Finite Group Databasebeta

← #11737833

Subgroups — #11737833

Order 896.

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

normal subgroups only

This group has 400 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#90407C211yesmaximal, characteristic, complemented
2#90271C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic, complemented
4#89945C212yesmaximal, complemented
5#89940C212yesmaximal, complemented
6#89844C211yesmaximal, characteristic, complemented
7#89803C211yesmaximal, characteristic, complemented
8#89613C211yesmaximal, characteristic, complemented
9#89605C211yesmaximal, characteristic, complemented
10#89553C212yesmaximal, complemented
11#89540C212yesmaximal, complemented
12C2×#1159C2212yes—
13C2×C7×SD16C2212yes—
14C7×#127C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6510C2211yescharacteristic
17#6509C2211yescharacteristic
18C2×#1144C2212yes—
19#6501—21no—
20#6501—21no—
21#6499C2212yes—
22D4×Dic7—21no—
23D4×Dic7—21no—
24C2×#1141C2212yes—
25C2×#1132C2212yes—
26#6461—21no—
27#6458C2211yescharacteristic
28#6457—21no—
29#6453C2211yescharacteristic
30C4×D28—21no—
31C4×D28C2211yescharacteristic
32#6435—21no—
33C42×D7C2211yescharacteristic
34#6432C2211yescharacteristic
35C7×C4⋊C8C2211yescharacteristic
36C7×#102C2212yes—
37#6406C2212yes—
38(C2×D7)⋊C8C2212yes—
39#6390C2212yes—
40C8×Dic7C2212yes—
41#6384C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3212—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C2×C7×D4—21no—
46C2×C7×D4—21no—
47C2×C7×D4C2312yes—
48C2×C22⋊D7—22no—
49C22×Dic7—22no—
50#1159—42no—
51#1159—42no—
52C2×D28—21no—
53C2×D28C2311yescharacteristic
54C2×C4×D7—21no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×Dic14C2312yes—
59C7×SD16—42no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4—21no—
62C7×C4⋊C4C2311yescharacteristic
63C42×C7—21no—
64C42×C7C2311yescharacteristic
65C22⋊Dic7—22no—
66C22⋊Dic7—22no—
67#1144—42no—
68#1141—42no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74C4⋊Dic7—21no—
75C4⋊Dic7—21no—
76C4⋊Dic7—21no—
77C4⋊Dic7C2311yescharacteristic
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84#1132—42no—
85#532—71no—
86#529—71no—
87#495—72no—
88#488—72no—
89#462—71no—
90#461—71no—
91#460D711yescharacteristic, complemented
92#459—71no—
93#452—72no—
94#449—72no—
95#431—71no—
96C23×C7—22no—
97C22×D7—21no—
98C22×D7C2×D411yescharacteristic
99C7×Q8—21no—
100C7×Q8—21no—
101C7×D4—22no—
102C7×D4—22no—
103C7×D4—42no—
104C7×D4—42no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2411yescharacteristic
110C22⋊D7—44no—
111C2×Dic7—21no—
112C2×Dic7—22no—
113C2×Dic7—44no—
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116C2×Dic7C2×D411yescharacteristic
117D28—21no—
118D28—21no—
119D28—42no—
120C4×D7—22no—
121C4×D7—21no—
122C4×D7—21no—
123C4×D7—41no—
124C4×D7—42no—
125Dic14—22no—
126Dic14—22no—
127C7×C8—22no—
128C7⋊C8—22no—
129C2×#41—72no—
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16C2×D712yes—
133C2×D8—72no—
134C4⋊Q8—71no—
135#127C2×D711yescharacteristic
136#126—141no—
137#125—71no—
138#124—71no—
139#124—72no—
140#123—141no—
141C22⋊Q8—71no—
142#121—71no—
143#121—141no—
144#121—141no—
