Finite Group Databasebeta

← #11732066

Subgroups — #11732066

Order 896.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×(C2×D7)⋊C4C211yesmaximal, characteristic
2C2×#6441C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic, complemented
4#89982C211yesmaximal, characteristic, complemented
5#89766C212yesmaximal, complemented
6#89764C212yesmaximal, complemented
7#89727C211yesmaximal, characteristic
8#89727C211yesmaximal, characteristic
9#89714C211yesmaximal, characteristic, complemented
10#89448C212yesmaximal, complemented
11#89445C212yesmaximal, complemented
12C24×D7C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic, complemented
16C2×C7×C4⋊C4C2212yescomplemented
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊Dic7C2211yescharacteristic
21C2×(C2×D7)⋊C4—24no—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic, complemented
29C2×Dic7⋊C4C2211yescharacteristic, complemented
30C2×Dic7⋊C4C2211yescharacteristic, complemented
31C2×Dic7⋊C4C2212yescomplemented
32C2×Dic7⋊C4C2212yescomplemented
33#6441—24no—
34C7×#95C2212yescomplemented
35C7⋊#95C2212yescomplemented
36C7⋊#95C2212yescomplemented
37#2626—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7—22no—
40C23×D7—22no—
41C23×D7—42no—
42C23×D7—42no—
43C23×D7Q814yescomplemented
44C23×D7C2312yes—
45C22×C4×C7—22no—
46C22×C4×C7—42no—
47C22×C4×C7C2312yes—
48C22×C4×C7C2312yes—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2311yescharacteristic
51C22×Dic7—42no—
52C22×Dic7D414yes—
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2311yescharacteristic
58C2×Dic14—22no—
59C2×Dic14—42no—
60C7×C4⋊C4—42no—
61C7×C22⋊C4—22no—
62C7×C22⋊C4—22no—
63C22⋊Dic7—22no—
64C22⋊Dic7—22no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—42no—
68(C2×D7)⋊C4—42no—
69(C2×D7)⋊C4—42no—
70(C2×D7)⋊C4—42no—
71C4⋊Dic7—22no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74Dic7⋊C4—42no—
75Dic7⋊C4—42no—
76C2×C22⋊Q8—71no—
77C22×C22⋊C4—71no—
78#399—72no—
79#396—72no—
80#396—72no—
81#393—72no—
82#385—71no—
83#385D711yescharacteristic, complemented
84#380—71no—
85#379—71no—
86#379—71no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7C2×D412yes—
90C23×C7C2×Q812yes—
91C23×C7#4112yes—
92C23×C7C2411yescharacteristic
93C22×D7—28no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—48no—
101C22×D7—44no—
102C22×D7—44no—
103C22×D7—48no—
104C22×D7C2×Q814yes—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—44no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—44no—
126C2×Dic7C2×D412yes—
127C2×Dic7C2×D412yes—
128Dic14—44no—
129C25—71no—
130C22×Q8—71no—
131C23×C4—71no—
132C23×C4C2×D711yescharacteristic
133C22⋊Q8—144no—
134C2×C4⋊C4—71no—
135C2×C4⋊C4—71no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—72no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4C2×D712yescomplemented
140C2×C22⋊C4—71no—
141C2×C22⋊C4—71no—
142C2×C22⋊C4—72no—
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×D711yescharacteristic
151#95—72no—
152#95—72no—
153#95C2×D712yescomplemented
154C22×C7—22no—
155C22×C7—22no—
156C22×C7—24no—
157C22×C7—24no—
158C22×C7C22⋊Q814yes—
159C22×C7C22×D411yescharacteristic
