Finite Group Databasebeta

← #11736686

Subgroups — #11736686

Order 896.

384 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 133 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#90213C211yesmaximal, characteristic
2C2×#6453C211yesmaximal, characteristic, complemented
3C7×#428C211yesmaximal, characteristic, complemented
4#89905C211yesmaximal, characteristic
5#89890C211yesmaximal, characteristic, complemented
6#89767C211yesmaximal, characteristic, complemented
7#89738C211yesmaximal, characteristic
8#89663C212yesmaximal, complemented
9#89649C212yesmaximal, complemented
10#89643C212yesmaximal, complemented
11#89629C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic7—21no—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4—21no—
19C2×C4×Dic7C2211yescharacteristic
20C2×#1136C2211yescharacteristic
21#6456C2211yescharacteristic
22#6456C2211yescharacteristic
23#6453—21no—
24#6453—21no—
25#6453C2212yes—
26#6453C2212yes—
27#6450C2211yescharacteristic
28#6450C2211yescharacteristic
29(C22⋊D7)⋊C4—21no—
30(C22⋊D7)⋊C4—21no—
31#6435—21no—
32C42×D7—21no—
33C7×#107C2212yes—
34C7×#106C2212yes—
35#6395C2212yes—
36#6390C2212yes—
37Dic7⋊C8C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42#6381C2212yes—
43#3080—71nomaximal, Sylow
44C22×C4×C7—21no—
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—21no—
47C22×Dic7—21no—
48C22×Dic7D411yescharacteristic
49C22×Dic7D411yescharacteristic
50#1157—22no—
51#1157Q814yes—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7—21no—
55C2×C4×D7—21no—
56C2×Dic14C2311yescharacteristic
57C7×M16—22no—
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2311yescharacteristic
61C7×C4⋊C4C2312yes—
62C7×C4⋊C4C2311yescharacteristic
63C7×C22⋊C4—21no—
64C42×C7—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7—22no—
70C4⋊Dic7C2312yes—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
76C4×Dic7—21no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C4×Dic7C2311yescharacteristic
81#1136—22no—
82C2×C7⋊C8C2312yes—
83C2×#125—71no—
84C4×#41—71no—
85#479—72no—
86#479—72no—
87#478—72no—
88#478—72no—
89#428—71no—
90#428D711yescharacteristic, complemented
91#423—71no—
92#417—71no—
93#417—71no—
94C23×C7C2×D411yescharacteristic
95C22×D7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—42no—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×Q812yes—
102C2×C4×C7#4112yes—
103C2×C4×C7C2411yescharacteristic
104C22⋊D7—24no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7—41no—
108C2×Dic7—41no—
109C2×Dic7—42no—
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114D28—21no—
115D28C2×Q812yes—
116C4×D7—24no—
117C4×D7—42no—
118C4×D7—42no—
119Dic14—21no—
120Dic14C2×Q812yes—
121C7×C8—22no—
122C7⋊C8—22no—
123C2×#41—71no—
124C2×M16—71no—
125C2×M16C2×D711yescharacteristic
126#125—72no—
127#125—72no—
128#125—141no—
129#125—141no—
130C4×Q8—71no—
131C4×Q8—71no—
132C4×D4—71no—
133C4×D4—71no—
134C4×D4—141no—
135C4×D4—141no—
136#117—141no—
137#117C2×D711yescharacteristic
138C2×C4⋊C4—71no—
139C2×C4⋊C4—141no—
140C2×C4⋊C4—141no—
141C2×C4⋊C4C2×D711yescharacteristic
142C2×C42—71no—
143C2×C42—141no—
144#107—72no—
145#107C2×D712yes—
146#106—72no—
147#106C2×D712yes—
148C4⋊C8—72no—
149C4⋊C8—72no—
150#103—72no—
151#103—72no—
152#102—72no—
153#102—72no—
154C22×C7—21no—
155C22×C7—21no—
156C22×C7C22⋊Q812yes—
157C22×C7C22×D411yescharacteristic
158C2×D7—22no—
159C2×D7—41no—
160C2×D7—41no—
161C4×C7—41no—
162C4×C7—41no—
163C4×C7—42no—
164C4×C7C22⋊Q812yes—
165C4×C7C22×Q811yescharacteristic
166C4×C7C2×#4111yescharacteristic
167Dic7—22no—
168Dic7—21no—
169Dic7—21no—
170Dic7—21no—
171Dic7—21no—
172Dic7—42no—
173#41—74no—
174#41—142no—
175C2×Q8—71no—
176C2×D4—71no—
177C2×D4—141no—
178C22×C4—21no—
179C22×C4—71no—
180C22×C4—71no—
181C22×C4—141no—
182C22×C4—141no—
183C22×C4—141no—
184C22×C4—141no—
185C22×C4C22×D711yescharacteristic
186M16—22no—
187M16—142no—
188C2×C8—72no—
189C2×C8C22×D712yes—
190C4⋊C4—21no—
191C4⋊C4—72no—
192C4⋊C4—142no—
193C4⋊C4—142no—
194C4⋊C4—141no—
195C4⋊C4—141no—
196C4⋊C4—142no—
197C4⋊C4—142no—
198C4⋊C4—141no—
199C4⋊C4C22×D712yes—
200C4⋊C4C22×D711yescharacteristic
201C4⋊C4C22×D711yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—141no—
204C22⋊C4—141no—
205C42—21no—
206C42—71no—
207C42—71no—
208C42—71no—
209C42—71no—
210C42—141no—
211C42—141no—
212C2×C7—22no—
213C2×C7C2×C22⋊Q811yescharacteristic
214C2×C7#57511yescharacteristic
215C2×C7#57511yescharacteristic
216D7—42no—
217C23—141no—
218C23D4×D711yescharacteristic
219Q8—72no—
220Q8—141no—
221D4—72no—
222D4—144no—
223D4—141no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—71no—
229C2×C4—71no—
230C2×C4—71no—
231C2×C4—71no—
232C2×C4—141no—
233C2×C4—144no—
234C2×C4—142no—
235C2×C4—282no—
236C2×C4—282no—
237C2×C4—281no—
238C2×C4—281no—
239C2×C4—282no—
240C2×C4#115712yes—
241C2×C4D4×D711yescharacteristic
242C2×C4Q8×D712yes—
243C2×C4C23×D711yescharacteristic
244C8—22no—
245C8—142no—
246C7#308011yescharacteristic, complemented, Sylow
247C22—21no—
248C22—21no—
249C22—142no—
250C22—281no—
251C22—281no—
252C22#645912yes—
253C22C2×D4×D711yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—142no—
258C4—141no—
259C4—141no—
260C4—141no—
261C4—141no—
262C4—282no—
263C4#645912yes—
264C4C2×#115711yescharacteristic
265C4C2×Q8×D711yescharacteristic
266C2—22no—
267C2—282no—
268C2C2×#645911yescharacteristic
269C2#9046411yescharacteristic
270C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#428D7
hypocenterC7#3080
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
2. C2×#6453C2semidirect
3. C7×#428C2semidirect
5. #89890C2semidirect
6. #89767C2semidirect
8. #89663C2semidirect
9. #89649C2semidirect
10. #89643C2semidirect
11. #89629C2semidirect
90. #428D7semidirect
246. C7#3080semidirect

Maximal subgroups

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

subgroupindex
#902132
C2×#64532
C7×#4282
#899052
#898902
#897672
#897382
#896632
#896492
#896432
#896292
#30807