—21no—
108C23×C173—21no—
109C23×C173C22×C412yes—
110C23×C173C2×D412yes—
111C23×C173C2×Q812yes—
112C23×C173C2411yescharacteristic
113C2×C4×C173—22no—
114C2×C4×C173—21no—
115C2×C4×C173—21no—
116C2×C4×C173—22no—
117C2×C4×C173—28no—
118C2×C4×C173—22no—
119C2×C4×C173—22no—
120C2×C4×C173—22no—
121C2×C4×C173—48no—
122C2×C4×C173C22×C412yes—
123C2×C4×C173C22×C412yes—
124C2×C4×C173C2×D414yes—
125C2×C4×C173C2×D412yes—
126C2×C4×C173#4112yes—
127C22×C173—22no—
128C22×C173—22no—
129C22×C173—22no—
130C22×C173—22no—
131C22×C173—22no—
132C22×C173—22no—
133C22×C173C2×C4⋊C414yescomplemented
134C22×C173C23×C411yescharacteristic
135C22×C173C22×D411yescharacteristic
136C22×C173C22×D411yescharacteristic
137C22×C173C22×Q811yescharacteristic
138C22×C173C2×#4111yescharacteristic
139C22×C173H(5,𝔽2)11yescharacteristic
140C22×C1732−1+411yescharacteristic
141C2×D173—22no—
142C2×D173—22no—
143C2×D173—22no—
144C2×D173—22no—
145C2×D173—22no—
146C2×D173—22no—
147C2×D173C2×C4⋊C414yescomplemented
148C4×C173—22no—
149C4×C173—24no—
150C4×C173—42no—
151C4×C173C4×D414yescomplemented
152Dic173—22no—
153Dic173—22no—
154Dic173—42no—
155Dic173—42no—
156Dic173C2×C4⋊C414yescomplemented
157C2×C173—24no—
158C2×C173C22×C4⋊C411yescharacteristic
159C2×C173C2×C4×D411yescharacteristic
160C2×C173#52011yescharacteristic
161C2×C173D4211yescharacteristic
162C2×C173#54711yescharacteristic
163C2×C173D4×Q811yescharacteristic
164C2×C173#55411yescharacteristic
165D173—24no—
166C173D4×C4⋊C411yescharacteristic, complemented, Sylow
167D4×C4⋊C4—1731nomaximal, Sylow
168C2×C4×D4—1732no—
169C2×C4×D4—1731no—
170C22×C4⋊C4—1731no—
171C22×C4⋊C4D17311yescharacteristic, complemented
172#389—1732no—
173#385—1732no—
174#385—1732no—
175#384—1731no—
176#380—1731no—
177#380—1731no—
178C4×C4⋊C4—1731no—
179C22×D4—1731no—
180C23×C4—1731no—
181C23×C4—1732no—
182C23×C4C2×D17312yes—
183C23×C4C2×D17311yescharacteristic
184C4×D4—3468no—
185C4×D4—1738no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—3464no—
188C2×C4⋊C4—1732no—
189C2×C4⋊C4—1731no—
190C2×C4⋊C4—1732no—
191C2×C4⋊C4—1731no—
192C2×C4⋊C4—1731no—
193C2×C4⋊C4—1732no—
194C2×C4⋊C4C2×D17311yescharacteristic
195C2×C4⋊C4C2×D17312yes—
196C2×C4⋊C4C2×D17311yescharacteristic
197C2×C22⋊C4—1732no—
198C2×C22⋊C4—1732no—
199C2×C22⋊C4—1731no—
200C2×C22⋊C4—1731no—
201C2×C42—1732no—
202C2×C42—1731no—
203#95—1732no—
204#95—1732no—
205#95—1732no—
206C24—1731no—
207C24C22×D17311yescharacteristic
208C2×D4—1734no—
209C2×D4—1734no—
210C2×D4—1734no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—3464no—
217C22×C4—3462no—
218C22×C4—3462no—
219C22×C4—3464no—
220C22×C4—1732no—
221C22×C4—1732no—
222C22×C4—1731no—
223C22×C4—1731no—
224C22×C4—1734no—
225C22×C4—1731no—
226C22×C4—6922no—
227C22×C4C4×D17314yescomplemented
228C22×C4C22×D17312yes—
229C22×C4C22×D17311yescharacteristic
230C22×C4C22×D17312yes—
231C22×C4C22×D17311yescharacteristic
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—3464no—
235C4⋊C4—3464no—
236C4⋊C4—3462no—
237C4⋊C4—3462no—
238C4⋊C4—1734no—
