Finite Group Databasebeta

← #11736619

Subgroups — #11736619

Order 896.

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

normal subgroups only

This group has 418 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
1#90216C211yesmaximal, characteristic
2#90214C211yesmaximal, characteristic, complemented
3C7×#427C211yesmaximal, characteristic, complemented
4C2×#6395C211yesmaximal, characteristic, complemented
5#89882C211yesmaximal, characteristic, complemented
6#89773C211yesmaximal, characteristic, complemented
7#89767C211yesmaximal, characteristic, complemented
8#89657C212yesmaximal, complemented
9#89654C212yesmaximal, complemented
10#89638C212yesmaximal, complemented
11#89635C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C22×C7×C8C2211yescharacteristic
15C7×#117C2211yescharacteristic
16C7×#117C2211yescharacteristic
17C2×C4⋊Dic7C2211yescharacteristic
18C2×#1136C2211yescharacteristic
19#6458C2212yes—
20#6458C2212yes—
21#6455C2212yescomplemented
22D7×C4⋊C4C2212yescomplemented
23#6445—22no—
24(C22⋊D7)⋊C4—22no—
25#6438—21no—
26#6437—21no—
27C4×D28—21no—
28C4×Dic14—21no—
29C7×#107C2211yescharacteristic, complemented
30C7×#107C2211yescharacteristic, complemented
31C7×#106C2211yescharacteristic, complemented
32C7×#106C2211yescharacteristic, complemented
33#6395C2212yes—
34#6395C2212yes—
35(C2×D7)⋊C8C2212yescomplemented
36(C2×D7)⋊C8C2212yescomplemented
37#6383C2212yes—
38#6383C2212yes—
39#6382C2212yescomplemented
40#6381C2212yescomplemented
41#3096—71nomaximal, Sylow
42C23×D7—21no—
43C22×C4×C7C2311yescharacteristic
44C2×C22⋊D7D412yes—
45C22×Dic7—21no—
46#1157—22no—
47#1157—22no—
48C2×D28—22no—
49C2×D28—22no—
50C2×D28—21no—
51C2×D28—22no—
52C2×D28—22no—
53C2×D28D411yescharacteristic
54C2×D28C2312yes—
55C2×C4×D7—22no—
56C2×C4×D7C2312yes—
57C2×Dic14D411yescharacteristic
58C2×C7×C8—22no—
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2311yescharacteristic
61C7×C4⋊C4C2311yescharacteristic
62C7×C4⋊C4C2311yescharacteristic
63C7×C4⋊C4C2311yescharacteristic
64C7×C4⋊C4C2311yescharacteristic
65C7×C22⋊C4—21no—
66C7×C22⋊C4—21no—
67C42×C7—21no—
68C42×C7—21no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72C4⋊Dic7—21no—
73C4⋊Dic7C2312yes—
74Dic7⋊C4—22no—
75C4×Dic7—22no—
76#1136—22no—
77C2×C7⋊C8C2312yes—
78#535—71no—
79#520—71no—
80#482—72no—
81#482—72no—
82#481—72no—
83#480—72no—
84#428—71no—
85#427D711yescharacteristic, complemented
86#418—71no—
87C2×#102—71no—
88#408—71no—
89C23×C7C2×D411yescharacteristic
90C22×D7—22no—
91C22×D7—42no—
92C22×D7—81no—
93C22×D7—81no—
94C22×D7C2×D412yes—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7#4112yes—
101C2×C4×C7#4112yes—
102C2×C4×C7C2411yescharacteristic
103C22⋊D7—24no—
104C2×Dic7—22no—
105C2×Dic7—81no—
106C2×Dic7C2×D412yes—
107D28—22no—
108D28—22no—
109D28—22no—
110D28—42no—
111D28—82no—
112D28—82no—
113C4×D7—22no—
114C4×D7—22no—
115C4×D7—82no—
116Dic14—22no—
117C7×C8—21no—
118C7×C8—21no—
119C7⋊C8—22no—
120C2×#41—71no—
121C22×D4—71no—
122C2×M16—71no—
123C22×C8C2×D711yescharacteristic
124#127—141no—
125#124—141no—
126#121—72no—
127#121—72no—
128#120—142no—
129C4×Q8—141no—
130C4×D4—141no—
131C4×D4—142no—
132#117—72no—
133#117C2×D711yescharacteristic
134#117C2×D711yescharacteristic
135C2×C4⋊C4—71no—
136C2×C4⋊C4—72no—
137#107—72no—
138#107C2×D711yescharacteristic, complemented
139#107C2×D711yescharacteristic, complemented
