Finite Group Databasebeta

← #11737439

Subgroups — #11737439

Order 896.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 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×#6503C211yesmaximal, characteristic, complemented
2#90287C211yesmaximal, characteristic, complemented
3C7×#447C211yesmaximal, characteristic, complemented
4#89989C211yesmaximal, characteristic, complemented
5#89923C212yesmaximal, complemented
6#89922C212yesmaximal, complemented
7#89808C211yesmaximal, characteristic, complemented
8#89558C212yesmaximal, complemented
9#89532C212yesmaximal, complemented
10#89504C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7—21no—
14C22×C22⋊D7—21no—
15C2×#1159—21no—
16C22×D28C2211yescharacteristic
17C2×C7×D8C2212yescomplemented
18C7×#121C2211yescharacteristic
19#6503—21no—
20#6503—21no—
21#6503C2212yes—
22#6503C2212yes—
23#6503C2211yescharacteristic, complemented
24#6502—21no—
25D4×Dic7C2211yescharacteristic, complemented
26C2×#1141C2212yescomplemented
27C2×#1141C2212yescomplemented
28#6493C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×D56C2212yescomplemented
31#6456C2211yescharacteristic, complemented
32#6451C2211yescharacteristic, complemented
33#6448—21no—
34(C22⋊D7)⋊C4—21no—
35C7×#102C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37#6406C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6395C2212yescomplemented
40Dic7⋊C8C2212yescomplemented
41C8×Dic7C2212yescomplemented
42#6383C2212yescomplemented
43#3205—71nomaximal, Sylow
44C24×C7—21no—
45C23×D7—21no—
46C2×C7×D4—21no—
47C2×C7×D4—21no—
48C2×C7×D4—42no—
49C2×C7×D4C2312yes—
50C2×C7×D4C2311yescharacteristic
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—21no—
53C2×C22⋊D7—21no—
54C2×C22⋊D7—21no—
55C2×C22⋊D7—22no—
56C2×C22⋊D7—22no—
57C2×C22⋊D7—22no—
58C2×C22⋊D7—22no—
59C2×C22⋊D7—44no—
60C2×C22⋊D7—44no—
61C22×Dic7—21no—
62C22×Dic7D411yescharacteristic
63C22×Dic7D411yescharacteristic
64#1159—44no—
65C2×D28—21no—
66C2×D28—42no—
67C2×D28C2311yescharacteristic
68C2×D28C2312yes—
69C2×C4×D7—21no—
70C2×Dic14—21no—
71C7×D8—42no—
72C2×C7×C8C2312yes—
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C22⋊Dic7—21no—
76#1141—42no—
77#1141—42no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80C4⋊Dic7C2311yescharacteristic
81Dic7⋊C4—21no—
82C4×Dic7D411yescharacteristic
83C4×Dic7D411yescharacteristic
84C4×Dic7C2311yescharacteristic
85C4×Dic7C2311yescharacteristic
86C2×C7⋊C8C2312yes—
87D56—42no—
88#532—71no—
89C2×#127—71no—
90#493—72no—
91#486—72no—
92#459—72no—
93#459—72no—
94#447—71no—
95#447—71no—
96#447—71no—
97#447D711yescharacteristic, complemented
98#431—71no—
99C23×C7—21no—
100C23×C7—22no—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7—42no—
104C23×C7C2×D411yescharacteristic
105C22×D7—21no—
106C22×D7—22no—
107C22×D7—42no—
108C22×D7—42no—
109C22×D7—42no—
110C7×D4—22no—
111C7×D4—22no—
112C7×D4—22no—
113C7×D4—42no—
114C7×D4—42no—
115C2×C4×C7—22no—
116C2×C4×C7—21no—
117C2×C4×C7C2×D411yescharacteristic
118C2×C4×C7C2411yescharacteristic
119C22⋊D7—44no—
120C22⋊D7—42no—
121C22⋊D7—42no—
122C22⋊D7—44no—
123C22⋊D7—44no—
124C22⋊D7—42no—
125C22⋊D7—44no—
126C2×Dic7—22no—
127C2×Dic7—21no—
128C2×Dic7—42no—
129C2×Dic7D814yescomplemented
130C2×Dic7C2×D411yescharacteristic
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134D28—22no—
135D28—22no—
136D28—22no—
137D28—42no—
138C4×D7—42no—
139Dic14—42no—
140C7×C8—22no—
141C7⋊C8—22no—
142C2×#41—141no—
143C22×D4—71no—
144C22×D4—141no—
145C22×D4—141no—
146C22×D4C2×D711yescharacteristic
147C2×D8—72no—
148C2×D8—72no—
149C2×D8—72no—
150C2×D8C2×D712yescomplemented
151C4⋊Q8—71no—
152#127—72no—
153#127—72no—
154#127—71no—
155#127—141no—
156#127—141no—
157#124—141no—
158#121—71no—
159#121—141no—
160#121C2×D711yescharacteristic
161C4×D4—71no—
162C4×D4—71no—
