Finite Group Databasebeta

← #11736660

Subgroups — #11736660

Order 896.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#90215C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C7×#428C211yesmaximal, characteristic, complemented
4#89905C211yesmaximal, characteristic, complemented
5#89898C211yesmaximal, characteristic, complemented
6#89767C211yesmaximal, characteristic, complemented
7#89738C211yesmaximal, characteristic, complemented
8#89656C212yesmaximal, complemented
9#89654C212yesmaximal, complemented
10#89635C212yesmaximal, complemented
11#89634C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C22×C4×D7—21no—
14C22×C4×D7C2211yescharacteristic
15C2×C7×M16C2211yescharacteristic
16C7×#117C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19C2×Dic7⋊C4—21no—
20C2×#1136C2211yescharacteristic
21#6460C2211yescharacteristic
22#6460C2211yescharacteristic
23#6458C2211yescharacteristic
24#6458C2211yescharacteristic
25D7×C4⋊C4—21no—
26D7×C4⋊C4—21no—
27D7×C4⋊C4C2212yes—
28D7×C4⋊C4C2212yes—
29#6449—21no—
30#6445—21no—
31C4×D28—21no—
32C4×D28—21no—
33C7×#107C2212yescomplemented
34C7×#106C2212yescomplemented
35#6395C2212yescomplemented
36(C2×D7)⋊C8C2212yes—
37(C2×D7)⋊C8C2212yes—
38#6390C2212yescomplemented
39#6384C2212yescomplemented
40#6383C2212yescomplemented
41#6382C2212yescomplemented
42#6381C2212yescomplemented
43#3092—71nomaximal, Sylow
44C23×D7D411yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C22×Dic7—21no—
49C22×Dic7D411yescharacteristic
50#1157—22no—
51#1157—22no—
52C2×D28—21no—
53C2×D28—21no—
54C2×D28C2311yescharacteristic
55C2×C4×D7—22no—
56C2×C4×D7—22no—
57C2×C4×D7—22no—
58C2×C4×D7—22no—
59C2×C4×D7—21no—
60C2×C4×D7—44no—
61C2×C4×D7D411yescharacteristic
62C2×C4×D7D411yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C2×C4×D7C2311yescharacteristic
65C2×Dic14C2311yescharacteristic
66C7×M16—22no—
67C2×C7×C8C2312yes—
68C7×C4⋊C4—21no—
69C7×C4⋊C4C2311yescharacteristic
70C7×C4⋊C4C2312yes—
71C7×C4⋊C4C2311yescharacteristic
72C7×C22⋊C4—21no—
73C42×C7—21no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—21no—
76(C2×D7)⋊C4—21no—
77(C2×D7)⋊C4—21no—
78C4⋊Dic7—21no—
79C4⋊Dic7—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7C2312yes—
82Dic7⋊C4—22no—
83Dic7⋊C4—22no—
84#1136—22no—
85C2×C7⋊C8C2312yes—
86#525—71no—
87C22×C4⋊C4—71no—
88#484—72no—
89#483—72no—
90#481—72no—
91#480—72no—
92#428—71no—
93#428D711yescharacteristic, complemented
94#417—71no—
95#417—71no—
96#407—71no—
97C23×C7C2×D411yescharacteristic
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—21no—
102C22×D7C2×D411yescharacteristic
103C22×D7C2×D411yescharacteristic
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—42no—
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7#4112yes—
110C2×C4×C7#4112yes—
111C2×C4×C7C2411yescharacteristic
112C22⋊D7—42no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—21no—
116C2×Dic7—42no—
117C2×Dic7C2×D411yescharacteristic
118C2×Dic7C2×D411yescharacteristic
119D28—21no—
120D28—21no—
121D28—42no—
122D28—42no—
123C4×D7—22no—
124C4×D7—22no—
125C4×D7—21no—
126C4×D7—21no—
127C4×D7—21no—
128C4×D7—21no—
129C4×D7—44no—
130C4×D7—44no—
131C4×D7—42no—
132Dic14—21no—
133Dic14—21no—
134C7×C8—22no—
135C7⋊C8—22no—
136C2×#41—71no—
137C23×C4—71no—
138C23×C4—141no—
139C2×M16—71no—
140C2×M16C2×D711yescharacteristic
141#123—141no—
142C22⋊Q8—71no—
143C22⋊Q8—71no—
144#121—71no—
145#121—71no—
146#120—141no—
147C4×D4—141no—
148C4×D4—141no—
149#117C2×D711yescharacteristic
150C2×C4⋊C4—71no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—72no—
153C2×C4⋊C4—141no—
154C2×C4⋊C4—141no—
155C2×C4⋊C4—141no—