145C4×Q8C2×D711yescharacteristic
146C4×D4—71no—
147C4×D4—141no—
148C4×D4—141no—
149C4×D4—141no—
150#117—141no—
151C2×C42—71no—
152C4⋊C8—71no—
153C4⋊C8C2×D711yescharacteristic
154#103—72no—
155#103—72no—
156#102—72no—
157#102C2×D712yes—
158C22⋊C8—72no—
159C4×C8—72no—
160C22×C7—42no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7C22×D411yescharacteristic
164C2×D7—21no—
165C2×D7—21no—
166C2×D7—21no—
167C2×D7—41no—
168C2×D7—41no—
169C2×D7—41no—
170C4×C7—21no—
171C4×C7—41no—
172C4×C7—41no—
173C4×C7#13512yes—
174C4×C7H(5,𝔽2)11yescharacteristic
175C4×C7H(5,𝔽2)11yescharacteristic
176Dic7—21no—
177Dic7—21no—
178Dic7—21no—
179Dic7—42no—
180Dic7—41no—
181#41—282no—
182#41—282no—
183C2×Q8—72no—
184C2×Q8C22×D711yescharacteristic
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—71no—
188C2×D4—141no—
189C2×D4—142no—
190C2×D4C22×D712yes—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—142no—
195C22×C4—141no—
196Q16—282no—
197SD16—42no—
198SD16—282no—
199D8—282no—
200C2×C8—72no—
201C2×C8C22×D712yes—
202C4⋊C4—21no—
203C4⋊C4—71no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4—141no—
207C4⋊C4—141no—
208C4⋊C4—141no—
209C4⋊C4C22×D711yescharacteristic
210C22⋊C4—141no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C22⋊C4—141no—
214C22⋊C4—142no—
215C22⋊C4—142no—
216C22⋊C4—141no—
217C42—21no—
218C42—71no—
219C42—71no—
220C42—71no—
221C42C22×D711yescharacteristic
222C2×C7—82no—
223C2×C7#53511yescharacteristic
224C2×C7C2×#13511yescharacteristic
225C2×C7#57511yescharacteristic
226D7—41no—
227D7—81no—
228C23—22no—
229C23—71no—
230C23—141no—
231Q8—21no—
232Q8—21no—
233Q8—142no—
234Q8—142no—
235D4—22no—
236D4—22no—
237D4—42no—
238D4—42no—
239D4—141no—
240D4—141no—
241D4—282no—
242D4—284no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—71no—
246C2×C4—71no—
247C2×C4—71no—
248C2×C4—142no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—142no—
253C2×C4—281no—
254C2×C4—284no—
255C2×C4—282no—
256C2×C4D4×D711yescharacteristic
257C2×C4D4×D711yescharacteristic
258C2×C4C23×D711yescharacteristic
259C8—22no—
260C8—142no—
261C7#321211yescharacteristic, complemented, Sylow
262C22—42no—
263C22—42no—
264C22—42no—
265C22—141no—
266C22—141no—
267C22—141no—
268C22—281no—
269C22—281no—
270C22—281no—
271C22C2×D4×D711yescharacteristic
272C4—21no—
273C4—41no—
274C4—41no—
275C4—141no—
276C4—141no—
277C4—141no—
278C4—281no—
279C4—282no—
280C4#647912yes—
281C4#654811yescharacteristic
282C4#655311yescharacteristic
283C2—82no—
284C2—281no—
285C2—561no—
286C2#9029811yescharacteristic
287C2C2×#647911yescharacteristic
288C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
hypercenter#460D7
hypocenterC7#3212
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #90407C2semidirect
2. #90271C2semidirect
3. C7×#460C2semidirect
4. #89945C2semidirect
5. #89940C2semidirect
6. #89844C2semidirect
7. #89803C2semidirect
8. #89613C2semidirect
9. #89605C2semidirect
10. #89553C2semidirect
11. #89540C2semidirect
91. #460D7semidirect
261. C7#3212semidirect

Maximal subgroups

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

subgroupindex
#904072
#902712
C7×#4602
#899452
#899402
#898442
#898032
#896132
#896052
#895532
#895402
#32127