Finite Group Databasebeta

← #11739024

Subgroups — #11739024

Order 896.

408 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 115 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#90425C211yesmaximal, characteristic, complemented
2#90409C211yesmaximal, characteristic, complemented
3C7×#486C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89844C211yesmaximal, characteristic, complemented
6#89553C212yesmaximal, complemented
7#89551C212yesmaximal, complemented
8#89547C212yesmaximal, complemented
9#89544C212yesmaximal, complemented
10C4×#1132C211yesmaximal, characteristic, complemented
11C8×D28C211yesmaximal, characteristic, complemented
12C2×#1159C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#127C2211yescharacteristic
15#6510—22no—
16#6503—21no—
17#6502—22no—
18#6501C2212yes—
19D4×Dic7—21no—
20C2×#1141C2212yes—
21C2×#1132C2211yescharacteristic
22C2×C8×D7C2212yes—
23#6458—21no—
24#6458C2212yes—
25#6455—21no—
26#6455C2212yes—
27C4×D28C2211yescharacteristic
28C4×D28C2211yescharacteristic
29C4×Dic14C2211yescharacteristic
30C7×#102C2212yes—
31C7×#102C2212yes—
32C4×C7×C8C2211yescharacteristic
33#6406C2212yes—
34#6395C2211yescharacteristic
35(C2×D7)⋊C8C2212yes—
36#6391C2211yescharacteristic
37#6390C2211yescharacteristic
38#6384C2212yes—
39#6381C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3305—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×D4—21no—
44C2×C7×D4—21no—
45C2×C7×D4C2312yes—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—22no—
48C22×Dic7—22no—
49#1159—24no—
50#1159—42no—
51C2×D28—22no—
52C2×D28D411yescharacteristic
53C2×D28C2311yescharacteristic
54C2×C4×D7—21no—
55C2×C4×D7C2312yes—
56C2×Dic14C2311yescharacteristic
57C2×C7×C8C2311yescharacteristic
58C2×C7×C8C2311yescharacteristic
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2312yes—
61C42×C7C2311yescharacteristic
62C22⋊Dic7—22no—
63#1141—42no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4D412yes—
68C4⋊Dic7D411yescharacteristic
69C4⋊Dic7C2311yescharacteristic
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72C4×Dic7—21no—
73C4×Dic7—22no—
74C2×C7⋊C8C2312yes—
75#1132—22no—
76C8×D7—42no—
77#555—71no—
78#550—71no—
79#486D711yescharacteristic, complemented
80#482—72no—
81#477—71no—
82#466—72no—
83#459—71no—
84#449—72no—
85C4×SD16—71no—
86C8×D4—71no—
87#416—72no—
88C23×C7—22no—
89C22×D7—21no—
90C22×D7C2×D412yes—
91C7×Q8—81no—
92C7×D4—22no—
93C7×D4—22no—
94C7×D4—42no—
95C7×D4—81no—
96C2×C4×C7—22no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2411yescharacteristic
100C22⋊D7—44no—
101C22⋊D7—82no—
102C2×Dic7—21no—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—44no—
106C2×Dic7C2×D412yes—
107D28—22no—
108D28—21no—
109D28—21no—
110D28—82no—
111C4×D7—22no—
112C4×D7—22no—
113C4×D7—42no—
114Dic14—21no—
115Dic14#4112yes—
116C7×C8—21no—
117C7×C8—21no—
118C7⋊C8—22no—
119C2×#41—72no—
120C2×SD16—71no—
121C2×D8—72no—
122C22×C8—72no—
123C4⋊Q8C2×D711yescharacteristic
124#127—141no—
125#127C2×D711yescharacteristic
126#124—142no—
127#121—72no—
