Finite Group Databasebeta

← #11738749

Subgroups — #11738749

Order 896.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#90297C211yesmaximal, characteristic
2C2×#6460C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89812C211yesmaximal, characteristic, complemented
5#89743C211yesmaximal, characteristic, complemented
6#89660C212yesmaximal, complemented
7#89659C212yesmaximal, complemented
8#89555C212yesmaximal, complemented
9#89550C212yesmaximal, complemented
10#89511C211yesmaximal, characteristic, complemented
11D7×C22⋊C8C211yesmaximal, characteristic, complemented
12C2×#1159C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C7×#121C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17#6501C2211yescharacteristic
18#6500—21no—
19D4×Dic7—21no—
20C2×#1142C2212yescomplemented
21C2×(C2×D7)⋊C4—21no—
22C2×C4⋊Dic7—21no—
23#6487C2211yescharacteristic
24C2×Dic28C2212yescomplemented
25C2×C8×D7C2212yes—
26#6460—21no—
27#6460—21no—
28#6460C2211yescharacteristic
29#6460C2212yes—
30#6460C2212yes—
31#6455C2211yescharacteristic
32#6446—21no—
33(C22⋊D7)⋊C4—21no—
34C7×#107C2212yescomplemented
35C7×#102C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37C22⋊(C7⋊C8)C2211yescharacteristic
38(C2×D7)⋊C8C2212yes—
39#6390C2212yescomplemented
40#6384C2212yescomplemented
41#6384C2212yescomplemented
42#6382C2212yescomplemented
43#3190—71nomaximal, Sylow
44C23×D7D411yescharacteristic
45C2×C7×D4—21no—
46C2×C7×D4C2311yescharacteristic
47C22×C4×C7—21no—
48C22×C4×C7C2311yescharacteristic
49C2×C22⋊D7—21no—
50C2×C22⋊D7—21no—
51C22×Dic7—21no—
52C22×Dic7—21no—
53C22×Dic7D411yescharacteristic
54#1159—42no—
55C2×C4×D7—42no—
56C2×C4×D7D411yescharacteristic
57C2×C4×D7D411yescharacteristic
58C2×C4×D7C2311yescharacteristic
59C2×C4×D7C2311yescharacteristic
60C2×Dic14—21no—
61C2×Dic14—42no—
62C2×Dic14C2312yes—
63C2×Dic14C2311yescharacteristic
64C2×C7×C8C2312yes—
65C7×C4⋊C4—21no—
66C7×C4⋊C4C2312yes—
67C7×C4⋊C4C2311yescharacteristic
68C7×C22⋊C4—21no—
69C22⋊Dic7—21no—
70C22⋊Dic7—21no—
71#1142—42no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76C4⋊Dic7—21no—
77C4⋊Dic7—22no—
78C4⋊Dic7—22no—
79C4⋊Dic7C2311yescharacteristic
80Dic7⋊C4—21no—
81C4×Dic7—21no—
82C2×C7⋊C8C2312yes—
83Dic28—42no—
84C8×D7—42no—
85#525—71no—
86C2×C22⋊Q8—71no—
87#485—72no—
88#483—71no—
89#480D711yescharacteristic, complemented
90#467—72no—
91#465—72no—
92#452—72no—
93#451—71no—
94#448—71no—
95C2×C22⋊C8—71no—
96C23×C7—21no—
97C23×C7C2×D411yescharacteristic
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—41no—
101C22×D7C2×D411yescharacteristic
102C22×D7C2×D411yescharacteristic
103C7×D4—22no—
104C7×D4—42no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—42no—
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2411yescharacteristic
111C22⋊D7—42no—
112C22⋊D7—42no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—41no—
117C2×Dic7—42no—
118C2×Dic7—42no—
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121C4×D7—22no—
122C4×D7—22no—
123C4×D7—41no—
124C4×D7—41no—
125Dic14—22no—
126Dic14—22no—
127Dic14—22no—
128Dic14—42no—
129C7×C8—22no—
130C7⋊C8—22no—
131C2×#41—71no—
132C22×Q8—71no—
133C23×C4—71no—
134C2×Q16—72no—
135C2×SD16—72no—
136C22×C8—72no—
137#123—141no—
138C22⋊Q8—71no—
139C22⋊Q8—72no—
140C22⋊Q8—72no—
141C22⋊Q8—71no—
142C22⋊Q8—141no—
143C22⋊Q8—141no—
144#121—71no—
145#121C2×D711yescharacteristic
146#120—141no—
147C4×D4—141no—
148C4×D4—141no—
