Finite Group Databasebeta

← #11739113

Subgroups — #11739113

Order 896.

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

normal subgroups only

This group has 410 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#90416C211yesmaximal, characteristic, complemented
2#90371C211yesmaximal, characteristic, complemented
3C7×#489C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89830C211yesmaximal, characteristic, complemented
6#89589C212yesmaximal, complemented
7#89579C212yesmaximal, complemented
8#89557C212yesmaximal, complemented
9#89547C212yesmaximal, complemented
10#89485C211yesmaximal, characteristic, complemented
11D7×#97C211yesmaximal, characteristic, complemented
12C2×#1161—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#124C2211yescharacteristic
15#6510—21no—
16#6509C2211yescharacteristic
17Q8×Dic7C2211yescharacteristic
18C2×#1143C2212yes—
19#6501C2211yescharacteristic
20D4×Dic7C2211yescharacteristic
21C2×#1141C2212yes—
22C2×C8×D7C2212yes—
23#6458—21no—
24#6458C2212yes—
25#6456—21no—
26#6456C2212yes—
27#6446—21no—
28(C22⋊D7)⋊C4—21no—
29#6442—21no—
30#6440—21no—
31#6437C2211yescharacteristic
32C42×D7C2211yescharacteristic
33#6433C2211yescharacteristic
34C7×#103C2212yes—
35C7×#102C2212yes—
36C7×#97C2211yescharacteristic
37#6395C2212yes—
38#6395C2212yes—
39#6389C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3180—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4C2311yescharacteristic
47C2×C22⋊D7—21no—
48C22×Dic7—21no—
49#1161—44no—
50C2×D28—22no—
51C2×D28—21no—
52C2×D28—21no—
53C2×D28C2312yes—
54C2×C4×D7—22no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2312yes—
61C7×C22⋊C4—21no—
62C7×C22⋊C4—21no—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—21no—
65C22⋊Dic7—21no—
66#1143—24no—
67#1141—24no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71C4⋊Dic7—21no—
72C4⋊Dic7C2311yescharacteristic
73C4⋊Dic7C2311yescharacteristic
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77C4×Dic7—21no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82C8×D7—24no—
83#532—71no—
84#529—71no—
85#501—71no—
86#496—71no—
87#490—71no—
88#489D711yescharacteristic, complemented
89#466—72no—
90#465—72no—
91#442—72no—
92#440—72no—
93C2×#97—71no—
94C23×C7—21no—
95C22×D7—22no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—22no—
98C7×Q8—42no—
99C7×D4—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2411yescharacteristic
105C22⋊D7—42no—
106C2×Dic7—21no—
107C2×Dic7—21no—
108C2×Dic7—42no—
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112D28—22no—
113D28—22no—
114D28—42no—
115D28—42no—
116D28—44no—
117C4×D7—22no—
118C4×D7—22no—
119C4×D7—41no—
120C4×D7—44no—
121C7×C8—22no—
122C7⋊C8#4114yescomplemented
123C2×#41—141no—
124C2×SD16—72no—
125C2×D8—72no—
126C22×C8—72no—
127C4⋊Q8C2×D711yescharacteristic
128#127—71no—
129#126—141no—
130#125—71no—
131#124—141no—
132#124C2×D711yescharacteristic
133#123—141no—
134C22⋊Q8—71no—
135#121—71no—
136#121—72no—
137#121—141no—
138C4×Q8—71no—
139C4×D4—71no—
140C4×D4—72no—
141C4×D4—141no—
142C4×D4—141no—
143#117—141no—
144C2×C42—71no—
145#107—72no—
146#106—72no—
147#103C2×D712yes—
148#102—72no—
149#102—72no—
150#102C2×D712yes—
151#97—72no—
152#97—71no—
153#97C2×D711yescharacteristic
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7C22×D411yescharacteristic
158C2×D7—21no—
159C2×D7—21no—
160C2×D7—42no—
161C2×D7—42no—
162C2×D7—42no—
163C2×D7AGL(1,ℤ/8)12yes—
164C4×C7—21no—
165C4×C7—21no—
166C4×C7—41no—
167C4×C7—42no—
168C4×C7C2×#4112yes—
169Dic7—21no—
170Dic7—21no—
171Dic7—21no—
172Dic7—41no—
173Dic7—41no—
174#41—284no—
175C2×Q8—21no—
176C2×Q8C22×D711yescharacteristic
177C2×D4—72no—
178C2×D4—142no—
179C2×D4—141no—
180C2×D4—141no—
181C2×D4—141no—
182C2×D4C22×D711yescharacteristic
183C22×C4—71no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—142no—
187C22×C4—141no—
188SD16—144no—
189D8—144no—
190C2×C8—72no—
191C2×C8—144no—
192C2×C8C22×D712yes—
193C4⋊C4—21no—
194C4⋊C4—71no—
195C4⋊C4—71no—
196C4⋊C4—141no—
197C4⋊C4—141no—
198C4⋊C4—141no—
199C4⋊C4—141no—
200C4⋊C4C22×D712yes—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—142no—
204C22⋊C4—142no—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C42—71no—
209C42—71no—
210C42—71no—
211C42—141no—
212C42C22×D711yescharacteristic
213C2×C7—81no—
214C2×C7#53211yescharacteristic
215C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C7#57511yescharacteristic
217D7—22no—
218D7—82no—
219C23—21no—
220C23—71no—
221C23—142no—
222Q8—22no—
223Q8—42no—
224D4—22no—
225D4—142no—
226D4—142no—
227D4—282no—
228D4—282no—
229D4—284no—
230D4—282no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—71no—
234C2×C4—71no—
235C2×C4—71no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—141no—
239C2×C4—141no—
240C2×C4—281no—
241C2×C4—284no—
242C2×C4—282no—
243C2×C4D4×D711yescharacteristic
244C2×C4D4×D711yescharacteristic
245C2×C4C23×D711yescharacteristic
246C8—22no—
247C8—74no—
248C7#318011yescharacteristic, complemented, Sylow
249C22—41no—
250C22—41no—
251C22—41no—
252C22—72no—
253C22—141no—
254C22—141no—
255C22—282no—
256C22—282no—
257C22—282no—
258C22C2×D4×D711yescharacteristic
259C4—21no—
260C4—21no—
261C4—42no—
262C4—41no—
263C4—141no—
264C4—141no—
265C4—141no—
266C4—281no—
267C4—281no—
268C4D7×#4112yes—
269C2—81no—
270C2—142no—
271C2—562no—
272C2#9035911yescharacteristic
273C2D7×AGL(1,ℤ/8)11yescharacteristic
274C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
hypercenter#489D7
hypocenterC7#3180
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90371C2semidirect
3. C7×#489C2semidirect
4. #89852C2semidirect
5. #89830C2semidirect
6. #89589C2semidirect
7. #89579C2semidirect
8. #89557C2semidirect
9. #89547C2semidirect
10. #89485C2semidirect
11. D7×#97C2semidirect
88. #489D7semidirect
122. C7⋊C8#41semidirect
248. C7#3180semidirect

Maximal subgroups

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

subgroupindex
#904162
#903712
C7×#4892
#898522
#898302
#895892
#895792
#895572
#895472
#894852
D7×#972
#31807