Finite Group Databasebeta

← #11736356

Subgroups — #11736356

Order 896.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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#90521C211yesmaximal, characteristic, complemented
2C2×#6513C211yesmaximal, characteristic, complemented
3C2×#6471C211yesmaximal, characteristic, complemented
4#90195C211yesmaximal, characteristic, complemented
5C7×#417C211yesmaximal, characteristic, complemented
6#89891C211yesmaximal, characteristic, complemented
7C2×#6383C211yesmaximal, characteristic, complemented
8#89591C212yesmaximal, complemented
9#89570C212yesmaximal, complemented
10#89558C212yesmaximal, complemented
11#89534C212yesmaximal, complemented
12C2×C7×#41C2211yescharacteristic
13C2×#1157C2211yescharacteristic
14C22×D28C2211yescharacteristic
15C2×C7×M16C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17#6513—24no—
18#6510C2212yes—
19C2×#1144C2211yescharacteristic, complemented
20C2×#1143C2211yescharacteristic, complemented
21#6503C2212yes—
22#6500—22no—
23C2×#1142C2211yescharacteristic, complemented
24C2×#1141C2211yescharacteristic, complemented
25#6493—21no—
26#6493—21no—
27C4×C22⋊D7—22no—
28#6489C2211yescharacteristic
29C22×C7⋊C8C2211yescharacteristic
30#6471—24no—
31C2×D56C2212yes—
32C2×#1132C2212yes—
33#6456C2212yescomplemented
34#6455—21no—
35D7×C4⋊C4—21no—
36#6450C2212yescomplemented
37C7×#103C2212yescomplemented
38C7×#102C2212yescomplemented
39Dic7⋊C8C2212yescomplemented
40Dic7⋊C8C2212yescomplemented
41#6383C2212yes—
42#6383C2212yes—
43#3045—71nomaximal, Sylow
44C23×D7—21no—
45C7×#41—22no—
46C7×#41—22no—
47C2×C7×Q8C2311yescharacteristic
48C2×C7×D4—21no—
49C2×C7×D4C2311yescharacteristic
50C22×C4×C7—21no—
51C22×C4×C7—21no—
52C22×C4×C7C2311yescharacteristic
53C2×C22⋊D7—21no—
54C2×C22⋊D7—22no—
55C2×C22⋊D7—22no—
56#1157—22no—
57#1157D414yescomplemented
58C2×D28—22no—
59C2×D28—22no—
60C2×D28—21no—
61C2×D28C2312yes—
62C2×D28C2311yescharacteristic
63C2×C4×D7—21no—
64C2×C4×D7—22no—
65C2×Dic14C2311yescharacteristic
66C7×M16—22no—
67C2×C7×C8C2312yes—
68C7×C4⋊C4—21no—
69C7×C4⋊C4C2312yes—
70C22⋊Dic7D412yes—
71#1144—22no—
72#1143—22no—
73#1142—22no—
74#1141—22no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—22no—
77(C2×D7)⋊C4—22no—
78C4⋊Dic7D411yescharacteristic
79C4⋊Dic7D411yescharacteristic
80Dic7⋊C4—22no—
81C4×Dic7—22no—
82C4×Dic7C2312yes—
83C2×C7⋊C8—22no—
84C2×C7⋊C8C2311yescharacteristic
85C2×C7⋊C8C2311yescharacteristic
86D56—24no—
87#1132—24no—
88C2×AGL(1,ℤ/8)—71no—
89C2×#135—71no—
90#535—71no—
91#520—71no—
92#464—72no—
93#463—72no—
94#460—72no—
95#459—72no—
96#424—71no—
97#417D711yescharacteristic, complemented
98C2×#102—71no—
99C23×C7—21no—
100C23×C7C2×D411yescharacteristic
101C22×D7—21no—
102C22×D7—22no—
103C22×D7—42no—
104C22×D7—81no—
105C22×D7—81no—
106C7×Q8—21no—
107C7×Q8—21no—
108C7×D4—21no—
109C7×D4—21no—
110C7×D4—42no—
111C2×C4×C7—22no—
112C2×C4×C7—21no—
113C2×C4×C7—42no—
114C2×C4×C7—81no—
115C2×C4×C7C2×D411yescharacteristic
116C2×C4×C7C2×D412yes—
117C2×C4×C7#4112yes—
118C2×C4×C7C2411yescharacteristic
119C22⋊D7—24no—
120C22⋊D7—82no—
121C22⋊D7—82no—
122C2×Dic7—21no—
123C2×Dic7C2×D412yes—
124C2×Dic7C2×D412yes—
125D28—22no—
126D28—21no—
127D28—22no—
128D28—42no—
129D28C2×D412yes—
130C4×D7—24no—
131C4×D7—82no—
132Dic14—21no—
133Dic14C2×D412yes—
134C7×C8—22no—
135C7⋊C8—21no—
136C7⋊C8—21no—
137C2×#41—71no—
138C2×#41C2×D711yescharacteristic
139C22×D4—71no—
140AGL(1,ℤ/8)—144no—
141#135—144no—
142C2×Q16—71no—
143C2×SD16—71no—
144C2×SD16—71no—
145C2×SD16—72no—
146C2×D8—71no—
147C2×D8—72no—
148C2×M16C2×D711yescharacteristic
149C22×C8—71no—
150#127—72no—
151#124—72no—
