Finite Group Databasebeta

← #11740042

Subgroups — #11740042

Order 896.

1000 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 157 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×C7×#120C211yesmaximal, characteristic, complemented
2C2×#6516C212yesmaximal, complemented
3C2×#6500C212yesmaximal, complemented
4C2×#6500C211yesmaximal, characteristic, complemented
5#89994C212yesmaximal, complemented
6#89993C211yesmaximal, characteristic, complemented
7#89733C212yesmaximal, complemented
8#89731C212yesmaximal, complemented
9#89729C212yesmaximal, complemented
10C25×C7C2211yescharacteristic, complemented
11C24×D7C2211yescharacteristic, complemented
12C22×C7×D4C2212yes—
13C22×C7×D4C2211yescharacteristic
14C22×C22⋊D7C2212yes—
15C22×C22⋊D7C2212yes—
16C22×C22⋊D7C2212yes—
17C7×#120—24no—
18C2×C7×C22⋊C4C2212yescomplemented
19C2×C7×C22⋊C4C2211yescharacteristic, complemented
20#6516—28no—
21C2×C22⋊Dic7C2212yescomplemented
22C2×C22⋊Dic7C2211yescharacteristic, complemented
23C2×C22⋊Dic7C2212yescomplemented
24C2×C22⋊Dic7C2212yescomplemented
25C2×C22⋊Dic7C2212yescomplemented
26#6500—24no—
27#6500—28no—
28C2×(C2×D7)⋊C4C2212yescomplemented
29C2×(C2×D7)⋊C4C2212yescomplemented
30C2×(C2×D7)⋊C4C2212yescomplemented
31C2×C4⋊Dic7C2211yescharacteristic, complemented
32C2×Dic7⋊C4C2212yescomplemented
33C2×C4×Dic7C2211yescharacteristic, complemented
34C7⋊#95C2212yescomplemented
35#2679—71nomaximal, Sylow
36C24×C7—24no—
37C24×C7—22no—
38C24×C7—42no—
39C24×C7—41no—
40C24×C7—41no—
41C24×C7C2311yescharacteristic
42C24×C7C2311yescharacteristic
43C24×C7C2312yes—
44C23×D7—22no—
45C23×D7—24no—
46C23×D7—41no—
47C23×D7—42no—
48C23×D7—41no—
49C23×D7C2312yes—
50C2×C7×D4—24no—
51C2×C7×D4—24no—
52C2×C7×D4—41no—
53C2×C7×D4—41no—
54C2×C7×D4—42no—
55C2×C7×D4D414yescomplemented
56C22×C4×C7C2312yes—
57C22×C4×C7C2311yescharacteristic
58C2×C22⋊D7—24no—
59C2×C22⋊D7—24no—
60C2×C22⋊D7—24no—
61C2×C22⋊D7—24no—
62C2×C22⋊D7—42no—
63C2×C22⋊D7—42no—
64C2×C22⋊D7—42no—
65C2×C22⋊D7—42no—
66C2×C22⋊D7D418yescomplemented
67C22×Dic7C2312yes—
68C22×Dic7C2312yes—
69C22×Dic7C2312yes—
70C7×C22⋊C4—22no—
71C7×C22⋊C4—24no—
72C22⋊Dic7—22no—
73C22⋊Dic7—24no—
74C22⋊Dic7—24no—
75C22⋊Dic7—24no—
76C22⋊Dic7—24no—
77(C2×D7)⋊C4—24no—
78(C2×D7)⋊C4—24no—
79(C2×D7)⋊C4—24no—
80C4⋊Dic7—41no—
81Dic7⋊C4—42no—
82C4×Dic7—41no—
83C2×#120—72no—
84C2×#120—72no—
85C2×#120—71no—
86C2×#120D711yescharacteristic, complemented
87#394—72no—
88#394—72no—
89#394—72no—
90#390—71no—
91#390—72no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—21no—
101C23×C7—21no—
102C23×C7—22no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—22no—
106C23×C7—44no—
