Finite Group Databasebeta

← #36363183

Subgroups — #36363183

Order 22144.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19149707C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic, complemented
3#26012003C211yesmaximal, characteristic, complemented
4C2×#19149654C211yesmaximal, characteristic, complemented
5#26011745C212yesmaximal, complemented
6#26011738C212yesmaximal, complemented
7#26011728C212yesmaximal, complemented
8#26011721C212yesmaximal, complemented
9#26011495C211yesmaximal, characteristic, complemented
10#26011490C211yesmaximal, characteristic, complemented
11#428×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×D692—21no—
14C22×D692C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16#19149707—24no—
17C2×#14583815C2212yescomplemented
18C2×D1384C2212yescomplemented
19C2×#14583810C2211yescharacteristic
20C2×#14583809C2212yescomplemented
21M16×D173—24no—
22C2×C8×D173C2212yescomplemented
23#19149663—21no—
24#19149659C2211yescharacteristic, complemented
25#19149659C2211yescharacteristic, complemented
26#19149654—21no—
27#19149654—21no—
28#19149654C2212yes—
29#19149654C2212yes—
30#19149654C2211yescharacteristic, complemented
31#19149654C2211yescharacteristic, complemented
32#19149649—21no—
33C4×D692—21no—
34C4×D692—21no—
35C2×#14583803—21no—
36#19149602C2212yescomplemented
37#19149599C2212yescomplemented
38#19149599C2212yescomplemented
39#19149599C2212yescomplemented
40C2×M16×C173C2211yescharacteristic
41#117×C173C2211yescharacteristic
42C2×C4⋊C4×C173C2211yescharacteristic
43#107×C173C2212yescomplemented
44#106×C173C2212yescomplemented
45C23×D173—21no—
46C23×D173D411yescharacteristic
47#14583826—22no—
48#14583826—22no—
49C2×Dic346C2311yescharacteristic
50C2×D692—21no—
51C2×D692—21no—
52C2×D692—22no—
53C2×D692—22no—
54C2×D692—22no—
55C2×D692—21no—
56C2×D692—22no—
57C2×D692—44no—
58C2×D692C2312yes—
59C2×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C22×Dic173D411yescharacteristic
62C2×C4×D173—22no—
63C2×C4×D173—21no—
64C2×C4×D173—22no—
65C2×C4×D173D411yescharacteristic
66C2×C4×D173D411yescharacteristic
67C2×C4×D173C2311yescharacteristic
68C2×C4×D173C2311yescharacteristic
69#14583815—24no—
70D1384—24no—
71#14583810—22no—
72#14583809—24no—
73C2×#13063436C2312yes—
74C8×D173—24no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77#14583803—21no—
78#14583803—21no—
79#14583803—22no—
80#14583803—22no—
81#14583803—21no—
82#14583803—21no—
83C22×C4×C173—21no—
84C22×C4×C173C2311yescharacteristic
85M16×C173D414yescomplemented
86C2×C8×C173C2312yes—
87C4⋊C4×C173—21no—
88C4⋊C4×C173C2311yescharacteristic
89C4⋊C4×C173C2311yescharacteristic
90C4⋊C4×C173C2312yes—
91C22⋊C4×C173—21no—
92C42×C173—21no—
93C22×D173—22no—
94C22×D173—22no—
95C22×D173—21no—
96C22×D173—22no—
97C22×D173—41no—
98C22×D173—42no—
99C22×D173—42no—
100C22×D173—42no—
101C22×D173C2×D411yescharacteristic
102C22×D173C2×D411yescharacteristic
103Dic346—21no—
104Dic346—21no—
105D692—21no—
106D692—22no—
107D692—22no—
108D692—21no—
109D692—42no—
110D692—44no—
111D692—42no—
112D692—44no—
113D692—42no—
114C22⋊D173—42no—
115C2×Dic173—22no—
116C2×Dic173—21no—
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C4×D173—22no—
120C4×D173—22no—
121C4×D173—21no—
122C4×D173—21no—
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—42no—
126#13063436—22no—
127C23×C173C2×D411yescharacteristic
128C2×C4×C173—22no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—42no—
132C2×C4×C173C2×D411yescharacteristic
133C2×C4×C173C2×D412yes—
134C2×C4×C173#4112yes—
135C2×C4×C173C2411yescharacteristic
136C8×C173C2×D414yescomplemented
137C22×C173—21no—
138C22×C173—21no—
139C22×C173#12112yes—
140C22×C173C22×D411yescharacteristic
141C2×D173—21no—
142C2×D173—21no—
143C2×D173—21no—
144C2×D173—21no—
145C2×D173—21no—
146C2×D173—44no—
147C2×D173—41no—
148C2×D173—42no—
149C2×D173—42no—
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—42no—
153C2×D173—84no—
154C2×D173AGL(1,ℤ/8)12yescomplemented
155C4×C173—42no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173#12112yes—
159C4×C173C22×D411yescharacteristic
160C4×C173C2×#4111yescharacteristic
161Dic173—21no—
162Dic173—21no—
163Dic173—41no—
164C2×C173—22no—
165C2×C173C2×#12111yescharacteristic
166C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C173#57511yescharacteristic
168D173—22no—
169D173—41no—
170D173—81no—
171D173—82no—
172C173#305511yescharacteristic, complemented, Sylow
