Finite Group Databasebeta

← #11739128

Subgroups — #11739128

Order 896.

424 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 121 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
1D7×C4⋊Q8C211yesmaximal, characteristic, complemented
2#90359C211yesmaximal, characteristic
3C7×#489C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89827C211yesmaximal, characteristic, complemented
6#89575C212yesmaximal, complemented
7#89564C212yesmaximal, complemented
8#89543C212yesmaximal, complemented
9#89532C212yesmaximal, complemented
10#89486C211yesmaximal, characteristic, complemented
11#89479C211yesmaximal, characteristic, complemented
12C2×#1161C2211yescharacteristic
13C2×Q8×D7—21no—
14C2×#1159C2211yescharacteristic
15C7×C4⋊Q8C2211yescharacteristic
16C7×#124C2211yescharacteristic
17#6507—21no—
18#6507C2211yescharacteristic
19#6503C2211yescharacteristic
20C2×#1131C2212yes—
21D7×C4⋊C4—21no—
22D7×C4⋊C4C2212yes—
23#6451—21no—
24#6451C2212yes—
25#6447—21no—
26#6447—21no—
27(C22⋊D7)⋊C4—21no—
28(C22⋊D7)⋊C4—21no—
29#6438C2211yescharacteristic
30C42×D7C2211yescharacteristic
31#6432C2211yescharacteristic
32C7×#103C2212yes—
33C7×#102C2212yes—
34C7×#97C2211yescharacteristic
35#6409C2212yes—
36#6406C2212yes—
37#6395C2212yes—
38#6390C2212yes—
39C8×Dic7C2212yes—
40#6384C2212yes—
41#6383C2212yes—
42#6377C2211yescharacteristic
43#3141—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×C22⋊D7—21no—
49C2×C22⋊D7—21no—
50C22×Dic7—21no—
51#1161—24no—
52Q8×D7—44no—
53#1159—24no—
54C2×D28—21no—
55C2×D28C2311yescharacteristic
56C2×C4×D7—22no—
57C2×C4×D7—21no—
58C2×C4×D7D411yescharacteristic
59C2×C4×D7D411yescharacteristic
60C2×C4×D7C2311yescharacteristic
61C2×Dic14—21no—
62C2×Dic14—22no—
63C2×Dic14C2311yescharacteristic
64C2×C7×C8C2312yes—
65C7×C4⋊C4—21no—
66C7×C4⋊C4C2312yes—
67C7×C22⋊C4—21no—
68C7×C22⋊C4—21no—
69C42×C7C2311yescharacteristic
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7C2312yes—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—22no—
77Dic7⋊C4—22no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82#1131—24no—
83#532—71no—
84C2×C4⋊Q8—71no—
85#489—71no—
86#489—71no—
87#489—71no—
88#489D711yescharacteristic, complemented
89#487—72no—
90#486—72no—
91#417—72no—
92#417—72no—
93C4×M16—71no—
94C23×C7—21no—
95C22×D7—21no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—22no—
98C7×Q8—42no—
99C7×D4—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2411yescharacteristic
105C22⋊D7—42no—
106C22⋊D7—42no—
107C22⋊D7—42no—
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7—42no—
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114D28—22no—
115D28—42no—
116C4×D7—22no—
117C4×D7—22no—
118C4×D7—44no—
119C4×D7—42no—
120C4×D7#4114yes—
121Dic14—22no—
122Dic14—42no—
123Dic14—44no—
124C7×C8—22no—
125C7⋊C8—22no—
126C2×#41—71no—
127C2×#41—71no—
128C22×Q8—141no—
129C2×M16—72no—
130C4⋊Q8—72no—
131C4⋊Q8—71no—
132C4⋊Q8—71no—
133C4⋊Q8—141no—
134C4⋊Q8—141no—
135C4⋊Q8C2×D711yescharacteristic
136#127—71no—
137#124—71no—
138#124C2×D711yescharacteristic
