Finite Group Databasebeta

← #11736690

Subgroups — #11736690

Order 896.

376 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 135 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#90208C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic
4#89892C211yesmaximal, characteristic
5#89890C211yesmaximal, characteristic
6#89776C211yesmaximal, characteristic
7#89736C211yesmaximal, characteristic
8#89649C212yesmaximal, complemented
9#89647C212yesmaximal, complemented
10#89629C212yesmaximal, complemented
11#89626C212yesmaximal, complemented
12C22×Dic14C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16#6489C2211yescharacteristic
17C2×Dic7⋊C4—21no—
18C2×C4×Dic7—21no—
19C2×C4×Dic7C2211yescharacteristic
20C2×#1136C2211yescharacteristic
21#6453C2211yescharacteristic
22#6453C2211yescharacteristic
23#6451C2211yescharacteristic
24#6451C2211yescharacteristic
25#6450—21no—
26#6450—21no—
27#6450C2212yes—
28#6450C2212yes—
29#6441—21no—
30#6441—21no—
31C4×Dic14—21no—
32C4×Dic14—21no—
33C7×#107C2212yes—
34C7×#106C2212yes—
35#6390C2212yes—
36#6390C2212yes—
37Dic7⋊C8C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6384C2212yes—
41#6382C2212yes—
42#6381C2212yes—
43#3078—71nomaximal, Sylow
44C22×C4×C7—21no—
45C22×C4×C7C2311yescharacteristic
46C22×Dic7—21no—
47C22×Dic7D411yescharacteristic
48C22×Dic7D411yescharacteristic
49C2×Dic14—21no—
50C2×Dic14—22no—
51C2×Dic14—21no—
52C2×Dic14—21no—
53C2×Dic14Q814yescomplemented
54C2×Dic14C2312yes—
55C7×M16—22no—
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4C2311yescharacteristic
59C7×C4⋊C4C2312yes—
60C7×C4⋊C4C2311yescharacteristic
61C7×C22⋊C4—21no—
62C42×C7—21no—
63C22⋊Dic7—21no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7C2311yescharacteristic
67C4⋊Dic7C2311yescharacteristic
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74C4×Dic7—21no—
75C4×Dic7—22no—
76C4×Dic7—22no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C4×Dic7C2311yescharacteristic
81#1136—22no—
82C2×C7⋊C8C2312yes—
83#533—71no—
84C2×C4×Q8—71no—
85#479—72no—
86#479—72no—
87#477—72no—
88#475—72no—
89#428—71no—
90#428D711yescharacteristic
91#423—71no—
92#419—71no—
93#419—71no—
94C23×C7C2×D411yescharacteristic
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—42no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×Q812yes—
101C2×C4×C7#4112yes—
102C2×C4×C7C2411yescharacteristic
103C2×Dic7—24no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—21no—
108C2×Dic7—41no—
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113Dic14—24no—
114Dic14—22no—
115Dic14—42no—
116Dic14—42no—
117Dic14C2×Q814yes—
118C7×C8—22no—
119C7⋊C8—22no—
120C22×Q8—71no—
121C2×M16—71no—
122C2×M16C2×D711yescharacteristic
123C4⋊Q8—71no—
124C4⋊Q8—71no—
125#125—71no—
126#125—71no—
127C22⋊Q8—141no—
128C22⋊Q8—141no—
129C4×Q8—72no—
130C4×Q8—72no—
131C4×Q8—141no—
132C4×Q8—141no—
133C4×Q8—141no—
134C4×Q8—141no—
135#117—71no—
136#117C2×D711yescharacteristic
137C2×C4⋊C4—141no—
138C2×C4⋊C4C2×D711yescharacteristic
139C2×C42—71no—
140C2×C42—141no—
141#107—72no—
142#107C2×D712yes—
143#106—72no—
144#106C2×D712yes—
145C4⋊C8—72no—
146C4⋊C8—72no—
147#103—72no—
148#103—72no—
149#103—72no—
150#103—72no—
151C22×C7—21no—
152C22×C7—21no—
153C22×C7C22⋊Q812yes—
154C22×C7C22×D411yescharacteristic
155C4×C7—41no—
156C4×C7—41no—
157C4×C7—42no—
158C4×C7C22⋊Q812yes—
159C4×C7C22×Q811yescharacteristic
160C4×C7C2×#4111yescharacteristic
161Dic7—24no—
162Dic7—21no—
163Dic7—21no—
164Dic7—21no—
165Dic7—41no—
166Dic7—41no—
167Dic7#13712yes—
168C2×Q8—74no—
169C2×Q8—72no—
170C2×Q8—141no—
171C2×Q8—141no—
172C2×Q8—142no—
173C2×Q8—141no—
174C22×C4—21no—
175C22×C4—71no—
176C22×C4—71no—
177C22×C4—141no—
178C22×C4C22×D711yescharacteristic
179M16—22no—
180M16—142no—
181C2×C8—72no—
182C2×C8C22×D712yes—
183C4⋊C4—21no—
184C4⋊C4—71no—
185C4⋊C4—71no—
186C4⋊C4—142no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4—141no—
193C4⋊C4—141no—
194C4⋊C4C22×D711yescharacteristic
195C4⋊C4C22×D711yescharacteristic
196C4⋊C4C22×D712yes—
197C22⋊C4—21no—
198C22⋊C4—141no—
199C42—21no—
200C42—71no—
201C42—71no—
202C42—71no—
203C42—71no—
204C42—142no—
205C42—142no—
206C42—141no—
207C2×C7—22no—
208C2×C7C2×C22⋊Q811yescharacteristic
209C2×C7C2×#13711yescharacteristic
210C2×C7#57511yescharacteristic
211C23D4×D711yescharacteristic
212Q8—74no—
213Q8—144no—
214Q8—142no—
215Q8—282no—
216Q8—282no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—71no—
222C2×C4—71no—
223C2×C4—71no—
224C2×C4—71no—
225C2×C4—142no—
226C2×C4—144no—
227C2×C4—142no—
228C2×C4—141no—
229C2×C4—141no—
230C2×C4—281no—
231C2×C4#115712yes—
232C2×C4D4×D711yescharacteristic
233C2×C4Q8×D712yes—
234C2×C4C23×D711yescharacteristic
235C8—22no—
236C8—142no—
237C7#307811yescharacteristic, complemented, Sylow
238C22—21no—
239C22—21no—
240C22#645912yes—
241C22C2×D4×D711yescharacteristic
242C4—42no—
243C4—41no—
244C4—41no—
245C4—72no—
246C4—144no—
247C4—141no—
248C4—141no—
249C4—141no—
250C4—281no—
251C4—281no—
252C4#645912yes—
253C4C2×#115711yescharacteristic
254C4C2×Q8×D711yescharacteristic
255C2—22no—
256C2C2×#645911yescharacteristic
257C2#9046411yescharacteristic
258C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#428D7
hypocenterC7#3078
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
8. #89649C2semidirect
9. #89647C2semidirect
10. #89629C2semidirect
11. #89626C2semidirect
53. C2×Dic14Q8semidirect
237. C7#3078semidirect

Maximal subgroups

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

subgroupindex
#902082
C2×#64502
C7×#4282
#898922
#898902
#897762
#897362
#896492
#896472
#896292
#896262
#30787