160C22×C7C22×Q811yescharacteristic
161C22×C7C2×#4111yescharacteristic
162C22×C7H(5,𝔽2)12yes—
163C22×C7H(5,𝔽2)12yes—
164C2×D7—24no—
165C2×D7—24no—
166C2×D7—24no—
167C2×D7—416no—
168C2×D7—42no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—42no—
172C4×C7—41no—
173C4×C7—41no—
174C4×C7—42no—
175C4×C7—42no—
176Dic7—22no—
177Dic7—22no—
178Dic7—41no—
179Dic7—41no—
180Dic7—42no—
181C24—72no—
182C24—74no—
183C24—142no—
184C24—142no—
185C24—282no—
186C24—282no—
187C24C22×D711yescharacteristic
188C2×Q8—142no—
189C2×Q8—282no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—74no—
195C22×C4—72no—
196C22×C4—71no—
197C22×C4—71no—
198C22×C4—282no—
199C22×C4C22×D712yes—
200C22×C4C22×D712yes—
201C22×C4C22×D711yescharacteristic
202C22×C4C22×D711yescharacteristic
203C4⋊C4—42no—
204C4⋊C4—142no—
205C4⋊C4—142no—
206C4⋊C4—142no—
207C4⋊C4—282no—
208C4⋊C4—282no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—144no—
212C22⋊C4—144no—
213C22⋊C4—142no—
214C22⋊C4—142no—
215C22⋊C4—282no—
216C22⋊C4—282no—
217C22⋊C4—282no—
218C22⋊C4—282no—
219C2×C7—24no—
220C2×C7C2×C22⋊Q811yescharacteristic
221C2×C7#53411yescharacteristic
222C2×C7#53511yescharacteristic
223C2×C7#53812yes—
224C2×C7#54312yes—
225D7—44no—
226C23—22no—
227C23—22no—
228C23—74no—
229C23—148no—
230C23—142no—
231C23—142no—
232C23—142no—
233C23—142no—
234C23—142no—
235C23—142no—
236C23—288no—
237C23—288no—
238C23—284no—
239C23—284no—
240C23#115712yes—
241C23D4×D712yes—
242C23Q8×D712yes—
243C23C23×D711yescharacteristic
244Q8—284no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—72no—
257C2×C4—72no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—284no—
268C7#262611yescharacteristic, complemented, Sylow
269C22—24no—
270C22—22no—
271C22—24no—
272C22—22no—
273C22—144no—
274C22—144no—
275C22—144no—
276C22—2816no—
277C22—282no—
278C22—282no—
279C22—282no—
280C22—282no—
281C22#645914yes—
282C22C2×#115711yescharacteristic
283C22C2×D4×D711yescharacteristic
284C22#654812yes—
285C22C2×Q8×D711yescharacteristic
286C22#655312yes—
287C4—42no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—142no—
292C4—142no—
293C4—282no—
294C4—281no—
295C4—281no—
296C2—24no—
297C2—284no—
298C2C2×#645911yescharacteristic
299C2#9025112yes—
300C2#9028311yescharacteristic
301C2#9032512yes—
302C2#9034411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×D7C22
hypercenter#385D7
hypocenterC7#2626
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6441C2semidirect
3. C7×#385C2semidirect
4. #89982C2semidirect
5. #89766C2semidirect
6. #89764C2semidirect
9. #89714C2semidirect
10. #89448C2semidirect
11. #89445C2semidirect
15. C22×Dic14C22semidirect
16. C2×C7×C4⋊C4C22semidirect
28. C2×C4⋊Dic7C22semidirect
29. C2×Dic7⋊C4C22semidirect
30. C2×Dic7⋊C4C22semidirect
31. C2×Dic7⋊C4C22semidirect
32. C2×Dic7⋊C4C22semidirect
34. C7×#95C22semidirect
35. C7⋊#95C22semidirect
36. C7⋊#95C22semidirect
43. C23×D7Q8semidirect
83. #385D7semidirect
139. C2×C4⋊C4C2×D7semidirect
153. #95C2×D7semidirect
268. C7#2626semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D7)⋊C42
C2×#64412
C7×#3852
#899822
#897662
#897642
#897272
#897142
#894482
#894452
#26267