Finite Group Databasebeta

← #11738713

Subgroups — #11738713

Order 896.

492 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 125 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#90287C211yesmaximal, characteristic, complemented
2C2×#6456C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89808C211yesmaximal, characteristic, complemented
5#89739C211yesmaximal, characteristic, complemented
6#89662C212yesmaximal, complemented
7#89644C212yesmaximal, complemented
8#89558C212yesmaximal, complemented
9#89528C212yesmaximal, complemented
10#89504C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C22×D28C2211yescharacteristic
14C22×C4×D7—21no—
15C7×#121C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17#6503C2211yescharacteristic, complemented
18#6502—21no—
19D4×Dic7C2211yescharacteristic, complemented
20C2×#1141C2212yes—
21#6493C2211yescharacteristic
22C2×(C2×D7)⋊C4—21no—
23C2×C4×Dic7C2211yescharacteristic
24C2×D56C2212yes—
25#6456—21no—
26#6456—21no—
27#6456C2212yes—
28#6456C2212yes—
29#6456C2211yescharacteristic, complemented
30#6451C2211yescharacteristic, complemented
31#6448—21no—
32(C22⋊D7)⋊C4—21no—
33C7×#107C2212yescomplemented
34C7×#102C2212yescomplemented
35C7×C22⋊C8C2211yescharacteristic
36C22⋊(C7⋊C8)C2211yescharacteristic
37#6395C2212yes—
38Dic7⋊C8C2212yescomplemented
39C8×Dic7C2212yescomplemented
40#6383C2212yes—
41#6383C2212yescomplemented
42#6381C2212yescomplemented
43#3100—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×D4—21no—
46C2×C7×D4C2311yescharacteristic
47C22×C4×C7—21no—
48C22×C4×C7C2311yescharacteristic
49C2×C22⋊D7—21no—
50C2×C22⋊D7—21no—
51C2×C22⋊D7—21no—
52C22×Dic7—21no—
53C22×Dic7D411yescharacteristic
54C22×Dic7D411yescharacteristic
55#1159—44no—
56C2×D28—21no—
57C2×D28—24no—
58C2×D28C2311yescharacteristic
59C2×D28C2312yes—
60C2×C4×D7—21no—
61C2×C4×D7—22no—
62C2×C4×D7—22no—
63C2×C4×D7—44no—
64C2×Dic14—21no—
65C2×C7×C8C2312yes—
66C7×C4⋊C4—21no—
67C7×C4⋊C4C2312yes—
68C7×C4⋊C4C2311yescharacteristic
69C7×C22⋊C4—21no—
70C22⋊Dic7—21no—
71#1141—24no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76C4⋊Dic7C2311yescharacteristic
77Dic7⋊C4—21no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C4×Dic7C2311yescharacteristic
82C2×C7⋊C8C2312yes—
83D56—24no—
84#532—71no—
85C2×C4×D4—71no—
86#480—71no—
87#480D711yescharacteristic, complemented
88#474—72no—
89#459—72no—
90#447—71no—
91#447—71no—
92C4×D8—72no—
93C4×D8—72no—
94#431—71no—
95C23×C7—21no—
96C23×C7C2×D411yescharacteristic
97C22×D7—24no—
98C22×D7—22no—
99C22×D7—21no—
100C22×D7—42no—
101C22×D7—42no—
102C7×D4—22no—
103C7×D4—42no—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—42no—
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2411yescharacteristic
110C22⋊D7—42no—
111C22⋊D7—42no—
112C22⋊D7—42no—
113C2×Dic7—21no—
114C2×Dic7—41no—
115C2×Dic7—42no—
116C2×Dic7D814yescomplemented
117C2×Dic7C2×D411yescharacteristic
118C2×Dic7C2×D411yescharacteristic
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121D28—24no—
122D28—22no—
123D28—22no—
124D28#4114yescomplemented
125C4×D7—44no—
126C4×D7—44no—
127C4×D7—42no—
128Dic14—42no—
129C7×C8—22no—
130C7⋊C8—22no—
131C2×#41—141no—
132C22×D4—71no—
133C23×C4—141no—
134C2×D8—72no—
135C2×D8—72no—
136C4⋊Q8—71no—
137#127—71no—
138#124—141no—
139#121—71no—
140#121—141no—
141#121C2×D711yescharacteristic
142C4×D4—71no—
143C4×D4—72no—
144C4×D4—72no—
145C4×D4—71no—
146C4×D4—141no—
147C4×D4—141no—
148C4×D4—141no—
149C2×C4⋊C4C2×D711yescharacteristic
