Finite Group Databasebeta

← #11734316

Subgroups — #11734316

Order 896.

668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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×#6509C211yesmaximal, characteristic, complemented
2C2×D7×C22⋊C4C211yesmaximal, characteristic
3C7×#393C211yesmaximal, characteristic, complemented
4#90003C211yesmaximal, characteristic, complemented
5#89727C212yesmaximal
6#89721C211yesmaximal, characteristic, complemented
7#89720C212yesmaximal, complemented
8#89449C211yesmaximal, characteristic, complemented
9#89448C212yesmaximal, complemented
10#89440C212yesmaximal, complemented
11#89439C211yesmaximal, characteristic, complemented
12C24×D7C2211yescharacteristic
13C22×C7×Q8C2211yescharacteristic, complemented
14C22×C4×D7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20#6509—24no—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×C4⋊Dic7C2211yescharacteristic, complemented
28C2×C4⋊Dic7C2212yescomplemented
29C2×Dic7⋊C4C2212yescomplemented
30C2×Dic7⋊C4C2212yescomplemented
31C2×Dic7⋊C4C2212yescomplemented
32D7×C22⋊C4—24no—
33C7×#95C2211yescharacteristic, complemented
34C7×#95C2212yescomplemented
35C7⋊#95C2211yescharacteristic, complemented
36C7⋊#95C2212yescomplemented
37#2626—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7—22no—
40C23×D7—22no—
41C23×D7—42no—
42C23×D7—42no—
43C23×D7Q814yescomplemented
44C23×D7C2311yescharacteristic
45C23×D7C2311yescharacteristic
46C2×C7×Q8—22no—
47C2×C7×Q8—42no—
48C22×C4×C7C2312yes—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2312yes—
51C22×C4×C7C2311yescharacteristic
52C22×Dic7C2312yes—
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2312yes—
55C22×Dic7C2311yescharacteristic
56C2×C4×D7—22no—
57C2×C4×D7—42no—
58C2×C4×D7—42no—
59C2×C4×D7D414yes—
60C7×C22⋊C4—22no—
61C7×C22⋊C4—42no—
62C22⋊Dic7—22no—
63C22⋊Dic7—42no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—42no—
69(C2×D7)⋊C4—42no—
70C4⋊Dic7—22no—
71C4⋊Dic7—42no—
72Dic7⋊C4—24no—
73Dic7⋊C4—42no—
74Dic7⋊C4—42no—
75C2×C22⋊Q8—71no—
76C22×C22⋊C4—71no—
77#399—71no—
78#399—71no—
79#396—72no—
80#396—72no—
81#393—71no—
82#393D711yescharacteristic, complemented
83#385—72no—
84#380—71no—
85#379—72no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7C2×Q812yes—
91C23×C7C2411yescharacteristic
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—21no—
95C22×D7—21no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—28no—
99C22×D7—48no—
100C22×D7—44no—
101C22×D7—44no—
102C22×D7—48no—
103C22×D7C2×D412yes—
104C22×D7C2×Q812yes—
105C22×D7C2×Q812yes—
106C22×D7#4112yes—
107C7×Q8—44no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7C2×D412yes—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—21no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7C2×D412yes—
128C4×D7—44no—
129C4×D7—44no—
130C25—71no—
131C22×Q8C2×D711yescharacteristic, complemented
132C23×C4—71no—
133C23×C4—71no—
134C22⋊Q8—144no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—72no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4—72no—
140C2×C22⋊C4—71no—
141C2×C22⋊C4—71no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—72no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4—144no—
149C2×C22⋊C4C2×D711yescharacteristic
150C2×C22⋊C4C2×D712yes—
151#95—71no—
152#95—72no—
153#95C2×D711yescharacteristic, complemented
154#95C2×D712yescomplemented
155C22×C7—22no—
