Finite Group Databasebeta

← #11731302

Subgroups — #11731302

Order 896.

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

normal subgroups only

This group has 816 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×#6516C212yesmaximal, complemented
2C2×#6447C212yesmaximal, complemented
3C7×#379C211yesmaximal, characteristic, complemented
4#89733C212yesmaximal, complemented
5#89732C211yesmaximal, characteristic, complemented
6#89728C212yesmaximal, complemented
7#89728C212yesmaximal, complemented
8#89724C212yesmaximal, complemented
9#89721C211yesmaximal, characteristic, complemented
10C25×C7C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×C22⋊D7C2212yes—
13C22×C22⋊D7C2212yes—
14C22×D28C2211yescharacteristic, complemented
15C22×C4×D7C2212yescomplemented
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20#6516—28no—
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24C2×(C2×D7)⋊C4C2212yescomplemented
25C2×(C2×D7)⋊C4C2212yescomplemented
26C2×C4⋊Dic7C2211yescharacteristic, complemented
27C2×Dic7⋊C4C2212yescomplemented
28#6447—28no—
29C7⋊#95C2211yescharacteristic, complemented
30C7⋊#95C2211yescharacteristic, complemented
31C7⋊#95C2212yescomplemented
32C7⋊#95C2212yescomplemented
33#2668—71nomaximal, Sylow
34C24×C7—24no—
35C24×C7—22no—
36C24×C7—41no—
37C24×C7—42no—
38C24×C7—41no—
39C24×C7C2312yes—
40C24×C7C2311yescharacteristic
41C23×D7C2312yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—24no—
46C2×C22⋊D7—24no—
47C2×C22⋊D7—24no—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—42no—
50C2×C22⋊D7—42no—
51C2×C22⋊D7—42no—
52C2×C22⋊D7—42no—
53C2×C22⋊D7D418yescomplemented
54C22×Dic7C2312yes—
55C22×Dic7C2312yes—
56C22×Dic7C2312yes—
57C2×D28—24no—
58C2×D28—41no—
59C2×C4×D7—24no—
60C2×C4×D7—42no—
61C2×C4×D7—42no—
62C7×C22⋊C4—24no—
63C7×C22⋊C4—41no—
64C7×C22⋊C4—42no—
65C7×C22⋊C4—41no—
66C22⋊Dic7—24no—
67C22⋊Dic7—24no—
68C22⋊Dic7—24no—
69(C2×D7)⋊C4—24no—
70(C2×D7)⋊C4—42no—
71C4⋊Dic7—41no—
72Dic7⋊C4—24no—
73C2×#121—72no—
74C2×#120—72no—
75#399—71no—
76#397—72no—
77#394—72no—
78#392—71no—
79#386—72no—
80#386—72no—
81#379D711yescharacteristic, complemented
82C23×C7—22no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—21no—
86C23×C7—21no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—44no—
93C23×C7—48no—
94C23×C7—48no—
95C23×C7—44no—
96C23×C7—44no—
97C23×C7C2×D414yes—
98C23×C7#4112yes—
99C23×C7C2411yescharacteristic
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7—22no—
104C22×D7—22no—
105C22×D7—22no—
106C22×D7C2×D414yes—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C22⋊D7—48no—
115C22⋊D7—48no—
116C22⋊D7—48no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7C2×D414yes—
126D28—44no—
127C4×D7—48no—
128C25C2×D711yescharacteristic
129C22×D4—72no—
130C22×D4—72no—
131C22×D4—72no—
132C22×D4—71no—
133C23×C4—72no—
134#121—148no—
135#120—148no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—71no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—72no—
142C2×C22⋊C4—72no—
143C2×C22⋊C4C2×D712yes—
144C2×C22⋊C4C2×D711yescharacteristic
145C2×C22⋊C4C2×D712yes—
146C2×C22⋊C4C2×D711yescharacteristic
147#95—72no—
148#95—71no—
149#95—72no—
150#95—71no—
151C22×C7—24no—
152C22×C7—22no—
153C22×C7—24no—
154C22×C7—24no—
155C22×C7—24no—