150C2×C22⋊C4—141no—
151C2×C42—71no—
152#107—72no—
153#107C2×D712yescomplemented
154C4⋊C8—72no—
155#102—72no—
156#102—72no—
157#102—72no—
158#102C2×D712yescomplemented
159C22⋊C8—71no—
160C22⋊C8C2×D711yescharacteristic
161C4×C8—72no—
162C22×C7—21no—
163C22×C7—21no—
164C22×C7—41no—
165C22×C7—41no—
166C22×C7—41no—
167C22×C7C2×D812yes—
168C22×C7C22×D411yescharacteristic
169C2×D7—24no—
170C2×D7—48no—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—41no—
174C2×D7—41no—
175C2×D7—41no—
176C4×C7—21no—
177C4×C7—41no—
178C4×C7—42no—
179C4×C7C2×#4112yes—
180Dic7—21no—
181Dic7—21no—
182Dic7—41no—
183Dic7C2×D812yes—
184Dic7C2×D812yes—
185C24—141no—
186#41—284no—
187C2×Q8—141no—
188C2×D4—21no—
189C2×D4—72no—
190C2×D4—71no—
191C2×D4—144no—
192C2×D4—141no—
193C2×D4—141no—
194C2×D4—141no—
195C2×D4—141no—
196C2×D4C22×D711yescharacteristic
197C22×C4—21no—
198C22×C4—71no—
199C22×C4—71no—
200C22×C4—142no—
201C22×C4—142no—
202C22×C4—141no—
203C22×C4—141no—
204C22×C4—284no—
205C22×C4C22×D711yescharacteristic
206D8—144no—
207D8—144no—
208C2×C8—72no—
209C2×C8C22×D712yes—
210C4⋊C4—21no—
211C4⋊C4—71no—
212C4⋊C4—141no—
213C4⋊C4C22×D711yescharacteristic
214C4⋊C4C22×D712yes—
215C22⋊C4—21no—
216C22⋊C4—142no—
217C22⋊C4—142no—
218C22⋊C4—141no—
219C22⋊C4—141no—
220C22⋊C4—141no—
221C42—71no—
222C42—71no—
223C42—71no—
224C42—71no—
225C2×C7—22no—
226C2×C7—81no—
227C2×C7#52511yescharacteristic
228C2×C7C22×D811yescharacteristic
229C2×C7#57511yescharacteristic
230D7—44no—
231D7—81no—
232C23—21no—
233C23—144no—
234C23—142no—
235C23—141no—
236C23—282no—
237C23—282no—
238C23D4×D711yescharacteristic
239Q8—282no—
240D4—22no—
241D4—42no—
242D4—74no—
243D4—144no—
244D4—142no—
245D4—142no—
246D4—282no—
247D4—282no—
248D4—282no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—74no—
254C2×C4—71no—
255C2×C4—71no—
256C2×C4—71no—
257C2×C4—71no—
258C2×C4—141no—
259C2×C4—284no—
260C2×C4—284no—
261C2×C4—282no—
262C2×C4—282no—
263C2×C4—281no—
264C2×C4D4×D711yescharacteristic
265C2×C4C23×D711yescharacteristic
266C8—22no—
267C8—142no—
268C7#310011yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—144no—
275C22—288no—
276C22—282no—
277C22—282no—
278C22—281no—
279C22—281no—
280C22—281no—
281C22D7×D812yes—
282C22C2×D4×D711yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—72no—
287C4—72no—
288C4—141no—
289C4—141no—
290C4—281no—
291C4D7×#4112yes—
292C2—22no—
293C2—81no—
294C2—284no—
295C2—561no—
296C2#9034711yescharacteristic
297C2C2×D7×D811yescharacteristic
298C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
hypercenter#480D7
hypocenterC7#3100
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #90287C2semidirect
2. C2×#6456C2semidirect
3. C7×#480C2semidirect
4. #89808C2semidirect
5. #89739C2semidirect
6. #89662C2semidirect
7. #89644C2semidirect
8. #89558C2semidirect
9. #89528C2semidirect
10. #89504C2semidirect
11. #89490C2semidirect
17. #6503C22semidirect
19. D4×Dic7C22semidirect
29. #6456C22semidirect
30. #6451C22semidirect
33. C7×#107C22semidirect
34. C7×#102C22semidirect
38. Dic7⋊C8C22semidirect
39. C8×Dic7C22semidirect
41. #6383C22semidirect
42. #6381C22semidirect
87. #480D7semidirect
116. C2×Dic7D8semidirect
124. D28#41semidirect
153. #107C2×D7semidirect
158. #102C2×D7semidirect
268. C7#3100semidirect

Maximal subgroups

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

subgroupindex
#902872
C2×#64562
C7×#4802
#898082
#897392
#896622
#896442
#895582
#895282
#895042
#894902
#31007