Finite Group Databasebeta

← #11734310

Subgroups — #11734310

Order 896.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#6509C212yesmaximal, complemented
2C2×D7×C22⋊C4C212yesmaximal
3C7×#393C211yesmaximal, characteristic, complemented
4#89727C211yesmaximal, characteristic
5#89721C211yesmaximal, characteristic, complemented
6#89720C212yesmaximal, complemented
7#89440C212yesmaximal, complemented
8#89440C212yesmaximal, complemented
9#89439C212yesmaximal, complemented
10C24×D7C2211yescharacteristic
11C22×C7×Q8C2211yescharacteristic, complemented
12C22×C4×D7C2212yes—
13C22×C4×D7C2212yes—
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18#6509—28no—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2212yescomplemented
24C2×C4⋊Dic7C2211yescharacteristic, complemented
25C2×Dic7⋊C4C2212yescomplemented
26C2×Dic7⋊C4C2212yescomplemented
27C2×Dic7⋊C4C2212yescomplemented
28D7×C22⋊C4—28no—
29C7×#95C2211yescharacteristic, complemented
30C7×#95C2212yescomplemented
31C7⋊#95C2212yescomplemented
32C7⋊#95C2211yescharacteristic, complemented
33#2437—71nomaximal, Sylow
34C24×C7C2311yescharacteristic
35C23×D7—24no—
36C23×D7—24no—
37C23×D7—41no—
38C23×D7—41no—
39C23×D7Q814yescomplemented
40C23×D7C2312yes—
41C2×C7×Q8—24no—
42C2×C7×Q8—41no—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C2×C4×D7—24no—
51C2×C4×D7—24no—
52C2×C4×D7—24no—
53C2×C4×D7—24no—
54C2×C4×D7—42no—
55C2×C4×D7—42no—
56C7×C22⋊C4—24no—
57C7×C22⋊C4—41no—
58C22⋊Dic7—24no—
59C22⋊Dic7—41no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4—24no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4D418yes—
64C4⋊Dic7—24no—
65C4⋊Dic7—41no—
66Dic7⋊C4—24no—
67Dic7⋊C4—24no—
68Dic7⋊C4—42no—
69C2×C22⋊Q8—72no—
70C22×C22⋊C4—72no—
71#399—71no—
72#396—72no—
73#393D711yescharacteristic, complemented
74#385—72no—
75#385—72no—
76#380—72no—
77#379—71no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7—21no—
81C23×C7—21no—
82C23×C7C2×Q812yes—
83C23×C7C2411yescharacteristic
84C22×D7—22no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—216no—
89C22×D7—44no—
90C22×D7—44no—
91C22×D7—48no—
92C22×D7—44no—
93C22×D7C2×D414yes—
94C22×D7C2×Q814yes—
95C22×D7#4114yes—
96C7×Q8—44no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7C2×D414yes—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7C2×D414yes—
116C4×D7—48no—
117C4×D7—48no—
118C25—71no—
119C22×Q8C2×D711yescharacteristic, complemented
120C23×C4—72no—
121C23×C4—72no—
122C22⋊Q8—148no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4—71no—
128C2×C22⋊C4—72no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—71no—
134C2×C22⋊C4—148no—
135C2×C22⋊C4C2×D712yes—
136C2×C22⋊C4C2×D711yescharacteristic
137#95—72no—
138#95—71no—
139#95C2×D711yescharacteristic, complemented
140#95C2×D712yescomplemented
141C22×C7—24no—
142C22×C7—22no—
143C22×C7—41no—
144C22×C7—42no—
145C22×C7—41no—
146C22×C7C22×D412yes—
147C22×C7C22×Q811yescharacteristic
148C22×C7C2×#4112yes—
149C22×C7H(5,𝔽2)11yescharacteristic