156C22×C7—416no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7—41no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—41no—
164C22×C7C22×D412yes—
165C22×C7C2×#4111yescharacteristic
166C22×C7H(5,𝔽2)11yescharacteristic
167C22×C7H(5,𝔽2)12yes—
168C22×C7H(5,𝔽2)11yescharacteristic
169C2×D7—24no—
170C2×D7—24no—
171C2×D7—24no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C4×C7—42no—
177C4×C7—41no—
178C4×C7—41no—
179Dic7—24no—
180Dic7—42no—
181Dic7—42no—
182C24—22no—
183C24—24no—
184C24—42no—
185C24—41no—
186C24—41no—
187C24—72no—
188C24C22×D711yescharacteristic
189C24C22×D712yes—
190C2×D4—78no—
191C2×D4—144no—
192C2×D4—144no—
193C2×D4—144no—
194C2×D4—144no—
195C2×D4—144no—
196C2×D4—281no—
197C2×D4—282no—
198C2×D4—282no—
199C2×D4—282no—
200C2×D4—282no—
201C22×C4—72no—
202C22×C4—72no—
203C22×C4—72no—
204C22×C4—144no—
205C22×C4—282no—
206C22×C4—282no—
207C22×C4C22×D712yes—
208C22×C4C22×D711yescharacteristic
209C22×C4C22×D711yescharacteristic
210C4⋊C4—144no—
211C4⋊C4—281no—
212C22⋊C4—24no—
213C22⋊C4—42no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—144no—
217C22⋊C4—144no—
218C22⋊C4—144no—
219C22⋊C4—144no—
220C22⋊C4—282no—
221C2×C7—44no—
222C2×C7—42no—
223C2×C7#53412yes—
224C2×C7#53811yescharacteristic
225C2×C7D4211yescharacteristic
226C2×C7#54612yes—
227C2×C7#56011yescharacteristic
228D7—44no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—48no—
240C23—44no—
241C23—44no—
242C23—48no—
243C23—44no—
244C23—74no—
245C23—142no—
246C23—142no—
247C23—142no—
248C23—142no—
249C23—142no—
250C23—142no—
251C23#115712yes—
252C23D4×D714yes—
253C23C23×D711yescharacteristic
254D4—284no—
255D4—288no—
256D4—288no—
257D4—288no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—74no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4—288no—
275C7#266811yescharacteristic, complemented, Sylow
276C22—24no—
277C22—24no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—416no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—144no—
290C22—144no—
291C22—144no—
292C22—282no—
293C22—282no—
294C22—282no—
295C22—282no—
296C22C2×#115711yescharacteristic
297C22C2×D4×D712yes—
298C22#654812yes—
299C22#654811yescharacteristic
300C22#654811yescharacteristic
301C4—42no—
302C4—41no—
303C4—41no—
304C4—144no—
305C4—282no—
306C4—282no—
307C2—42no—
308C2—44no—
309C2—284no—
310C2#9018112yes—
311C2#9018511yescharacteristic
312C2#9028012yes—
313C2#9028111yescharacteristic
314C2#9028611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#379C2
hypercenter#379D7
hypocenterC7#2668
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#6516C2semidirect
2. C2×#6447C2semidirect
3. C7×#379C2semidirect
4. #89733C2semidirect
5. #89732C2semidirect
6. #89728C2semidirect
7. #89728C2semidirect
8. #89724C2semidirect
9. #89721C2semidirect
14. C22×D28C22semidirect
15. C22×C4×D7C22semidirect
24. C2×(C2×D7)⋊C4C22semidirect
25. C2×(C2×D7)⋊C4C22semidirect
26. C2×C4⋊Dic7C22semidirect
27. C2×Dic7⋊C4C22semidirect
29. C7⋊#95C22semidirect
30. C7⋊#95C22semidirect
31. C7⋊#95C22semidirect
32. C7⋊#95C22semidirect
53. C2×C22⋊D7D4semidirect
81. #379D7semidirect
275. C7#2668semidirect

Maximal subgroups

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

subgroupindex
C2×#65162
C2×#64472
C7×#3792
#897332
#897322
#897282
#897242
#897212
#26687