Finite Group Databasebeta

← #11737451

Subgroups — #11737451

Order 896.

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

normal subgroups only

This group has 520 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
1C2×#6501C211yesmaximal, characteristic, complemented
2D7×#121C211yesmaximal, characteristic, complemented
3C7×#447C211yesmaximal, characteristic, complemented
4#89989C211yesmaximal, characteristic, complemented
5#89929C212yesmaximal, complemented
6#89928C212yesmaximal, complemented
7#89805C211yesmaximal, characteristic, complemented
8#89547C212yesmaximal, complemented
9#89546C212yesmaximal, complemented
10#89508C211yesmaximal, characteristic, complemented
11D7×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic, complemented
13C22×C22⋊D7—21no—
14C2×D4×D7—21no—
15C2×D4×D7C2211yescharacteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C2×C7×D8C2212yescomplemented
18C7×#121C2211yescharacteristic
19C2×C22⋊Dic7—21no—
20#6502—21no—
21#6501—21no—
22#6501—21no—
23#6501C2212yes—
24#6501C2211yescharacteristic, complemented
25#6501C2212yescomplemented
26C2×#1141C2212yescomplemented
27#6493C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic, complemented
29C2×C8×D7C2212yescomplemented
30#6458C2211yescharacteristic, complemented
31D7×C4⋊C4C2211yescharacteristic, complemented
32#6447—21no—
33D7×C22⋊C4—21no—
34C7×#102C2212yescomplemented
35C7×C22⋊C8C2211yescharacteristic, complemented
36#6406C2212yescomplemented
37#6406C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic, complemented
39#6395C2212yescomplemented
40(C2×D7)⋊C8C2212yescomplemented
41#6392C2212yescomplemented
42#6381C2212yescomplemented
43#3186—71nomaximal, Sylow
44C24×C7—21no—
45C23×D7—21no—
46C23×D7D411yescharacteristic
47C2×C7×D4—21no—
48C2×C7×D4—21no—
49C2×C7×D4—42no—
50C2×C7×D4C2311yescharacteristic
51C2×C7×D4C2312yes—
52C22×C4×C7C2311yescharacteristic
53C2×C22⋊D7—21no—
54C2×C22⋊D7—21no—
55C2×C22⋊D7—22no—
56C2×C22⋊D7—22no—
57C2×C22⋊D7—21no—
58C2×C22⋊D7—44no—
59C22×Dic7—21no—
60C22×Dic7D411yescharacteristic
61D4×D7—42no—
62D4×D7—44no—
63C2×D28—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—42no—
67C2×C4×D7D411yescharacteristic
68C2×C4×D7D411yescharacteristic
69C2×C4×D7C2311yescharacteristic
70C2×C4×D7C2311yescharacteristic
71C7×D8—42no—
72C2×C7×C8C2312yes—
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C22⋊Dic7—21no—
76C22⋊Dic7—22no—
77C22⋊Dic7—22no—
78#1141—42no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7C2311yescharacteristic
83C4⋊Dic7C2312yes—
84Dic7⋊C4—21no—
85C2×C7⋊C8C2312yes—
86C8×D7—42no—
87C2×#121—71no—
88C2×#121—71no—
89#480—71no—
90#480—72no—
91#480—71no—
92#466—72no—
93#466—72no—
94#447—71no—
95#447—72no—
96#447D711yescharacteristic, complemented
97C2×C22⋊C8—71no—
98C23×C7—22no—
99C23×C7—21no—
100C23×C7—42no—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7C2×D411yescharacteristic
104C22×D7—21no—
105C22×D7—41no—
106C22×D7—41no—
107C22×D7—42no—
108C22×D7—42no—
109C22×D7—42no—
110C22×D7D814yescomplemented
111C22×D7C2×D411yescharacteristic
112C22×D7C2×D411yescharacteristic
113C7×D4—22no—
114C7×D4—22no—
115C7×D4—22no—
116C7×D4—42no—
117C7×D4—42no—
118C2×C4×C7—22no—
119C2×C4×C7—21no—
120C2×C4×C7C2×D411yescharacteristic
121C2×C4×C7C2411yescharacteristic
122C22⋊D7—42no—
123C22⋊D7—42no—
124C22⋊D7—42no—
125C22⋊D7—44no—
126C22⋊D7—44no—
127C2×Dic7—21no—
128C2×Dic7—22no—
129C2×Dic7—41no—
130C2×Dic7—42no—
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133D28—22no—
134D28—42no—
135C4×D7—22no—
136C4×D7—22no—
137C4×D7—41no—
138C4×D7—41no—
139C4×D7—42no—
140C7×C8—22no—
141C7⋊C8—22no—
142C22×D4—71no—
143C22×D4—141no—
144C22×D4—141no—
145C22×D4C2×D711yescharacteristic, complemented
146C23×C4—71no—
147C2×D8—72no—
148C2×D8C2×D712yescomplemented
149C22×C8—72no—
150#121—72no—
151#121—71no—
152#121—72no—
153#121—71no—
154#121—71no—
155#121—141no—
156#121—141no—
157#121—141no—
158#121—141no—
159#121C2×D711yescharacteristic
160C2×C4⋊C4—71no—
161C2×C4⋊C4—71no—
162C2×C22⋊C4—141no—
