Finite Group Databasebeta

← #11737503

Subgroups — #11737503

Order 896.

444 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 113 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×#6510C211yesmaximal, characteristic, complemented
2#90287C211yesmaximal, characteristic, complemented
3C7×#448C211yesmaximal, characteristic, complemented
4#89999C211yesmaximal, characteristic, complemented
5#89948C212yesmaximal, complemented
6#89937C212yesmaximal, complemented
7#89808C211yesmaximal, characteristic, complemented
8#89591C212yesmaximal, complemented
9#89567C212yesmaximal, complemented
10#89504C211yesmaximal, characteristic, complemented
11#89491C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C2×#1159—21no—
14C22×D28C2211yescharacteristic
15C2×C7×SD16C2212yescomplemented
16C7×#121C2211yescharacteristic
17#6510—21no—
18#6510—21no—
19#6510C2212yes—
20#6510C2212yes—
21C2×#1143C2212yescomplemented
22#6503C2211yescharacteristic, complemented
23#6502—21no—
24D4×Dic7C2211yescharacteristic, complemented
25C2×#1141C2212yes—
26#6493C2211yescharacteristic
27C2×(C2×D7)⋊C4—21no—
28C2×(C2×D7)⋊C4—21no—
29C2×C4×Dic7C2211yescharacteristic
30C2×D56C2212yes—
31#6456C2211yescharacteristic, complemented
32#6451C2211yescharacteristic, complemented
33#6448—21no—
34(C22⋊D7)⋊C4—21no—
35C7×#103C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37#6409C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6395C2212yes—
40#6389C2212yescomplemented
41Dic7⋊C8C2212yescomplemented
42#6383C2212yes—
43#3213—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8—21no—
46C2×C7×Q8—42no—
47C2×C7×Q8C2312yes—
48C2×C7×D4—21no—
49C2×C7×D4C2311yescharacteristic
50C22×C4×C7—21no—
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—21no—
53C2×C22⋊D7—21no—
54C2×C22⋊D7—21no—
55C22×Dic7—21no—
56C22×Dic7D411yescharacteristic
57C22×Dic7D411yescharacteristic
58#1159—44no—
59C2×D28—21no—
60C2×D28—42no—
61C2×D28C2311yescharacteristic
62C2×D28C2312yes—
63C2×C4×D7—21no—
64C2×Dic14—21no—
65C7×SD16—42no—
66C2×C7×C8C2312yes—
67C7×C4⋊C4C2311yescharacteristic
68C7×C22⋊C4—21no—
69C22⋊Dic7—21no—
70#1143—42no—
71#1141—42no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—22no—
77(C2×D7)⋊C4—22no—
78C4⋊Dic7C2311yescharacteristic
79Dic7⋊C4—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C4×Dic7C2311yescharacteristic
84C2×C7⋊C8C2312yes—
85D56—42no—
86#532—71no—
87C2×#124—71no—
88#496—72no—
89#489—72no—
90#463—72no—
91#463—72no—
92#448—71no—
93#448D711yescharacteristic, complemented
94#447—71no—
95#447—71no—
96#432—71no—
97C23×C7—21no—
98C23×C7C2×D411yescharacteristic
99C22×D7—22no—
100C22×D7—21no—
101C22×D7—42no—
102C22×D7—42no—
103C22×D7—42no—
104C7×Q8—22no—
105C7×Q8—22no—
106C7×Q8—42no—
107C7×D4—22no—
108C7×D4—42no—
109C2×C4×C7—22no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—42no—
113C2×C4×C7C2×D411yescharacteristic
114C2×C4×C7C2411yescharacteristic
115C22⋊D7—42no—
116C22⋊D7—42no—
117C22⋊D7—42no—
118C2×Dic7—22no—
119C2×Dic7—21no—
120C2×Dic7—41no—
121C2×Dic7—42no—
122C2×Dic7C2×D411yescharacteristic
123C2×Dic7C2×D411yescharacteristic
124C2×Dic7C2×D411yescharacteristic
125C2×Dic7C2×D411yescharacteristic
126D28—22no—
127D28—22no—
128D28—22no—
129D28—42no—
130C4×D7—42no—
131Dic14—42no—
132C7×C8—22no—
133C7⋊C8—22no—
134C2×#41—141no—
135C22×Q8C2×D711yescharacteristic
136C22×D4—71no—
137C2×SD16—72no—
138C2×SD16C2×D712yescomplemented
139C2×D8—72no—
140C2×D8—72no—
141C4⋊Q8—71no—
142#127—71no—
143#124—72no—
144#124—72no—
145#124—141no—
146#124—141no—
147#124—141no—
148#121—71no—
149#121—141no—
150#121C2×D711yescharacteristic
