Finite Group Databasebeta

← #11736835

Subgroups — #11736835

Order 896.

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

normal subgroups only

This group has 420 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#90226C211yesmaximal, characteristic, complemented
2#90216C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89616C212yesmaximal, complemented
5#89615C212yesmaximal, complemented
6#89506C212yesmaximal, complemented
7#89504C212yesmaximal, complemented
8#89485C211yesmaximal, characteristic, complemented
9#89484C211yesmaximal, characteristic, complemented
10#89469C211yesmaximal, characteristic, complemented
11#89467C211yesmaximal, characteristic, complemented
12C2×#1157C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C7×#117C2211yescharacteristic
15#6493C2212yes—
16C2×D56C2212yes—
17C2×D56C2212yes—
18C2×D56C2212yescomplemented
19C2×#1132C2212yescomplemented
20#6458—22no—
21#6457—22no—
22#6447—22no—
23#6445—22no—
24#6438C2211yescharacteristic
25#6438C2211yescharacteristic
26#6437C2211yescharacteristic
27#6437C2211yescharacteristic
28#6437C2211yescharacteristic, complemented
29C4×D28C2212yescomplemented
30#6433C2211yescharacteristic, complemented
31C7×C4⋊C8C2212yescomplemented
32C7×C22⋊C8C2212yes—
33C7×#97C2211yescharacteristic, complemented
34C4×C7×C8C2211yescharacteristic, complemented
35#6395C2212yes—
36#6395C2212yes—
37#6395C2212yescomplemented
38#6390C2212yescomplemented
39#3220—71nomaximal, Sylow
40C23×D7—21no—
41C22×C4×C7C2311yescharacteristic
42C2×C22⋊D7—21no—
43C2×C22⋊D7—22no—
44#1157—42no—
45C2×D28—21no—
46C2×D28—22no—
47C2×D28—22no—
48C2×D28—22no—
49C2×D28—42no—
50C2×D28C2312yes—
51C2×D28C2311yescharacteristic
52C2×D28C2311yescharacteristic
53C2×D28C2311yescharacteristic
54C2×C4×D7—21no—
55C2×C4×D7—22no—
56C2×Dic14C2311yescharacteristic
57C2×C7×C8C2312yes—
58C2×C7×C8C2312yes—
59C7×C4⋊C4D412yes—
60C7×C22⋊C4D412yes—
61C42×C7C2311yescharacteristic
62C42×C7C2311yescharacteristic
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—22no—
67C4⋊Dic7C2312yes—
68Dic7⋊C4—22no—
69D56—42no—
70D56—42no—
71D56—42no—
72#1132—42no—
73#540—71no—
74#535—71no—
75#496—71no—
76#493—71no—
77#490—71no—
78#488—71no—
79#463—72no—
80#459—72no—
81#449—72no—
82#447—72no—
83#433D711yescharacteristic, complemented
84C23×C7C2×D411yescharacteristic
85C22×D7—22no—
86C22×D7—21no—
87C22×D7—21no—
88C22×D7—21no—
89C22×D7—42no—
90C22×D7—82no—
91C2×C4×C7—22no—
92C2×C4×C7C2×D412yes—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2411yescharacteristic
96C22⋊D7—42no—
97C22⋊D7—82no—
98C2×Dic7—22no—
99C2×Dic7—21no—
100D28—22no—
101D28—22no—
102D28—22no—
103D28—22no—
104D28—22no—
105D28—42no—
106D28—82no—
107D28—82no—
108D28—82no—
109C4×D7—42no—
110C4×D7—82no—
111Dic14—22no—
112C7×C8—22no—
113C7×C8—22no—
114C2×#41—71no—
115C22×D4—71no—
116C2×SD16—72no—
117C2×D8—72no—
118C2×D8—72no—
119C2×D8—72no—
120#127—71no—
121#127—71no—
122#127—71no—
123#125—71no—
124#124—71no—
125#124—71no—
126#123—142no—
127#121—72no—
128#121—142no—
129#121—142no—
130#120—142no—
131C4×D4—72no—
132#117C2×D711yescharacteristic
