Finite Group Databasebeta

← #11741247

Subgroups — #11741247

Order 896.

504 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 149 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×C7×#131C211yesmaximal, characteristic, complemented
2#90524C211yesmaximal, characteristic, complemented
3#90521C211yesmaximal, characteristic, complemented
4C2×#6471C211yesmaximal, characteristic, complemented
5#89907C212yesmaximal, complemented
6#89906C212yesmaximal, complemented
7#89893C211yesmaximal, characteristic, complemented
8#89887C212yesmaximal, complemented
9#89886C212yesmaximal, complemented
10#89885C211yesmaximal, characteristic, complemented
11C2×#6395C211yesmaximal, characteristic, complemented
12C2×C7×#41C2211yescharacteristic
13C2×#1157C2211yescharacteristic
14C22×D28C2211yescharacteristic
15C7×#131—24no—
16C2×C7×M16C2211yescharacteristic
17C2×C7×M16C2212yescomplemented
18C22×C7×C8C2211yescharacteristic
19C22×C7×C8C2212yescomplemented
20#6510—21no—
21#6509—21no—
22#6503—21no—
23#6502—22no—
24#6501—21no—
25#6500—22no—
26#6493C2212yes—
27#6493C2212yes—
28#6493C2212yescomplemented
29#6489C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31#6487C2212yescomplemented
32#6471—24no—
33C2×D56C2212yes—
34C2×#1132C2212yes—
35#6395C2212yes—
36#6395C2212yes—
37#6395C2211yescharacteristic, complemented
38#6395C2211yescharacteristic, complemented
39#6392C2212yescomplemented
40#6391C2212yescomplemented
41#6390C2211yescharacteristic, complemented
42#6390C2211yescharacteristic, complemented
43#3062—71nomaximal, Sylow
44C23×D7—21no—
45C7×#41—22no—
46C7×#41—22no—
47C2×C7×Q8D411yescharacteristic
48C2×C7×D4D412yes—
49C2×C7×D4D411yescharacteristic
50C22×C4×C7C2312yes—
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—21no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7—22no—
55C2×C22⋊D7—22no—
56C22×Dic7—21no—
57#1157—22no—
58#1157—22no—
59C2×D28—22no—
60C2×D28—22no—
61C2×D28—21no—
62C2×D28C2312yes—
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×Dic14C2311yescharacteristic
66C7×M16—24no—
67C7×M16D414yescomplemented
68C2×C7×C8—22no—
69C2×C7×C8—24no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2312yes—
73C22⋊Dic7—22no—
74C22⋊Dic7—21no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—22no—
77(C2×D7)⋊C4—22no—
78C4⋊Dic7—21no—
79C4⋊Dic7C2312yes—
80C4⋊Dic7C2311yescharacteristic
81C4⋊Dic7C2311yescharacteristic
82Dic7⋊C4—22no—
83C4×Dic7—21no—
84D56—24no—
85#1132—24no—
86C2×AGL(1,ℤ/8)—71no—
87C2×#131D711yescharacteristic, complemented
88#537—71no—
89#535—71no—
90#469—72no—
91#468—72no—
92#466—72no—
93#465—72no—
94#428—71no—
95#416—71no—
96C2×#102—71no—
97C23×C7C2×D412yes—
98C23×C7C2×D411yescharacteristic
99C22×D7—21no—
100C22×D7—22no—
101C22×D7—42no—
102C22×D7—81no—
103C22×D7—81no—
104C7×Q8—22no—
105C7×D4—24no—
106C7×D4—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2×D411yescharacteristic
111C2×C4×C7C2×D412yes—
112C2×C4×C7#4112yes—
113C2×C4×C7C2411yescharacteristic
114C22⋊D7—42no—
115C22⋊D7—82no—
116C22⋊D7—82no—
117C22⋊D7—82no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—22no—
122C2×Dic7—81no—
123D28—22no—
124D28—21no—
125D28—22no—
126D28—21no—
127D28—42no—
128C4×D7—42no—
129Dic14—21no—
130Dic14—21no—
131C7×C8—21no—
132C7×C8—21no—
133C7×C8C2×D414yes—
134C2×#41—71no—
135C2×#41C2×D711yescharacteristic
136C22×D4—71no—
137AGL(1,ℤ/8)—144no—
138C2×SD16—72no—
139C2×D8—72no—
140#131—24no—
141C2×M16C2×D711yescharacteristic
142C2×M16C2×D712yescomplemented
143C22×C8C2×D711yescharacteristic
144C22×C8C2×D712yescomplemented
145#127—141no—
146#124—141no—
147C22⋊Q8—72no—
148C22⋊Q8—141no—
149#121—72no—
150#121—72no—
151#121—72no—
