Finite Group Databasebeta

← #11740583

Subgroups — #11740583

Order 896.

366 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C7×#529C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3#89841C211yesmaximal, characteristic, complemented
4#89837C211yesmaximal, characteristic, complemented
5#89832C211yesmaximal, characteristic, complemented
6#89827C211yesmaximal, characteristic, complemented
7#89823C212yesmaximal, complemented
8#89813C212yesmaximal
9#89802C212yesmaximal, complemented
10#89790C212yesmaximal, complemented
11C2×#6377C211yesmaximal, characteristic, complemented
12C7×#126—21no—
13C7×#125C2211yescharacteristic
14C7×#124C2211yescharacteristic
15C7×#123—21no—
16C7×C22⋊Q8C2211yescharacteristic
17C7×#121C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4—21no—
20C4×C7×D4C2211yescharacteristic
21C7×#117—21no—
22C2×C42×C7C2211yescharacteristic
23C2×#1144C2212yes—
24C2×#1142C2212yes—
25#6489—21no—
26#6487—21no—
27#6487C2212yes—
28C22×C7⋊C8C2212yes—
29#6433—21no—
30#6432C2211yescharacteristic
31#6432C2211yescharacteristic
32C4×Dic14—21no—
33C4×Dic14C2212yes—
34#6409C2212yes—
35#6406C2212yes—
36#6384C2212yes—
37#6384C2212yes—
38#6382C2212yes—
39#6381C2212yes—
40#6377C2212yes—
41#6377C2211yescharacteristic
42#6377C2211yescharacteristic
43#3181—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C2×C7×D4—21no—
46C2×C7×D4C2311yescharacteristic
47C22×C4×C7—21no—
48C22×C4×C7D411yescharacteristic
49C22×C4×C7D411yescharacteristic
50C22×C4×C7C2311yescharacteristic
51C2×Dic14—21no—
52C2×Dic14C2312yes—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2311yescharacteristic
58C7×C4⋊C4C2311yescharacteristic
59C7×C22⋊C4—21no—
60C7×C22⋊C4—21no—
61C7×C22⋊C4—21no—
62C7×C22⋊C4—21no—
63C7×C22⋊C4—21no—
64C42×C7—21no—
65C42×C7D411yescharacteristic
66C42×C7D411yescharacteristic
67C42×C7C2311yescharacteristic
68C42×C7C2311yescharacteristic
69C22⋊Dic7—22no—
70#1144—24no—
71#1142—24no—
72C4⋊Dic7—21no—
73C4⋊Dic7—21no—
74C4⋊Dic7—21no—
75C4⋊Dic7C2312yes—
76Dic7⋊C4—22no—
77Dic7⋊C4—22no—
78C4×Dic7—22no—
79C2×C7⋊C8—24no—
80C2×C7⋊C8C2312yes—
81C2×C7⋊C8C2312yes—
82#533—71no—
83#529D711yescharacteristic
84#501—71no—
85#497—71no—
86#491—71no—
87#489—71no—
88#467—72no—
89#465—72no—
90#441—72no—
91#440—72no—
92C2×#97—71no—
93C23×C7—21no—
94C23×C7C2×D411yescharacteristic
95C7×Q8—22no—
96C7×D4—22no—
97C7×D4—42no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2411yescharacteristic
111C2×Dic7—22no—
112C2×Dic7—22no—
113Dic14—22no—
114Dic14—22no—
115Dic14—42no—
116C7⋊C8—22no—
117C7⋊C8#4114yes—
118C2×Q16—72no—
119C2×SD16—72no—
120C22×C8—72no—
121C4⋊Q8—71no—
122C4⋊Q8—71no—
123#126—21no—
124#125—141no—
125#125C2×D711yescharacteristic
126#124C2×D711yescharacteristic
127#123—21no—
128C22⋊Q8—72no—
129C22⋊Q8—141no—
130C22⋊Q8C2×D711yescharacteristic
131#121C2×D711yescharacteristic
132C4×Q8—72no—
133C4×Q8—141no—
134C4×Q8C2×D711yescharacteristic
135C4×D4—21no—
136C4×D4C2×D711yescharacteristic
137#117—21no—
138#117—141no—
139C2×C42C2×D711yescharacteristic
140#107—72no—
141#106—72no—
142#103—72no—
143#103—72no—
144#103—72no—
145#102—72no—
146#97—72no—
147#97—71no—
148#97—71no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7#13712yes—
155C22×C7C22×D411yescharacteristic
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—21no—
161C4×C7—41no—
162C4×C7—41no—
163C4×C7—41no—
164C4×C7C2×#4112yes—
165Dic7—42no—
166Dic7—42no—
167C2×Q8—72no—
168C2×Q8—141no—
169C2×Q8C22×D711yescharacteristic
170C2×D4—21no—
171C2×D4C22×D711yescharacteristic
172C22×C4—21no—
173C22×C4C22⋊D711yescharacteristic
174C22×C4C22⋊D711yescharacteristic
175C22×C4C22×D711yescharacteristic
176Q16—144no—
177SD16—144no—
178C2×C8—72no—
179C2×C8—72no—
180C2×C8—144no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—72no—
186C4⋊C4—142no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4C22×D711yescharacteristic
192C4⋊C4C22×D711yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—142no—
199C42—21no—
200C42—142no—
201C42C22⋊D711yescharacteristic
202C42C22⋊D711yescharacteristic
203C42C22×D711yescharacteristic
204C42C22×D711yescharacteristic
205C2×C7—22no—
206C2×C7—81no—
207C2×C7#53211yescharacteristic
208C2×C7C2×#13711yescharacteristic
209C2×C7#57511yescharacteristic
210C23—21no—
211C23C2×C22⋊D711yescharacteristic
212Q8—22no—
213Q8—142no—
214Q8—142no—
215Q8—282no—
216D4—22no—
217D4—42no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4C2×C22⋊D711yescharacteristic
228C2×C4C2×C22⋊D711yescharacteristic
229C2×C4C2×C22⋊D711yescharacteristic
230C2×C4C2×C22⋊D711yescharacteristic
231C2×C4C2×C22⋊D711yescharacteristic
232C2×C4C23×D711yescharacteristic
233C8—74no—
234C8—142no—
235C7#318111yescharacteristic, complemented, Sylow
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22#651412yes—
242C22C22×C22⋊D711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—282no—
252C4—282no—
253C4D7×#4112yes—
254C2—22no—
255C2—81no—
256C2#9051311yescharacteristic
257C2C2×#651411yescharacteristic
258C2#9052311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#529C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#529D7
hypocenterC7#3181
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. #89841C2semidirect
4. #89837C2semidirect
5. #89832C2semidirect
6. #89827C2semidirect
7. #89823C2semidirect
9. #89802C2semidirect
10. #89790C2semidirect
11. C2×#6377C2semidirect
235. C7#3181semidirect

Maximal subgroups

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

subgroupindex
C7×#5292
#901612
#898412
#898372
#898322
#898272
#898232
#898132
#898022
#897902
C2×#63772
#31817