140#106—72no—
141#106C2×D711yescharacteristic, complemented
142#106C2×D711yescharacteristic, complemented
143#102—72no—
144#102—72no—
145#102—72no—
146#102—72no—
147C22⋊C8—72no—
148C22⋊C8—72no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7#12312yes—
152C22×C7C22×D411yescharacteristic
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—22no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—84no—
160C4×C7—41no—
161C4×C7—41no—
162C4×C7—41no—
163C4×C7—41no—
164C4×C7#12312yes—
165C4×C7C2×#4111yescharacteristic
166C4×C7C2×#4111yescharacteristic
167Dic7—22no—
168Dic7—42no—
169C24—141no—
170#41—142no—
171#41—142no—
172C2×Q8—71no—
173C2×D4—72no—
174C2×D4—71no—
175C2×D4—72no—
176C2×D4—142no—
177C2×D4—142no—
178C2×D4—142no—
179C2×D4—141no—
180C2×D4—142no—
181C22×C4—72no—
182C22×C4—142no—
183C22×C4—141no—
184C22×C4C22×D711yescharacteristic
185M16—142no—
186C2×C8—22no—
187C2×C8—72no—
188C2×C8C22×D711yescharacteristic
189C2×C8C22×D711yescharacteristic
190C4⋊C4—72no—
191C4⋊C4—142no—
192C4⋊C4—141no—
193C4⋊C4C22×D711yescharacteristic
194C4⋊C4C22×D711yescharacteristic
195C4⋊C4C22×D711yescharacteristic
196C4⋊C4C22×D711yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—142no—
202C42—21no—
203C42—21no—
204C42—142no—
205C2×C7—22no—
206C2×C7C2×#12311yescharacteristic
207C2×C7#57611yescharacteristic
208C2×C7#57711yescharacteristic
209D7—42no—
210D7—82no—
211C23—72no—
212C23—142no—
213C23—282no—
214C23—561no—
215C23—561no—
216C23D4×D711yescharacteristic
217Q8—142no—
218D4—144no—
219D4—142no—
220D4—142no—
221D4—142no—
222D4—282no—
223D4—562no—
224D4—562no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—72no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—562no—
234C2×C4—561no—
235C2×C4#115712yes—
236C2×C4D4×D711yescharacteristic
237C2×C4#116112yes—
238C2×C4C23×D711yescharacteristic
239C8—21no—
240C8—21no—
241C8—142no—
242C7#309611yescharacteristic, complemented, Sylow
243C22—21no—
244C22—21no—
245C22—142no—
246C22—142no—
247C22—142no—
248C22—282no—
249C22—282no—
250C22—282no—
251C22—564no—
252C22#645712yes—
253C22C2×D4×D711yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—142no—
259C4—282no—
260C4#645712yes—
261C4C2×#115711yescharacteristic
262C4C2×#116111yescharacteristic
263C2—22no—
264C2—282no—
265C2—562no—
266C2C2×#645711yescharacteristic
267C2#9046011yescharacteristic
268C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#427C2
socleC22×C7C22×D4
prime radical#90216C2
hypercenter#427D7
hypocenterC7#3096
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #90214C2semidirect
3. C7×#427C2semidirect
4. C2×#6395C2semidirect
5. #89882C2semidirect
6. #89773C2semidirect
7. #89767C2semidirect
8. #89657C2semidirect
9. #89654C2semidirect
10. #89638C2semidirect
11. #89635C2semidirect
21. #6455C22semidirect
22. D7×C4⋊C4C22semidirect
29. C7×#107C22semidirect
30. C7×#107C22semidirect
31. C7×#106C22semidirect
32. C7×#106C22semidirect
35. (C2×D7)⋊C8C22semidirect
36. (C2×D7)⋊C8C22semidirect
39. #6382C22semidirect
40. #6381C22semidirect
85. #427D7semidirect
138. #107C2×D7semidirect
139. #107C2×D7semidirect
141. #106C2×D7semidirect
142. #106C2×D7semidirect
242. C7#3096semidirect

Maximal subgroups

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

subgroupindex
#902162
#902142
C7×#4272
C2×#63952
#898822
#897732
#897672
#896572
#896542
#896382
#896352
#30967