107C23×C7—48no—
108C23×C7—48no—
109C23×C7—44no—
110C23×C7—44no—
111C23×C7C2×D414yes—
112C23×C7C2×D412yes—
113C23×C7C2×D412yes—
114C23×C7C2411yescharacteristic
115C22×D7—22no—
116C22×D7—22no—
117C22×D7—22no—
118C22×D7—22no—
119C22×D7—22no—
120C22×D7—22no—
121C22×D7—44no—
122C22×D7—48no—
123C22×D7—48no—
124C22×D7—44no—
125C22×D7—44no—
126C22×D7C2×D414yes—
127C7×D4—44no—
128C7×D4—48no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7C2×D412yes—
135C22⋊D7—48no—
136C22⋊D7—48no—
137C22⋊D7—48no—
138C2×Dic7—22no—
139C2×Dic7—22no—
140C2×Dic7—22no—
141C2×Dic7—22no—
142C2×Dic7—22no—
143C2×Dic7—22no—
144C2×Dic7—22no—
145C2×Dic7—22no—
146C2×Dic7C2×D414yes—
147C25—71no—
148C25C2×D711yescharacteristic, complemented
149C22×D4—72no—
150C22×D4—72no—
151C22×D4—72no—
152C22×D4C2×D712yes—
153C22×D4C2×D711yescharacteristic
154#120—24no—
155#120—148no—
156#120—148no—
157#120—144no—
158C2×C4⋊C4—72no—
159C2×C4⋊C4—71no—
160C2×C22⋊C4—72no—
161C2×C22⋊C4—72no—
162C2×C22⋊C4—72no—
163C2×C22⋊C4—72no—
164C2×C22⋊C4—71no—
165C2×C22⋊C4—72no—
166C2×C22⋊C4—72no—
167C2×C22⋊C4—72no—
168C2×C22⋊C4C2×D712yescomplemented
169C2×C22⋊C4C2×D711yescharacteristic, complemented
170C2×C42—71no—
171#95—72no—
172C22×C7—24no—
173C22×C7—22no—
174C22×C7—22no—
175C22×C7—24no—
176C22×C7—24no—
177C22×C7—24no—
178C22×C7—24no—
179C22×C7—416no—
180C22×C7—42no—
181C22×C7—42no—
182C22×C7—42no—
183C22×C7—42no—
184C22×C7—41no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—42no—
189C22×C7—41no—
190C22×C7C22×D412yes—
191C22×C7C22×D411yescharacteristic
192C22×C7H(5,𝔽2)11yescharacteristic
193C22×C7H(5,𝔽2)12yes—
194C22×C7H(5,𝔽2)11yescharacteristic
195C2×D7—24no—
196C2×D7—24no—
197C2×D7—24no—
198C2×D7—416no—
199C2×D7—42no—
200C2×D7—42no—
201C2×D7—42no—
202C2×D7—42no—
203C4×C7—22no—
204C4×C7—42no—
205Dic7—24no—
206Dic7—42no—
207Dic7—42no—
208C24—24no—
209C24—22no—
210C24—42no—
211C24—41no—
212C24—41no—
213C24—72no—
214C24—144no—
215C24—142no—
216C24—282no—
217C24—281no—
218C24—281no—
219C24C22×D712yes—
220C24C22×D711yescharacteristic
221C24C22×D711yescharacteristic
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—78no—
228C2×D4—144no—
229C2×D4—144no—
230C2×D4—144no—
231C2×D4—144no—
232C2×D4—282no—
233C2×D4—282no—
234C2×D4—282no—
235C2×D4—282no—
236C2×D4C22⋊D714yescomplemented
237C22×C4—72no—
238C22×C4—72no—
239C22×C4—72no—
240C22×C4C22×D712yes—
241C22×C4C22×D711yescharacteristic
242C4⋊C4—282no—
243C4⋊C4—281no—
244C22⋊C4—22no—
245C22⋊C4—24no—
246C22⋊C4—144no—
247C22⋊C4—144no—
248C22⋊C4—144no—
249C22⋊C4—144no—
250C22⋊C4—144no—
251C22⋊C4—144no—
252C22⋊C4—142no—
253C22⋊C4—144no—
254C42—281no—
255C2×C7—42no—
