Finite Group Databasebeta

← #11734318

Subgroups — #11734318

Order 896.

796 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 141 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×#6510C212yesmaximal, complemented
2C2×#6445C212yesmaximal, complemented
3C7×#393C211yesmaximal, characteristic, complemented
4#89732C211yesmaximal, characteristic, complemented
5#89729C212yesmaximal, complemented
6#89727C211yesmaximal, characteristic, complemented
7#89444C212yesmaximal, complemented
8#89443C212yesmaximal, complemented
9#89441C212yesmaximal, complemented
10C24×D7C2211yescharacteristic
11C22×C7×Q8C2211yescharacteristic, complemented
12C22×C22⋊D7C2212yes—
13C22×D28C2212yes—
14C22×D28C2212yes—
15C22×D28C2211yescharacteristic, complemented
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C22⋊Dic7C2211yescharacteristic
19#6510—28no—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yescomplemented
25C2×(C2×D7)⋊C4C2212yescomplemented
26C2×(C2×D7)⋊C4C2212yescomplemented
27C2×Dic7⋊C4C2212yescomplemented
28C2×C4×Dic7C2212yescomplemented
29#6445—28no—
30C7×#95C2212yescomplemented
31C7×#95C2211yescharacteristic, complemented
32C7⋊#95C2211yescharacteristic, complemented
33#2670—71nomaximal, Sylow
34C24×C7C2311yescharacteristic
35C23×D7—22no—
36C23×D7—24no—
37C23×D7—42no—
38C23×D7—41no—
39C23×D7—41no—
40C23×D7C2312yes—
41C23×D7C2312yes—
42C2×C7×Q8—24no—
43C2×C7×Q8—41no—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2312yes—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—42no—
51C22×Dic7C2312yes—
52C22×Dic7C2312yes—
53C2×D28—24no—
54C2×D28—24no—
55C2×D28—24no—
56C2×D28—42no—
57C2×D28—42no—
58C2×D28—42no—
59C2×D28—42no—
60C2×D28—41no—
61C7×C22⋊C4—24no—
62C7×C22⋊C4—41no—
63C22⋊Dic7—41no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—42no—
69(C2×D7)⋊C4—42no—
70(C2×D7)⋊C4D418yescomplemented
71Dic7⋊C4—42no—
72C4×Dic7—24no—
73C2×#124—72no—
74C2×#120—72no—
75#394—72no—
76#393D711yescharacteristic, complemented
77#392—71no—
78#392—72no—
79#390—72no—
80#388—72no—
81#379—71no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7—22no—
86C23×C7#4112yes—
87C23×C7C2411yescharacteristic
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—44no—
102C22×D7—48no—
103C22×D7—48no—
104C22×D7—44no—
105C22×D7—44no—
106C22×D7C2×D414yes—
107C7×Q8—44no—
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—21no—
116C2×C4×C7—21no—
117C2×C4×C7C2×D414yes—
118C22⋊D7—48no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7C2×D414yes—
125D28—48no—
126D28—48no—
127D28—44no—
128C25—71no—
129C22×Q8C2×D711yescharacteristic, complemented
130C22×D4—72no—
131C22×D4—72no—
132C22×D4—72no—
133C22×D4—71no—
134#124—148no—
135#120—148no—
136C2×C4⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—72no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—72no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4C2×D712yes—
146C2×C22⋊C4C2×D711yescharacteristic
147C2×C42—72no—
148#95—71no—
149#95C2×D712yescomplemented
150#95C2×D711yescharacteristic, complemented
151C22×C7—22no—
152C22×C7—24no—
153C22×C7—42no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7C22×D412yes—
157C22×C7C2×#4111yescharacteristic
158C22×C7H(5,𝔽2)11yescharacteristic
159C22×C7H(5,𝔽2)11yescharacteristic
160C22×C7H(5,𝔽2)12yes—
