Finite Group Databasebeta

← #11738310

Subgroups — #11738310

Order 896.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D7×D8C211yesmaximal, characteristic, complemented
2C2×#6470C211yesmaximal, characteristic, complemented
3#90298C212yesmaximal, complemented
4C7×#469C211yesmaximal, characteristic, complemented
5#89931C211yesmaximal, characteristic, complemented
6#89907C211yesmaximal, characteristic, complemented
7#89810C212yesmaximal, complemented
8#89637C211yesmaximal, characteristic, complemented
9#89632C211yesmaximal, characteristic, complemented
10#89547C212yesmaximal, complemented
11#89542C212yesmaximal, complemented
12C2×D4×D7C2212yes—
13C2×#1157C2211yescharacteristic
14C2×C7×D8C2211yescharacteristic
15C2×C7×M16C2211yescharacteristic, complemented
16C7×#121C2212yescomplemented
17#6503—22no—
18#6501C2212yescomplemented
19#6500—22no—
20C2×#1141C2212yes—
21#6493C2212yescomplemented
22C22×C7⋊C8C2211yescharacteristic, complemented
23D7×D8—24no—
24#6470—24no—
25C2×D56C2211yescharacteristic
26C2×#1131C2211yescharacteristic
27C2×#1131C2211yescharacteristic, complemented
28C2×C8×D7C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic, complemented
30#6458C2212yescomplemented
31#6455C2212yes—
32#6448—22no—
33#6445—22no—
34C7×#106C2211yescharacteristic, complemented
35C7×#102C2212yescomplemented
36#6406C2212yescomplemented
37#6395C2212yescomplemented
38#6391C2211yescharacteristic, complemented
39#6383C2212yescomplemented
40#6381C2212yescomplemented
41#3058—71nomaximal, Sylow
42C23×D7—22no—
43C2×C7×D4—22no—
44C2×C7×D4C2312yes—
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—22no—
47C2×C22⋊D7—22no—
48C2×C22⋊D7—22no—
49C2×C22⋊D7D411yescharacteristic
50C2×C22⋊D7D411yescharacteristic
51D4×D7—24no—
52D4×D7—24no—
53#1157—22no—
54#1157—22no—
55C2×D28—22no—
56C2×D28D411yescharacteristic
57C2×D28C2312yes—
58C2×C4×D7C2311yescharacteristic
59C2×C4×D7C2311yescharacteristic
60C2×Dic14D411yescharacteristic
61C7×D8—22no—
62C7×M16—22no—
63C2×C7×C8C2311yescharacteristic
64C2×C7×C8C2311yescharacteristic
65C7×C4⋊C4C2312yes—
66C7×C22⋊C4—22no—
67C22⋊Dic7—22no—
68#1141—24no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72C4⋊Dic7C2312yes—
73C4×Dic7—22no—
74C2×C7⋊C8—22no—
75C2×C7⋊C8C2311yescharacteristic
76C2×C7⋊C8C2311yescharacteristic
77D56—22no—
78#1131—22no—
79#1131—22no—
80C8×D7—22no—
81C8×D7D414yescomplemented
82C22×D8—71no—
83C2×#131—71no—
84#535—72no—
85#469—71no—
86#469—71no—
87#469—71no—
88#469D711yescharacteristic, complemented
89#466—72no—
90#466—72no—
91#427—71no—
92#418—72no—
93C23×C7—22no—
94C23×C7C2×D411yescharacteristic
95C22×D7—22no—
96C22×D7—44no—
97C22×D7—82no—
98C22×D7—82no—
99C22×D7C2×D411yescharacteristic
100C22×D7C2×D411yescharacteristic
101C7×D4—22no—
102C7×D4—22no—
103C7×D4—82no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2411yescharacteristic
109C22⋊D7—22no—
110C22⋊D7—22no—
111C22⋊D7—44no—
112C22⋊D7—82no—
113C22⋊D7—82no—
114C2×Dic7—22no—
115C2×Dic7C2×D411yescharacteristic
116C2×Dic7C2×D411yescharacteristic
117D28—22no—
118D28—22no—
119D28—22no—
120D28—82no—
121C4×D7—21no—
122C4×D7—21no—
123C4×D7C2×D412yes—
124C4×D7#4112yes—
125Dic14—22no—
126C7×C8—21no—
127C7×C8C2×D412yes—
128C7⋊C8—21no—
129C7⋊C8C2×D412yes—
130C2×#41—71no—
131C22×D4—72no—
132C2×D8—72no—
133C2×D8—71no—
134C2×D8—144no—
135C2×D8C2×D711yescharacteristic
136#131—144no—
137C2×M16—71no—
138C2×M16—71no—
139C2×M16C2×D711yescharacteristic, complemented
140C22×C8—71no—
141C22×C8—71no—
142C22×C8—71no—
143#127—142no—
144#124—142no—
145#121—72no—
146#121—72no—
147#121—72no—
148#121C2×D712yescomplemented
149#120—142no—
150#120—142no—
