Finite Group Databasebeta

← #11739186

Subgroups — #11739186

Order 896.

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

normal subgroups only

This group has 386 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
1D7×C4⋊Q8C211yesmaximal, characteristic, complemented
2#90382C211yesmaximal, characteristic
3C7×#491C211yesmaximal, characteristic, complemented
4#89858C211yesmaximal, characteristic, complemented
5#89841C211yesmaximal, characteristic, complemented
6#89590C212yesmaximal, complemented
7#89580C212yesmaximal, complemented
8#89578C212yesmaximal
9#89565C212yesmaximal, complemented
10#89487C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#125C2211yescharacteristic
15#6507C2212yes—
16C2×#1144C2212yes—
17C2×#1143C2212yes—
18C2×Dic28C2212yes—
19C2×#1132C2212yes—
20#6461—21no—
21#6460C2211yescharacteristic
22#6458C2211yescharacteristic
23#6457—21no—
24#6456—21no—
25#6456C2211yescharacteristic
26#6455—21no—
27D7×C4⋊C4—21no—
28D7×C4⋊C4—21no—
29#6451—21no—
30#6451—21no—
31#6450C2211yescharacteristic
32#6438C2211yescharacteristic
33C42×D7C2211yescharacteristic
34#6432C2211yescharacteristic
35C7×#103C2212yes—
36C7×#103C2212yes—
37C7×#97C2211yescharacteristic
38(C2×D7)⋊C8C2212yes—
39Dic7⋊C8C2212yes—
40#6384C2212yes—
41#6384C2212yes—
42#6377C2211yescharacteristic
43#3218—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45Q8×D7—42no—
46Q8×D7—42no—
47C2×D28—21no—
48C2×D28C2311yescharacteristic
49C2×C4×D7—22no—
50C2×C4×D7—21no—
51C2×C4×D7D411yescharacteristic
52C2×C4×D7D411yescharacteristic
53C2×C4×D7C2311yescharacteristic
54C2×Dic14—22no—
55C2×Dic14C2312yes—
56C2×Dic14C2311yescharacteristic
57C2×C7×C8C2312yes—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4C2311yescharacteristic
63C7×C4⋊C4C2311yescharacteristic
64C42×C7C2311yescharacteristic
65#1144—42no—
66#1143—42no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7—21no—
74C4⋊Dic7—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—22no—
77Dic7⋊C4—22no—
78C4×Dic7—21no—
79C4×Dic7D411yescharacteristic
80C4×Dic7D411yescharacteristic
81C4×Dic7C2311yescharacteristic
82C2×C7⋊C8C2312yes—
83Dic28—42no—
84#1132—42no—
85C2×C4⋊Q8—71no—
86#529—71no—
87#497—71no—
88#497—71no—
89#491—71no—
90#491D711yescharacteristic, complemented
91#462—72no—
92#461—72no—
93#451—72no—
94#448—72no—
95#432—71no—
96C22×D7—21no—
97C22×D7C2×D411yescharacteristic
98C7×Q8—22no—
99C7×Q8—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2411yescharacteristic
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111D28—22no—
112D28—42no—
113C4×D7—22no—
114C4×D7—22no—
115C4×D7—22no—
116C4×D7—44no—
117C4×D7—42no—
118Dic14—22no—
119Dic14—22no—
120Dic14—22no—
121Dic14—44no—
122C7×C8—22no—
123C7⋊C8—22no—
124C22×Q8—72no—
125C2×Q16—72no—
126C2×Q16—72no—
127C2×SD16—72no—
128C2×SD16—72no—
129C4⋊Q8—72no—
130C4⋊Q8—71no—
131C4⋊Q8—141no—
132C4⋊Q8—141no—
133C4⋊Q8C2×D711yescharacteristic
134#126—141no—
135#125C2×D711yescharacteristic
136#124—71no—
137#123—141no—
138C22⋊Q8—71no—
139#121—71no—
140C4×Q8—71no—
141C4×D4—71no—
142C4×D4—141no—
143#117—141no—
144C2×C4⋊C4—141no—
145C2×C4⋊C4—141no—
146C2×C42—71no—
147C4⋊C8—72no—
148#103—72no—
149#103—72no—
150#103C2×D712yes—
151#103C2×D712yes—
152C22⋊C8—72no—
153#97—71no—
154#97C2×D711yescharacteristic
155C22×C7C22×D411yescharacteristic
156C2×D7—21no—
157C2×D7—21no—
158C2×D7—41no—
159C2×D7—41no—
160C2×D7—41no—
161C2×D7#13712yes—
162C4×C7—21no—
163C4×C7—21no—
164C4×C7—42no—
165C4×C7—41no—
166C4×C7—41no—
167C4×C7H(5,𝔽2)12yes—
168Dic7—21no—
169Dic7—21no—
170Dic7—42no—
171Dic7—41no—
172Dic7#13712yes—
173C2×Q8—72no—
174C2×Q8—71no—
175C2×Q8—142no—
176C2×Q8—282no—
177C2×Q8—282no—
178C2×Q8C22×D712yes—
179C2×D4—71no—
180C2×D4—141no—
181C22×C4—71no—
182C22×C4—71no—
183C22×C4—71no—
184C22×C4—141no—
185C22×C4—142no—
186Q16—282no—
187Q16—282no—
188SD16—282no—
189SD16—282no—
190C2×C8—72no—
191C2×C8C22×D712yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—142no—
197C4⋊C4—142no—
198C4⋊C4—141no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4C22×D711yescharacteristic
203C4⋊C4C22×D711yescharacteristic
204C22⋊C4—141no—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C22⋊C4—141no—
209C42—71no—
210C42—71no—
211C42—71no—
212C42—141no—
213C42C22×D711yescharacteristic
214C2×C7#53511yescharacteristic
215C2×C7C2×#13711yescharacteristic
216C2×C7C2×#13711yescharacteristic
217D7—22no—
218D7—81no—
219C23—71no—
220C23—141no—
221Q8—22no—
222Q8—22no—
223Q8—142no—
224Q8—142no—
225Q8—142no—
226Q8—284no—
227D4—142no—
228D4—282no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—71no—
233C2×C4—71no—
234C2×C4—71no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—141no—
239C2×C4—142no—
240C2×C4—282no—
241C2×C4—284no—
242C2×C4D4×D711yescharacteristic
243C2×C4D4×D711yescharacteristic
244C2×C4C23×D711yescharacteristic
245C8—22no—
246C8—142no—
247C7#321811yescharacteristic, complemented, Sylow
248C22—72no—
249C22—141no—
250C22—141no—
251C22—281no—
252C22—281no—
253C22—281no—
254C22C2×D4×D711yescharacteristic
255C4—21no—
256C4—21no—
257C4—41no—
258C4—41no—
259C4—42no—
260C4—72no—
261C4—141no—
262C4—141no—
263C4—282no—
264C4—281no—
265C4#655312yes—
266C2—142no—
267C2—561no—
268C2#9036511yescharacteristic
269C2D7×#13711yescharacteristic
270C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#491D7
hypocenterC7#3218
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊Q8C2semidirect
3. C7×#491C2semidirect
4. #89858C2semidirect
5. #89841C2semidirect
6. #89590C2semidirect
7. #89580C2semidirect
9. #89565C2semidirect
10. #89487C2semidirect
11. #89477C2semidirect
90. #491D7semidirect
247. C7#3218semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#903822
C7×#4912
#898582
#898412
#895902
#895802
#895782
#895652
#894872
#894772
#32187