Finite Group Databasebeta

← #11741418

Subgroups — #11741418

Order 896.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×C7×#137C211yesmaximal, characteristic, complemented
2#90523C211yesmaximal, characteristic, complemented
3C2×#6512C211yesmaximal, characteristic, complemented
4C2×#6510C211yesmaximal, characteristic, complemented
5C2×#6505C211yesmaximal, characteristic, complemented
6C2×#6471C211yesmaximal, characteristic, complemented
7#89963C212yesmaximal, complemented
8#89962C212yesmaximal, complemented
9#89948C212yesmaximal, complemented
10#89940C212yesmaximal, complemented
11M16×Dic7C211yesmaximal, characteristic, complemented
12C2×C7×#41C2211yescharacteristic
13C22×C7×Q8C2211yescharacteristic
14C2×#1157C2211yescharacteristic
15C22×D28C2211yescharacteristic
16C7×#137—24no—
17C2×C7×Q16C2212yescomplemented
18C2×C7×SD16C2212yescomplemented
19C2×C7×M16C2211yescharacteristic
20#6512—24no—
21#6510—22no—
22#6510C2212yes—
23#6510C2212yes—
24#6510C2211yescharacteristic, complemented
25#6507C2211yescharacteristic, complemented
26C2×#1144C2212yescomplemented
27#6505—24no—
28C2×#1143C2212yes—
29C2×#1143C2212yescomplemented
30#6503C2211yescharacteristic, complemented
31#6502—22no—
32#6499C2211yescharacteristic, complemented
33C2×#1141C2212yes—
34C4×C22⋊D7—22no—
35C2×(C2×D7)⋊C4—22no—
36C2×C4×Dic7C2211yescharacteristic
37C2×#1136C2211yescharacteristic
38#6471—24no—
39C2×D56C2212yes—
40C2×#1132C2212yes—
41#6389C2212yescomplemented
42C8×Dic7C2212yescomplemented
43#3153—71nomaximal, Sylow
44C23×D7—21no—
45C7×#41—22no—
46C7×#41—22no—
47C2×C7×Q8—21no—
48C2×C7×Q8—22no—
49C2×C7×Q8—22no—
50C2×C7×Q8C2311yescharacteristic
51C2×C7×Q8C2312yes—
52C2×C7×D4—21no—
53C2×C7×D4C2311yescharacteristic
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7C2311yescharacteristic
57C2×C22⋊D7—21no—
58C2×C22⋊D7—22no—
59C22×Dic7D412yes—
60#1157—22no—
61#1157—22no—
62C2×D28—22no—
63C2×D28—22no—
64C2×D28—21no—
65C2×D28C2312yes—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×Dic14C2311yescharacteristic
69C7×Q16—24no—
70C7×SD16—24no—
71C7×M16D414yescomplemented
72C2×C7×C8C2312yes—
73C22⋊Dic7—22no—
74#1144—24no—
75#1143—24no—
76#1143—24no—
77#1141—24no—
78(C2×D7)⋊C4—24no—
79(C2×D7)⋊C4—24no—
80(C2×D7)⋊C4—22no—
81Dic7⋊C4—22no—
82C4×Dic7D412yes—
83C4×Dic7C2311yescharacteristic
84C4×Dic7C2311yescharacteristic
85#1136D414yescomplemented
86C2×C7⋊C8C2312yes—
87D56—24no—
88#1132—24no—
89C2×#137—71no—
90C2×#137D711yescharacteristic, complemented
91C2×AGL(1,ℤ/8)—71no—
92C2×AGL(1,ℤ/8)—71no—
93#532—71no—
94C2×#124—71no—
95#497—72no—
96#496—72no—
97#495—72no—
98#495—72no—
99C4×M16—71no—
100C23×C7—21no—
101C23×C7C2×D411yescharacteristic
102C22×D7—22no—
103C22×D7—21no—
104C22×D7—44no—
105C22×D7—42no—
106C7×Q8—21no—
107C7×Q8—22no—
108C7×Q8—22no—
109C7×Q8—21no—
110C7×Q8—42no—
111C7×D4—21no—
112C7×D4—21no—
113C7×D4—42no—
114C2×C4×C7—21no—
115C2×C4×C7—22no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7C2×D411yescharacteristic
119C2×C4×C7C2×D412yes—
120C2×C4×C7C2×D412yes—
121C2×C4×C7C2411yescharacteristic
122C22⋊D7—44no—
123C22⋊D7—42no—
124C2×Dic7—21no—
125C2×Dic7—42no—
126C2×Dic7C2×D412yes—
127C2×Dic7C2×D412yes—
128D28—22no—
129D28—21no—
130D28—22no—
131D28—21no—
132D28—42no—
133C4×D7—42no—
134Dic14—21no—
135Dic14—21no—
136C7×C8C2×D414yes—
137C7⋊C8C2×D414yes—
138C2×#41—71no—
139C2×#41C2×D711yescharacteristic
140C22×Q8C2×D711yescharacteristic
141C22×D4—71no—
142#137—24no—
143#137—144no—
144AGL(1,ℤ/8)—144no—
145AGL(1,ℤ/8)—144no—
146C2×Q16—72no—
147C2×Q16C2×D712yescomplemented
148C2×SD16—72no—
149C2×SD16—72no—