139#121—141no—
140#121—141no—
141C4×D4—141no—
142C4×D4—141no—
143C2×C4⋊C4—72no—
144C2×C4⋊C4—141no—
145C2×C42—71no—
146#103—72no—
147#103—72no—
148#103—72no—
149#103C2×D712yes—
150#102—72no—
151#102—72no—
152#102—72no—
153#102C2×D712yes—
154#97—71no—
155#97C2×D711yescharacteristic
156C4×C8—72no—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7C22×D411yescharacteristic
161C2×D7—21no—
162C2×D7—21no—
163C2×D7—41no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7#12412yes—
167C4×C7—21no—
168C4×C7—21no—
169C4×C7—41no—
170C4×C7—42no—
171C4×C7C2×#4112yes—
172Dic7—41no—
173Dic7—42no—
174Dic7#12412yes—
175Dic7AGL(1,ℤ/8)12yes—
176Dic7#13712yes—
177#41—144no—
178#41—144no—
179C2×Q8—21no—
180C2×Q8—71no—
181C2×Q8—142no—
182C2×Q8—141no—
183C2×Q8—284no—
184C2×Q8C22×D711yescharacteristic
185C2×D4—71no—
186C2×D4—141no—
187C2×D4—141no—
188C2×D4—141no—
189C2×D4—141no—
190C2×D4C22×D711yescharacteristic
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—141no—
195C22×C4—142no—
196C22×C4—141no—
197M16—144no—
198C2×C8—72no—
199C2×C8C22×D712yes—
200C4⋊C4—21no—
201C4⋊C4—72no—
202C4⋊C4—142no—
203C4⋊C4—142no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4—141no—
207C4⋊C4C22×D712yes—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C42—71no—
213C42—71no—
214C42—71no—
215C42C22×D711yescharacteristic
216C2×C7—81no—
217C2×C7C2×#12411yescharacteristic
218C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C7C2×#13711yescharacteristic
220D7—22no—
221D7—81no—
222C23—21no—
223C23—71no—
224C23—141no—
225Q8—22no—
226Q8—42no—
227Q8—142no—
228Q8—282no—
229Q8—284no—
230D4—22no—
231D4—142no—
232D4—282no—
233D4—282no—
234D4—282no—
235D4—282no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—71no—
239C2×C4—71no—
240C2×C4—71no—
241C2×C4—74no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—141no—
245C2×C4—142no—
246C2×C4—282no—
247C2×C4—284no—
248C2×C4—282no—
249C2×C4D4×D711yescharacteristic
250C2×C4D4×D711yescharacteristic
251C2×C4C23×D711yescharacteristic
252C8—22no—
253C8—142no—
254C7#314111yescharacteristic, complemented, Sylow
255C22—41no—
256C22—41no—
257C22—41no—
258C22—72no—
259C22—141no—
260C22—141no—
261C22—281no—
262C22—281no—
263C22—281no—
264C22C2×D4×D711yescharacteristic
265C4—21no—
266C4—21no—
267C4—42no—
268C4—41no—
269C4—72no—
270C4—72no—
271C4—72no—
272C4—282no—
273C4—281no—
274C4D7×#4112yes—
275C2—81no—
276C2—142no—
277C2—561no—
278C2D7×#12411yescharacteristic
279C2D7×AGL(1,ℤ/8)11yescharacteristic
280C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radical#90359C2
hypercenter#489D7
hypocenterC7#3141
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊Q8C2semidirect
3. C7×#489C2semidirect
4. #89861C2semidirect
5. #89827C2semidirect
6. #89575C2semidirect
7. #89564C2semidirect
8. #89543C2semidirect
9. #89532C2semidirect
10. #89486C2semidirect
11. #89479C2semidirect
88. #489D7semidirect
254. C7#3141semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#903592
C7×#4892
#898612
#898272
#895752
#895642
#895432
#895322
#894862
#894792
#31417