Finite Group Databasebeta

← #11736752

Subgroups — #11736752

Order 896.

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

normal subgroups only

This group has 410 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
1#90170C211yesmaximal, characteristic, complemented
2C2×#6438C211yesmaximal, characteristic, complemented
3C7×#431C211yesmaximal, characteristic, complemented
4#89617C212yesmaximal, complemented
5#89616C212yesmaximal, complemented
6#89511C211yesmaximal, characteristic, complemented
7#89510C211yesmaximal, characteristic, complemented
8#89504C211yesmaximal, characteristic
9#89503C211yesmaximal, characteristic, complemented
10#89469C212yesmaximal, complemented
11#89468C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15#6493C2211yescharacteristic
16#6492—21no—
17C4×C22⋊D7—21no—
18C2×(C2×D7)⋊C4—21no—
19C2×(C2×D7)⋊C4—21no—
20#6487C2211yescharacteristic
21C2×Dic28C2212yescomplemented
22C2×D56C2212yes—
23C2×#1132C2212yes—
24C2×#1132C2212yescomplemented
25#6439—21no—
26#6438—21no—
27#6438—21no—
28#6438C2212yes—
29#6438C2212yes—
30#6438C2211yescharacteristic, complemented
31C4×D28C2211yescharacteristic, complemented
32#6435—21no—
33#6433C2211yescharacteristic, complemented
34C4×Dic14C2211yescharacteristic, complemented
35C7×C4⋊C8C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37C7×C22⋊C8C2211yescharacteristic
38C4×C7×C8C2212yescomplemented
39#6395C2212yes—
40#6395C2212yes—
41#6390C2212yescomplemented
42#6390C2212yescomplemented
43#3211—71nomaximal, Sylow
44C23×D7—21no—
45C22×C4×C7D411yescharacteristic
46C22×C4×C7D411yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—21no—
49C22×Dic7—21no—
50C2×D28—21no—
51C2×D28—42no—
52C2×D28C2311yescharacteristic
53C2×D28C2312yes—
54C2×C4×D7—21no—
55C2×Dic14—21no—
56C2×Dic14—42no—
57C2×Dic14C2311yescharacteristic
58C2×Dic14C2312yes—
59C2×C7×C8C2312yes—
60C2×C7×C8C2312yes—
61C42×C7D411yescharacteristic
62C42×C7D411yescharacteristic
63C42×C7C2311yescharacteristic
64C42×C7C2311yescharacteristic
65C22⋊Dic7—21no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—21no—
74C4⋊Dic7C2311yescharacteristic
75C4⋊Dic7C2311yescharacteristic
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81Dic28—42no—
82D56—42no—
83#1132—42no—
84#1132—42no—
85#529—71no—
86C2×#124—71no—
87#495—72no—
88#488—72no—
89#464—72no—
90#463—72no—
91#451—71no—
92#450—71no—
93#448—71no—
94#447—71no—
95#431D711yescharacteristic, complemented
96C23×C7C2×D411yescharacteristic
97C22×D7—21no—
98C22×D7—22no—
99C22×D7—42no—
100C22×D7—42no—
101C22×D7—42no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—41no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2411yescharacteristic
111C22⋊D7—42no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—42no—
117D28—22no—
118D28—22no—
119D28—22no—
120D28—42no—
121C4×D7—42no—
122Dic14—22no—
123Dic14—22no—
124Dic14—22no—
125Dic14—42no—
126C7×C8—22no—
127C7×C8—22no—
128C22×Q8—71no—
129C22×D4—71no—
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16—72no—
133C2×D8—72no—
134#126—141no—
135#125—71no—
136#124—72no—
137#124—72no—
138#124—71no—
139#124—141no—
140#124—141no—
141#123—141no—
142C22⋊Q8—71no—
143#121—71no—
144C4×Q8—71no—
145C4×D4—71no—
146C4×D4—141no—
147#117—141no—
148C2×C22⋊C4—141no—
149C2×C22⋊C4—141no—
150C2×C42C2×D711yescharacteristic
151C4⋊C8C2×D712yescomplemented
152#103—72no—
