Finite Group Databasebeta

← #11735151

Subgroups — #11735151

Order 896.

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

normal subgroups only

This group has 560 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
1C7×#397C211yesmaximal, characteristic, complemented
2#89762C212yesmaximal, complemented
3#89727C212yesmaximal
4#89727C211yesmaximal, characteristic
5#89723C212yesmaximal, complemented
6#89719C211yesmaximal, characteristic, complemented
7#89446C212yesmaximal, complemented
8#89445C212yesmaximal, complemented
9#89442C212yesmaximal, complemented
10C24×D7C2211yescharacteristic
11C22×Dic14C2211yescharacteristic, complemented
12C2×C7×C4⋊C4C2211yescharacteristic, complemented
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2212yescomplemented
24C2×C4×Dic7C2212yescomplemented
25C7×#95C2212yescomplemented
26C7×#95C2211yescharacteristic, complemented
27C7⋊#95C2211yescharacteristic, complemented
28C7⋊#95C2212yescomplemented
29C7⋊#95C2212yescomplemented
30C7⋊#95C2212yescomplemented
31#2736—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C23×D7—22no—
34C23×D7—24no—
35C23×D7—41no—
36C23×D7—42no—
37C23×D7—41no—
38C23×D7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C2×Dic14—41no—
47C2×Dic14—42no—
48C7×C4⋊C4—41no—
49C7×C22⋊C4—24no—
50C7×C22⋊C4—41no—
51C22⋊Dic7—41no—
52C22⋊Dic7—42no—
53(C2×D7)⋊C4—24no—
54(C2×D7)⋊C4—24no—
55(C2×D7)⋊C4—42no—
56(C2×D7)⋊C4—42no—
57(C2×D7)⋊C4—42no—
58(C2×D7)⋊C4—42no—
59C4⋊Dic7—42no—
60C4×Dic7—42no—
61#397—72no—
62#397D711yescharacteristic, complemented
63#393—71no—
64#393—72no—
65#388—72no—
66#388—72no—
67#388—72no—
68#379—71no—
69#379—72no—
70C23×C7—22no—
71C23×C7—21no—
72C23×C7—22no—
73C23×C7—21no—
74C23×C7#4112yes—
75C23×C7C2411yescharacteristic
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C22×D7—44no—
83C22×D7—48no—
84C22×D7—48no—
85C22×D7—44no—
86C22×D7—44no—
87C22×D7#4114yes—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107Dic14—44no—
108C25—71no—
109C22×Q8—71no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4C2×D711yescharacteristic, complemented
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—71no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C22⋊C4—72no—
118C2×C22⋊C4—72no—
119C2×C22⋊C4—72no—
120C2×C22⋊C4C2×D712yes—
121C2×C22⋊C4C2×D711yescharacteristic
122C2×C42—72no—
123#95—72no—
124#95—71no—
125#95—72no—
126#95—72no—
127#95C2×D712yescomplemented
128#95C2×D711yescharacteristic, complemented
129C22×C7—22no—
130C22×C7—24no—
131C22×C7—42no—
132C22×C7—41no—
133C22×C7—41no—
134C22×C7C2×#4111yescharacteristic
135C22×C7C2×#4112yes—
136C22×C7H(5,𝔽2)11yescharacteristic
137C22×C7H(5,𝔽2)11yescharacteristic
138C22×C7H(5,𝔽2)12yes—
139C2×D7—24no—
140C2×D7—24no—
141C2×D7—24no—
142C2×D7—416no—
143C2×D7—42no—
144C2×D7—42no—
145C2×D7—42no—
146C2×D7—42no—
147C4×C7—42no—
148C4×C7—42no—
149C4×C7—41no—
150C4×C7—41no—
151Dic7—42no—
152Dic7—42no—
153Dic7—42no—
154C24—72no—
155C24—144no—
156C24—142no—
157C24—281no—
158C24—282no—
159C24—281no—
160C24C22×D711yescharacteristic
161C2×Q8—281no—
162C2×Q8—282no—
163C22×C4—72no—
164C22×C4—72no—
165C22×C4—72no—
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D712yes—
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D712yes—
170C4⋊C4—41no—
171C4⋊C4—282no—
172C22⋊C4—24no—
173C22⋊C4—41no—
174C22⋊C4—144no—
175C22⋊C4—144no—
176C22⋊C4—282no—
177C22⋊C4—282no—
178C22⋊C4—282no—
179C22⋊C4—282no—
180C22⋊C4—281no—
181C22⋊C4—282no—
182C42—282no—
183C2×C7—42no—
184C2×C7#53812yes—
185C2×C7#53811yescharacteristic
186C2×C7#55112yes—
187C2×C7#55111yescharacteristic
188C2×C7H(3,𝔽4)11yescharacteristic
189D7—44no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—74no—
195C23—142no—
196C23—142no—
197C23—142no—
198C23—142no—
199C23—142no—
200C23—142no—
201C23—288no—
202C23—284no—
203C23—284no—
204C23—288no—
205C23—284no—
206C23#115912yes—
207C23C23×D711yescharacteristic
208Q8—284no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C7#273611yescharacteristic, complemented, Sylow
229C22—22no—
230C22—24no—
231C22—41no—
232C22—42no—
233C22—41no—
234C22—144no—
235C22—144no—
236C22—144no—
237C22—2816no—
238C22—282no—
239C22—282no—
240C22—282no—
241C22—282no—
242C22C2×#115911yescharacteristic
243C22#654811yescharacteristic
244C22D7×#4112yes—
245C22#655312yes—
246C22#655311yescharacteristic
247C4—41no—
248C4—42no—
249C4—42no—
250C4—41no—
251C4—282no—
252C4—282no—
253C4—282no—
254C2—42no—
255C2—284no—
256C2#9030811yescharacteristic
257C2#9034912yes—
258C2#9035011yescharacteristic
259C2#9037411yescharacteristic
260C2#9039512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×D7C22
hypercenter#397D7
hypocenterC7#2736
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C7×#397C2semidirect
2. #89762C2semidirect
5. #89723C2semidirect
6. #89719C2semidirect
7. #89446C2semidirect
8. #89445C2semidirect
9. #89442C2semidirect
11. C22×Dic14C22semidirect
12. C2×C7×C4⋊C4C22semidirect
23. C2×C4⋊Dic7C22semidirect
24. C2×C4×Dic7C22semidirect
25. C7×#95C22semidirect
26. C7×#95C22semidirect
27. C7⋊#95C22semidirect
28. C7⋊#95C22semidirect
29. C7⋊#95C22semidirect
30. C7⋊#95C22semidirect
62. #397D7semidirect
111. C2×C4⋊C4C2×D7semidirect
127. #95C2×D7semidirect
128. #95C2×D7semidirect
228. C7#2736semidirect

Maximal subgroups

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

subgroupindex
C7×#3972
#897622
#897272
#897232
#897192
#894462
#894452
#894422
#27367