Finite Group Databasebeta

← #11737502

Subgroups — #11737502

Order 896.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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×#6510C211yesmaximal, characteristic, complemented
2#90291C211yesmaximal, characteristic, complemented
3C7×#448C211yesmaximal, characteristic, complemented
4#90000C211yesmaximal, characteristic, complemented
5#89940C212yesmaximal, complemented
6#89937C212yesmaximal, complemented
7#89808C211yesmaximal, characteristic
8#89572C212yesmaximal, complemented
9#89570C212yesmaximal, complemented
10#89503C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C2×C7×SD16C2212yescomplemented
15C7×#121C2211yescharacteristic
16#6510—21no—
17#6510—21no—
18#6510C2212yes—
19#6510C2212yes—
20C2×#1144C2212yescomplemented
21#6499C2211yescharacteristic, complemented
22#6498—21no—
23D4×Dic7—21no—
24D4×Dic7C2211yescharacteristic, complemented
25C2×#1141C2212yes—
26C2×(C2×D7)⋊C4—21no—
27C2×(C2×D7)⋊C4—21no—
28#6487C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×#1132C2212yes—
31#6453C2211yescharacteristic, complemented
32#6450C2211yescharacteristic, complemented
33#6442—21no—
34#6440—21no—
35C7×#103C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37#6409C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6395C2212yes—
40Dic7⋊C8C2212yescomplemented
41C8×Dic7C2212yescomplemented
42#6383C2212yes—
43#3211—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8—21no—
46C2×C7×Q8—42no—
47C2×C7×Q8C2312yes—
48C2×C7×D4—21no—
49C2×C7×D4C2311yescharacteristic
50C22×C4×C7—21no—
51C22×C4×C7C2311yescharacteristic
52C22×Dic7—21no—
53C22×Dic7D411yescharacteristic
54C22×Dic7D411yescharacteristic
55C2×D28—21no—
56C2×D28—42no—
57C2×D28C2312yes—
58C2×Dic14C2311yescharacteristic
59C7×SD16—42no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4C2311yescharacteristic
62C7×C22⋊C4—21no—
63C22⋊Dic7—21no—
64C22⋊Dic7—21no—
65C22⋊Dic7—21no—
66C22⋊Dic7—21no—
67#1144—42no—
68#1141—42no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—22no—
73C4⋊Dic7—21no—
74C4⋊Dic7C2311yescharacteristic
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78C4×Dic7—21no—
79C4×Dic7D411yescharacteristic
80C4×Dic7D411yescharacteristic
81C4×Dic7C2311yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
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#448D711yescharacteristic, complemented
94#447—71no—
95#431—71no—
96C23×C7—21no—
97C23×C7C2×D411yescharacteristic
98C22×D7—22no—
99C22×D7—42no—
100C22×D7—42no—
101C22×D7—42no—
102C7×Q8—22no—
103C7×Q8—22no—
104C7×Q8—42no—
105C7×D4—22no—
106C7×D4—42no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—22no—
110C2×C4×C7—42no—
111C2×C4×C7C2×D411yescharacteristic
112C2×C4×C7C2411yescharacteristic
113C2×Dic7—22no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—41no—
117C2×Dic7—42no—
118C2×Dic7C2×D411yescharacteristic
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121C2×Dic7C2×D411yescharacteristic
122D28—22no—
123D28—22no—
124D28—42no—
125Dic14—22no—
126C7×C8—22no—
127C7⋊C8—22no—
128C22×Q8C2×D711yescharacteristic
129C22×D4—71no—
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16C2×D712yescomplemented
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#121C2×D711yescharacteristic
144C4×Q8—71no—
145C4×D4—71no—
146C4×D4—141no—
147#117—141no—
148C2×C22⋊C4—141no—
149C2×C22⋊C4—141no—
150C2×C42—71no—
151C4⋊C8—72no—
152#103—72no—
153#103C2×D712yescomplemented
154#102—72no—
155#102—72no—
156C22⋊C8—71no—
157C22⋊C8C2×D711yescharacteristic
158C4×C8—72no—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7—41no—
162C22×C7—41no—
163C22×C7—41no—
164C22×C7#13512yes—
165C22×C7C22×D411yescharacteristic
166C2×D7—42no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—84no—
170C4×C7—21no—
171C4×C7—42no—
172C4×C7—41no—
173C4×C7H(5,𝔽2)12yes—
174Dic7—21no—
175Dic7—21no—
176Dic7—21no—
177Dic7—21no—
178Dic7—41no—
179Dic7—41no—
180C24—141no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×Q8—71no—
184C2×Q8C22×D712yes—
185C2×D4—21no—
186C2×D4—72no—
187C2×D4—141no—
188C2×D4—282no—
189C2×D4C22×D711yescharacteristic
190C22×C4—21no—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—141no—
194C22×C4C22×D711yescharacteristic
195Q16—282no—
196SD16—42no—
197SD16—282no—
198D8—282no—
199C2×C8—72no—
200C2×C8C22×D712yes—
201C4⋊C4—71no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4C22×D711yescharacteristic
207C22⋊C4—21no—
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—71no—
217C42—71no—
218C42—71no—
219C42—71no—
220C42—141no—
221C2×C7—22no—
222C2×C7—81no—
223C2×C7#53511yescharacteristic
224C2×C7C2×#13511yescharacteristic
225C2×C7#57511yescharacteristic
226D7—82no—
227C23—21no—
228C23—142no—
229C23—282no—
230C23—282no—
231C23—282no—
232C23D4×D711yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—142no—
237D4—22no—
238D4—42no—
239D4—142no—
240D4—142no—
241D4—282no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—71no—
247C2×C4—71no—
248C2×C4—71no—
249C2×C4—71no—
250C2×C4—142no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—281no—
254C2×C4—282no—
255C2×C4D4×D711yescharacteristic
256C2×C4C23×D711yescharacteristic
257C8—22no—
258C8—142no—
259C7#321111yescharacteristic, complemented, Sylow
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—282no—
266C22—282no—
267C22—282no—
268C22—564no—
269C22#647912yes—
270C22C2×D4×D711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—141no—
275C4—141no—
276C4—141no—
277C4—141no—
278C4—281no—
279C4—281no—
280C4#654812yes—
281C2—22no—
282C2—81no—
283C2—562no—
284C2#9028311yescharacteristic
285C2C2×#647911yescharacteristic
286C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#448C2
socleC22×C7C22×D4
prime radical#89808C2
hypercenter#448D7
hypocenterC7#3211
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#6510C2semidirect
2. #90291C2semidirect
3. C7×#448C2semidirect
4. #90000C2semidirect
5. #89940C2semidirect
6. #89937C2semidirect
8. #89572C2semidirect
9. #89570C2semidirect
10. #89503C2semidirect
11. #89490C2semidirect
14. C2×C7×SD16C22semidirect
20. C2×#1144C22semidirect
21. #6499C22semidirect
24. D4×Dic7C22semidirect
31. #6453C22semidirect
32. #6450C22semidirect
35. C7×#103C22semidirect
37. #6409C22semidirect
40. Dic7⋊C8C22semidirect
41. C8×Dic7C22semidirect
93. #448D7semidirect
132. C2×SD16C2×D7semidirect
153. #103C2×D7semidirect
259. C7#3211semidirect

Maximal subgroups

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

subgroupindex
C2×#65102
#902912
C7×#4482
#900002
#899402
#899372
#898082
#895722
#895702
#895032
#894902
#32117