Finite Group Databasebeta

← #22619406

Subgroups — #22619406

Order 9344.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556615C212yesmaximal, complemented
2#127×D73C211yesmaximal, characteristic, complemented
3C4×#11924027C211yesmaximal, characteristic, complemented
4#17988788C211yesmaximal, characteristic, complemented
5#17988786C212yesmaximal, complemented
6#17988784C211yesmaximal, characteristic, complemented
7#17988782C211yesmaximal, characteristic, complemented
8#17988780C212yesmaximal, complemented
9#17988609C211yesmaximal, characteristic, complemented
10#17988590C212yesmaximal, complemented
11#459×C73C211yesmaximal, characteristic, complemented
12C2×#11924027C2211yescharacteristic
13C2×D4×D73—21no—
14C2×D4×D73—21no—
15C2×D4×D73C2212yes—
16#14556615—28no—
17C2×#11924017C2212yescomplemented
18C2×#11924016C2212yescomplemented
19C2×#11924013C2212yescomplemented
20C2×#11924011C2212yes—
21#14556588C2211yescharacteristic
22#14556587—21no—
23#14556587—21no—
24#14556587C2212yes—
25C4×Dic146C2211yescharacteristic, complemented
26#14556556C2211yescharacteristic, complemented
27D4×Dic73C2211yescharacteristic, complemented
28C4×C22⋊D73—21no—
29#14556551—21no—
30#14556549C2211yescharacteristic
31#14556547—21no—
32C2×C4×Dic73—21no—
33C42×D73C2211yescharacteristic
34C4⋊#10596831C2211yescharacteristic, complemented
35#14556520C2212yescomplemented
36#14556513C2212yescomplemented
37#14556512C2212yescomplemented
38#14556511C2212yescomplemented
39C2×D8×C73C2212yescomplemented
40#127×C73C2211yescharacteristic
41C4×D4×C73C2211yescharacteristic, complemented
42C4⋊C8×C73C2211yescharacteristic, complemented
43#102×C73C2212yescomplemented
44C23×D73—22no—
45#11924027—22no—
46#11924027D414yescomplemented
47C2×Dic146C2311yescharacteristic
48C2×D292—21no—
49C2×D292—21no—
50C2×D292C2312yes—
51C2×C22⋊D73—21no—
52C2×C22⋊D73—22no—
53C2×C22⋊D73—22no—
54C2×C22⋊D73—22no—
55D4×D73—24no—
56D4×D73—24no—
57D4×D73—44no—
58D4×D73—44no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C2×C4×D73D411yescharacteristic
62C2×C4×D73D411yescharacteristic
63C2×C4×D73C2311yescharacteristic
64#11924017—24no—
65#11924016—24no—
66#11924013—24no—
67#11924011—24no—
68C2×#10596831C2312yes—
69C4⋊Dic73C2311yescharacteristic
70#11924007—21no—
71#11924007—21no—
72C22⋊Dic73—21no—
73#11924005—21no—
74C4×Dic73—21no—
75C4×Dic73—21no—
76C4×Dic73—21no—
77C4×Dic73D411yescharacteristic
78C4×Dic73D411yescharacteristic
79C4×Dic73C2311yescharacteristic
80C2×D4×C73—21no—
81C2×D4×C73—21no—
82C2×D4×C73C2312yes—
83C2×D4×C73C2311yescharacteristic
84C22×C4×C73—21no—
85D8×C73—24no—
86C2×C8×C73C2312yes—
87C4⋊C4×C73C2311yescharacteristic
88C22⋊C4×C73—21no—
89C42×C73C2311yescharacteristic
90C23×C73—21no—
91C23×C73—22no—
92C22×D73—22no—
93C22×D73—44no—
94C22×D73—44no—
95C22×D73—44no—
96C22×D73C2×D411yescharacteristic
97D4×C73—22no—
98D4×C73—21no—
99D4×C73—22no—
100D4×C73—42no—
101D4×C73—42no—
102D4×C73C2×D412yescomplemented
103C2×C4×C73—21no—
104C2×C4×C73—42no—
105C2×C4×C73C2×D411yescharacteristic
106C2×C4×C73C2×D411yescharacteristic
107C2×C4×C73C2411yescharacteristic
108C22⋊D73—24no—
109C22⋊D73—44no—
110C22⋊D73—44no—
111C22⋊D73—44no—
112C2×Dic73—21no—
113C2×Dic73—24no—
114C2×Dic73—21no—
115C2×Dic73—42no—
116C2×Dic73C2×D411yescharacteristic
117C2×Dic73C2×D411yescharacteristic
118C2×Dic73C2×D411yescharacteristic
119D292—22no—
120D292—22no—
121D292—42no—
122D292—42no—
123C4×D73—22no—
124C4×D73—22no—
125C4×D73C2×D412yes—
126C4×D73#4112yes—
127Dic146—21no—
128Dic146C2×D412yescomplemented
129C8×C73—22no—
130#10596831—22no—
131C22×C73—22no—
132C22×C73—41no—
133C22×C73—41no—
134C22×C73—42no—
135C22×C73—42no—
136C22×C73—42no—
137C22×C73C22×D411yescharacteristic
138C2×D73—21no—
139C2×D73—21no—
140C2×D73—42no—
141C2×D73—42no—
142C2×D73—42no—
143C2×D73—88no—
144C2×D73#12112yes—
145C4×C73—21no—
146C4×C73—41no—
147C4×C73AGL(1,ℤ/8)12yescomplemented
148C4×C73C22×D411yescharacteristic
149C4×C73C2×#4111yescharacteristic
150Dic73—22no—
151Dic73—21no—
152Dic73—41no—
153Dic73#12112yes—
154Dic73AGL(1,ℤ/8)12yescomplemented
155C2×C73—42no—
156C2×C73—82no—
157C2×C73C2×#12111yescharacteristic
158C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
160D73—22no—
