Finite Group Databasebeta

← #11731287

Subgroups — #11731287

Order 896.

552 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 125 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×#379C211yesmaximal, characteristic
2#90019C212yesmaximal
3#89725C211yesmaximal, characteristic, complemented
4#89724C212yesmaximal, complemented
5#89723C212yesmaximal, complemented
6#89721C211yesmaximal, characteristic, complemented
7#89720C212yesmaximal, complemented
8#89717C212yesmaximal, complemented
9#89716C212yesmaximal, complemented
10C25×C7C2211yescharacteristic
11C2×C7×C22⋊C4C2212yes—
12C2×C7×C22⋊C4C2212yes—
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic, complemented
22C2×C4⋊Dic7C2211yescharacteristic, complemented
23C2×Dic7⋊C4C2212yescomplemented
24C2×Dic7⋊C4C2212yescomplemented
25C2×C4×Dic7C2212yescomplemented
26C7⋊#95C2211yescharacteristic, complemented
27C7⋊#95C2212yescomplemented
28C7⋊#95C2212yescomplemented
29C7⋊#95C2211yescharacteristic, complemented
30C7⋊#95C2212yescomplemented
31#2735—71nomaximal, Sylow
32C24×C7—22no—
33C24×C7—24no—
34C24×C7—42no—
35C24×C7—41no—
36C24×C7—41no—
37C24×C7C2312yes—
38C24×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C7×C22⋊C4—24no—
47C7×C22⋊C4—42no—
48C7×C22⋊C4—41no—
49C7×C22⋊C4—41no—
50C22⋊Dic7—24no—
51C22⋊Dic7—24no—
52C22⋊Dic7—42no—
53C22⋊Dic7—42no—
54C22⋊Dic7—42no—
55C22⋊Dic7—42no—
56C4⋊Dic7—41no—
57C4⋊Dic7—41no—
58Dic7⋊C4—42no—
59Dic7⋊C4—42no—
60C4×Dic7—42no—
61#399—71no—
62#397—72no—
63#397—71no—
64#396—72no—
65#388—72no—
66#388—72no—
67#388—72no—
68#379—72no—
69#379D711yescharacteristic
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—22no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—21no—
78C23×C7—21no—
79C23×C7—22no—
80C23×C7—48no—
81C23×C7—44no—
82C23×C7—44no—
83C23×C7—48no—
84C23×C7—44no—
85C23×C7#4112yes—
86C23×C7#4114yes—
87C23×C7C2411yescharacteristic
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
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—
107C25C2×D711yescharacteristic
108C2×C4⋊C4—72no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—71no—
111C2×C4⋊C4—71no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C22⋊C4—72no—
118C2×C22⋊C4C2×D712yes—
119C2×C22⋊C4C2×D711yescharacteristic
120C2×C22⋊C4C2×D712yes—
121C2×C22⋊C4C2×D711yescharacteristic
122C2×C42—72no—
123#95—72no—
124#95—71no—
125#95—72no—
126#95—72no—
127#95—71no—
128C22×C7—22no—
129C22×C7—24no—
130C22×C7—24no—
131C22×C7—24no—
132C22×C7—24no—
133C22×C7—416no—
134C22×C7—42no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7—42no—
138C22×C7—41no—
139C22×C7—42no—
140C22×C7—41no—
141C22×C7C2×#4111yescharacteristic
142C22×C7C2×#4112yes—
143C22×C7H(5,𝔽2)11yescharacteristic
144C22×C7H(5,𝔽2)11yescharacteristic
145C22×C7H(5,𝔽2)12yes—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—41no—
149Dic7—42no—
150Dic7—42no—
151Dic7—42no—
152Dic7—42no—
153C24—22no—
154C24—24no—
155C24—41no—
156C24—42no—
157C24—41no—
158C24C22×D711yescharacteristic
159C24C22×D712yes—
160C22×C4—72no—
161C22×C4—72no—
162C22×C4—72no—
163C22×C4—72no—
164C22×C4C22×D711yescharacteristic
165C22×C4C22×D712yes—
166C22×C4C22×D711yescharacteristic
167C4⋊C4—282no—
168C4⋊C4—282no—
169C4⋊C4—281no—
170C4⋊C4—281no—
171C22⋊C4—24no—
172C22⋊C4—42no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—144no—
176C22⋊C4—144no—
177C22⋊C4—282no—
178C22⋊C4—282no—
179C22⋊C4—282no—
180C22⋊C4—282no—
181C42—282no—
182C2×C7—42no—
183C2×C7—44no—
184C2×C7#53811yescharacteristic
185C2×C7#53812yes—
186C2×C7#55112yes—
187C2×C7#55111yescharacteristic
188C2×C7#56011yescharacteristic
189C23—22no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—48no—
200C23—44no—
201C23—44no—
202C23—48no—
203C23—44no—
204C23#115712yes—
205C23#115914yes—
206C23C23×D711yescharacteristic
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C7#273511yescharacteristic, complemented, Sylow
227C22—22no—
228C22—24no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—416no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—42no—
240C22C2×#115711yescharacteristic
241C22C2×#115912yes—
242C22#654811yescharacteristic
243C22#654811yescharacteristic
244C22#654812yes—
245C4—41no—
246C4—42no—
247C4—41no—
248C4—282no—
249C4—282no—
250C4—282no—
251C4—282no—
252C2—42no—
253C2—44no—
254C2#9018511yescharacteristic
255C2#9018612yes—
256C2#9027811yescharacteristic
257C2#9028412yes—
258C2#9028611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#379C2
prime radicalC25×C7C22
hypercenter#379D7
hypocenterC7#2735
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
3. #89725C2semidirect
4. #89724C2semidirect
5. #89723C2semidirect
6. #89721C2semidirect
7. #89720C2semidirect
8. #89717C2semidirect
9. #89716C2semidirect
21. C2×C4⋊Dic7C22semidirect
22. C2×C4⋊Dic7C22semidirect
23. C2×Dic7⋊C4C22semidirect
24. C2×Dic7⋊C4C22semidirect
25. C2×C4×Dic7C22semidirect
26. C7⋊#95C22semidirect
27. C7⋊#95C22semidirect
28. C7⋊#95C22semidirect
29. C7⋊#95C22semidirect
30. C7⋊#95C22semidirect
226. C7#2735semidirect

Maximal subgroups

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

subgroupindex
C7×#3792
#900192
#897252
#897242
#897232
#897212
#897202
#897172
#897162
#27357