Finite Group Databasebeta

← #11736836

Subgroups — #11736836

Order 896.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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#90216C211yesmaximal, characteristic
2#90209C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89620C212yesmaximal, complemented
5#89617C212yesmaximal, complemented
6#89505C212yesmaximal, complemented
7#89503C212yesmaximal, complemented
8#89487C211yesmaximal, characteristic, complemented
9#89485C211yesmaximal, characteristic, complemented
10#89469C211yesmaximal, characteristic, complemented
11#89465C211yesmaximal, characteristic, complemented
12C2×#1157C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C7×#117C2211yescharacteristic
15#6487C2212yes—
16C2×Dic28C2212yescomplemented
17C2×#1132C2212yes—
18C2×#1132C2212yes—
19C2×#1132C2212yescomplemented
20#6458—22no—
21#6452—22no—
22#6445—22no—
23#6442—22no—
24#6438C2211yescharacteristic
25#6438C2211yescharacteristic
26#6437C2211yescharacteristic
27#6437C2211yescharacteristic
28#6433C2211yescharacteristic, complemented
29#6432C2211yescharacteristic, complemented
30C4×Dic14C2212yescomplemented
31C7×C4⋊C8C2212yescomplemented
32C7×C22⋊C8C2212yes—
33C7×#97C2211yescharacteristic, complemented
34C4×C7×C8C2211yescharacteristic, complemented
35#6395C2212yes—
36#6395C2212yes—
37#6395C2212yescomplemented
38#6390C2212yescomplemented
39#3222—71nomaximal, Sylow
40C23×D7—21no—
41C22×C4×C7C2311yescharacteristic
42C2×C22⋊D7—21no—
43#1157—42no—
44C2×D28—21no—
45C2×D28—22no—
46C2×D28—22no—
47C2×D28—42no—
48C2×D28C2311yescharacteristic
49C2×D28C2311yescharacteristic
50C2×D28C2311yescharacteristic
51C2×C4×D7—21no—
52C2×Dic14C2312yes—
53C2×Dic14C2311yescharacteristic
54C2×C7×C8C2312yes—
55C2×C7×C8C2312yes—
56C7×C4⋊C4D412yes—
57C7×C22⋊C4D412yes—
58C42×C7C2311yescharacteristic
59C42×C7C2311yescharacteristic
60C22⋊Dic7—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63C4⋊Dic7—22no—
64C4⋊Dic7C2312yes—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4—22no—
68C4×Dic7—22no—
69Dic28—42no—
70#1132—42no—
71#1132—42no—
72#1132—42no—
73#541—71no—
74#535—71no—
75#497—71no—
76#492—71no—
77#490—71no—
78#488—71no—
79#464—72no—
80#460—72no—
81#452—72no—
82#450—72no—
83#433D711yescharacteristic, complemented
84C23×C7C2×D411yescharacteristic
85C22×D7—21no—
86C22×D7—21no—
87C22×D7—21no—
88C22×D7—42no—
89C22×D7—82no—
90C2×C4×C7—22no—
91C2×C4×C7C2×D412yes—
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2411yescharacteristic
95C22⋊D7—42no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—21no—
99D28—22no—
100D28—22no—
101D28—22no—
102D28—42no—
103D28—82no—
104D28—82no—
105C4×D7—42no—
106Dic14—22no—
107Dic14—22no—
108Dic14—22no—
109C7×C8—22no—
110C7×C8—22no—
111C2×#41—71no—
112C22×D4—71no—
113C2×Q16—72no—
114C2×SD16—72no—
115C2×SD16—72no—
116C2×SD16—72no—
117C4⋊Q8—71no—
118#127—71no—
119#127—71no—
120#126—142no—
121#125—71no—
122#125—142no—
123#124—71no—
124#124—71no—
125C22⋊Q8—72no—
126#121—142no—
127#120—142no—
128C4×Q8—72no—