133C4⋊C8C2×D712yescomplemented
134#103—72no—
135#102—72no—
136#102—72no—
137#102—72no—
138C22⋊C8C2×D712yes—
139#97C2×D711yescharacteristic, complemented
140C4×C8C2×D711yescharacteristic, complemented
141C22×C7—21no—
142C22×C7—22no—
143C22×C7C22×D411yescharacteristic
144C2×D7—41no—
145C2×D7—41no—
146C2×D7—41no—
147C2×D7—42no—
148C2×D7—42no—
149C2×D7—42no—
150C2×D7—42no—
151C2×D7—42no—
152C2×D7—42no—
153C2×D7—84no—
154C4×C7—21no—
155C4×C7—22no—
156C4×C7—22no—
157C4×C7H(5,𝔽2)12yes—
158Dic7—41no—
159Dic7—42no—
160C24—141no—
161#41—282no—
162C2×Q8—71no—
163C2×D4—71no—
164C2×D4—71no—
165C2×D4—71no—
166C2×D4—72no—
167C2×D4—141no—
168C2×D4—142no—
169C2×D4—141no—
170C2×D4—142no—
171C2×D4—142no—
172C2×D4—142no—
173C2×D4—282no—
174C22×C4—141no—
175C22×C4—142no—
176C22×C4C22×D711yescharacteristic
177SD16—282no—
178D8—282no—
179D8—282no—
180D8—282no—
181C2×C8C22×D712yes—
182C2×C8C22×D712yes—
183C4⋊C4—72no—
184C4⋊C4—142no—
185C4⋊C4D2812yes—
186C22⋊C4—142no—
187C22⋊C4—142no—
188C22⋊C4—142no—
189C22⋊C4—142no—
190C22⋊C4D2812yes—
191C42C22×D711yescharacteristic
192C42C22×D711yescharacteristic
193C2×C7—41no—
194C2×C7#53511yescharacteristic
195C2×C7#57612yes—
196D7—81no—
197D7—82no—
198D7—81no—
199D7—81no—
200C23—142no—
201C23—141no—
202C23—141no—
203C23—141no—
204C23—282no—
205C23—562no—
206C23C2×D2811yescharacteristic
207Q8—142no—
208D4—142no—
209D4—142no—
210D4—142no—
211D4—142no—
212D4—142no—
213D4—282no—
214D4—282no—
215D4—562no—
216D4—562no—
217D4—562no—
218D4—562no—
219C2×C4—22no—
220C2×C4—141no—
221C2×C4—142no—
222C2×C4—282no—
223C2×C4—562no—
224C2×C4C2×D2812yes—
225C2×C4C2×D2812yes—
226C2×C4C2×D2811yescharacteristic
227C2×C4C23×D711yescharacteristic
228C8—22no—
229C8—22no—
230C7#322011yescharacteristic, complemented, Sylow
231C22—22no—
232C22—21no—
233C22—282no—
234C22—282no—
235C22—282no—
236C22—282no—
237C22—282no—
238C22—281no—
239C22—281no—
240C22—281no—
241C22—282no—
242C22—564no—
243C22C22×D2811yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—282no—
248C4—281no—
249C4#655312yes—
250C2—41no—
251C2—561no—
252C2—561no—
253C2—562no—
254C2—561no—
255C2#9021611yescharacteristic
256C2#9044312yes—
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
hypercenter#433D7
hypocenterC7#3220
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
1. #90226C2semidirect
2. #90216C2semidirect
3. C7×#433C2semidirect
4. #89616C2semidirect
5. #89615C2semidirect
6. #89506C2semidirect
7. #89504C2semidirect
8. #89485C2semidirect
9. #89484C2semidirect
10. #89469C2semidirect
11. #89467C2semidirect
18. C2×D56C22semidirect
19. C2×#1132C22semidirect
28. #6437C22semidirect
29. C4×D28C22semidirect
30. #6433C22semidirect
31. C7×C4⋊C8C22semidirect
33. C7×#97C22semidirect
34. C4×C7×C8C22semidirect
37. #6395C22semidirect
38. #6390C22semidirect
83. #433D7semidirect
133. C4⋊C8C2×D7semidirect
139. #97C2×D7semidirect
140. C4×C8C2×D7semidirect
230. C7#3220semidirect

Maximal subgroups

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

subgroupindex
#902262
#902162
C7×#4332
#896162
#896152
#895062
#895042
#894852
#894842
#894692
#894672
#32207