149#117—71no—
150C2×C4⋊C4—141no—
151C2×C4⋊C4C2×D711yescharacteristic
152C2×C22⋊C4—141no—
153#107C2×D712yescomplemented
154#106—72no—
155#103—72no—
156#103—72no—
157#103—72no—
158#102C2×D712yescomplemented
159C22⋊C8—71no—
160C22⋊C8—72no—
161C22⋊C8C2×D711yescharacteristic
162C22×C7—21no—
163C22×C7—21no—
164C22×C7—41no—
165C22×C7—41no—
166C22×C7—41no—
167C22×C7#13512yes—
168C22×C7C22×D411yescharacteristic
169C2×D7—21no—
170C2×D7—21no—
171C2×D7—21no—
172C2×D7—21no—
173C2×D7—21no—
174C2×D7—44no—
175C2×D7#13712yes—
176C4×C7—21no—
177C4×C7—42no—
178C4×C7—41no—
179C4×C7H(5,𝔽2)12yes—
180Dic7—21no—
181Dic7—21no—
182Dic7—42no—
183Dic7—41no—
184Dic7—41no—
185C24—71no—
186#41—282no—
187C2×Q8—72no—
188C2×Q8—71no—
189C2×Q8—141no—
190C2×Q8—282no—
191C2×D4—21no—
192C2×D4—141no—
193C2×D4—141no—
194C2×D4C22×D711yescharacteristic
195C22×C4—21no—
196C22×C4—71no—
197C22×C4—71no—
198C22×C4—71no—
199C22×C4—71no—
200C22×C4—71no—
201C22×C4—141no—
202C22×C4—141no—
203C22×C4—282no—
204C22×C4C22×D711yescharacteristic
205Q16—282no—
206SD16—282no—
207C2×C8—72no—
208C2×C8—282no—
209C2×C8C22×D712yes—
210C4⋊C4—21no—
211C4⋊C4—71no—
212C4⋊C4—142no—
213C4⋊C4—142no—
214C4⋊C4—141no—
215C4⋊C4—141no—
216C4⋊C4C22×D712yes—
217C4⋊C4C22×D711yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—142no—
220C22⋊C4—142no—
221C22⋊C4—141no—
222C22⋊C4—141no—
223C22⋊C4—141no—
224C22⋊C4—141no—
225C42—141no—
226C2×C7—22no—
227C2×C7—81no—
228C2×C7#53411yescharacteristic
229C2×C7C2×#13511yescharacteristic
230C2×C7C2×#13711yescharacteristic
231D7—22no—
232D7—41no—
233C23—21no—
234C23—71no—
235C23—71no—
236C23—142no—
237C23—142no—
238C23—281no—
239C23D4×D711yescharacteristic
240Q8—142no—
241Q8—142no—
242Q8—142no—
243Q8—282no—
244D4—22no—
245D4—42no—
246D4—282no—
247D4—282no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—71no—
253C2×C4—71no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—142no—
259C2×C4—281no—
260C2×C4—281no—
261C2×C4—281no—
262C2×C4—282no—
263C2×C4—282no—
264C2×C4D4×D711yescharacteristic
265C2×C4C23×D711yescharacteristic
266C8—22no—
267C8—142no—
268C7#319011yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—72no—
275C22—141no—
276C22—141no—
277C22—141no—
278C22—141no—
279C22—141no—
280C22—284no—
281C22#647512yes—
282C22C2×D4×D711yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—141no—
287C4—141no—
288C4—281no—
289C4—281no—
290C4—282no—
291C4#655312yes—
292C2—22no—
293C2—81no—
294C2—142no—
295C2—281no—
296C2#9034411yescharacteristic
297C2C2×#647511yescharacteristic
298C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
prime radical#90297C2
hypercenter#480D7
hypocenterC7#3190
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#6460C2semidirect
3. C7×#480C2semidirect
4. #89812C2semidirect
5. #89743C2semidirect
6. #89660C2semidirect
7. #89659C2semidirect
8. #89555C2semidirect
9. #89550C2semidirect
10. #89511C2semidirect
11. D7×C22⋊C8C2semidirect
20. C2×#1142C22semidirect
24. C2×Dic28C22semidirect
34. C7×#107C22semidirect
35. C7×#102C22semidirect
39. #6390C22semidirect
40. #6384C22semidirect
41. #6384C22semidirect
42. #6382C22semidirect
89. #480D7semidirect
153. #107C2×D7semidirect
158. #102C2×D7semidirect
268. C7#3190semidirect

Maximal subgroups

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

subgroupindex
#902972
C2×#64602
C7×#4802
#898122
#897432
#896602
#896592
#895552
#895502
#895112
D7×C22⋊C82
#31907