129#117C2×D711yescharacteristic
130C4⋊C8C2×D712yescomplemented
131#103—72no—
132#102—72no—
133#102—72no—
134#102—72no—
135C22⋊C8C2×D712yes—
136#97C2×D711yescharacteristic, complemented
137C4×C8C2×D711yescharacteristic, complemented
138C22×C7—21no—
139C22×C7—22no—
140C22×C7C22×D411yescharacteristic
141C2×D7—41no—
142C2×D7—41no—
143C2×D7—41no—
144C2×D7—42no—
145C2×D7—42no—
146C2×D7—42no—
147C2×D7—84no—
148C4×C7—21no—
149C4×C7—22no—
150C4×C7—22no—
151C4×C7H(5,𝔽2)12yes—
152Dic7—41no—
153Dic7—42no—
154Dic7—42no—
155C24—141no—
156#41—282no—
157C2×Q8—71no—
158C2×Q8—72no—
159C2×D4—71no—
160C2×D4—71no—
161C2×D4—71no—
162C2×D4—141no—
163C2×D4—141no—
164C2×D4—142no—
165C2×D4—142no—
166C2×D4—282no—
167C22×C4—141no—
168C22×C4C22×D711yescharacteristic
169Q16—282no—
170SD16—282no—
171SD16—282no—
172SD16—282no—
173C2×C8C22×D712yes—
174C2×C8C22×D712yes—
175C4⋊C4—72no—
176C4⋊C4—142no—
177C4⋊C4—142no—
178C4⋊C4—142no—
179C4⋊C4—142no—
180C4⋊C4D2812yes—
181C22⋊C4—142no—
182C22⋊C4—142no—
183C22⋊C4—142no—
184C22⋊C4D2812yes—
185C42—142no—
186C42C22×D711yescharacteristic
187C42C22×D711yescharacteristic
188C2×C7—41no—
189C2×C7#53511yescharacteristic
190C2×C7#57712yes—
191D7—81no—
192D7—81no—
193D7—81no—
194C23—141no—
195C23—141no—
196C23—141no—
197C23—282no—
198C23—562no—
199C23C2×D2811yescharacteristic
200Q8—142no—
201Q8—142no—
202Q8—142no—
203D4—142no—
204D4—142no—
205D4—142no—
206D4—282no—
207D4—282no—
208D4—562no—
209D4—562no—
210C2×C4—22no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—141no—
214C2×C4—282no—
215C2×C4C2×D2812yes—
216C2×C4C2×D2812yes—
217C2×C4C2×D2811yescharacteristic
218C2×C4C23×D711yescharacteristic
219C8—22no—
220C8—22no—
221C7#322211yescharacteristic, complemented, Sylow
222C22—22no—
223C22—21no—
224C22—282no—
225C22—282no—
226C22—282no—
227C22—281no—
228C22—281no—
229C22—281no—
230C22—564no—
231C22C22×D2811yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—282no—
236C4—281no—
237C4—282no—
238C4#655312yes—
239C2—41no—
240C2—561no—
241C2—561no—
242C2—561no—
243C2#9021611yescharacteristic
244C2#9044212yes—
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
prime radical#90216C2
hypercenter#433D7
hypocenterC7#3222
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #90209C2semidirect
3. C7×#433C2semidirect
4. #89620C2semidirect
5. #89617C2semidirect
6. #89505C2semidirect
7. #89503C2semidirect
8. #89487C2semidirect
9. #89485C2semidirect
10. #89469C2semidirect
11. #89465C2semidirect
16. C2×Dic28C22semidirect
19. C2×#1132C22semidirect
28. #6433C22semidirect
29. #6432C22semidirect
30. C4×Dic14C22semidirect
31. C7×C4⋊C8C22semidirect
33. C7×#97C22semidirect
34. C4×C7×C8C22semidirect
37. #6395C22semidirect
38. #6390C22semidirect
83. #433D7semidirect
130. C4⋊C8C2×D7semidirect
136. #97C2×D7semidirect
137. C4×C8C2×D7semidirect
221. C7#3222semidirect

Maximal subgroups

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

subgroupindex
#902162
#902092
C7×#4332
#896202
#896172
#895052
#895032
#894872
#894852
#894692
#894652
#32227