Finite Group Databasebeta

← #11736719

Subgroups — #11736719

Order 896.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#90170C211yesmaximal, characteristic
2C2×#6433C211yesmaximal, characteristic, complemented
3C7×#431C211yesmaximal, characteristic, complemented
4#89619C212yesmaximal, complemented
5#89601C212yesmaximal, complemented
6#89508C211yesmaximal, characteristic
7#89507C211yesmaximal, characteristic
8#89494C211yesmaximal, characteristic, complemented
9#89493C211yesmaximal, characteristic, complemented
10#89469C212yesmaximal, complemented
11#89455C212yesmaximal, complemented
12C2×C42×C7C2211yescharacteristic
13#6493C2211yescharacteristic
14#6492—21no—
15C4×C22⋊D7—21no—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2211yescharacteristic
18#6487C2211yescharacteristic
19C2×Dic7⋊C4—21no—
20C2×Dic7⋊C4—21no—
21#6439—21no—
22#6438C2211yescharacteristic
23C4×D28C2211yescharacteristic
24#6435—21no—
25#6433—21no—
26#6433—21no—
27#6433C2212yes—
28#6433C2212yes—
29#6433C2211yescharacteristic
30C4×Dic14C2211yescharacteristic
31C7×C4⋊C8C2212yes—
32C7×C22⋊C8C2211yescharacteristic
33C7×C22⋊C8C2211yescharacteristic
34C4×C7×C8C2212yes—
35#6395C2212yes—
36#6395C2212yes—
37#6392C2212yes—
38#6392C2212yes—
39#6391C2212yes—
40#6391C2212yes—
41#6390C2212yes—
42#6390C2212yes—
43#3232—71nomaximal, Sylow
44C22×C4×C7D411yescharacteristic
45C22×C4×C7D411yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C22×Dic7—21no—
49C22×Dic7—21no—
50C2×D28C2311yescharacteristic
51C2×C4×D7—21no—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2312yes—
54C2×C7×C8C2312yes—
55C42×C7D411yescharacteristic
56C42×C7D411yescharacteristic
57C42×C7C2311yescharacteristic
58C42×C7C2311yescharacteristic
59C22⋊Dic7—21no—
60(C2×D7)⋊C4—21no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7C2311yescharacteristic
67C4⋊Dic7C2312yes—
68C4⋊Dic7C2312yes—
69C4⋊Dic7C2311yescharacteristic
70Dic7⋊C4—21no—
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78C4×Dic7—21no—
79#529—71no—
80C2×#125—71no—
81#499—72no—
82#488—72no—
83#484—71no—
84#483—71no—
85#481—71no—
86#480—71no—
87#479—72no—
88#478—72no—
89#431D711yescharacteristic, complemented
90C23×C7C2×D411yescharacteristic
91C22×D7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—41no—
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2411yescharacteristic
101C22⋊D7—42no—
102C2×Dic7—21no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109D28—22no—
110C4×D7—42no—
111Dic14—22no—
112C7×C8—22no—
113C7×C8—22no—
114#126—141no—
115#125—72no—
116#125—72no—
117#125—71no—
118#125—141no—
119#125—141no—
120#124—71no—
121#123—141no—
122C22⋊Q8—71no—
123#121—71no—
124C4×Q8—71no—
125C4×D4—71no—
126C4×D4—141no—
127#117—141no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—141no—
131C2×C4⋊C4—141no—
132C2×C42C2×D711yescharacteristic
133#107—72no—
134#107—72no—
135#106—72no—
136#106—72no—
137C4⋊C8C2×D712yes—
138#103—72no—
139#103—72no—
140#102—72no—
141#102—72no—
142C22⋊C8C2×D711yescharacteristic
143C22⋊C8C2×D711yescharacteristic
144C4×C8C2×D712yes—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7#13512yes—
148C22×C7C22×D411yescharacteristic
149C2×D7—41no—
150C2×D7—41no—
151C2×D7—41no—
152C4×C7—21no—
153C4×C7—21no—
154C4×C7—21no—
155C4×C7—21no—
156C4×C7—21no—
157C4×C72−1+412yes—
158Dic7—41no—
159Dic7—42no—
160Dic7—41no—
161Dic7—41no—
162Dic7—42no—
163C2×Q8—71no—
164C2×D4—71no—
165C2×D4—141no—
166C22×C4—141no—
167C22×C4—141no—
168C22×C4—141no—
169C22×C4D2811yescharacteristic
170C22×C4D2811yescharacteristic
171C22×C4C22×D711yescharacteristic
172C2×C8C22×D712yes—
173C2×C8C22×D712yes—
174C4⋊C4—72no—
175C4⋊C4—72no—
176C4⋊C4—71no—
177C4⋊C4—71no—
178C4⋊C4—142no—
179C4⋊C4—142no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C22⋊C4—141no—
189C22⋊C4—141no—
190C22⋊C4—141no—
191C22⋊C4—141no—
192C22⋊C4—141no—
193C42—141no—
194C42D2811yescharacteristic
195C42D2811yescharacteristic
196C42C22×D711yescharacteristic
197C42C22×D711yescharacteristic
198C2×C7—22no—
199C2×C7#53611yescharacteristic
200C2×C7C2×#13511yescharacteristic
201C2×C7#57511yescharacteristic
202D7—81no—
203C23—141no—
204C23C2×D2811yescharacteristic
205Q8—142no—
206D4—142no—
207D4—282no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—282no—
217C2×C4—282no—
218C2×C4—282no—
219C2×C4C2×D2811yescharacteristic
220C2×C4C2×D2811yescharacteristic
221C2×C4C2×D2811yescharacteristic
222C2×C4C2×D2811yescharacteristic
223C2×C4C2×D2811yescharacteristic
224C2×C4C23×D711yescharacteristic
225C8—22no—
226C8—22no—
227C7#323211yescharacteristic, complemented, Sylow
228C22—21no—
229C22—21no—
230C22—281no—
231C22—281no—
232C22—281no—
233C22#646712yes—
234C22C22×D2811yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—282no—
241C4—281no—
242C4—281no—
243C4—282no—
244C4—281no—
245C4#655412yes—
246C2—22no—
247C2—561no—
248C2#9021711yescharacteristic
249C2C2×#646711yescharacteristic
250C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenter#431D7
hypocenterC7#3232
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. C2×#6433C2semidirect
3. C7×#431C2semidirect
4. #89619C2semidirect
5. #89601C2semidirect
8. #89494C2semidirect
9. #89493C2semidirect
10. #89469C2semidirect
11. #89455C2semidirect
89. #431D7semidirect
227. C7#3232semidirect

Maximal subgroups

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

subgroupindex
#901702
C2×#64332
C7×#4312
#896192
#896012
#895082
#895072
#894942
#894932
#894692
#894552
#32327