Finite Group Databasebeta

← #36363194

Subgroups — #36363194

Order 22144.

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

normal subgroups only

This group has 496 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×#19149712C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic, complemented
3#26012003C211yesmaximal, characteristic
4C2×#19149659C211yesmaximal, characteristic, complemented
5#26011750C212yesmaximal, complemented
6#26011738C212yesmaximal, complemented
7#26011733C212yesmaximal, complemented
8#26011721C212yesmaximal, complemented
9#26011499C211yesmaximal, characteristic, complemented
10#26011490C211yesmaximal, characteristic, complemented
11#428×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×Dic346C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15#19149712—24no—
16C2×Dic692C2212yescomplemented
17C2×#14583815C2212yescomplemented
18C2×#14583810C2211yescharacteristic
19C2×#14583809C2212yes—
20M16×D173—24no—
21C2×C8×D173C2212yes—
22#19149663—21no—
23#19149659—21no—
24#19149659—21no—
25#19149659C2212yes—
26#19149659C2211yescharacteristic
27#19149659C2212yes—
28#19149659C2211yescharacteristic
29#19149654C2211yescharacteristic
30#19149654C2211yescharacteristic
31#19149649—21no—
32C4×D692—21no—
33C4×D692—21no—
34C2×C4⋊Dic173—21no—
35C2×#14583803—21no—
36#19149602C2212yescomplemented
37#19149602C2212yescomplemented
38#19149602C2212yescomplemented
39#19149599C2212yescomplemented
40C2×M16×C173C2211yescharacteristic
41#117×C173C2211yescharacteristic
42C2×C4⋊C4×C173C2211yescharacteristic
43#107×C173C2212yescomplemented
44#106×C173C2212yescomplemented
45C23×D173D411yescharacteristic
46#14583826—22no—
47#14583826—22no—
48C2×Dic346—21no—
49C2×Dic346—22no—
50C2×Dic346—22no—
51C2×Dic346C2312yes—
52C2×Dic346C2311yescharacteristic
53C2×D692—21no—
54C2×D692—21no—
55C2×D692C2311yescharacteristic
56C2×C22⋊D173—21no—
57C22×Dic173—21no—
58C22×Dic173D411yescharacteristic
59C2×C4×D173—22no—
60C2×C4×D173—21no—
61C2×C4×D173—22no—
62C2×C4×D173D411yescharacteristic
63C2×C4×D173D411yescharacteristic
64C2×C4×D173C2311yescharacteristic
65C2×C4×D173C2311yescharacteristic
66Dic692—24no—
67#14583815—24no—
68#14583810—22no—
69#14583809—24no—
70C2×#13063436C2312yes—
71C8×D173—24no—
72C4⋊Dic173—22no—
73C4⋊Dic173—22no—
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76#14583803—22no—
77#14583803—22no—
78#14583803—21no—
79#14583803—21no—
80#14583803—21no—
81#14583803—21no—
82C22×C4×C173—21no—
83C22×C4×C173C2311yescharacteristic
84M16×C173D414yescomplemented
85C2×C8×C173C2312yes—
86C4⋊C4×C173—21no—
87C4⋊C4×C173C2312yes—
88C4⋊C4×C173C2311yescharacteristic
89C4⋊C4×C173C2311yescharacteristic
90C22⋊C4×C173—21no—
91C42×C173—21no—
92C22×D173—22no—
93C22×D173—22no—
94C22×D173—21no—
95C22×D173—41no—
96C22×D173C2×D411yescharacteristic
97C22×D173C2×D411yescharacteristic
98Dic346—22no—
99Dic346—21no—
100Dic346—22no—
101Dic346—21no—
102Dic346—42no—
103D692—21no—
104D692—21no—
105D692—42no—
106D692—42no—
107C22⋊D173—42no—
108C2×Dic173—22no—
109C2×Dic173—21no—
110C2×Dic173—22no—
111C2×Dic173—42no—
112C2×Dic173C2×D411yescharacteristic
113C2×Dic173C2×D411yescharacteristic
114C4×D173—22no—
115C4×D173—22no—
116C4×D173—21no—
117C4×D173—21no—
118C4×D173—21no—
119C4×D173—21no—
120C4×D173—42no—
121#13063436—22no—
122C23×C173C2×D411yescharacteristic
123C2×C4×C173—21no—
124C2×C4×C173—21no—
125C2×C4×C173—22no—
126C2×C4×C173—42no—
127C2×C4×C173C2×D411yescharacteristic
128C2×C4×C173C2×D412yes—
129C2×C4×C173#4112yes—
130C2×C4×C173C2411yescharacteristic
131C8×C173C2×D414yes—
132C22×C173—21no—
133C22×C173—21no—
134C22×C173#12112yes—
135C22×C173C22×D411yescharacteristic
136C2×D173—21no—
137C2×D173—21no—
138C2×D173—21no—
139C2×D173—21no—
140C2×D173—21no—
141C2×D173—44no—
142C2×D173—41no—
143C2×D173—41no—
144C2×D173—41no—
145C2×D173#13712yescomplemented
146C4×C173—41no—
147C4×C173—42no—
148C4×C173—41no—
149C4×C173#12112yes—
150C4×C173C22×D411yescharacteristic
151C4×C173C2×#4111yescharacteristic
152Dic173—21no—
153Dic173—21no—
154Dic173—41no—
155Dic173—42no—
156C2×C173—22no—
157C2×C173C2×#12111yescharacteristic
158C2×C173C2×#13711yescharacteristic
159C2×C173#57511yescharacteristic
