Finite Group Databasebeta

← #11738759

Subgroups — #11738759

Order 896.

404 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 123 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#90287C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic, complemented
3C7×#481C211yesmaximal, characteristic, complemented
4#89806C211yesmaximal, characteristic, complemented
5#89740C211yesmaximal, characteristic, complemented
6#89642C212yesmaximal, complemented
7#89627C212yesmaximal, complemented
8#89533C212yesmaximal, complemented
9#89532C212yesmaximal, complemented
10#89507C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C22×Dic14—21no—
14C7×#121C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16#6503C2211yescharacteristic
17#6502—21no—
18D4×Dic7C2211yescharacteristic
19#6493C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4—21no—
22C2×C4×Dic7C2211yescharacteristic
23#6456C2211yescharacteristic
24#6451—21no—
25#6451—21no—
26#6451C2211yescharacteristic
27#6451C2212yes—
28#6451C2212yes—
29#6448—21no—
30(C22⋊D7)⋊C4—21no—
31C7×#106C2212yes—
32C7×#102C2212yes—
33C7×C22⋊C8C2211yescharacteristic
34#6406C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36#6395C2212yes—
37#6391C2212yes—
38Dic7⋊C8C2212yes—
39C8×Dic7C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42#6382C2212yes—
43#3227—71nomaximal, Sylow
44C2×C7×D4—21no—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—21no—
49C2×C22⋊D7—21no—
50C2×C22⋊D7—21no—
51C22×Dic7—21no—
52C22×Dic7—21no—
53C22×Dic7D411yescharacteristic
54C22×Dic7D411yescharacteristic
55#1159—44no—
56C2×D28C2311yescharacteristic
57C2×C4×D7—21no—
58C2×Dic14—21no—
59C2×Dic14—22no—
60C2×Dic14—22no—
61C2×Dic14—44no—
62C2×C7×C8C2312yes—
63C7×C4⋊C4—21no—
64C7×C4⋊C4C2312yes—
65C7×C4⋊C4C2311yescharacteristic
66C7×C22⋊C4—21no—
67C22⋊Dic7—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7C2312yes—
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—21no—
74Dic7⋊C4—22no—
75Dic7⋊C4—22no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81#532—71no—
82C2×C4⋊Q8—71no—
83#498—72no—
84#486—72no—
85#481—71no—
86#481—71no—
87#481—71no—
88#481D711yescharacteristic, complemented
89#476—72no—
90#476—72no—
91#431—71no—
92C23×C7—21no—
93C23×C7C2×D411yescharacteristic
94C22×D7—21no—
95C7×D4—22no—
96C7×D4—42no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—21no—
100C2×C4×C7—42no—
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2411yescharacteristic
103C22⋊D7—42no—
104C22⋊D7—42no—
105C22⋊D7—42no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7SD1614yes—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116D28—22no—
117C4×D7—42no—
118Dic14—44no—
119Dic14—42no—
120Dic14—44no—
121C7×C8—22no—
122C7⋊C8—22no—
123C2×#41—141no—
124C22×Q8—141no—
125C4⋊Q8—72no—
126C4⋊Q8—72no—
127C4⋊Q8—71no—
128C4⋊Q8—141no—
129C4⋊Q8—141no—
130#127—71no—
131#124—141no—
132#121—71no—
133#121—141no—
134#121C2×D711yescharacteristic
135C4×D4—71no—
136C4×D4—71no—
137C4×D4—141no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4—141no—
140C2×C4⋊C4C2×D711yescharacteristic
141C2×C42—71no—
142#106—72no—
143#106—72no—
144#106—72no—
145#106C2×D712yes—
146C4⋊C8—72no—
147#102—72no—
148#102—72no—
149#102—72no—
150#102C2×D712yes—
151C22⋊C8—71no—
152C22⋊C8C2×D711yescharacteristic
153C4×C8—72no—
154C22×C7—21no—
155C22×C7—21no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7C2×SD1612yes—
160C22×C7C22×D411yescharacteristic
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—41no—
164C4×C7—21no—
165C4×C7—41no—
166C4×C7—42no—
167C4×C72−1+412yes—
168Dic7—21no—
169Dic7—41no—
170Dic7—42no—
171Dic7C2×SD1612yes—
172Dic7C2×SD1612yes—
173Dic7AGL(1,ℤ/8)12yes—
174#41—284no—
175C2×Q8—141no—
176C2×Q8—142no—
177C2×Q8—142no—
178C2×Q8—284no—
179C2×D4—21no—
180C2×D4—71no—
181C2×D4—141no—
182C2×D4—141no—
183C2×D4—141no—
184C2×D4C22×D711yescharacteristic
185C22×C4—21no—
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—141no—
189C22×C4—141no—
190C22×C4—141no—
191C22×C4C22×D711yescharacteristic
192C2×C8—72no—
193C2×C8C22×D712yes—
194C4⋊C4—21no—
195C4⋊C4—72no—
196C4⋊C4—71no—
197C4⋊C4—142no—
198C4⋊C4—142no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4C22×D712yes—
202C4⋊C4C22×D711yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—141no—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C42—71no—
208C42—71no—
209C42—71no—
210C42—71no—
211C2×C7—22no—
212C2×C7—81no—
213C2×C7#53611yescharacteristic
214C2×C7C22×SD1611yescharacteristic
215C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
216D7—81no—
217C23—21no—
218C23—141no—
219C23D4×D711yescharacteristic
220Q8—284no—
221Q8—282no—
222Q8—284no—
223D4—22no—
224D4—42no—
225D4—142no—
226D4—282no—
227D4—282no—
228D4—282no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—74no—
234C2×C4—71no—
235C2×C4—71no—
236C2×C4—71no—
237C2×C4—71no—
238C2×C4—142no—
239C2×C4—141no—
240C2×C4—142no—
241C2×C4—282no—
242C2×C4—282no—
243C2×C4—282no—
244C2×C4D4×D711yescharacteristic
245C2×C4C23×D711yescharacteristic
246C8—22no—
247C8—142no—
248C7#322711yescharacteristic, complemented, Sylow
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—281no—
255C22—281no—
256C22—281no—
257C22D7×SD1612yes—
258C22C2×D4×D711yescharacteristic
259C4—21no—
260C4—41no—
261C4—42no—
262C4—72no—
263C4—72no—
264C4—72no—
265C4—141no—
266C4—281no—
267C4—282no—
268C4#655412yes—
269C2—22no—
270C2—81no—
271C2—561no—
272C2#9033911yescharacteristic
273C2C2×D7×SD1611yescharacteristic
274C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#481C2
socleC22×C7C22×D4
prime radical#90287C2
hypercenter#481D7
hypocenterC7#3227
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#6451C2semidirect
3. C7×#481C2semidirect
4. #89806C2semidirect
5. #89740C2semidirect
6. #89642C2semidirect
7. #89627C2semidirect
8. #89533C2semidirect
9. #89532C2semidirect
10. #89507C2semidirect
11. #89490C2semidirect
88. #481D7semidirect
248. C7#3227semidirect

Maximal subgroups

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

subgroupindex
#902872
C2×#64512
C7×#4812
#898062
#897402
#896422
#896272
#895332
#895322
#895072
#894902
#32277