173#3055—1731nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1731no—
175C22×M16—1731no—
176#525—1731no—
177C2×#121—1731no—
178#469—1732no—
179#468—1732no—
180#466—1732no—
181#465—1732no—
182#428D17311yescharacteristic, complemented
183#418—1731no—
184#417—1731no—
185C2×#41—1731no—
186C22×D4—3461no—
187C22×D4—1731no—
188C23×C4—1731no—
189AGL(1,ℤ/8)—3464no—
190C2×SD16—1732no—
191C2×D8—1732no—
192C2×M16—3464no—
193C2×M16—1731no—
194C2×M16—1732no—
195C2×M16C2×D17311yescharacteristic
196C22×C8—1732no—
197#123—3461no—
198C22⋊Q8—1731no—
199C22⋊Q8—1731no—
200#121—3461no—
201#121—3461no—
202#121—1732no—
203#121—1732no—
204#121—1731no—
205#121—1731no—
206#120—3461no—
207C4×D4—3461no—
208C4×D4—3461no—
209#117C2×D17311yescharacteristic
210C2×C4⋊C4C2×D17311yescharacteristic
211C2×C22⋊C4—3461no—
212#107C2×D17312yescomplemented
213#106C2×D17312yescomplemented
214#103—1732no—
215#102—1732no—
216#102—1732no—
217#102—1732no—
218C24—3461no—
219C24—1731no—
220#41—3462no—
221#41—3462no—
222C2×Q8—1731no—
223C2×D4—3461no—
224C2×D4—3461no—
225C2×D4—3462no—
226C2×D4—3462no—
227C2×D4—3461no—
228C2×D4—3461no—
229C2×D4—3462no—
230C2×D4—3462no—
231C2×D4—1731no—
232C2×D4—1732no—
233C2×D4—6924no—
234C22×C4—21no—
235C22×C4—3462no—
236C22×C4—3462no—
237C22×C4—3461no—
238C22×C4—1731no—
239C22×C4—1731no—
240C22×C4—1731no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4C22×D17311yescharacteristic
244SD16—3464no—
245D8—3464no—
246M16—3462no—
247M16—3464no—
248M16D69214yescomplemented
249C2×C8—3464no—
250C2×C8—1732no—
251C2×C8C22×D17312yes—
252C4⋊C4—21no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4C22×D17311yescharacteristic
256C4⋊C4C22×D17311yescharacteristic
257C4⋊C4C22×D17312yes—
258C22⋊C4—21no—
259C22⋊C4—3461no—
260C22⋊C4—3461no—
261C22⋊C4—3462no—
262C22⋊C4—3462no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C42—21no—
266C23—3462no—
267C23—3462no—
268C23—3461no—
269C23—3462no—
270C23—1731no—
271C23—1731no—
272C23—6921no—
273C23—6922no—
274C23—6922no—
275C23—6922no—
276C23D4×D17311yescharacteristic
277Q8—3461no—
278Q8—3461no—
279D4—3461no—
280D4—3462no—
281D4—3462no—
282D4—3461no—
283D4—6922no—
284D4—6924no—
285D4—6922no—
286D4—6924no—
287D4—6922no—
288D4—6922no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—3462no—
294C2×C4—3462no—
295C2×C4—3462no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—6922no—
304C2×C4D4×D17311yescharacteristic
305C2×C4C2×D69212yes—
306C2×C4#1458382712yes—
307C2×C4C23×D17311yescharacteristic
308C8—3462no—
309C8C2×D69214yescomplemented
310C22—21no—
311C22—21no—
312C22—3461no—
313C22—3461no—
314C22—3461no—
315C22—3461no—
316C22—3461no—
317C22—13844no—
318C22—1732no—
319C22—6924no—
320C22—6921no—
321C22—6922no—
322C22—6922no—
323C22—6921no—
324C22—6921no—
325C22—6922no—
326C22#1914965412yes—
327C22C2×D4×D17311yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—3461no—
332C4—3461no—
333C4—6921no—
334C4#1914965412yes—
335C4C22×D69211yescharacteristic
336C4C2×#1458382711yescharacteristic
337C2—22no—
338C2—3462no—
339C2—13842no—
340C2—13841no—
341C2—6921no—
342C2C2×#1914965411yescharacteristic
343C2AGL(1,ℤ/8)×D17311yescharacteristic
344C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#428×C173C2
socleC22×C173C22×D4
hypercenter#428D173
hypocenterC173#3055
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. C2×M16×D173C2semidirect
3. #26012003C2semidirect
4. C2×#19149654C2semidirect
5. #26011745C2semidirect
6. #26011738C2semidirect
7. #26011728C2semidirect
8. #26011721C2semidirect
9. #26011495C2semidirect
10. #26011490C2semidirect
11. #428×C173C2semidirect
17. C2×#14583815C22semidirect
18. C2×D1384C22semidirect
20. C2×#14583809C22semidirect
22. C2×C8×D173C22semidirect
24. #19149659C22semidirect
25. #19149659C22semidirect
30. #19149654C22semidirect
31. #19149654C22semidirect
36. #19149602C22semidirect
37. #19149599C22semidirect
38. #19149599C22semidirect
39. #19149599C22semidirect
43. #107×C173C22semidirect
44. #106×C173C22semidirect
85. M16×C173D4semidirect
136. C8×C173C2×D4semidirect
154. C2×D173AGL(1,ℤ/8)semidirect
172. C173#3055semidirect
182. #428D173semidirect
212. #107C2×D173semidirect
213. #106C2×D173semidirect
248. M16D692semidirect
309. C8C2×D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C2×M16×D1732
#260120032
C2×#191496542
#260117452
#260117382
#260117282
#260117212
#260114952
#260114902
#428×C1732
#3055173