Finite Group Databasebeta

← #22622161

Subgroups — #22622161

Order 9344.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#17989311C211yesmaximal, characteristic, complemented
2#17989222C211yesmaximal, characteristic
3#17988910C212yesmaximal, complemented
4#17988864C211yesmaximal, characteristic, complemented
5#17988832C212yesmaximal, complemented
6#17988784C211yesmaximal, characteristic, complemented
7#17988739C212yesmaximal, complemented
8C4×#11924017C211yesmaximal, characteristic, complemented
9#17988666C211yesmaximal, characteristic, complemented
10#17988590C212yesmaximal, complemented
11#486×C73C211yesmaximal, characteristic, complemented
12C2×D4×D73—21no—
13C2×D4×D73C2212yes—
14C2×#11924017C2211yescharacteristic
15C2×#11924016C2212yescomplemented
16C2×#11924011C2212yescomplemented
17#14556586—22no—
18#14556580—21no—
19#14556571C2212yescomplemented
20#14556562—21no—
21#14556562C2212yes—
22#14556559—22no—
23Q8×Dic73—21no—
24C4×Dic146C2211yescharacteristic, complemented
25C4×D292C2211yescharacteristic
26C4×D292C2211yescharacteristic
27#14556549C2212yescomplemented
28#14556524C2212yescomplemented
29#14556523C2211yescharacteristic, complemented
30C4⋊#10596831C2211yescharacteristic, complemented
31#14556516C2211yescharacteristic, complemented
32#14556513C2211yescharacteristic
33#14556512C2212yescomplemented
34#14556511C2212yes—
35#14556503C2212yescomplemented
36C4⋊Q8×C73C2211yescharacteristic, complemented
37#127×C73C2211yescharacteristic
38#102×C73C2212yescomplemented
39#102×C73C2212yescomplemented
40C4×C8×C73C2211yescharacteristic, complemented
41C23×D73—22no—
42C2×Dic146C2311yescharacteristic
43C2×D292D411yescharacteristic
44C2×D292C2311yescharacteristic
45C2×C22⋊D73—22no—
46C2×C22⋊D73—22no—
47D4×D73—24no—
48D4×D73—42no—
49D4×D73—44no—
50C2×C4×D73—21no—
51C2×C4×D73C2312yes—
52#11924017—22no—
53#11924016—42no—
54#11924011—42no—
55C2×#10596831C2312yes—
56C4⋊Dic73—22no—
57C4⋊Dic73D411yescharacteristic
58C4⋊Dic73C2311yescharacteristic
59C4⋊Dic73C2311yescharacteristic
60#11924007—21no—
61C22⋊Dic73—22no—
62#11924005—22no—
63#11924005—22no—
64#11924005—21no—
65#11924005D412yes—
66C4×Dic73—21no—
67C4×Dic73—22no—
68C2×Q8×C73—21no—
69C2×D4×C73—21no—
70C2×D4×C73—21no—
71C2×D4×C73C2312yes—
72C2×C8×C73C2311yescharacteristic
73C2×C8×C73C2311yescharacteristic
74C4⋊C4×C73—21no—
75C4⋊C4×C73C2312yes—
76C42×C73C2311yescharacteristic
77C23×C73—22no—
78C22×D73—21no—
79C22×D73—44no—
80C22×D73—44no—
81C22×D73—82no—
82C22×D73C2×D412yes—
83Q8×C73—42no—
84D4×C73—22no—
85D4×C73—22no—
86D4×C73—42no—
87D4×C73—81no—
88C2×C4×C73—22no—
89C2×C4×C73C2×D411yescharacteristic
90C2×C4×C73C2×D411yescharacteristic
91C2×C4×C73C2411yescharacteristic
92C22⋊D73—44no—
93C22⋊D73—44no—
94C2×Dic73—21no—
95C2×Dic73—21no—
96C2×Dic73—21no—
97C2×Dic73C2×D412yes—
98D292—22no—
99D292—21no—
100D292#4112yes—
101C4×D73—22no—
102C4×D73—22no—
103C4×D73—42no—
104Dic146—21no—
105Dic146—21no—
106C8×C73—21no—
107C8×C73—21no—
108#10596831—22no—
109C22×C73—42no—
110C22×C73—42no—
111C22×C73—42no—
112C22×C73C22×D411yescharacteristic
113C2×D73—22no—
114C2×D73—22no—
115C2×D73—22no—
116C2×D73—22no—
117C2×D73—41no—
118C2×D73—41no—
119C2×D73—88no—
120C4×C73—21no—
121C4×C73—42no—
122C4×C73AGL(1,ℤ/8)12yes—
123C4×C73C2×#4111yescharacteristic
124C4×C73H(5,𝔽2)11yescharacteristic
125Dic73—22no—
126Dic73—41no—
127Dic73—41no—
128Dic73—41no—
129C2×C73—82no—
130C2×C73#54611yescharacteristic
131C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C73#57711yescharacteristic
133D73—42no—
134D73—42no—
135#3324—731nomaximal, Sylow
