Finite Group Databasebeta

← #11740937

Subgroups — #11740937

Order 896.

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

normal subgroups only

This group has 556 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
1C7×D42C211yesmaximal, characteristic, complemented
2C2×#6495C212yesmaximal, complemented
3#90244C211yesmaximal, characteristic, complemented
4#89989C212yesmaximal, complemented
5#89847C211yesmaximal, characteristic, complemented
6#89845C211yesmaximal, characteristic, complemented
7#89811C212yesmaximal, complemented
8#89809C212yesmaximal, complemented
9C4×#1142C211yesmaximal, characteristic, complemented
10#89787C211yesmaximal, characteristic, complemented
11#89784C211yesmaximal, characteristic, complemented
12C22×C7×D4—21no—
13C22×C7×D4C2212yes—
14C2×#1157C2212yescomplemented
15C7×#127C2211yescharacteristic
16C7×#121—21no—
17C7×#121C2212yes—
18C7×#120—22no—
19C4×C7×D4C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic
21#6503—21no—
22C2×#1142C2211yescharacteristic
23C2×#1142C2212yescomplemented
24#6495—28no—
25C2×#1141C2212yes—
26C2×#1141C2212yescomplemented
27C2×#1141C2212yescomplemented
28#6493C2212yescomplemented
29C2×#1136C2212yescomplemented
30#6456—21no—
31#6447—22no—
32#6437C2211yescharacteristic, complemented
33C4×Dic14C2211yescharacteristic, complemented
34#6406C2211yescharacteristic
35#6406C2212yescomplemented
36C22⋊(C7⋊C8)C2212yescomplemented
37#6384C2211yescharacteristic, complemented
38#6383C2212yescomplemented
39#6382C2211yescharacteristic, complemented
40C4⋊(C7⋊C8)C2211yescharacteristic, complemented
41C4×C7⋊C8C2211yescharacteristic, complemented
42#3274—71nomaximal, Sylow
43C24×C7—22no—
44C2×C7×D4—24no—
45C2×C7×D4—22no—
46C2×C7×D4—22no—
47C2×C7×D4—21no—
48C2×C7×D4—21no—
49C2×C7×D4—42no—
50C2×C7×D4—44no—
51C2×C7×D4D411yescharacteristic
52C2×C7×D4C2312yes—
53C2×C7×D4C2311yescharacteristic
54C22×C4×C7—21no—
55C22×C4×C7C2312yes—
56C2×C22⋊D7—22no—
57C2×C22⋊D7—22no—
58#1157—24no—
59#1157—42no—
60C2×D28—21no—
61C2×D28—21no—
62C2×D28C2312yes—
63C2×C4×D7—22no—
64C2×Dic14C2311yescharacteristic
65C7×C4⋊C4D411yescharacteristic
66C7×C4⋊C4C2311yescharacteristic
67C7×C22⋊C4—21no—
68C7×C22⋊C4—22no—
69C7×C22⋊C4D412yes—
70C42×C7C2311yescharacteristic
71#1142—24no—
72#1142D414yescomplemented
73#1141—24no—
74#1141—24no—
75#1141—42no—
76(C2×D7)⋊C4—22no—
77C4⋊Dic7C2311yescharacteristic
78Dic7⋊C4—21no—
79C4×Dic7—21no—
80#1136—24no—
81C2×C7⋊C8C2311yescharacteristic
82C2×C7⋊C8C2312yes—
83C2×C7⋊C8C2311yescharacteristic
84C2×AGL(1,ℤ/8)—72no—
85#550—71no—
86D42D711yescharacteristic, complemented
87#493—71no—
88#469—72no—
89#460—71no—
90#459—71no—
91#449—72no—
92#447—72no—
93C4×SD16—71no—
94#436—71no—
95C23×C7—21no—
96C23×C7—22no—
97C23×C7—44no—
98C23×C7—44no—
99C23×C7—82no—
100C23×C7C2×D412yes—
101C22×D7—22no—
102C7×D4—22no—
103C7×D4—22no—
104C7×D4—21no—
105C7×D4—22no—
106C7×D4—44no—
107C7×D4—44no—
108C7×D4—42no—
109C7×D4—81no—
110C7×D4C2×D412yes—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C2×C4×C7—42no—
115C2×C4×C7C2×D411yescharacteristic
116C2×C4×C7C2×D412yes—
117C2×C4×C7C2×D411yescharacteristic
118C2×C4×C7C2411yescharacteristic
119C22⋊D7—44no—
120C22⋊D7—82no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123D28—22no—
124D28—22no—
125D28—42no—
126D28—81no—
127C4×D7—44no—
128Dic14—21no—
129Dic14C2×D412yescomplemented
130C7⋊C8—22no—
131C7⋊C8—21no—
132C7⋊C8C2×D412yescomplemented
133C2×#41—72no—
134C22×D4—21no—
135C22×D4C2×D712yes—
136AGL(1,ℤ/8)—148no—
137C2×SD16—71no—
138C2×SD16—72no—
139C2×D8—72no—
140C2×D8—72no—
141C2×D8—72no—
142C2×M16—72no—
143#127—71no—
144#127—141no—
145#127C2×D711yescharacteristic
146#121—21no—
147#121—72no—
148#121—142no—
