Finite Group Databasebeta

← #11739430

Subgroups — #11739430

Order 896.

328 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 109 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#90419C211yesmaximal, characteristic, complemented
2#90382C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89837C211yesmaximal, characteristic, complemented
6#89657C212yesmaximal
7#89649C212yesmaximal, complemented
8#89643C212yesmaximal, complemented
9#89639C212yesmaximal, complemented
10#89486C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C7×#125C2211yescharacteristic
14#6509—21no—
15Q8×Dic7—21no—
16#6461—21no—
17#6460—21no—
18#6460C2211yescharacteristic
19#6458C2211yescharacteristic
20#6457—21no—
21#6456—21no—
22#6456C2211yescharacteristic
23#6455—21no—
24#6455C2212yes—
25#6453C2212yes—
26#6450—21no—
27#6450C2211yescharacteristic
28#6438C2211yescharacteristic
29C42×D7C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#107C2212yes—
32C7×#106C2212yes—
33C7×#97C2211yescharacteristic
34#6395C2212yes—
35(C2×D7)⋊C8C2212yes—
36#6390C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3240—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×D28—21no—
46C2×D28C2311yescharacteristic
47C2×C4×D7—21no—
48C2×C4×D7D411yescharacteristic
49C2×C4×D7D411yescharacteristic
50C2×C4×D7C2311yescharacteristic
51C2×Dic14—21no—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2311yescharacteristic
58C7×C4⋊C4C2312yes—
59C7×C4⋊C4C2311yescharacteristic
60C42×C7C2311yescharacteristic
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7—21no—
70C4⋊Dic7C2312yes—
71Dic7⋊C4—22no—
72Dic7⋊C4—21no—
73C4×Dic7—22no—
74C4×Dic7—21no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C2×C7⋊C8C2312yes—
79#533—71no—
80#529—71no—
81#501—71no—
82#501D711yescharacteristic, complemented
83#489—71no—
84#489—71no—
85#485—72no—
86#482—72no—
87#479—72no—
88#478—72no—
89#432—71no—
90C22×D7—21no—
91C22×D7C2×D411yescharacteristic
92C7×Q8—42no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2411yescharacteristic
99C2×Dic7—22no—
100C2×Dic7—21no—
101C2×Dic7C2×D411yescharacteristic
102C2×Dic7C2×D411yescharacteristic
103C2×Dic7C2×D411yescharacteristic
104D28—22no—
105D28—42no—
106C4×D7—22no—
107C4×D7—41no—
108C4×D7—41no—
109C4×D7—42no—
110Dic14—22no—
111Dic14—42no—
112C7×C8—22no—
113C7⋊C8—22no—
114C4⋊Q8—71no—
115C4⋊Q8C2×D711yescharacteristic
116#126—141no—
117#125—72no—
118#125C2×D711yescharacteristic
119#124—71no—
120#123—141no—
121C22⋊Q8—71no—
122C22⋊Q8—141no—
123C22⋊Q8—141no—
124#121—71no—
125C4×Q8—71no—
126C4×Q8—141no—
127C4×Q8—141no—
128C4×D4—71no—
129C4×D4—141no—
130#117—72no—
131#117—141no—
132C2×C42—71no—
133#107—72no—
134#107C2×D712yes—
135#106—72no—
136#106C2×D712yes—
137C4⋊C8—72no—
138#103—72no—
139#103—72no—
140#102—72no—
141#102—72no—
142C22⋊C8—72no—
143#97—71no—
144#97C2×D711yescharacteristic
145C22×C7C22×D411yescharacteristic
146C2×D7—21no—
147C2×D7—21no—
148C2×D7—21no—
149C2×D7—41no—
150C2×D7—41no—
151C2×D7—41no—
152C4×C7—21no—
153C4×C7—21no—
154C4×C7—41no—
155C4×C7—41no—
156C4×C7—42no—
157C4×C72−1+412yes—
158Dic7—21no—
159Dic7—21no—
160Dic7—21no—
161Dic7—41no—
162Dic7—42no—
163C2×Q8—21no—
164C2×Q8—71no—
165C2×Q8—141no—
166C2×D4—71no—
167C2×D4—141no—
168C22×C4—71no—
169C22×C4—71no—
170C22×C4—71no—
171C22×C4—141no—
172C2×C8—72no—
173C2×C8C22×D712yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—72no—
178C4⋊C4—141no—
179C4⋊C4—142no—
180C4⋊C4—141no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4C22×D712yes—
184C4⋊C4C22×D711yescharacteristic
185C4⋊C4C22×D711yescharacteristic
186C22⋊C4—141no—
187C22⋊C4—141no—
188C22⋊C4—142no—
189C22⋊C4—141no—
190C22⋊C4—141no—
191C22⋊C4—141no—
192C42—71no—
193C42—71no—
194C42—71no—
195C42—142no—
196C42—141no—
197C42C22×D711yescharacteristic
198C2×C7#53611yescharacteristic
199C2×C7#57511yescharacteristic
200C2×C7#57511yescharacteristic
201D7—41no—
202D7—81no—
203C23—71no—
204C23—141no—
205Q8—42no—
206Q8—142no—
207Q8—282no—
208D4—142no—
209D4—282no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—71no—
214C2×C4—71no—
215C2×C4—71no—
216C2×C4—142no—
217C2×C4—141no—
218C2×C4—142no—
219C2×C4—281no—
220C2×C4—281no—
221C2×C4—282no—
222C2×C4D4×D711yescharacteristic
223C2×C4D4×D711yescharacteristic
224C2×C4C23×D711yescharacteristic
225C8—22no—
226C8—142no—
227C7#324011yescharacteristic, complemented, Sylow
228C22—141no—
229C22—141no—
230C22—141no—
231C22—281no—
232C22—281no—
233C22—281no—
234C22C2×D4×D711yescharacteristic
235C4—21no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—141no—
241C4—141no—
242C4—141no—
243C4—281no—
244C4—282no—
245C4#655112yes—
246C2—281no—
247C2—561no—
248C2#9041511yescharacteristic
249C2#9046411yescharacteristic
250C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#501D7
hypocenterC7#3240
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#501C2semidirect
4. #89861C2semidirect
5. #89837C2semidirect
7. #89649C2semidirect
8. #89643C2semidirect
9. #89639C2semidirect
10. #89486C2semidirect
11. #89477C2semidirect
82. #501D7semidirect
227. C7#3240semidirect

Maximal subgroups

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

subgroupindex
#904192
#903822
C7×#5012
#898612
#898372
#896572
#896492
#896432
#896392
#894862
#894772
#32407