256C2×C7—44no—
257C2×C7—42no—
258C2×C7#53411yescharacteristic
259C2×C7#53412yes—
260C2×C7D4212yes—
261C2×C7D4211yescharacteristic
262C2×C7H(3,𝔽4)11yescharacteristic
263D7—44no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—44no—
280C23—44no—
281C23—48no—
282C23—44no—
283C23—74no—
284C23—142no—
285C23—142no—
286C23—142no—
287C23—142no—
288C23—142no—
289C23—142no—
290C23—288no—
291C23—284no—
292C23—284no—
293C23—288no—
294C23—284no—
295C23D4×D714yes—
296C23C2×C22⋊D712yes—
297C23C2×C22⋊D712yes—
298C23C23×D711yescharacteristic
299D4—44no—
300D4—48no—
301D4—288no—
302D4—288no—
303D4—288no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—74no—
310C2×C4—142no—
311C2×C4—142no—
312C2×C4—142no—
313C2×C4—142no—
314C2×C4—142no—
315C2×C4—142no—
316C2×C4—142no—
317C2×C4—142no—
318C2×C4C2×C22⋊D712yes—
319C7#267911yescharacteristic, complemented, Sylow
320C22—24no—
321C22—24no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—416no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—144no—
339C22—144no—
340C22—144no—
341C22—2816no—
342C22—282no—
343C22—282no—
344C22—282no—
345C22—282no—
346C22C2×D4×D712yes—
347C22#654812yes—
348C22#654811yescharacteristic
349C22#654811yescharacteristic
350C22C22×C22⋊D711yescharacteristic
351C4—22no—
352C4—42no—
353C4—144no—
354C4—282no—
355C4—282no—
356C2—44no—
357C2—42no—
358C2—42no—
359C2—284no—
360C2#9028012yes—
361C2#9028111yescharacteristic
362C2#9028511yescharacteristic
363C2D4×C22⋊D712yes—
364C2#9049511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#120C2
hypercenterC2×#120D7
hypocenterC7#2679
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×C7×#120C2semidirect
2. C2×#6516C2semidirect
3. C2×#6500C2semidirect
4. C2×#6500C2semidirect
5. #89994C2semidirect
6. #89993C2semidirect
7. #89733C2semidirect
8. #89731C2semidirect
9. #89729C2semidirect
10. C25×C7C22semidirect
11. C24×D7C22semidirect
18. C2×C7×C22⋊C4C22semidirect
19. C2×C7×C22⋊C4C22semidirect
21. C2×C22⋊Dic7C22semidirect
22. C2×C22⋊Dic7C22semidirect
23. C2×C22⋊Dic7C22semidirect
24. C2×C22⋊Dic7C22semidirect
25. C2×C22⋊Dic7C22semidirect
28. C2×(C2×D7)⋊C4C22semidirect
29. C2×(C2×D7)⋊C4C22semidirect
30. C2×(C2×D7)⋊C4C22semidirect
31. C2×C4⋊Dic7C22semidirect
32. C2×Dic7⋊C4C22semidirect
33. C2×C4×Dic7C22semidirect
34. C7⋊#95C22semidirect
55. C2×C7×D4D4semidirect
66. C2×C22⋊D7D4semidirect
86. C2×#120D7semidirect
148. C25C2×D7semidirect
168. C2×C22⋊C4C2×D7semidirect
169. C2×C22⋊C4C2×D7semidirect
236. C2×D4C22⋊D7semidirect
319. C7#2679semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1202
C2×#65162
C2×#65002
#899942
#899932
#897332
#897312
#897292
#26797