149#121C2×D712yes—
150#120—22no—
151C4×Q8—71no—
152C4×D4—141no—
153C4×D4C2×D711yescharacteristic
154C4×D4C2×D711yescharacteristic
155#106—71no—
156C4⋊C8—71no—
157#103—71no—
158#102—71no—
159#102—72no—
160#102—72no—
161C22⋊C8—72no—
162C4×C8—71no—
163C22×C7—22no—
164C22×C7—22no—
165C22×C7—22no—
166C22×C7—22no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7—42no—
172C22×C7—88no—
173C22×C7C22×D411yescharacteristic
174C2×D7—42no—
175C2×D7—42no—
176C2×D7—42no—
177C4×C7—22no—
178C4×C7—21no—
179C4×C7—41no—
180C4×C7AGL(1,ℤ/8)12yescomplemented
181C4×C7C22×D411yescharacteristic
182C4×C7H(5,𝔽2)11yescharacteristic
183Dic7—22no—
184Dic7—41no—
185C24—22no—
186#41—144no—
187#41—282no—
188C2×Q8—71no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—72no—
197C2×D4—142no—
198C2×D4—142no—
199C2×D4—141no—
200C2×D4—141no—
201C2×D4C22⋊D711yescharacteristic
202C2×D4C22×D712yes—
203C2×D4C22×D711yescharacteristic
204C22×C4—21no—
205C22×C4—142no—
206C22×C4C22×D712yes—
207SD16—74no—
208SD16—144no—
209D8—144no—
210D8—144no—
211D8—282no—
212M16—144no—
213C2×C8—71no—
214C2×C8—71no—
215C2×C8—72no—
216C4⋊C4—71no—
217C4⋊C4—141no—
218C4⋊C4C22⋊D711yescharacteristic
219C4⋊C4C22×D711yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—22no—
222C22⋊C4—142no—
223C22⋊C4C22⋊D712yes—
224C42—141no—
225C42C22×D711yescharacteristic
226C2×C7—42no—
227C2×C7—42no—
228C2×C7—82no—
229C2×C7D4211yescharacteristic
230C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C7#57611yescharacteristic
232D7—82no—
233C23—21no—
234C23—22no—
235C23—44no—
236C23—44no—
237C23—82no—
238C23—142no—
239C23C2×C22⋊D712yes—
240Q8—72no—
241Q8—141no—
242D4—22no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—44no—
247D4—44no—
248D4—42no—
249D4—81no—
250D4—142no—
251D4—142no—
252D4—284no—
253D4—282no—
254D4—561no—
255D4—562no—
256D4D4×D712yes—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—284no—
264C2×C4C2×C22⋊D711yescharacteristic
265C2×C4C2×C22⋊D712yes—
266C2×C4C2×C22⋊D711yescharacteristic
267C2×C4C23×D711yescharacteristic
268C8—72no—
269C8—142no—
270C8—141no—
271C7#327411yescharacteristic, complemented, Sylow
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—282no—
283C22—282no—
284C22—282no—
285C22C22×C22⋊D711yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—142no—
290C4—281no—
291C4#649512yescomplemented
292C4C2×D4×D711yescharacteristic
293C4#654811yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—562no—
298C2C2×#649511yescharacteristic
299C2D4×C22⋊D711yescharacteristic
300C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×D42C2
socleC22×C7C22×D4
hypercenterD42D7
hypocenterC7#3274
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C7×D42C2semidirect
2. C2×#6495C2semidirect
3. #90244C2semidirect
4. #89989C2semidirect
5. #89847C2semidirect
6. #89845C2semidirect
7. #89811C2semidirect
8. #89809C2semidirect
9. C4×#1142C2semidirect
10. #89787C2semidirect
11. #89784C2semidirect
14. C2×#1157C22semidirect
23. C2×#1142C22semidirect
26. C2×#1141C22semidirect
27. C2×#1141C22semidirect
28. #6493C22semidirect
29. C2×#1136C22semidirect
32. #6437C22semidirect
33. C4×Dic14C22semidirect
35. #6406C22semidirect
36. C22⋊(C7⋊C8)C22semidirect
37. #6384C22semidirect
38. #6383C22semidirect
39. #6382C22semidirect
40. C4⋊(C7⋊C8)C22semidirect
41. C4×C7⋊C8C22semidirect
72. #1142D4semidirect
86. D42D7semidirect
129. Dic14C2×D4semidirect
132. C7⋊C8C2×D4semidirect
180. C4×C7AGL(1,ℤ/8)semidirect
271. C7#3274semidirect
291. C4#6495semidirect

Maximal subgroups

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

subgroupindex
C7×D422
C2×#64952
#902442
#899892
#898472
#898452
#898112
#898092
C4×#11422
#897872
#897842
#32747