163C2×C22⋊C4—141no—
164#107—72no—
165#107—72no—
166#102—72no—
167#102—72no—
168#102—72no—
169#102C2×D712yescomplemented
170C22⋊C8—71no—
171C22⋊C8—72no—
172C22⋊C8C2×D711yescharacteristic, complemented
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—41no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—42no—
181C22×C7—84no—
182C22×C7C2×D812yes—
183C22×C7C22×D411yescharacteristic
184C2×D7—21no—
185C2×D7—21no—
186C2×D7—21no—
187C2×D7—21no—
188C2×D7—44no—
189C2×D7—41no—
190C2×D7—41no—
191C2×D7—41no—
192C2×D7—84no—
193C2×D7C2×D812yes—
194C2×D7C2×D812yes—
195C4×C7—21no—
196C4×C7—41no—
197C4×C7H(5,𝔽2)12yes—
198Dic7—21no—
199Dic7—21no—
200Dic7—41no—
201Dic7—42no—
202C24—21no—
203C24—71no—
204C24—141no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—71no—
209C2×D4—141no—
210C2×D4—141no—
211C2×D4—141no—
212C2×D4—142no—
213C2×D4—142no—
214C2×D4—141no—
215C2×D4—282no—
216C2×D4—284no—
217C2×D4—284no—
218C2×D4C22×D711yescharacteristic
219C2×D4C22×D712yes—
220C22×C4—71no—
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—141no—
226C22×C4—141no—
227C22×C4—282no—
228C22×C4C22×D711yescharacteristic
229D8—42no—
230D8—282no—
231C2×C8—72no—
232C2×C8—282no—
233C2×C8C22×D712yes—
234C4⋊C4—71no—
235C4⋊C4—72no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4C22×D711yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—141no—
241C22⋊C4—141no—
242C22⋊C4—142no—
243C22⋊C4—142no—
244C22⋊C4—141no—
245C2×C7—22no—
246C2×C7—81no—
247C2×C7—82no—
248C2×C7#53411yescharacteristic
249C2×C7C22×D811yescharacteristic
250C2×C7#57511yescharacteristic
251D7—22no—
252D7—22no—
253D7—81no—
254C23—21no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—71no—
260C23—71no—
261C23—74no—
262C23—141no—
263C23—281no—
264C23—281no—
265C23—282no—
266C23—282no—
267C23—282no—
268C23D4×D711yescharacteristic
269D4—22no—
270D4—22no—
271D4—22no—
272D4—42no—
273D4—42no—
274D4—142no—
275D4—282no—
276D4—282no—
277D4—282no—
278D4—284no—
279D4—284no—
280D4—282no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—142no—
286C2×C4—142no—
287C2×C4—142no—
288C2×C4—141no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—282no—
292C2×C4—281no—
293C2×C4—282no—
294C2×C4D4×D711yescharacteristic
295C2×C4C23×D711yescharacteristic
296C8—22no—
297C8—142no—
298C7#318611yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—72no—
308C22—72no—
309C22—84no—
310C22—141no—
311C22—141no—
312C22—141no—
313C22—141no—
314C22—284no—
315C22—281no—
316C22—281no—
317C22—281no—
318C22—564no—
319C22D7×D812yes—
320C22C2×D4×D711yescharacteristic
321C4—21no—
322C4—41no—
323C4—141no—
324C4—141no—
325C4—281no—
326C4—282no—
327C4#654812yes—
328C2—22no—
329C2—82no—
330C2—81no—
331C2—142no—
332C2—142no—
333C2—561no—
334C2#9028011yescharacteristic
335C2C2×D7×D811yescharacteristic
336C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#447C2
socleC22×C7C22×D4
hypercenter#447D7
hypocenterC7#3186
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×#6501C2semidirect
2. D7×#121C2semidirect
3. C7×#447C2semidirect
4. #89989C2semidirect
5. #89929C2semidirect
6. #89928C2semidirect
7. #89805C2semidirect
8. #89547C2semidirect
9. #89546C2semidirect
10. #89508C2semidirect
11. D7×C22⋊C8C2semidirect
12. C22×C7×D4C22semidirect
15. C2×D4×D7C22semidirect
17. C2×C7×D8C22semidirect
24. #6501C22semidirect
25. #6501C22semidirect
26. C2×#1141C22semidirect
28. C2×C4⋊Dic7C22semidirect
29. C2×C8×D7C22semidirect
30. #6458C22semidirect
31. D7×C4⋊C4C22semidirect
34. C7×#102C22semidirect
35. C7×C22⋊C8C22semidirect
36. #6406C22semidirect
37. #6406C22semidirect
38. C22⋊(C7⋊C8)C22semidirect
39. #6395C22semidirect
40. (C2×D7)⋊C8C22semidirect
41. #6392C22semidirect
42. #6381C22semidirect
96. #447D7semidirect
110. C22×D7D8semidirect
145. C22×D4C2×D7semidirect
148. C2×D8C2×D7semidirect
169. #102C2×D7semidirect
172. C22⋊C8C2×D7semidirect
298. C7#3186semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
D7×#1212
C7×#4472
#899892
#899292
#899282
#898052
#895472
#895462
#895082
D7×C22⋊C82
#31867