152#121—141no—
153#121—141no—
154#120—142no—
155C4×Q8—72no—
156C4×D4—72no—
157C4×D4—142no—
158#117—71no—
159#117—141no—
160C2×C4⋊C4—141no—
161C2×C4⋊C4C2×D711yescharacteristic
162C4⋊C8—72no—
163C4⋊C8—72no—
164#103C2×D712yescomplemented
165#102—72no—
166#102—72no—
167#102C2×D712yescomplemented
168C22×C7—21no—
169C22×C7—21no—
170C22×C7—41no—
171C22×C7—41no—
172C22×C7—41no—
173C22×C7#12112yes—
174C22×C7C22×D411yescharacteristic
175C2×D7—22no—
176C2×D7—41no—
177C2×D7—41no—
178C2×D7—42no—
179C2×D7—42no—
180C2×D7—42no—
181C2×D7—84no—
182C4×C7—41no—
183C4×C7—42no—
184C4×C7#12112yes—
185C4×C7C22×D411yescharacteristic
186C4×C7C2×#4111yescharacteristic
187Dic7—22no—
188Dic7—22no—
189Dic7—22no—
190C24—141no—
191#41—22no—
192#41—22no—
193#41—74no—
194#41—142no—
195C2×Q8—71no—
196C2×Q8C22×D711yescharacteristic
197C2×D4—21no—
198C2×D4—71no—
199C2×D4—72no—
200C2×D4—142no—
201C2×D4—142no—
202C2×D4—141no—
203C2×D4—142no—
204C2×D4—142no—
205C2×D4—141no—
206C2×D4C22×D711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—142no—
210C22×C4—141no—
211C22×C4C22×D711yescharacteristic
212Q16—142no—
213SD16—144no—
214SD16—142no—
215SD16—142no—
216D8—144no—
217D8—142no—
218M16—22no—
219C2×C8—71no—
220C2×C8—71no—
221C2×C8—142no—
222C2×C8C22×D712yes—
223C4⋊C4—21no—
224C4⋊C4—71no—
225C4⋊C4—71no—
226C4⋊C4—142no—
227C4⋊C4C22×D712yes—
228C22⋊C4—72no—
229C22⋊C4—142no—
230C22⋊C4—142no—
231C22⋊C4—142no—
232C42—72no—
233C42—142no—
234C2×C7—22no—
235C2×C7—81no—
236C2×C7C2×#12111yescharacteristic
237C2×C7#57611yescharacteristic
238C2×C7#57711yescharacteristic
239D7—42no—
240D7—82no—
241C23—21no—
242C23—141no—
243C23—142no—
244C23—282no—
245C23—561no—
246C23—561no—
247C23D4×D711yescharacteristic
248Q8—21no—
249Q8—21no—
250Q8—72no—
251Q8—141no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—72no—
256D4—144no—
257D4—141no—
258D4—142no—
259D4—142no—
260D4—282no—
261D4—562no—
262D4—562no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—72no—
267C2×C4—72no—
268C2×C4—81no—
269C2×C4—141no—
270C2×C4—144no—
271C2×C4—562no—
272C2×C4#115712yes—
273C2×C4D4×D711yescharacteristic
274C2×C4D4×D712yes—
275C2×C4C23×D711yescharacteristic
276C8—22no—
277C8—141no—
278C8—141no—
279C7#304511yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—142no—
286C22—281no—
287C22—281no—
288C22—282no—
289C22—282no—
290C22—282no—
291C22—564no—
292C22#644712yes—
293C22C2×D4×D711yescharacteristic
294C4—41no—
295C4—42no—
296C4—142no—
297C4—142no—
298C4—142no—
299C4#644712yes—
300C4C2×#115711yescharacteristic
301C4C2×D4×D711yescharacteristic
302C2—22no—
303C2—81no—
304C2—282no—
305C2—562no—
306C2C2×#644711yescharacteristic
307C2#9046511yescharacteristic
308C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
hypercenter#417D7
hypocenterC7#3045
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #90521C2semidirect
2. C2×#6513C2semidirect
3. C2×#6471C2semidirect
4. #90195C2semidirect
5. C7×#417C2semidirect
6. #89891C2semidirect
7. C2×#6383C2semidirect
8. #89591C2semidirect
9. #89570C2semidirect
10. #89558C2semidirect
11. #89534C2semidirect
19. C2×#1144C22semidirect
20. C2×#1143C22semidirect
23. C2×#1142C22semidirect
24. C2×#1141C22semidirect
33. #6456C22semidirect
36. #6450C22semidirect
37. C7×#103C22semidirect
38. C7×#102C22semidirect
39. Dic7⋊C8C22semidirect
40. Dic7⋊C8C22semidirect
57. #1157D4semidirect
97. #417D7semidirect
164. #103C2×D7semidirect
167. #102C2×D7semidirect
279. C7#3045semidirect

Maximal subgroups

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

subgroupindex
#905212
C2×#65132
C2×#64712
#901952
C7×#4172
#898912
C2×#63832
#895912
#895702
#895582
#895342
#30457