156C2×C4⋊C4C2×D711yescharacteristic
157#107—72no—
158#107C2×D712yescomplemented
159#106—72no—
160#106C2×D712yescomplemented
161#103—72no—
162#103—72no—
163#102—72no—
164#102—72no—
165C22⋊C8—72no—
166C22⋊C8—72no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7#12312yes—
170C22×C7C22×D411yescharacteristic
171C2×D7—28no—
172C2×D7—21no—
173C2×D7—21no—
174C2×D7—21no—
175C2×D7—21no—
176C2×D7—41no—
177C2×D7—41no—
178C2×D7—41no—
179C2×D7AGL(1,ℤ/8)12yes—
180C2×D7#13712yes—
181C4×C7—42no—
182C4×C7—41no—
183C4×C7—41no—
184C4×C7#12312yes—
185C4×C7C2×#4111yescharacteristic
186C4×C7C2×#4111yescharacteristic
187Dic7—21no—
188Dic7—21no—
189Dic7—41no—
190Dic7—42no—
191C24—71no—
192#41—142no—
193#41—142no—
194C2×Q8—71no—
195C2×D4—71no—
196C2×D4—141no—
197C2×D4—141no—
198C2×D4—141no—
199C22×C4—21no—
200C22×C4—71no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—71no—
205C22×C4—142no—
206C22×C4—142no—
207C22×C4—142no—
208C22×C4—142no—
209C22×C4—141no—
210C22×C4—141no—
211C22×C4—284no—
212C22×C4C22×D711yescharacteristic
213M16—22no—
214M16—142no—
215C2×C8—72no—
216C2×C8C22×D712yes—
217C4⋊C4—21no—
218C4⋊C4—72no—
219C4⋊C4—142no—
220C4⋊C4—142no—
221C4⋊C4—141no—
222C4⋊C4—141no—
223C4⋊C4—141no—
224C4⋊C4C22×D712yes—
225C4⋊C4C22×D711yescharacteristic
226C4⋊C4C22×D711yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—141no—
229C22⋊C4—141no—
230C22⋊C4—141no—
231C22⋊C4—141no—
232C42—21no—
233C2×C7—22no—
234C2×C7C2×#12311yescharacteristic
235C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C7C2×#13711yescharacteristic
237D7—22no—
238D7—22no—
239D7—81no—
240C23—71no—
241C23—71no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—141no—
246C23D4×D711yescharacteristic
247Q8—141no—
248Q8—141no—
249D4—141no—
250D4—141no—
251D4—282no—
252D4—282no—
253D4—282no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—71no—
259C2×C4—71no—
260C2×C4—142no—
261C2×C4—142no—
262C2×C4—142no—
263C2×C4—141no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—141no—
267C2×C4—141no—
268C2×C4—142no—
269C2×C4—284no—
270C2×C4—284no—
271C2×C4—282no—
272C2×C4—282no—
273C2×C4#115712yes—
274C2×C4D4×D711yescharacteristic
275C2×C4#116112yes—
276C2×C4C23×D711yescharacteristic
277C8—22no—
278C8—142no—
279C7#309211yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—72no—
283C22—72no—
284C22—148no—
285C22—141no—
286C22—141no—
287C22—141no—
288C22—141no—
289C22—281no—
290C22—281no—
291C22—281no—
292C22#645712yes—
293C22C2×D4×D711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—141no—
298C4—141no—
299C4—281no—
300C4—282no—
301C4#645712yes—
302C4C2×#115711yescharacteristic
303C4C2×#116111yescharacteristic
304C2—22no—
305C2—142no—
306C2—142no—
307C2—561no—
308C2C2×#645711yescharacteristic
309C2D7×AGL(1,ℤ/8)11yescharacteristic
310C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radical#90215C2
hypercenter#428D7
hypocenterC7#3092
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×D7×C4⋊C4C2semidirect
3. C7×#428C2semidirect
4. #89905C2semidirect
5. #89898C2semidirect
6. #89767C2semidirect
7. #89738C2semidirect
8. #89656C2semidirect
9. #89654C2semidirect
10. #89635C2semidirect
11. #89634C2semidirect
33. C7×#107C22semidirect
34. C7×#106C22semidirect
35. #6395C22semidirect
38. #6390C22semidirect
39. #6384C22semidirect
40. #6383C22semidirect
41. #6382C22semidirect
42. #6381C22semidirect
93. #428D7semidirect
158. #107C2×D7semidirect
160. #106C2×D7semidirect
279. C7#3092semidirect

Maximal subgroups

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

subgroupindex
#902152
C2×D7×C4⋊C42
C7×#4282
#899052
#898982
#897672
#897382
#896562
#896542
#896352
#896342
#30927