161D73—82no—
162#3038—731nomaximal, Sylow
163C73#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—732no—
165C2×#127—731no—
166C4×#41—731no—
167#460—731no—
168#460—731no—
169#460—732no—
170#459—732no—
171#459—731no—
172#459D7311yescharacteristic, complemented
173#423—731no—
174#418—732no—
175C2×#41—731no—
176C22×D4—732no—
177C22×D4—1461no—
178C22×D4—1461no—
179AGL(1,ℤ/8)—1468no—
180C2×SD16—732no—
181C2×SD16—732no—
182C2×D8—732no—
183C2×D8C2×D7312yescomplemented
184C2×M16—732no—
185#127—732no—
186#127—731no—
187#127—1461no—
188#127—1461no—
189#127C2×D7311yescharacteristic
190C4×Q8—731no—
191C4×Q8—731no—
192C4×D4—731no—
193C4×D4—1461no—
194C4×D4—1461no—
195C4×D4C2×D7311yescharacteristic, complemented
196#117—731no—
197#117—1461no—
198C2×C42—731no—
199C2×C42—1461no—
200C4⋊C8—732no—
201C4⋊C8—731no—
202C4⋊C8C2×D7311yescharacteristic, complemented
203#102—732no—
204#102—732no—
205#102—732no—
206#102C2×D7312yescomplemented
207C24—1462no—
208#41—734no—
209#41—1462no—
210C2×Q8—731no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—2924no—
214C2×D4—2924no—
215C2×D4—732no—
216C2×D4—1464no—
217C2×D4—1464no—
218C2×D4—1462no—
219C2×D4—1462no—
220C2×D4—1462no—
221C2×D4—1461no—
222C2×D4—1461no—
223C2×D4—1461no—
224C2×D4C22×D7312yes—
225C2×D4C22×D7311yescharacteristic
226C22×C4—21no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—1461no—
231C22×C4—1461no—
232SD16—1464no—
233SD16—1464no—
234D8—24no—
235D8—1464no—
236M16—1464no—
237C2×C8—732no—
238C2×C8C22×D7312yes—
239C4⋊C4—731no—
240C4⋊C4—1461no—
241C4⋊C4—1461no—
242C4⋊C4C22×D7311yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—1461no—
245C22⋊C4—1461no—
246C42—731no—
247C42—731no—
248C42—731no—
249C42—1461no—
250C42—1461no—
251C42—1461no—
252C42C22×D7311yescharacteristic
253C23—21no—
254C23—22no—
255C23—2924no—
256C23—2924no—
257C23—2924no—
258C23—731no—
259C23—1462no—
260Q8—732no—
261Q8—1461no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—2922no—
268D4—2924no—
269D4—2924no—
270D4—2924no—
271D4—2922no—
272D4—1464no—
273D4—1462no—
274D4—1462no—
275D4D4×D7312yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—2922no—
279C2×C4—731no—
280C2×C4—731no—
281C2×C4—732no—
282C2×C4—732no—
283C2×C4—731no—
284C2×C4—1462no—
285C2×C4—1461no—
286C2×C4—1464no—
287C2×C4—1461no—
288C2×C4—1462no—
289C2×C4D4×D7311yescharacteristic
290C2×C4D4×D7311yescharacteristic
291C2×C4C23×D7311yescharacteristic
292C8—22no—
293C8—1462no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—2922no—
301C22—2922no—
302C22—2922no—
303C22—5848no—
304C22—732no—
305C22—1461no—
306C22—1461no—
307C22C2×D4×D7311yescharacteristic
308C4—21no—
309C4—41no—
310C4—2921no—
311C4—732no—
312C4—732no—
313C4—1462no—
314C4—1461no—
315C4#1455661512yescomplemented
316C4C2×D4×D7311yescharacteristic
317C4#41×D7311yescharacteristic
318C2—42no—
319C2—82no—
320C2—5842no—
321C2—1462no—
322C2#121×D7311yescharacteristic
323C2AGL(1,ℤ/8)×D7311yescharacteristic
324C2C2×#1455661511yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#459×C73C2
socleC22×C73C22×D4
hypercenter#459D73
hypocenterC73#3038
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14556615C2semidirect
2. #127×D73C2semidirect
3. C4×#11924027C2semidirect
4. #17988788C2semidirect
5. #17988786C2semidirect
6. #17988784C2semidirect
7. #17988782C2semidirect
8. #17988780C2semidirect
9. #17988609C2semidirect
10. #17988590C2semidirect
11. #459×C73C2semidirect
17. C2×#11924017C22semidirect
18. C2×#11924016C22semidirect
19. C2×#11924013C22semidirect
25. C4×Dic146C22semidirect
26. #14556556C22semidirect
27. D4×Dic73C22semidirect
34. C4⋊#10596831C22semidirect
35. #14556520C22semidirect
36. #14556513C22semidirect
37. #14556512C22semidirect
38. #14556511C22semidirect
39. C2×D8×C73C22semidirect
41. C4×D4×C73C22semidirect
42. C4⋊C8×C73C22semidirect
43. #102×C73C22semidirect
46. #11924027D4semidirect
102. D4×C73C2×D4semidirect
128. Dic146C2×D4semidirect
147. C4×C73AGL(1,ℤ/8)semidirect
154. Dic73AGL(1,ℤ/8)semidirect
163. C73#3038semidirect
172. #459D73semidirect
183. C2×D8C2×D73semidirect
195. C4×D4C2×D73semidirect
202. C4⋊C8C2×D73semidirect
206. #102C2×D73semidirect
275. D4D4×D73semidirect
315. C4#14556615semidirect