150C2×SD16—72no—
151C2×SD16C2×D712yescomplemented
152C2×D8—72no—
153C2×D8—72no—
154C2×M16—71no—
155C2×M16C2×D711yescharacteristic
156C4⋊Q8—71no—
157#127—71no—
158#124—72no—
159#124—72no—
160#124—71no—
161#124—71no—
162#124—142no—
163#121—142no—
164C4×D4—142no—
165C2×C22⋊C4—142no—
166C2×C42—71no—
167#97—72no—
168C4×C8—72no—
169C22×C7—22no—
170C22×C7—41no—
171C22×C7—42no—
172C22×C7#12712yes—
173C22×C7C22×D411yescharacteristic
174C2×D7—41no—
175C2×D7—42no—
176C2×D7—42no—
177C2×D7—44no—
178C2×D7—84no—
179C4×C7—41no—
180C4×C7—42no—
181C4×C7#12712yes—
182C4×C7C22×D411yescharacteristic
183C4×C7C22×D411yescharacteristic
184Dic7—22no—
185Dic7—22no—
186Dic7—41no—
187C24—141no—
188#41—22no—
189#41—22no—
190#41—142no—
191#41—142no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—71no—
196C2×Q8C22×D712yes—
197C2×Q8C22×D711yescharacteristic
198C2×D4—21no—
199C2×D4—72no—
200C2×D4—71no—
201C2×D4—142no—
202C2×D4—141no—
203C2×D4—142no—
204C2×D4—141no—
205C2×D4—142no—
206C2×D4C22×D711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—72no—
210C22×C4—141no—
211C22×C4C22×D711yescharacteristic
212Q16—24no—
213Q16—144no—
214SD16—24no—
215SD16—144no—
216SD16—144no—
217SD16—144no—
218D8—144no—
219D8—144no—
220M16—74no—
221M16C22⋊D714yescomplemented
222C2×C8—72no—
223C2×C8C22×D712yes—
224C4⋊C4—142no—
225C22⋊C4—144no—
226C22⋊C4—144no—
227C22⋊C4—142no—
228C22⋊C4—142no—
229C42—72no—
230C42—71no—
231C42—71no—
232C2×C7—22no—
233C2×C7—81no—
234C2×C7C2×#12711yescharacteristic
235C2×C7#57512yes—
236D7—81no—
237D7—82no—
238C23—21no—
239C23—142no—
240C23—141no—
241C23—284no—
242C23—282no—
243C23D4×D711yescharacteristic
244Q8—21no—
245Q8—22no—
246Q8—22no—
247Q8—21no—
248Q8—42no—
249Q8—141no—
250Q8—141no—
251D4—21no—
252D4—21no—
253D4—42no—
254D4—142no—
255D4—141no—
256D4—141no—
257D4—142no—
258D4—284no—
259D4—282no—
260D4—282no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—72no—
266C2×C4—72no—
267C2×C4—141no—
268C2×C4—282no—
269C2×C4—282no—
270C2×C4D4×D711yescharacteristic
271C2×C4D4×D712yes—
272C2×C4C2×C22⋊D712yes—
273C2×C4C23×D711yescharacteristic
274C8—74no—
275C8C2×C22⋊D714yes—
276C7#315311yescharacteristic, complemented, Sylow
277C22—22no—
278C22—42no—
279C22—41no—
280C22—282no—
281C22—284no—
282C22—281no—
283C22—282no—
284C22—564no—
285C22#650312yes—
286C22C2×D4×D711yescharacteristic
287C4—41no—
288C4—42no—
289C4—142no—
290C4—142no—
291C4—281no—
292C4#650312yes—
293C4C2×D4×D711yescharacteristic
294C4C22×C22⋊D711yescharacteristic
295C2—22no—
296C2—81no—
297C2—562no—
298C2—561no—
299C2#9046812yes—
300C2C2×#650311yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#137C2
socleC22×C7C22×D4
hypercenterC2×#137D7
hypocenterC7#3153
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×C7×#137C2semidirect
2. #90523C2semidirect
3. C2×#6512C2semidirect
4. C2×#6510C2semidirect
5. C2×#6505C2semidirect
6. C2×#6471C2semidirect
7. #89963C2semidirect
8. #89962C2semidirect
9. #89948C2semidirect
10. #89940C2semidirect
11. M16×Dic7C2semidirect
17. C2×C7×Q16C22semidirect
18. C2×C7×SD16C22semidirect
24. #6510C22semidirect
25. #6507C22semidirect
26. C2×#1144C22semidirect
29. C2×#1143C22semidirect
30. #6503C22semidirect
32. #6499C22semidirect
41. #6389C22semidirect
42. C8×Dic7C22semidirect
71. C7×M16D4semidirect
85. #1136D4semidirect
90. C2×#137D7semidirect
147. C2×Q16C2×D7semidirect
151. C2×SD16C2×D7semidirect
221. M16C22⋊D7semidirect
276. C7#3153semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1372
#905232
C2×#65122
C2×#65102
C2×#65052
C2×#64712
#899632
#899622
#899482
#899402
M16×Dic72
#31537