151C4×D4—71no—
152C4×D4—71no—
153C4×D4—141no—
154C2×C22⋊C4—141no—
155C2×C22⋊C4—141no—
156C2×C42—71no—
157C4⋊C8—72no—
158#103—72no—
159#103C2×D712yescomplemented
160#102—72no—
161#102—72no—
162C22⋊C8—71no—
163C22⋊C8C2×D711yescharacteristic
164#97—72no—
165C22×C7—21no—
166C22×C7—21no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7—41no—
170C22×C7AGL(1,ℤ/8)12yes—
171C22×C7C22×D411yescharacteristic
172C2×D7—42no—
173C2×D7—41no—
174C2×D7—41no—
175C2×D7—42no—
176C2×D7—41no—
177C2×D7—42no—
178C2×D7—84no—
179C4×C7—21no—
180C4×C7—42no—
181C4×C7—41no—
182C4×C7H(5,𝔽2)12yes—
183Dic7—21no—
184Dic7—21no—
185Dic7—21no—
186Dic7—21no—
187Dic7—41no—
188C24—141no—
189#41—284no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—141no—
193C2×Q8C22×D712yes—
194C2×D4—21no—
195C2×D4—72no—
196C2×D4—71no—
197C2×D4—141no—
198C2×D4—141no—
199C2×D4—141no—
200C2×D4—141no—
201C2×D4—282no—
202C2×D4C22×D711yescharacteristic
203C22×C4—21no—
204C22×C4—71no—
205C22×C4—71no—
206C22×C4—141no—
207C22×C4—141no—
208C22×C4C22×D711yescharacteristic
209SD16—42no—
210SD16—282no—
211D8—282no—
212D8—282no—
213C2×C8—72no—
214C2×C8C22×D712yes—
215C4⋊C4—71no—
216C4⋊C4—141no—
217C4⋊C4C22×D711yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—142no—
220C22⋊C4—142no—
221C22⋊C4—142no—
222C22⋊C4—142no—
223C22⋊C4—141no—
224C22⋊C4—141no—
225C22⋊C4—141no—
226C42—71no—
227C42—71no—
228C42—71no—
229C42—71no—
230C2×C7—22no—
231C2×C7—81no—
232C2×C7#53511yescharacteristic
233C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
234C2×C7#57511yescharacteristic
235D7—81no—
236D7—82no—
237C23—21no—
238C23—142no—
239C23—141no—
240C23—282no—
241C23—282no—
242C23—282no—
243C23D4×D711yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—42no—
247Q8—282no—
248D4—22no—
249D4—42no—
250D4—142no—
251D4—142no—
252D4—142no—
253D4—282no—
254D4—282no—
255D4—282no—
256D4—282no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—71no—
262C2×C4—71no—
263C2×C4—71no—
264C2×C4—71no—
265C2×C4—142no—
266C2×C4—141no—
267C2×C4—282no—
268C2×C4—281no—
269C2×C4—282no—
270C2×C4D4×D711yescharacteristic
271C2×C4C23×D711yescharacteristic
272C8—22no—
273C8—142no—
274C7#321311yescharacteristic, complemented, Sylow
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—281no—
281C22—282no—
282C22—281no—
283C22—282no—
284C22—282no—
285C22—281no—
286C22—564no—
287C22#647712yes—
288C22C2×D4×D711yescharacteristic
289C4—21no—
290C4—41no—
291C4—42no—
292C4—141no—
293C4—141no—
294C4—141no—
295C4—141no—
296C4—281no—
297C4#654812yes—
298C2—22no—
299C2—81no—
300C2—562no—
301C2—561no—
302C2#9028311yescharacteristic
303C2C2×#647711yescharacteristic
304C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#448C2
socleC22×C7C22×D4
hypercenter#448D7
hypocenterC7#3213
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#6510C2semidirect
2. #90287C2semidirect
3. C7×#448C2semidirect
4. #89999C2semidirect
5. #89948C2semidirect
6. #89937C2semidirect
7. #89808C2semidirect
8. #89591C2semidirect
9. #89567C2semidirect
10. #89504C2semidirect
11. #89491C2semidirect
15. C2×C7×SD16C22semidirect
21. C2×#1143C22semidirect
22. #6503C22semidirect
24. D4×Dic7C22semidirect
31. #6456C22semidirect
32. #6451C22semidirect
35. C7×#103C22semidirect
37. #6409C22semidirect
40. #6389C22semidirect
41. Dic7⋊C8C22semidirect
93. #448D7semidirect
138. C2×SD16C2×D7semidirect
159. #103C2×D7semidirect
274. C7#3213semidirect

Maximal subgroups

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

subgroupindex
C2×#65102
#902872
C7×#4482
#899992
#899482
#899372
#898082
#895912
#895672
#895042
#894912
#32137