Finite Group Databasebeta

← #22619408

Subgroups — #22619408

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×#14556618C212yesmaximal, complemented
2#127×D73C211yesmaximal, characteristic, complemented
3C4×#11924029C211yesmaximal, characteristic, complemented
4#17988787C211yesmaximal, characteristic, complemented
5#17988785C212yesmaximal, complemented
6#17988783C211yesmaximal, characteristic, complemented
7#17988781C212yesmaximal, complemented
8#17988779C211yesmaximal, characteristic, complemented
9#17988609C211yesmaximal, characteristic, complemented
10#17988590C212yesmaximal, complemented
11#460×C73C211yesmaximal, characteristic, complemented
12C2×#11924029C2211yescharacteristic
13C2×D4×D73—21no—
14C2×D4×D73—21no—
15C2×D4×D73C2212yes—
16#14556618—28no—
17C2×#11924015C2212yescomplemented
18C2×D584C2212yescomplemented
19C2×#11924013C2212yescomplemented
20C2×#11924011C2212yes—
21#14556588C2211yescharacteristic
22#14556587—21no—
23#14556587—21no—
24#14556587C2212yes—
25Q8×Dic73C2211yescharacteristic, complemented
26C4×D292—21no—
27C4×D292C2211yescharacteristic, complemented
28#14556553—21no—
29#14556553C2211yescharacteristic, complemented
30#14556549—21no—
31#14556549C2211yescharacteristic
32C42×D73—21no—
33C42×D73C2211yescharacteristic
34C4⋊#10596831C2211yescharacteristic, complemented
35#14556520C2212yescomplemented
36#14556513C2212yescomplemented
37#14556512C2212yescomplemented
38#14556511C2212yescomplemented
39C2×SD16×C73C2212yescomplemented
40#127×C73C2211yescharacteristic
41C4×Q8×C73C2211yescharacteristic, complemented
42C4⋊C8×C73C2211yescharacteristic, complemented
43#102×C73C2212yescomplemented
44C23×D73—22no—
45#11924029—22no—
46#11924029D414yescomplemented
47C2×D292—21no—
48C2×D292—21no—
49C2×D292—21no—
50C2×D292C2312yes—
51C2×D292C2311yescharacteristic
52C2×C22⋊D73—22no—
53C2×C22⋊D73—22no—
54C2×C22⋊D73—22no—
55D4×D73—24no—
56D4×D73—24no—
57D4×D73—44no—
58D4×D73—44no—
59C2×C4×D73—21no—
60C2×C4×D73—21no—
61C2×C4×D73—21no—
62C2×C4×D73D411yescharacteristic
63C2×C4×D73D411yescharacteristic
64C2×C4×D73C2311yescharacteristic
65#11924015—24no—
66D584—24no—
67#11924013—24no—
68#11924011—24no—
69C2×#10596831C2312yes—
70C4⋊Dic73—21no—
71C4⋊Dic73C2311yescharacteristic
72#11924005—21no—
73#11924005—21no—
74#11924005—21no—
75C4×Dic73—21no—
76C4×Dic73—21no—
77C4×Dic73D411yescharacteristic
78C4×Dic73D411yescharacteristic
79C4×Dic73C2311yescharacteristic
80C2×Q8×C73C2311yescharacteristic
81C2×D4×C73—21no—
82C2×D4×C73—21no—
83C2×D4×C73C2312yes—
84SD16×C73—24no—
85C2×C8×C73C2312yes—
86C4⋊C4×C73—21no—
87C4⋊C4×C73C2311yescharacteristic
88C42×C73—21no—
89C42×C73C2311yescharacteristic
90C23×C73—22no—
91C22×D73—21no—
92C22×D73—22no—
93C22×D73—44no—
94C22×D73—44no—
95C22×D73—44no—
96C22×D73C2×D411yescharacteristic
97Q8×C73—21no—
98Q8×C73C2×D412yescomplemented
99D4×C73—22no—
100D4×C73—22no—
101D4×C73—42no—
102D4×C73—42no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73C2×D411yescharacteristic
106C2×C4×C73C2×D411yescharacteristic
107C2×C4×C73C2411yescharacteristic
108C22⋊D73—44no—
109C22⋊D73—44no—
110C22⋊D73—44no—
111C2×Dic73—21no—
112C2×Dic73C2×D411yescharacteristic
113C2×Dic73C2×D411yescharacteristic
114C2×Dic73C2×D411yescharacteristic
115D292—24no—
116D292—22no—
117D292—21no—
118D292—22no—
119D292—42no—
120D292—42no—
121D292C2×D412yescomplemented
122C4×D73—22no—
123C4×D73—24no—
124C4×D73—22no—
125C4×D73—42no—
126C4×D73—42no—
127C4×D73C2×D412yes—
128C4×D73#4112yes—
129C8×C73—22no—
130#10596831—22no—
131C22×C73—42no—
132C22×C73—42no—
133C22×C73—42no—
134C22×C73C22×D411yescharacteristic
135C2×D73—21no—
136C2×D73—21no—
137C2×D73—22no—
138C2×D73—41no—
139C2×D73—41no—
140C2×D73—42no—
141C2×D73—42no—
142C2×D73—42no—
143C2×D73—88no—
144C2×D73#12112yes—
145C4×C73—22no—
146C4×C73—21no—
147C4×C73—41no—
148C4×C73AGL(1,ℤ/8)12yescomplemented
149C4×C73C22×D411yescharacteristic
150C4×C73C2×#4111yescharacteristic
151Dic73—21no—
152Dic73—41no—
153Dic73#12112yes—
154Dic73AGL(1,ℤ/8)12yescomplemented
155C2×C73—82no—