163C4×D4—141no—
164C2×C42—71no—
165C4⋊C8—72no—
166#102—72no—
167#102—72no—
168#102—72no—
169#102C2×D712yescomplemented
170C22⋊C8—71no—
171C22⋊C8C2×D711yescharacteristic
172C4×C8—72no—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—42no—
176C22×C7—41no—
177C22×C7—41no—
178C22×C7—42no—
179C22×C7—42no—
180C22×C7—41no—
181C22×C7—84no—
182C22×C7C2×D812yes—
183C22×C7C22×D411yescharacteristic
184C2×D7—42no—
185C2×D7—41no—
186C2×D7—41no—
187C2×D7—42no—
188C2×D7—42no—
189C2×D7—41no—
190C2×D7—84no—
191C4×C7—21no—
192C4×C7—41no—
193C4×C7H(5,𝔽2)12yes—
194Dic7—21no—
195Dic7—41no—
196Dic7C2×D812yes—
197Dic7C2×D812yes—
198Dic7AGL(1,ℤ/8)12yes—
199C24—21no—
200C24—141no—
201#41—284no—
202C2×Q8—141no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—72no—
207C2×D4—71no—
208C2×D4—141no—
209C2×D4—141no—
210C2×D4—142no—
211C2×D4—142no—
212C2×D4—142no—
213C2×D4—142no—
214C2×D4—141no—
215C2×D4—141no—
216C2×D4—282no—
217C2×D4—284no—
218C2×D4—284no—
219C2×D4C22×D712yes—
220C2×D4C22×D711yescharacteristic
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—141no—
224C22×C4—141no—
225C22×C4C22×D711yescharacteristic
226D8—42no—
227D8—282no—
228D8—282no—
229D8—282no—
230C2×C8—72no—
231C2×C8C22×D712yes—
232C4⋊C4—71no—
233C4⋊C4—141no—
234C4⋊C4C22×D711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—141no—
237C22⋊C4—141no—
238C22⋊C4—141no—
239C42—71no—
240C42—71no—
241C42—71no—
242C42—71no—
243C2×C7—22no—
244C2×C7—81no—
245C2×C7—82no—
246C2×C7#53511yescharacteristic
247C2×C7C22×D811yescharacteristic
248C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
249D7—81no—
250D7—82no—
251C23—21no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—141no—
257C23—142no—
258C23—282no—
259C23—282no—
260C23—282no—
261C23D4×D711yescharacteristic
262Q8—282no—
263D4—22no—
264D4—22no—
265D4—22no—
266D4—42no—
267D4—42no—
268D4—142no—
269D4—142no—
270D4—142no—
271D4—282no—
272D4—282no—
273D4—284no—
274D4—284no—
275D4—284no—
276D4—282no—
277D4—284no—
278D4—282no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—74no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—142no—
287C2×C4—141no—
288C2×C4—282no—
289C2×C4—282no—
290C2×C4D4×D711yescharacteristic
291C2×C4C23×D711yescharacteristic
292C8—22no—
293C8—142no—
294C7#320511yescharacteristic, complemented, Sylow
295C22—21no—
296C22—21no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—84no—
304C22—281no—
305C22—282no—
306C22—282no—
307C22—281no—
308C22—282no—
309C22—281no—
310C22—564no—
311C22D7×D812yes—
312C22C2×D4×D711yescharacteristic
313C4—21no—
314C4—41no—
315C4—72no—
316C4—72no—
317C4—72no—
318C4—141no—
319C4—281no—
320C4#654812yes—
321C2—22no—
322C2—82no—
323C2—81no—
324C2—562no—
325C2—561no—
326C2#9028311yescharacteristic
327C2C2×D7×D811yescharacteristic
328C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#447C2
socleC22×C7C22×D4
hypercenter#447D7
hypocenterC7#3205
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#6503C2semidirect
2. #90287C2semidirect
3. C7×#447C2semidirect
4. #89989C2semidirect
5. #89923C2semidirect
6. #89922C2semidirect
7. #89808C2semidirect
8. #89558C2semidirect
9. #89532C2semidirect
10. #89504C2semidirect
11. #89490C2semidirect
17. C2×C7×D8C22semidirect
23. #6503C22semidirect
25. D4×Dic7C22semidirect
26. C2×#1141C22semidirect
27. C2×#1141C22semidirect
30. C2×D56C22semidirect
31. #6456C22semidirect
32. #6451C22semidirect
35. C7×#102C22semidirect
37. #6406C22semidirect
39. #6395C22semidirect
40. Dic7⋊C8C22semidirect
41. C8×Dic7C22semidirect
42. #6383C22semidirect
97. #447D7semidirect
129. C2×Dic7D8semidirect
150. C2×D8C2×D7semidirect
169. #102C2×D7semidirect
294. C7#3205semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
#902872
C7×#4472
#899892
#899232
#899222
#898082
#895582
#895322
#895042
#894902
#32057