Finite Group Databasebeta

← #11735283

Subgroups — #11735283

Order 896.

598 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6458C211yesmaximal, characteristic, complemented
2C2×#6457C211yesmaximal, characteristic, complemented
3C7×#398C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89763C211yesmaximal, characteristic, complemented
6#89763C212yesmaximal, complemented
7#89760C212yesmaximal, complemented
8#89443C211yesmaximal, characteristic, complemented
9#89443C212yesmaximal, complemented
10#89441C212yesmaximal, complemented
11#89438C211yesmaximal, characteristic, complemented
12C22×D28C2211yescharacteristic
13C22×D28C2212yes—
14C22×D28C2211yescharacteristic
15C22×D28C2212yes—
16C22×C4×D7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2211yescharacteristic
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×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×Dic7⋊C4C2211yescharacteristic
30C2×C4×Dic7C2212yes—
31#6458—24no—
32#6457—24no—
33C7×#95C2211yescharacteristic
34C7×#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2661—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2312yes—
40C23×D7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×D28—22no—
50C2×D28—24no—
51C2×D28—22no—
52C2×D28—24no—
53C2×D28—42no—
54C2×D28—42no—
55C2×D28—42no—
56C2×D28—42no—
57C2×D28—42no—
58C2×D28—42no—
59C2×C4×D7—42no—
60C2×C4×D7D414yescomplemented
61C7×C4⋊C4—22no—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—42no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—24no—
70(C2×D7)⋊C4—42no—
71(C2×D7)⋊C4—42no—
72(C2×D7)⋊C4—42no—
73Dic7⋊C4—22no—
74C4×Dic7—42no—
75C2×#123—71no—
76C2×#121—71no—
77#398D711yescharacteristic, complemented
78#397—71no—
79#394—71no—
80#394—72no—
81#392—71no—
82#392—72no—
83#390—72no—
84#388—72no—
85C4×C22⋊C4—71no—
86C23×C7C2411yescharacteristic
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—21no—
90C22×D7—21no—
91C22×D7—21no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—21no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—21no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7C2×D412yes—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—22no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7C2×D412yes—
116C2×C4×C7#4114yes—
117C2×Dic7—22no—
118C2×Dic7—21no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7C2×D412yes—
124D28—44no—
125D28—48no—
126D28—44no—
127D28—48no—
128C4×D7—44no—
129C22×D4—72no—
130C22×D4—71no—
131C22×D4—71no—
132C22×D4—72no—
133C23×C4—71no—
134#123—144no—
135#121—144no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4C2×D711yescharacteristic
138C2×C4⋊C4C2×D711yescharacteristic
139C2×C4⋊C4C2×D712yes—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—71no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4—72no—
149C2×C42—72no—
150#95—71no—
151#95—72no—
152#95C2×D711yescharacteristic
153#95C2×D712yes—
154C22×C7C22×D411yescharacteristic
155C22×C7C2×#4112yes—
156C22×C7H(5,𝔽2)11yescharacteristic
157C22×C7H(5,𝔽2)12yes—
158C22×C7H(5,𝔽2)11yescharacteristic
159C2×D7—22no—
160C2×D7—22no—
161C2×D7—22no—
162C2×D7—42no—
163C2×D7—42no—
164C2×D7—41no—
165C2×D7—42no—
166C2×D7—42no—
167C2×D7—41no—
168C2×D7—42no—
169C2×D7—41no—
170C2×D7—42no—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C4×C7—22no—
177C4×C7—24no—
178C4×C7—41no—
179C4×C7—42no—
180C4×C7—41no—
181Dic7—22no—
182Dic7—41no—
183Dic7—42no—
184C24—71no—
185C24—71no—
186C24—72no—
187C2×D4—144no—
188C2×D4—142no—
189C2×D4—142no—
190C2×D4—144no—
191C2×D4—282no—
192C2×D4—282no—
193C2×D4—282no—
194C2×D4—282no—
195C2×D4—282no—
196C2×D4—282no—
197C22×C4—71no—
198C22×C4—74no—
199C22×C4—72no—
200C22×C4—71no—
201C22×C4—282no—
202C22×C4C22×D712yes—
203C22×C4C22×D711yescharacteristic
204C22×C4C22×D711yescharacteristic
205C22×C4C22×D711yescharacteristic
206C22×C4C22×D712yes—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—142no—
211C22⋊C4—144no—
212C22⋊C4—142no—
213C22⋊C4—142no—
214C22⋊C4—142no—
215C22⋊C4—144no—
216C22⋊C4—144no—
217C22⋊C4—282no—
218C22⋊C4—282no—
219C22⋊C4—282no—
220C42—282no—
221C2×C7#53211yescharacteristic
222C2×C7#53411yescharacteristic
223C2×C7#53511yescharacteristic
224C2×C7#53812yes—
225C2×C7#54012yes—
226D7—42no—
227D7—82no—
228D7—81no—
229C23—72no—
230C23—141no—
231C23—142no—
232C23—142no—
233C23—141no—
234C23—141no—
235C23—142no—
236C23—142no—
237C23—141no—
238C23—142no—
239C23—142no—
240C23—142no—
241C23—141no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—142no—
246C23C23×D711yescharacteristic
247D4—288no—
248D4—284no—
249D4—288no—
250D4—284no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—72no—
263C2×C4—142no—
264C2×C4—141no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—284no—
270C2×C4D4×D712yes—
271C2×C4#116114yes—
272C7#266111yescharacteristic, complemented, Sylow
273C22—142no—
274C22—142no—
275C22—142no—
276C22—282no—
277C22—282no—
278C22—282no—
279C22—282no—
280C22—282no—
281C22—281no—
282C22—282no—
283C22—282no—
284C22—281no—
285C22—282no—
286C22—282no—
287C22—282no—
288C22—282no—
289C22—281no—
290C22C2×D4×D711yescharacteristic
291C22#654811yescharacteristic
292C22C2×#116112yes—
293C22#655311yescharacteristic
294C22#655312yes—
295C4—24no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—142no—
301C4—282no—
302C4—281no—
303C2—282no—
304C2—562no—
305C2—561no—
306C2#9029811yescharacteristic
307C2#9033512yes—
308C2#9034411yescharacteristic
309C2#9038912yes—
310C2#9041611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2661
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#6458C2semidirect
2. C2×#6457C2semidirect
3. C7×#398C2semidirect
4. #89765C2semidirect
5. #89763C2semidirect
6. #89763C2semidirect
7. #89760C2semidirect
8. #89443C2semidirect
9. #89443C2semidirect
10. #89441C2semidirect
11. #89438C2semidirect
60. C2×C4×D7D4semidirect
77. #398D7semidirect
272. C7#2661semidirect

Maximal subgroups

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

subgroupindex
C2×#64582
C2×#64572
C7×#3982
#897652
#897632
#897602
#894432
#894412
#894382
#26617