156C22×C7—24no—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—42no—
160C22×C7C22×D411yescharacteristic
161C22×C7C22×Q811yescharacteristic
162C22×C7C2×#4111yescharacteristic
163C22×C7H(5,𝔽2)12yes—
164C22×C7H(5,𝔽2)12yes—
165C2×D7—22no—
166C2×D7—22no—
167C2×D7—24no—
168C2×D7—24no—
169C2×D7—22no—
170C2×D7—24no—
171C2×D7—416no—
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—42no—
175C2×D7C22⋊Q814yes—
176C4×C7—22no—
177C4×C7—42no—
178C4×C7—41no—
179C4×C7—42no—
180Dic7—22no—
181Dic7—41no—
182Dic7—42no—
183Dic7—42no—
184C24—71no—
185C24—71no—
186C24—74no—
187C24—142no—
188C24—142no—
189C24—282no—
190C24—282no—
191C24C22×D711yescharacteristic
192C2×Q8—22no—
193C2×Q8—42no—
194C22×C4—71no—
195C22×C4—74no—
196C22×C4—72no—
197C22×C4—71no—
198C22×C4—72no—
199C22×C4—142no—
200C22×C4—282no—
201C22×C4—282no—
202C22×C4C22×D712yes—
203C22×C4C22×D712yes—
204C22×C4C22×D711yescharacteristic
205C22×C4C22×D711yescharacteristic
206C4⋊C4—144no—
207C4⋊C4—142no—
208C4⋊C4—282no—
209C4⋊C4—282no—
210C4⋊C4—282no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—142no—
214C22⋊C4—144no—
215C22⋊C4—144no—
216C22⋊C4—142no—
217C22⋊C4—142no—
218C22⋊C4—282no—
219C22⋊C4—282no—
220C22⋊C4—282no—
221C2×C7—42no—
222C2×C7C2×C22⋊Q811yescharacteristic
223C2×C7#53411yescharacteristic
224C2×C7#53511yescharacteristic
225C2×C7#53812yes—
226C2×C7#54312yes—
227D7—24no—
228D7—42no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—72no—
234C23—72no—
235C23—72no—
236C23—72no—
237C23—148no—
238C23—142no—
239C23—142no—
240C23—141no—
241C23—142no—
242C23—141no—
243C23—142no—
244C23—288no—
245C23—288no—
246C23—284no—
247C23—284no—
248C23Q8×D712yes—
249C23C23×D711yescharacteristic
250Q8—44no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—72no—
261C2×C4—142no—
262C2×C4—141no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—284no—
271C2×C4—284no—
272C2×C4D4×D712yes—
273C7#262611yescharacteristic, complemented, Sylow
274C22—22no—
275C22—24no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—74no—
280C22—144no—
281C22—142no—
282C22—144no—
283C22—142no—
284C22—142no—
285C22—144no—
286C22—2816no—
287C22—282no—
288C22—281no—
289C22—281no—
290C22C2×D4×D711yescharacteristic
291C22#654812yes—
292C22C2×Q8×D711yescharacteristic
293C22D7×#4111yescharacteristic
294C22#655312yes—
295C4—22no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—142no—
300C4—282no—
301C4—281no—
302C4—282no—
303C2—42no—
304C2—144no—
305C2—282no—
306C2#9028011yescharacteristic
307C2D7×C22⋊Q811yescharacteristic
308C2#9032512yes—
309C2#9036511yescharacteristic
310C2#9036912yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC24×D7C22
hypercenter#393D7
hypocenterC7#2626
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
4. #90003C2semidirect
6. #89721C2semidirect
7. #89720C2semidirect
8. #89449C2semidirect
9. #89448C2semidirect
10. #89440C2semidirect
11. #89439C2semidirect
13. C22×C7×Q8C22semidirect
27. C2×C4⋊Dic7C22semidirect
28. C2×C4⋊Dic7C22semidirect
29. C2×Dic7⋊C4C22semidirect
30. C2×Dic7⋊C4C22semidirect
31. C2×Dic7⋊C4C22semidirect
33. C7×#95C22semidirect
34. C7×#95C22semidirect
35. C7⋊#95C22semidirect
36. C7⋊#95C22semidirect
43. C23×D7Q8semidirect
82. #393D7semidirect
131. C22×Q8C2×D7semidirect
153. #95C2×D7semidirect
154. #95C2×D7semidirect
273. C7#2626semidirect

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
#900032
#897272
#897212
#897202
#894492
#894482
#894402
#894392
#26267