151#117—72no—
152#107—72no—
153#106—71no—
154#106C2×D711yescharacteristic, complemented
155#102—72no—
156#102—72no—
157#102—72no—
158#102C2×D712yescomplemented
159C22×C7—21no—
160C22×C7—22no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—42no—
164C22×C7C22×D411yescharacteristic
165C2×D7—22no—
166C2×D7—21no—
167C2×D7—22no—
168C2×D7—42no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—88no—
172C2×D7#12112yes—
173C4×C7—21no—
174C4×C7—42no—
175C4×C7C22×D411yescharacteristic
176C4×C7C2×#4111yescharacteristic
177Dic7—21no—
178Dic7—42no—
179Dic7#12112yes—
180C24—142no—
181#41—142no—
182#41—142no—
183C2×Q8—71no—
184C2×D4—22no—
185C2×D4—71no—
186C2×D4—71no—
187C2×D4—71no—
188C2×D4—72no—
189C2×D4—142no—
190C2×D4—144no—
191C2×D4—144no—
192C2×D4—142no—
193C2×D4—142no—
194C2×D4—142no—
195C2×D4C22×D712yes—
196C22×C4—71no—
197C22×C4—71no—
198C22×C4C22×D711yescharacteristic
199D8—22no—
200D8—144no—
201D8—142no—
202M16—22no—
203M16—142no—
204M16—142no—
205C2×C8—71no—
206C2×C8—74no—
207C2×C8—71no—
208C2×C8—142no—
209C2×C8—142no—
210C2×C8C22×D711yescharacteristic
211C2×C8C22×D711yescharacteristic
212C4⋊C4—72no—
213C4⋊C4C22×D712yes—
214C22⋊C4—22no—
215C22⋊C4—142no—
216C22⋊C4—142no—
217C22⋊C4—142no—
218C22⋊C4—142no—
219C42—142no—
220C2×C7—41no—
221C2×C7—82no—
222C2×C7C2×#12111yescharacteristic
223C2×C7#57612yes—
224D7—22no—
225D7—41no—
226D7—82no—
227C23—22no—
228C23—71no—
229C23—71no—
230C23—142no—
231C23—284no—
232C23—562no—
233C23—562no—
234C23D4×D711yescharacteristic
235Q8—142no—
236D4—22no—
237D4—22no—
238D4—82no—
239D4—142no—
240D4—142no—
241D4—142no—
242D4—142no—
243D4—142no—
244D4—284no—
245D4—562no—
246D4—562no—
247D4—562no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—71no—
252C2×C4—72no—
253C2×C4—72no—
254C2×C4—71no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—142no—
258C2×C4D4×D711yescharacteristic
259C2×C4C23×D711yescharacteristic
260C8—21no—
261C8—72no—
262C8—141no—
263C8D4×D712yes—
264C7#305811yescharacteristic, complemented, Sylow
265C22—22no—
266C22—21no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—72no—
271C22—142no—
272C22—142no—
273C22—141no—
274C22—282no—
275C22—282no—
276C22—282no—
277C22—568no—
278C22C2×D4×D711yescharacteristic
279C4—21no—
280C4—42no—
281C4—72no—
282C4—141no—
283C4—282no—
284C4C2×D4×D711yescharacteristic
285C4D7×#4111yescharacteristic
286C2—41no—
287C2—82no—
288C2—142no—
289C2—281no—
290C2—562no—
291C2D7×#12111yescharacteristic
292C2#9046512yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#469C2
socleC22×C7C22×D4
hypercenter#469D7
hypocenterC7#3058
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D7×D8C2semidirect
2. C2×#6470C2semidirect
3. #90298C2semidirect
4. C7×#469C2semidirect
5. #89931C2semidirect
6. #89907C2semidirect
7. #89810C2semidirect
8. #89637C2semidirect
9. #89632C2semidirect
10. #89547C2semidirect
11. #89542C2semidirect
15. C2×C7×M16C22semidirect
16. C7×#121C22semidirect
18. #6501C22semidirect
21. #6493C22semidirect
22. C22×C7⋊C8C22semidirect
27. C2×#1131C22semidirect
29. C2×C8×D7C22semidirect
30. #6458C22semidirect
34. C7×#106C22semidirect
35. C7×#102C22semidirect
36. #6406C22semidirect
37. #6395C22semidirect
38. #6391C22semidirect
39. #6383C22semidirect
40. #6381C22semidirect
81. C8×D7D4semidirect
88. #469D7semidirect
139. C2×M16C2×D7semidirect
148. #121C2×D7semidirect
154. #106C2×D7semidirect
158. #102C2×D7semidirect
264. C7#3058semidirect

Maximal subgroups

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

subgroupindex
C2×D7×D82
C2×#64702
#902982
C7×#4692
#899312
#899072
#898102
#896372
#896322
#895472
#895422
#30587