161C2×D7—24no—
162C2×D7—24no—
163C2×D7—24no—
164C2×D7—416no—
165C2×D7—42no—
166C2×D7—42no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C4×C7—24no—
177C4×C7—42no—
178C4×C7—41no—
179C4×C7—41no—
180Dic7—24no—
181Dic7—42no—
182C24—72no—
183C24—72no—
184C24—144no—
185C24—142no—
186C24—281no—
187C24—281no—
188C24—282no—
189C24C22×D711yescharacteristic
190C2×Q8—24no—
191C2×Q8—41no—
192C2×D4—144no—
193C2×D4—144no—
194C2×D4—144no—
195C2×D4—144no—
196C2×D4—144no—
197C2×D4—282no—
198C2×D4—281no—
199C2×D4—282no—
200C2×D4—282no—
201C2×D4—282no—
202C2×D4—282no—
203C22×C4—72no—
204C22×C4—72no—
205C22×C4C22×D711yescharacteristic
206C22×C4C22×D712yes—
207C22×C4C22×D711yescharacteristic
208C22×C4C22×D712yes—
209C4⋊C4—282no—
210C22⋊C4—24no—
211C22⋊C4—41no—
212C22⋊C4—78no—
213C22⋊C4—144no—
214C22⋊C4—144no—
215C22⋊C4—144no—
216C22⋊C4—144no—
217C22⋊C4—282no—
218C22⋊C4—282no—
219C22⋊C4—281no—
220C42—144no—
221C2×C7—42no—
222C2×C7#53512yes—
223C2×C7#53811yescharacteristic
224C2×C7D4211yescharacteristic
225C2×C7#54612yes—
226C2×C7H(3,𝔽4)11yescharacteristic
227D7—44no—
228D7—82no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—74no—
234C23—142no—
235C23—142no—
236C23—142no—
237C23—142no—
238C23—142no—
239C23—142no—
240C23—142no—
241C23—142no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—142no—
246C23—142no—
247C23—288no—
248C23—284no—
249C23—284no—
250C23—288no—
251C23—284no—
252C23#116112yes—
253C23C23×D711yescharacteristic
254Q8—44no—
255D4—288no—
256D4—288no—
257D4—284no—
258D4—288no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—74no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4D4×D714yes—
275C7#267011yescharacteristic, complemented, Sylow
276C22—24no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—144no—
282C22—144no—
283C22—144no—
284C22—2816no—
285C22—282no—
286C22—282no—
287C22—282no—
288C22—282no—
289C22—282no—
290C22—282no—
291C22—282no—
292C22—282no—
293C22—282no—
294C22—282no—
295C22—282no—
296C22C2×D4×D712yes—
297C22#654811yescharacteristic
298C22C2×#116111yescharacteristic
299C22#655312yes—
300C22#655311yescharacteristic
301C4—24no—
302C4—41no—
303C4—41no—
304C4—42no—
305C4—144no—
306C4—282no—
307C2—42no—
308C2—284no—
309C2—562no—
310C2#9028111yescharacteristic
311C2#9032112yes—
312C2#9033511yescharacteristic
313C2#9036512yes—
314C2#9037411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
hypercenter#393D7
hypocenterC7#2670
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#6510C2semidirect
2. C2×#6445C2semidirect
3. C7×#393C2semidirect
4. #89732C2semidirect
5. #89729C2semidirect
6. #89727C2semidirect
7. #89444C2semidirect
8. #89443C2semidirect
9. #89441C2semidirect
11. C22×C7×Q8C22semidirect
15. C22×D28C22semidirect
24. C2×(C2×D7)⋊C4C22semidirect
25. C2×(C2×D7)⋊C4C22semidirect
26. C2×(C2×D7)⋊C4C22semidirect
27. C2×Dic7⋊C4C22semidirect
28. C2×C4×Dic7C22semidirect
30. C7×#95C22semidirect
31. C7×#95C22semidirect
32. C7⋊#95C22semidirect
70. (C2×D7)⋊C4D4semidirect
76. #393D7semidirect
129. C22×Q8C2×D7semidirect
149. #95C2×D7semidirect
150. #95C2×D7semidirect
275. C7#2670semidirect

Maximal subgroups

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

subgroupindex
C2×#65102
C2×#64452
C7×#3932
#897322
#897292
#897272
#894442
#894432
#894412
#26707