160D173—22no—
161D173—41no—
162D173—81no—
163C173#305611yescharacteristic, complemented, Sylow
164#3056—1731nomaximal, Sylow
165C2×#137—1731no—
166C22×M16—1731no—
167#525—1731no—
168C2×C22⋊Q8—1731no—
169#470—1732no—
170#468—1732no—
171#467—1732no—
172#465—1732no—
173#428D17311yescharacteristic, complemented
174#419—1731no—
175#417—1731no—
176C2×#41—1731no—
177C22×Q8—1731no—
178C23×C4—1731no—
179#137—3464no—
180C2×Q16—1732no—
181C2×SD16—1732no—
182C2×M16—3464no—
183C2×M16—1731no—
184C2×M16—1732no—
185C2×M16C2×D17311yescharacteristic
186C22×C8—1732no—
187#123—3461no—
188C22⋊Q8—3461no—
189C22⋊Q8—3461no—
190C22⋊Q8—1732no—
191C22⋊Q8—1732no—
192C22⋊Q8—1731no—
193C22⋊Q8—1731no—
194#121—1731no—
195#121—1731no—
196#120—3461no—
197C4×D4—3461no—
198C4×D4—3461no—
199#117C2×D17311yescharacteristic
200C2×C4⋊C4—3461no—
201C2×C4⋊C4C2×D17311yescharacteristic
202C2×C22⋊C4—3461no—
203#107C2×D17312yescomplemented
204#106C2×D17312yescomplemented
205#103—1732no—
206#103—1732no—
207#103—1732no—
208#102—1732no—
209C24—1731no—
210#41—3462no—
211#41—3462no—
212C2×Q8—3461no—
213C2×Q8—3462no—
214C2×Q8—3462no—
215C2×Q8—1732no—
216C2×Q8—1731no—
217C2×D4—3461no—
218C2×D4—3461no—
219C2×D4—3461no—
220C2×D4—1731no—
221C22×C4—21no—
222C22×C4—3462no—
223C22×C4—3462no—
224C22×C4—3461no—
225C22×C4—3461no—
226C22×C4—1731no—
227C22×C4—1731no—
228C22×C4—1731no—
229C22×C4—1731no—
230C22×C4—1731no—
231C22×C4C22×D17311yescharacteristic
232Q16—3464no—
233SD16—3464no—
234M16—3462no—
235M16—3464no—
236M16D69214yescomplemented
237C2×C8—3464no—
238C2×C8—1732no—
239C2×C8C22×D17312yes—
240C4⋊C4—21no—
241C4⋊C4—3462no—
242C4⋊C4—3462no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4C22×D17312yes—
246C4⋊C4C22×D17311yescharacteristic
247C4⋊C4C22×D17311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—3462no—
250C22⋊C4—3462no—
251C22⋊C4—3461no—
252C22⋊C4—3461no—
253C22⋊C4—3461no—
254C22⋊C4—3461no—
255C42—21no—
256C23—3462no—
257C23—3462no—
258C23—3461no—
259C23—1731no—
260C23—1731no—
261C23—6921no—
262C23D4×D17311yescharacteristic
263Q8—3462no—
264Q8—3461no—
265Q8—3462no—
266Q8—3461no—
267Q8—6922no—
268D4—3461no—
269D4—3461no—
270D4—6922no—
271D4—6922no—
272D4—6922no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3462no—
278C2×C4—3462no—
279C2×C4—3462no—
280C2×C4—3461no—
281C2×C4—3461no—
282C2×C4—3461no—
283C2×C4—3461no—
284C2×C4—3461no—
285C2×C4—3462no—
286C2×C4—1731no—
287C2×C4—1731no—
288C2×C4—6922no—
289C2×C4—6922no—
290C2×C4D4×D17311yescharacteristic
291C2×C4C2×D69212yes—
292C2×C4#1458382712yes—
293C2×C4C23×D17311yescharacteristic
294C8—3462no—
295C8C2×D69214yes—
296C22—21no—
297C22—21no—
298C22—3461no—
299C22—3461no—
300C22—3461no—
301C22—3461no—
302C22—3461no—
303C22—1732no—
304C22—6924no—
305C22—6921no—
306C22—6921no—
307C22—6921no—
308C22#1914965412yes—
309C22C2×D4×D17311yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3461no—
314C4—3461no—
315C4—6921no—
316C4—6922no—
317C4#1914965412yes—
318C4C22×D69211yescharacteristic
319C4C2×#1458382711yescharacteristic
320C2—22no—
321C2—3462no—
322C2—13841no—
323C2—6921no—
324C2C2×#1914965411yescharacteristic
325C2#137×D17311yescharacteristic
326C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#428×C173C2
socleC22×C173C22×D4
prime radical#26012003C2
hypercenter#428D173
hypocenterC173#3056
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×#19149712C2semidirect
2. C2×M16×D173C2semidirect
4. C2×#19149659C2semidirect
5. #26011750C2semidirect
6. #26011738C2semidirect
7. #26011733C2semidirect
8. #26011721C2semidirect
9. #26011499C2semidirect
10. #26011490C2semidirect
11. #428×C173C2semidirect
16. C2×Dic692C22semidirect
17. C2×#14583815C22semidirect
36. #19149602C22semidirect
37. #19149602C22semidirect
38. #19149602C22semidirect
39. #19149599C22semidirect
43. #107×C173C22semidirect
44. #106×C173C22semidirect
84. M16×C173D4semidirect
145. C2×D173#137semidirect
163. C173#3056semidirect
173. #428D173semidirect
203. #107C2×D173semidirect
204. #106C2×D173semidirect
236. M16D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×M16×D1732
#260120032
C2×#191496592
#260117502
#260117382
#260117332
#260117212
#260114992
#260114902
#428×C1732
#3056173