239C4⋊C4—6922no—
240C4⋊C4C22⋊D17314yesdirect product factor
241C22⋊C4—3464no—
242C22⋊C4—3464no—
243C22⋊C4—1734no—
244C22⋊C4—1734no—
245C42—3464no—
246C42—1734no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—3461no—
252C23—3461no—
253C23—3461no—
254C23—3461no—
255C23—1732no—
256C23—1732no—
257C23—1732no—
258C23C2×C4×D17312yes—
259C23D4×D17312yes—
260C23Q8×D17312yes—
261C23C23×D17311yescharacteristic
262D4—17316no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—28no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—48no—
272C2×C4—3461no—
273C2×C4—3461no—
274C2×C4—3461no—
275C2×C4—3461no—
276C2×C4—3468no—
277C2×C4—3462no—
278C2×C4—3462no—
279C2×C4—3462no—
280C2×C4—3462no—
281C2×C4—3462no—
282C2×C4—3462no—
283C2×C4—1732no—
284C2×C4—1732no—
285C2×C4—1732no—
286C2×C4—1732no—
287C2×C4—1732no—
288C2×C4—6928no—
289C2×C4C2×C4×D17312yes—
290C2×C4C2×C4×D17312yes—
291C2×C4C2×C22⋊D17314yes—
292C2×C4C2×C22⋊D17312yes—
293C2×C4#1458382612yes—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—3462no—
301C22—3462no—
302C22—3462no—
303C22—3462no—
304C22—3462no—
305C22—3462no—
306C22—1734no—
307C22C4⋊C4×D17314yescomplemented
308C22C22×C4×D17311yescharacteristic
309C22C2×D4×D17311yescharacteristic
310C22C22×C22⋊D17311yescharacteristic
311C22C2×Q8×D17311yescharacteristic
312C22C2×#1458382611yescharacteristic
313C22#1914972411yescharacteristic
314C22#1914972711yescharacteristic
315C4—22no—
316C4—24no—
317C4—42no—
318C4—3462no—
319C4—3462no—
320C4—1734no—
321C4—6922no—
322C4—6922no—
323C4C4×C22⋊D17314yescomplemented
324C2—24no—
325C2—3464no—
326C2C2×C4⋊C4×D17311yescharacteristic
327C2C2×C4×C22⋊D17311yescharacteristic
328C2#2601197211yescharacteristic
329C2D4×C22⋊D17311yescharacteristic
330C2#2601214211yescharacteristic
331C2Q8×C22⋊D17311yescharacteristic
332C2#2601219611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
FittingC22×C4⋊C4×C173C2
socle, cosocleC23×C173C24
prime radicalC22×C22⋊D173C22
hypercenterC22×C4⋊C4D173
hypocenterC173D4×C4⋊C4
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D173C2semidirect
4. #26011386C2semidirect
5. #26011364C2semidirect
6. #26011362C2semidirect
7. #26011360C2semidirect
8. #26011351C2semidirect
9. #26011348C2semidirect
10. C4⋊C4×Dic173C2semidirect
11. C22×C4⋊C4×C173C2semidirect
16. C4×C22⋊D173C4semidirect
53. C2×C4×D173C2×C4semidirect
59. #14583805C2×C4semidirect
61. C22⋊Dic173C2×C4semidirect
63. #14583803C2×C4semidirect
65. C4×Dic173C2×C4semidirect
72. C22×C4×C173C2×C4semidirect
79. C4⋊C4×C173D4semidirect
87. C22⋊D173C4⋊C4direct (N × H)
133. C22×C173C2×C4⋊C4semidirect
147. C2×D173C2×C4⋊C4semidirect
151. C4×C173C4×D4semidirect
156. Dic173C2×C4⋊C4semidirect
166. C173D4×C4⋊C4semidirect
171. C22×C4⋊C4D173semidirect
227. C22×C4C4×D173semidirect
240. C4⋊C4C22⋊D173direct (N × H)
307. C22C4⋊C4×D173semidirect
323. C4C4×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1732
C2×C4⋊C4×D1732
#260113862
#260113642
#260113622
#260113602
#260113512
#260113482
C4⋊C4×Dic1732
C22×C4⋊C4×C1732
D4×C4⋊C4173