Finite Group Databasebeta

← #11739059

Subgroups — #11739059

Order 896.

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

normal subgroups only

This group has 386 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#90423C211yesmaximal, characteristic, complemented
2#90417C211yesmaximal, characteristic
3C7×#487C211yesmaximal, characteristic, complemented
4#89862C211yesmaximal, characteristic, complemented
5#89859C211yesmaximal, characteristic, complemented
6#89587C212yesmaximal, complemented
7#89585C212yesmaximal, complemented
8#89582C212yesmaximal, complemented
9#89576C212yesmaximal, complemented
10C4×#1132C211yesmaximal, characteristic, complemented
11C8×D28C211yesmaximal, characteristic, complemented
12C2×#1161C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic
15#6510—21no—
16#6509C2212yes—
17Q8×Dic7—21no—
18C2×#1144C2212yes—
19C2×#1132C2211yescharacteristic
20C2×C8×D7C2212yes—
21#6461—22no—
22#6460C2212yes—
23#6458—21no—
24#6457—22no—
25#6455C2212yes—
26D7×C4⋊C4—21no—
27C4×D28C2211yescharacteristic
28C4×D28C2211yescharacteristic
29C4×Dic14C2211yescharacteristic
30C7×#103C2212yes—
31C7×#103C2212yes—
32C4×C7×C8C2211yescharacteristic
33#6409C2212yes—
34#6395C2211yescharacteristic
35(C2×D7)⋊C8C2212yes—
36#6391C2211yescharacteristic
37#6390C2211yescharacteristic
38#6383C2212yes—
39#6381C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3302—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2312yes—
44#1161—24no—
45#1161—42no—
46C2×D28—22no—
47C2×D28D411yescharacteristic
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×C4×D7—22no—
51C2×C4×D7C2312yes—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2311yescharacteristic
54C2×C7×C8C2311yescharacteristic
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4—21no—
58C7×C4⋊C4C2312yes—
59C42×C7C2311yescharacteristic
60#1144—42no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4D412yes—
66C4⋊Dic7—22no—
67C4⋊Dic7D411yescharacteristic
68C4⋊Dic7C2311yescharacteristic
69C4⋊Dic7C2311yescharacteristic
70Dic7⋊C4—21no—
71Dic7⋊C4—22no—
72C4×Dic7—21no—
73C4×Dic7—22no—
74C2×C7⋊C8C2312yes—
75#1132—22no—
76C8×D7—42no—
77#556—71no—
78#547—71no—
79#487D711yescharacteristic, complemented
80#485—72no—
81#474—71no—
82#467—72no—
83#462—71no—
84#452—72no—
85C4×SD16—71no—
86C8×D4—71no—
87#416—72no—
88C22×D7—21no—
89C22×D7C2×D412yes—
90C7×Q8—22no—
91C7×Q8—22no—
92C7×Q8—42no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2411yescharacteristic
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7C2×D412yes—
102D28—22no—
103D28—21no—
104D28—44no—
105D28#4112yes—
106C4×D7—22no—
107C4×D7—22no—
108C4×D7—44no—
109C4×D7—42no—
110Dic14—21no—
111Dic14—21no—
112C7×C8—21no—
113C7×C8—21no—
114C7⋊C8—22no—
115C2×#41—72no—
116C2×Q16—72no—
117C2×SD16—71no—
118C22×C8—72no—
119C4⋊Q8C2×D711yescharacteristic
120C4⋊Q8C2×D711yescharacteristic
121#126—142no—
122#124—141no—
123#123—142no—
124C22⋊Q8—72no—
125C22⋊Q8—72no—
126#121—141no—
127C4×Q8—71no—
128C4×Q8—141no—
129C4×D4—71no—
130C4×D4—71no—
131#117—72no—
132C2×C4⋊C4—141no—
133#107—72no—
134#106—71no—
135C4⋊C8—71no—
136#103—72no—
137#103—71no—
138#103C2×D712yes—
139#103C2×D712yes—
140#102—71no—
141#102—72no—
142C22⋊C8—72no—
143C4×C8C2×D711yescharacteristic
144C22×C7C22×D411yescharacteristic
145C2×D7—22no—
146C2×D7—22no—
147C2×D7—22no—
148C2×D7—22no—
149C2×D7—41no—
150C2×D7—41no—
151C4×C7—21no—
152C4×C7—42no—
153C4×C7—42no—
154C4×C7#13512yes—
155C4×C7C2×#4111yescharacteristic
156C4×C7H(5,𝔽2)11yescharacteristic
157Dic7—22no—
158Dic7—41no—
159Dic7—41no—
160Dic7—41no—
161#41—144no—
162#41—282no—
163C2×Q8—21no—
164C2×Q8—71no—
165C2×Q8C22×D712yes—
166C2×D4—71no—
167C2×D4—71no—
168C2×D4—142no—
169C22×C4—72no—
170C22×C4—142no—
171C22×C4—141no—
172Q16—282no—
173SD16—142no—
174C2×C8—72no—
175C2×C8—282no—
176C2×C8C22×D711yescharacteristic
177C2×C8C22×D711yescharacteristic
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—71no—
182C4⋊C4—71no—
183C4⋊C4—71no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—141no—
187C4⋊C4C22×D712yes—
188C22⋊C4—72no—
189C22⋊C4—142no—
190C22⋊C4—142no—
191C22⋊C4—142no—
192C22⋊C4—141no—
193C42—142no—
194C42—141no—
195C42C22×D711yescharacteristic
196C2×C7#54611yescharacteristic
197C2×C7C2×#13511yescharacteristic
198C2×C7#57811yescharacteristic
199D7—42no—
200D7—42no—
201C23—72no—
202C23—141no—
203Q8—22no—
204Q8—22no—
205Q8—42no—
206Q8—141no—
207Q8—141no—
208D4—72no—
209D4—142no—
210D4—141no—
211D4—284no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—72no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4—141no—
220C2×C4—284no—
221C2×C4—282no—
222C2×C4D4×D711yescharacteristic
223C2×C4D4×D711yescharacteristic
224C2×C4C23×D711yescharacteristic
225C8—21no—
226C8—21no—
227C8—142no—
228C7#330211yescharacteristic, complemented, Sylow
229C22—142no—
230C22—142no—
231C22—142no—
232C22—142no—
233C22—281no—
234C22—281no—
235C22C2×D4×D711yescharacteristic
236C4—21no—
237C4—42no—
238C4—42no—
239C4—142no—
240C4—281no—
241C4—281no—
242C4—281no—
243C4#647912yes—
244C4D7×#4111yescharacteristic
245C4#655311yescharacteristic
246C2—282no—
247C2—282no—
248C2#9036711yescharacteristic
249C2C2×#647911yescharacteristic
250C2#9045711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radical#90417C2
hypercenter#487D7
hypocenterC7#3302
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #90423C2semidirect
3. C7×#487C2semidirect
4. #89862C2semidirect
5. #89859C2semidirect
6. #89587C2semidirect
7. #89585C2semidirect
8. #89582C2semidirect
9. #89576C2semidirect
10. C4×#1132C2semidirect
11. C8×D28C2semidirect
79. #487D7semidirect
228. C7#3302semidirect

Maximal subgroups

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

subgroupindex
#904232
#904172
C7×#4872
#898622
#898592
#895872
#895852
#895822
#895762
C4×#11322
C8×D282
#33027