153#103—72no—
154#102—72no—
155#102—72no—
156C22⋊C8C2×D711yescharacteristic
157C22⋊C8C2×D711yescharacteristic
158C4×C8C2×D712yescomplemented
159C22×C7—21no—
160C22×C7—21no—
161C22×C7#13512yes—
162C22×C7C22×D411yescharacteristic
163C2×D7—42no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—42no—
167C2×D7—41no—
168C2×D7—42no—
169C2×D7—84no—
170C4×C7—21no—
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—21no—
174C4×C7—21no—
175C4×C7H(5,𝔽2)12yes—
176Dic7—42no—
177Dic7—41no—
178Dic7—41no—
179Dic7—41no—
180C24—141no—
181C2×Q8—72no—
182C2×Q8—71no—
183C2×Q8—141no—
184C2×Q8—282no—
185C2×D4—72no—
186C2×D4—71no—
187C2×D4—141no—
188C2×D4—141no—
189C2×D4—282no—
190C22×C4—141no—
191C22×C4—141no—
192C22×C4D2811yescharacteristic
193C22×C4D2811yescharacteristic
194C22×C4C22×D711yescharacteristic
195Q16—282no—
196SD16—282no—
197SD16—282no—
198D8—282no—
199C2×C8C22×D712yes—
200C2×C8C22×D712yes—
201C4⋊C4—71no—
202C4⋊C4—71no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4—141no—
207C22⋊C4—141no—
208C22⋊C4—142no—
209C22⋊C4—142no—
210C22⋊C4—142no—
211C22⋊C4—142no—
212C22⋊C4—141no—
213C22⋊C4—141no—
214C22⋊C4—141no—
215C22⋊C4—141no—
216C42—141no—
217C42D2811yescharacteristic
218C42D2811yescharacteristic
219C42C22×D711yescharacteristic
220C42C22×D711yescharacteristic
221C2×C7—22no—
222C2×C7#53511yescharacteristic
223C2×C7C2×#13511yescharacteristic
224C2×C7#57511yescharacteristic
225D7—81no—
226D7—82no—
227C23—141no—
228C23—142no—
229C23—282no—
230C23—282no—
231C23—282no—
232C23C2×D2811yescharacteristic
233Q8—142no—
234Q8—142no—
235Q8—142no—
236Q8—282no—
237D4—142no—
238D4—142no—
239D4—142no—
240D4—282no—
241D4—282no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—142no—
248C2×C4—141no—
249C2×C4—282no—
250C2×C4—282no—
251C2×C4C2×D2811yescharacteristic
252C2×C4C2×D2811yescharacteristic
253C2×C4C2×D2811yescharacteristic
254C2×C4C2×D2811yescharacteristic
255C2×C4C2×D2811yescharacteristic
256C2×C4C23×D711yescharacteristic
257C8—22no—
258C8—22no—
259C7#321111yescharacteristic, complemented, Sylow
260C22—21no—
261C22—21no—
262C22—281no—
263C22—282no—
264C22—281no—
265C22—282no—
266C22—282no—
267C22—281no—
268C22—564no—
269C22#646712yes—
270C22C22×D2811yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—281no—
277C4—281no—
278C4—282no—
279C4—281no—
280C4#655312yes—
281C2—22no—
282C2—562no—
283C2—561no—
284C2#9021611yescharacteristic
285C2C2×#646711yescharacteristic
286C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#89504C2
hypercenter#431D7
hypocenterC7#3211
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #90170C2semidirect
2. C2×#6438C2semidirect
3. C7×#431C2semidirect
4. #89617C2semidirect
5. #89616C2semidirect
6. #89511C2semidirect
7. #89510C2semidirect
9. #89503C2semidirect
10. #89469C2semidirect
11. #89468C2semidirect
21. C2×Dic28C22semidirect
24. C2×#1132C22semidirect
30. #6438C22semidirect
31. C4×D28C22semidirect
33. #6433C22semidirect
34. C4×Dic14C22semidirect
35. C7×C4⋊C8C22semidirect
38. C4×C7×C8C22semidirect
41. #6390C22semidirect
42. #6390C22semidirect
95. #431D7semidirect
151. C4⋊C8C2×D7semidirect
158. C4×C8C2×D7semidirect
259. C7#3211semidirect

Maximal subgroups

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

subgroupindex
#901702
C2×#64382
C7×#4312
#896172
#896162
#895112
#895102
#895042
#895032
#894692
#894682
#32117