128#121—72no—
129#121—142no—
130#121—141no—
131C4×Q8—71no—
132C4×D4—71no—
133C4×D4—71no—
134C4×D4—141no—
135#117—72no—
136#117—141no—
137#107—72no—
138#106—71no—
139C4⋊C8—71no—
140#103—71no—
141#103—72no—
142#102—72no—
143#102—71no—
144#102C2×D712yes—
145#102C2×D712yes—
146C22⋊C8—72no—
147C4×C8C2×D711yescharacteristic
148C22×C7—42no—
149C22×C7—42no—
150C22×C7—42no—
151C22×C7C22×D411yescharacteristic
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C4×C7—21no—
159C4×C7—42no—
160C4×C7#13512yes—
161C4×C7C2×#4111yescharacteristic
162C4×C7H(5,𝔽2)11yescharacteristic
163Dic7—22no—
164Dic7—22no—
165Dic7—41no—
166Dic7—41no—
167#41—144no—
168#41—282no—
169C2×Q8—21no—
170C2×Q8—71no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—71no—
174C2×D4—71no—
175C2×D4—142no—
176C2×D4—142no—
177C2×D4—142no—
178C2×D4C22×D712yes—
179C22×C4—72no—
180C22×C4—142no—
181C22×C4—141no—
182SD16—142no—
183D8—282no—
184C2×C8—72no—
185C2×C8—282no—
186C2×C8C22×D711yescharacteristic
187C2×C8C22×D711yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—71no—
190C4⋊C4—71no—
191C4⋊C4—71no—
192C4⋊C4—141no—
193C4⋊C4C22×D712yes—
194C22⋊C4—72no—
195C22⋊C4—142no—
196C22⋊C4—142no—
197C22⋊C4—142no—
198C22⋊C4—141no—
199C42—142no—
200C42—141no—
201C42C22×D711yescharacteristic
202C2×C7—82no—
203C2×C7#54611yescharacteristic
204C2×C7C2×#13511yescharacteristic
205C2×C7#57611yescharacteristic
206D7—42no—
207D7—81no—
208C23—22no—
209C23—72no—
210C23—141no—
211Q8—72no—
212Q8—81no—
213Q8—141no—
214D4—22no—
215D4—22no—
216D4—42no—
217D4—81no—
218D4—142no—
219D4—141no—
220D4—141no—
221D4—284no—
222D4—562no—
223D4—562no—
224C2×C4—22no—
225C2×C4—72no—
226C2×C4—141no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—141no—
230C2×C4—141no—
231C2×C4—284no—
232C2×C4—282no—
233C2×C4D4×D711yescharacteristic
234C2×C4D4×D711yescharacteristic
235C2×C4C23×D711yescharacteristic
236C8—21no—
237C8—21no—
238C8—142no—
239C7#330511yescharacteristic, complemented, Sylow
240C22—42no—
241C22—42no—
242C22—42no—
243C22—142no—
244C22—142no—
245C22—142no—
246C22—281no—
247C22—281no—
248C22—281no—
249C22C2×D4×D711yescharacteristic
250C4—21no—
251C4—42no—
252C4—142no—
253C4—142no—
254C4—281no—
255C4—281no—
256C4#647912yes—
257C4D7×#4111yescharacteristic
258C4#655311yescharacteristic
259C2—82no—
260C2—282no—
261C2—561no—
262C2#9036711yescharacteristic
263C2#9044811yescharacteristic
264C2C2×#647911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#486C2
socleC22×C7C22×D4
hypercenter#486D7
hypocenterC7#3305
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #90425C2semidirect
2. #90409C2semidirect
3. C7×#486C2semidirect
4. #89863C2semidirect
5. #89844C2semidirect
6. #89553C2semidirect
7. #89551C2semidirect
8. #89547C2semidirect
9. #89544C2semidirect
10. C4×#1132C2semidirect
11. C8×D28C2semidirect
79. #486D7semidirect
239. C7#3305semidirect

Maximal subgroups

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

subgroupindex
#904252
#904092
C7×#4862
#898632
#898442
#895532
#895512
#895472
#895442
C4×#11322
C8×D282
#33057