152#121—141no—
153#121—142no—
154#120—142no—
155#117—71no—
156C2×C4⋊C4—71no—
157#107—72no—
158#106—72no—
159#103—71no—
160#103—71no—
161#102—72no—
162#102—72no—
163#102—71no—
164#102—71no—
165C22×C7—21no—
166C22×C7—21no—
167C22×C7—22no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7#12112yes—
171C22×C7C22×D411yescharacteristic
172C2×D7—41no—
173C2×D7—42no—
174C2×D7—41no—
175C2×D7—42no—
176C2×D7—42no—
177C2×D7—41no—
178C2×D7—84no—
179C4×C7—22no—
180C4×C7#12112yes—
181C4×C7C22×D411yescharacteristic
182C4×C7C2×#4111yescharacteristic
183Dic7—41no—
184Dic7—42no—
185Dic7—41no—
186Dic7—41no—
187C24—141no—
188#41—22no—
189#41—22no—
190#41—142no—
191#41—142no—
192C2×Q8—71no—
193C2×Q8D2811yescharacteristic
194C2×D4—71no—
195C2×D4—72no—
196C2×D4—141no—
197C2×D4—142no—
198C2×D4—142no—
199C2×D4—142no—
200C2×D4—141no—
201C2×D4—142no—
202C2×D4—142no—
203C2×D4D2812yes—
204C2×D4D2811yescharacteristic
205C22×C4—141no—
206C22×C4—141no—
207C22×C4C22×D712yes—
208C22×C4C22×D711yescharacteristic
209SD16—144no—
210D8—144no—
211M16—24no—
212M16C22⋊D714yescomplemented
213C2×C8—22no—
214C2×C8—24no—
215C2×C8C22×D711yescharacteristic
216C2×C8C22×D711yescharacteristic
217C2×C8C22×D712yes—
218C4⋊C4—72no—
219C4⋊C4—71no—
220C4⋊C4—71no—
221C4⋊C4—142no—
222C4⋊C4—141no—
223C22⋊C4—141no—
224C22⋊C4—142no—
225C22⋊C4—142no—
226C22⋊C4—142no—
227C22⋊C4—142no—
228C42—141no—
229C2×C7—22no—
230C2×C7—42no—
231C2×C7C2×#12111yescharacteristic
232C2×C7#57611yescharacteristic
233C2×C7#57711yescharacteristic
234D7—81no—
235D7—82no—
236C23—141no—
237C23—142no—
238C23—282no—
239C23—561no—
240C23—561no—
241C23C2×D2812yes—
242C23C2×D2811yescharacteristic
243Q8—22no—
244Q8—141no—
245Q8—141no—
246D4—24no—
247D4—22no—
248D4—142no—
249D4—141no—
250D4—141no—
251D4—142no—
252D4—282no—
253D4—282no—
254D4—562no—
255D4—562no—
256D4—562no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—141no—
260C2×C4—141no—
261C2×C4—142no—
262C2×C4—141no—
263C2×C4—282no—
264C2×C4—561no—
265C2×C4C2×D2812yes—
266C2×C4C2×D2811yescharacteristic
267C2×C4#115712yes—
268C2×C4C2×C22⋊D712yes—
269C2×C4C23×D711yescharacteristic
270C8—21no—
271C8—21no—
272C8C2×C22⋊D714yes—
273C7#306211yescharacteristic, complemented, Sylow
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—282no—
280C22—281no—
281C22—281no—
282C22—282no—
283C22—281no—
284C22—282no—
285C22—564no—
286C22#649312yes—
287C22C22×D2811yescharacteristic
288C4—22no—
289C4—281no—
290C4—282no—
291C4—281no—
292C4—281no—
293C4#649312yes—
294C4C2×#115711yescharacteristic
295C4C22×C22⋊D711yescharacteristic
296C2—22no—
297C2—42no—
298C2—562no—
299C2—561no—
300C2#9044211yescharacteristic
301C2#9044311yescharacteristic
302C2C2×#649311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#131C2
socleC22×C7C22×D4
hypercenterC2×#131D7
hypocenterC7#3062
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×C7×#131C2semidirect
2. #90524C2semidirect
3. #90521C2semidirect
4. C2×#6471C2semidirect
5. #89907C2semidirect
6. #89906C2semidirect
7. #89893C2semidirect
8. #89887C2semidirect
9. #89886C2semidirect
10. #89885C2semidirect
11. C2×#6395C2semidirect
17. C2×C7×M16C22semidirect
19. C22×C7×C8C22semidirect
28. #6493C22semidirect
31. #6487C22semidirect
37. #6395C22semidirect
38. #6395C22semidirect
39. #6392C22semidirect
40. #6391C22semidirect
41. #6390C22semidirect
42. #6390C22semidirect
67. C7×M16D4semidirect
87. C2×#131D7semidirect
142. C2×M16C2×D7semidirect
144. C22×C8C2×D7semidirect
212. M16C22⋊D7semidirect
273. C7#3062semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1312
#905242
#905212
C2×#64712
#899072
#899062
#898932
#898872
#898862
#898852
C2×#63952
#30627