Finite Group Databasebeta

← #11738764

Subgroups — #11738764

Order 896.

368 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 113 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×#6453C211yesmaximal, characteristic, complemented
3C7×#481C211yesmaximal, characteristic, complemented
4#89805C211yesmaximal, characteristic, complemented
5#89740C211yesmaximal, characteristic, complemented
6#89643C212yesmaximal, complemented
7#89628C212yesmaximal, complemented
8#89538C212yesmaximal, complemented
9#89536C212yesmaximal, complemented
10#89508C211yesmaximal, characteristic, complemented
11#89491C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C7×#121C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15#6503C2211yescharacteristic
16#6502—21no—
17D4×Dic7C2211yescharacteristic
18#6493C2211yescharacteristic
19C2×C4⋊Dic7—21no—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4—21no—
22C2×C4×Dic7C2211yescharacteristic
23#6456C2211yescharacteristic
24#6453—21no—
25#6453—21no—
26#6453C2212yes—
27#6453C2212yes—
28#6451C2211yescharacteristic
29#6448—21no—
30(C22⋊D7)⋊C4—21no—
31C7×#106C2212yes—
32C7×#102C2212yes—
33C7×C22⋊C8C2211yescharacteristic
34#6406C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36#6395C2212yes—
37#6392C2212yes—
38#6389C2212yes—
39Dic7⋊C8C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42#6381C2212yes—
43#3234—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×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2312yes—
62C7×C4⋊C4C2311yescharacteristic
63C7×C22⋊C4—21no—
64C22⋊Dic7—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7—22no—
70C4⋊Dic7C2312yes—
71C4⋊Dic7C2311yescharacteristic
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#532—71no—
81C2×#125—71no—
82#501—72no—
83#490—72no—
84#481—71no—
85#481D711yescharacteristic, complemented
86#480—71no—
87#480—71no—
88#478—72no—
89#478—72no—
90#432—71no—
91C23×C7—21no—
92C23×C7C2×D411yescharacteristic
93C22×D7—21no—
94C7×D4—22no—
95C7×D4—42no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—42no—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—42no—
103C22⋊D7—42no—
104C22⋊D7—42no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7—41no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115D28—22no—
116C4×D7—42no—
117Dic14—42no—
118C7×C8—22no—
119C7⋊C8—22no—
120C2×#41—141no—
121C4⋊Q8—71no—
122#127—71no—
123#125—72no—
124#125—72no—
125#125—141no—
126#125—141no—
127#124—141no—
128#121—71no—
129#121—141no—
130#121C2×D711yescharacteristic
131C4×D4—71no—
132C4×D4—71no—
133C4×D4—141no—
134C2×C4⋊C4—71no—
135C2×C4⋊C4—141no—
136C2×C4⋊C4—141no—
137C2×C4⋊C4C2×D711yescharacteristic
138C2×C42—71no—
139#107—72no—
140#107—72no—
141#106—72no—
142#106C2×D712yes—
143C4⋊C8—72no—
144#102—72no—
145#102—72no—
146#102—72no—
147#102C2×D712yes—
148C22⋊C8—71no—
149C22⋊C8C2×D711yescharacteristic
150#97—72no—
151C22×C7—21no—
152C22×C7—21no—
153C22×C7—41no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7AGL(1,ℤ/8)12yes—
157C22×C7C22×D411yescharacteristic
158C2×D7—41no—
159C2×D7—41no—
160C2×D7—41no—
161C4×C7—21no—
162C4×C7—41no—
163C4×C7—42no—
164C4×C72−1+412yes—
165Dic7—21no—
166Dic7—21no—
167Dic7—21no—
168Dic7—21no—
169Dic7—41no—
170Dic7—42no—
171#41—284no—
172C2×Q8—141no—
173C2×D4—21no—
174C2×D4—71no—
175C2×D4—141no—
176C2×D4—141no—
177C2×D4—141no—
178C2×D4C22×D711yescharacteristic
179C22×C4—21no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4—141no—
183C22×C4—141no—
184C22×C4—141no—
185C22×C4C22×D711yescharacteristic
186C2×C8—72no—
187C2×C8C22×D712yes—
188C4⋊C4—21no—
189C4⋊C4—72no—
190C4⋊C4—71no—
191C4⋊C4—142no—
192C4⋊C4—142no—
193C4⋊C4—141no—
194C4⋊C4—142no—
195C4⋊C4—142no—
196C4⋊C4—141no—
197C4⋊C4C22×D712yes—
198C4⋊C4C22×D711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—141no—
201C22⋊C4—141no—
202C22⋊C4—141no—
203C42—71no—
204C42—71no—
205C42—71no—
206C42—71no—
207C2×C7—22no—
208C2×C7—81no—
209C2×C7#53611yescharacteristic
210C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
211C2×C7#57511yescharacteristic
212D7—81no—
213C23—21no—
214C23—141no—
215C23D4×D711yescharacteristic
216Q8—282no—
217D4—22no—
218D4—42no—
219D4—142no—
220D4—282no—
221D4—282no—
222D4—282no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—71no—
228C2×C4—71no—
229C2×C4—71no—
230C2×C4—71no—
231C2×C4—142no—
232C2×C4—141no—
233C2×C4—142no—
234C2×C4—282no—
235C2×C4—281no—
236C2×C4—282no—
237C2×C4—282no—
238C2×C4D4×D711yescharacteristic
239C2×C4C23×D711yescharacteristic
240C8—22no—
241C8—142no—
242C7#323411yescharacteristic, complemented, Sylow
243C22—21no—
244C22—21no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—281no—
249C22—281no—
250C22—281no—
251C22#647712yes—
252C22C2×D4×D711yescharacteristic
253C4—21no—
254C4—41no—
255C4—42no—
256C4—141no—
257C4—141no—
258C4—141no—
259C4—141no—
260C4—281no—
261C4—282no—
262C4#655412yes—
263C2—22no—
264C2—81no—
265C2—561no—
266C2#9033911yescharacteristic
267C2C2×#647711yescharacteristic
268C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#481C2
socleC22×C7C22×D4
prime radical#90287C2
hypercenter#481D7
hypocenterC7#3234
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×#6453C2semidirect
3. C7×#481C2semidirect
4. #89805C2semidirect
5. #89740C2semidirect
6. #89643C2semidirect
7. #89628C2semidirect
8. #89538C2semidirect
9. #89536C2semidirect
10. #89508C2semidirect
11. #89491C2semidirect
85. #481D7semidirect
242. C7#3234semidirect

Maximal subgroups

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

subgroupindex
#902872
C2×#64532
C7×#4812
#898052
#897402
#896432
#896282
#895382
#895362
#895082
#894912
#32347