136C73#332411yescharacteristic, complemented, Sylow
137#556—731no—
138D42—731no—
139#486D7311yescharacteristic, complemented
140#482—732no—
141#474—731no—
142#468—732no—
143#460—731no—
144#450—732no—
145C4×SD16—731no—
146#436—731no—
147#418—732no—
148C22×D4—732no—
149C22×D4—1461no—
150C2×SD16—732no—
151C2×SD16—731no—
152C2×M16—732no—
153C4⋊Q8C2×D7311yescharacteristic, complemented
154#127C2×D7311yescharacteristic
155#126—1462no—
156#124—1461no—
157C22⋊Q8—732no—
158#121—732no—
159#121—1461no—
160#120—1462no—
161C4×Q8—731no—
162C4×Q8—1461no—
163C4×D4—731no—
164C4×D4—731no—
165#117—732no—
166#107—732no—
167#106—731no—
168C4⋊C8—731no—
169#103—731no—
170#102—732no—
171#102—731no—
172#102—732no—
173#102C2×D7312yescomplemented
174#102C2×D7312yescomplemented
175C22⋊C8—732no—
176C4×C8C2×D7311yescharacteristic, complemented
177C24—1462no—
178C2×Q8—21no—
179C2×Q8—731no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—2924no—
183C2×D4—2922no—
184C2×D4—731no—
185C2×D4—731no—
186C2×D4—1462no—
187C2×D4—1464no—
188C2×D4—1462no—
189C2×D4C22×D7312yes—
190C22×C4—732no—
191C22×C4—1461no—
192SD16—2922no—
193SD16—1462no—
194M16—2922no—
195C2×C8—732no—
196C2×C8C22×D7311yescharacteristic
197C2×C8C22×D7311yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—731no—
200C4⋊C4—731no—
201C4⋊C4—731no—
202C4⋊C4—1462no—
203C4⋊C4—1461no—
204C4⋊C4C22×D7312yes—
205C22⋊C4—732no—
206C22⋊C4—1462no—
207C22⋊C4—1462no—
208C22⋊C4—1462no—
209C22⋊C4—1461no—
210C42—1462no—
211C42—1461no—
212C42C22×D7311yescharacteristic
213C23—22no—
214C23—2924no—
215C23—2924no—
216C23—5842no—
217C23—732no—
218C23—1461no—
219Q8—42no—
220Q8—1461no—
221Q8—1461no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—81no—
226D4—2924no—
227D4—2924no—
228D4—732no—
229D4—1462no—
230D4—1461no—
231C2×C4—22no—
232C2×C4—2922no—
233C2×C4—732no—
234C2×C4—1461no—
235C2×C4—1461no—
236C2×C4—1461no—
237C2×C4—1462no—
238C2×C4—1462no—
239C2×C4D4×D7311yescharacteristic
240C2×C4D4×D7311yescharacteristic
241C2×C4C23×D7311yescharacteristic
242C8—21no—
243C8—21no—
244C8—1462no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—2921no—
249C22—2921no—
250C22—5848no—
251C22—1462no—
252C22—1462no—
253C22—1462no—
254C22—1462no—
255C22C2×D4×D7311yescharacteristic
256C4—21no—
257C4—42no—
258C4—2921no—
259C4—2921no—
260C4—2921no—
261C4—1462no—
262C4#1455661512yes—
263C4#41×D7311yescharacteristic
264C4#1455663711yescharacteristic
265C2—82no—
266C2—2922no—
267C2—2922no—
268C2#1798923011yescharacteristic
269C2C2×#1455661511yescharacteristic
270C2#1798939311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#486×C73C2
socleC22×C73C22×D4
hypercenter#486D73
hypocenterC73#3324
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #17989311C2semidirect
3. #17988910C2semidirect
4. #17988864C2semidirect
5. #17988832C2semidirect
6. #17988784C2semidirect
7. #17988739C2semidirect
8. C4×#11924017C2semidirect
9. #17988666C2semidirect
10. #17988590C2semidirect
11. #486×C73C2semidirect
15. C2×#11924016C22semidirect
16. C2×#11924011C22semidirect
19. #14556571C22semidirect
24. C4×Dic146C22semidirect
27. #14556549C22semidirect
28. #14556524C22semidirect
29. #14556523C22semidirect
30. C4⋊#10596831C22semidirect
31. #14556516C22semidirect
33. #14556512C22semidirect
35. #14556503C22semidirect
36. C4⋊Q8×C73C22semidirect
38. #102×C73C22semidirect
39. #102×C73C22semidirect
40. C4×C8×C73C22semidirect
136. C73#3324semidirect
139. #486D73semidirect
153. C4⋊Q8C2×D73semidirect
173. #102C2×D73semidirect
174. #102C2×D73semidirect
176. C4×C8C2×D73semidirect