150C22×C7H(5,𝔽2)11yescharacteristic
151C2×D7—24no—
152C2×D7—24no—
153C2×D7—24no—
154C2×D7—24no—
155C2×D7—24no—
156C2×D7—24no—
157C2×D7—416no—
158C2×D7C22⋊Q818yescomplemented
159C4×C7—24no—
160C4×C7—41no—
161C4×C7—42no—
162C4×C7—41no—
163Dic7—24no—
164Dic7—42no—
165Dic7—42no—
166C24—72no—
167C24—74no—
168C24—144no—
169C24—144no—
170C24—281no—
171C24—281no—
172C24C22×D711yescharacteristic
173C2×Q8—24no—
174C2×Q8—41no—
175C22×C4—72no—
176C22×C4—72no—
177C22×C4—72no—
178C22×C4—144no—
179C22×C4—144no—
180C22×C4—144no—
181C22×C4—144no—
182C22×C4—282no—
183C22×C4—282no—
184C22×C4C22×D711yescharacteristic
185C22×C4C22×D712yes—
186C22×C4C22×D711yescharacteristic
187C22×C4C22×D712yes—
188C4⋊C4—144no—
189C4⋊C4—144no—
190C4⋊C4—144no—
191C4⋊C4—282no—
192C4⋊C4—281no—
193C22⋊C4—24no—
194C22⋊C4—41no—
195C22⋊C4—78no—
196C22⋊C4—144no—
197C22⋊C4—144no—
198C22⋊C4—144no—
199C22⋊C4—144no—
200C22⋊C4—281no—
201C2×C7—42no—
202C2×C7C2×C22⋊Q812yes—
203C2×C7#54311yescharacteristic
204C2×C7D4211yescharacteristic
205C2×C7#54612yes—
206C2×C7#55111yescharacteristic
207D7—28no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—74no—
213C23—74no—
214C23—74no—
215C23—1416no—
216C23—142no—
217C23—142no—
218C23—142no—
219C23—142no—
220C23—284no—
221C23—284no—
222C23—284no—
223C23—288no—
224C23Q8×D712yes—
225C23C23×D711yescharacteristic
226Q8—44no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—74no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—288no—
246C2×C4—288no—
247C2×C4D4×D714yes—
248C7#243711yescharacteristic, complemented, Sylow
249C22—24no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—78no—
255C22—144no—
256C22—144no—
257C22—144no—
258C22—144no—
259C22—144no—
260C22—144no—
261C22—2816no—
262C22C2×D4×D712yes—
263C22#654811yescharacteristic
264C22C2×Q8×D711yescharacteristic
265C22D7×#4112yes—
266C22#655311yescharacteristic
267C4—24no—
268C4—41no—
269C4—41no—
270C4—42no—
271C4—144no—
272C4—282no—
273C4—282no—
274C2—42no—
275C2—148no—
276C2#9028111yescharacteristic
277C2D7×C22⋊Q812yes—
278C2#9032511yescharacteristic
279C2#9036712yes—
280C2#9037011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC24×D7C22
hypercenter#393D7
hypocenterC7#2437
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#6509C2semidirect
3. C7×#393C2semidirect
5. #89721C2semidirect
6. #89720C2semidirect
7. #89440C2semidirect
8. #89440C2semidirect
9. #89439C2semidirect
11. C22×C7×Q8C22semidirect
23. C2×C4⋊Dic7C22semidirect
24. C2×C4⋊Dic7C22semidirect
25. C2×Dic7⋊C4C22semidirect
26. C2×Dic7⋊C4C22semidirect
27. C2×Dic7⋊C4C22semidirect
29. C7×#95C22semidirect
30. C7×#95C22semidirect
31. C7⋊#95C22semidirect
32. C7⋊#95C22semidirect
39. C23×D7Q8semidirect
73. #393D7semidirect
119. C22×Q8C2×D7semidirect
139. #95C2×D7semidirect
140. #95C2×D7semidirect
158. C2×D7C22⋊Q8semidirect
248. C7#2437semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×D7×C22⋊C42
C7×#3932
#897272
#897212
#897202
#894402
#894392
#24377