156C2×C73C2×#12111yescharacteristic
157C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
159D73—22no—
160D73—42no—
161D73—82no—
162#3038—731nomaximal, Sylow
163C73#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—732no—
165C2×#127—731no—
166C4×#41—731no—
167#460—732no—
168#460—731no—
169#460D7311yescharacteristic, complemented
170#459—731no—
171#459—731no—
172#459—732no—
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×SD16C2×D7312yescomplemented
182C2×D8—732no—
183C2×D8—732no—
184C2×M16—732no—
185#127—732no—
186#127—731no—
187#127—1461no—
188#127—1461no—
189#127C2×D7311yescharacteristic
190C4×Q8—731no—
191C4×Q8C2×D7311yescharacteristic, complemented
192C4×D4—731no—
193C4×D4—731no—
194C4×D4—1461no—
195C4×D4—1461no—
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×Q8C22×D7311yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—2924no—
214C2×D4—2924no—
215C2×D4—731no—
216C2×D4—732no—
217C2×D4—1461no—
218C2×D4—1464no—
219C2×D4—1464no—
220C2×D4—1462no—
221C2×D4—1462no—
222C2×D4—1462no—
223C2×D4—1461no—
224C2×D4—1461no—
225C2×D4C22×D7312yes—
226C22×C4—731no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—1461no—
230C22×C4—1461no—
231C22×C4—1461no—
232SD16—24no—
233SD16—1464no—
234D8—1464no—
235D8—1464no—
236M16—1464no—
237C2×C8—732no—
238C2×C8C22×D7312yes—
239C4⋊C4—21no—
240C4⋊C4—731no—
241C4⋊C4—1461no—
242C4⋊C4C22×D7311yescharacteristic
243C22⋊C4—1461no—
244C22⋊C4—1461no—
245C22⋊C4—1461no—
246C42—21no—
247C42—731no—
248C42—731no—
249C42—731no—
250C42—1461no—
251C42—1461no—
252C42C22×D7311yescharacteristic
253C23—22no—
254C23—2924no—
255C23—2924no—
256C23—2924no—
257C23—731no—
258C23—1462no—
259C23—1461no—
260Q8—21no—
261Q8D4×D7312yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2922no—
267D4—2924no—
268D4—2924no—
269D4—2924no—
270D4—2922no—
271D4—732no—
272D4—1464no—
273D4—1462no—
274D4—1462no—
275D4—1461no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—2922no—
279C2×C4—2922no—
280C2×C4—731no—
281C2×C4—731no—
282C2×C4—732no—
283C2×C4—732no—
284C2×C4—731no—
285C2×C4—1462no—
286C2×C4—1464no—
287C2×C4—1461no—
288C2×C4—1462no—
289C2×C4D4×D7311yescharacteristic
290C2×C4D4×D7311yescharacteristic
291C2×C4C23×D7311yescharacteristic
292C8—22no—
293C8—1462no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—2921no—
298C22—2921no—
299C22—2922no—
300C22—2922no—
301C22—2922no—
302C22—5848no—
303C22—732no—
304C22—1462no—
305C22—1461no—
306C22—1461no—
307C22C2×D4×D7311yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—2921no—
312C4—732no—
313C4—732no—
314C4—1461no—
315C4#1455661812yescomplemented
316C4C2×D4×D7311yescharacteristic
317C4#41×D7311yescharacteristic
318C2—82no—
319C2—2922no—
320C2—5842no—
321C2—1462no—
322C2#121×D7311yescharacteristic
323C2AGL(1,ℤ/8)×D7311yescharacteristic
324C2C2×#1455661811yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#460×C73C2
socleC22×C73C22×D4
hypercenter#460D73
hypocenterC73#3038
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#14556618C2semidirect
2. #127×D73C2semidirect
3. C4×#11924029C2semidirect
4. #17988787C2semidirect
5. #17988785C2semidirect
6. #17988783C2semidirect
7. #17988781C2semidirect
8. #17988779C2semidirect
9. #17988609C2semidirect
10. #17988590C2semidirect
11. #460×C73C2semidirect
17. C2×#11924015C22semidirect
18. C2×D584C22semidirect
19. C2×#11924013C22semidirect
25. Q8×Dic73C22semidirect
27. C4×D292C22semidirect
29. #14556553C22semidirect
34. C4⋊#10596831C22semidirect
35. #14556520C22semidirect
36. #14556513C22semidirect
37. #14556512C22semidirect
38. #14556511C22semidirect
39. C2×SD16×C73C22semidirect
41. C4×Q8×C73C22semidirect
42. C4⋊C8×C73C22semidirect
43. #102×C73C22semidirect
46. #11924029D4semidirect
98. Q8×C73C2×D4semidirect
121. D292C2×D4semidirect
148. C4×C73AGL(1,ℤ/8)semidirect
154. Dic73AGL(1,ℤ/8)semidirect
163. C73#3038semidirect
169. #460D73semidirect
181. C2×SD16C2×D73semidirect
191. C4×Q8C2×D73semidirect
202. C4⋊C8C2×D73semidirect
206. #102C2×D73semidirect
261. Q8D4×D73semidirect
315. C4#14556618semidirect