Finite Group Databasebeta

← #11740345

Subgroups — #11740345

Order 896.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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×#123C211yesmaximal, characteristic, complemented
2C2×#6500C211yesmaximal, characteristic
3C2×D4×Dic7C211yesmaximal, characteristic, complemented
4C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
5#89983C211yesmaximal, characteristic, complemented
6#89766C212yesmaximal, complemented
7#89762C212yesmaximal, complemented
8#89731C212yesmaximal, complemented
9#89730C212yesmaximal, complemented
10#89727C211yesmaximal, characteristic, complemented
11#89714C211yesmaximal, characteristic, complemented
12C24×D7C2211yescharacteristic
13C22×C7×D4C2211yescharacteristic
14C23×C4×C7C2211yescharacteristic
15C22×C22⋊D7C2212yes—
16C23×Dic7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic, complemented
18C7×#123—24no—
19C2×C7×C4⋊C4C2212yescomplemented
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2212yescomplemented
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic, complemented
25#6500—24no—
26D4×Dic7—24no—
27C2×(C2×D7)⋊C4—24no—
28C2×(C2×D7)⋊C4C2212yes—
29C2×(C2×D7)⋊C4C2212yes—
30C2×(C2×D7)⋊C4C2212yes—
31C2×(C2×D7)⋊C4C2212yes—
32C2×(C2×D7)⋊C4C2212yes—
33C2×C4⋊Dic7C2211yescharacteristic, complemented
34C2×Dic7⋊C4C2212yescomplemented
35C2×C4×Dic7C2211yescharacteristic, complemented
36C7⋊#95C2212yescomplemented
37C7⋊#95C2212yescomplemented
38C7⋊#95C2212yescomplemented
39#2534—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C24×C7C2311yescharacteristic
42C23×D7—22no—
43C23×D7—24no—
44C23×D7—22no—
45C23×D7—22no—
46C23×D7—42no—
47C23×D7C2312yes—
48C2×C7×D4—22no—
49C2×C7×D4—22no—
50C2×C7×D4D414yes—
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7C2312yes—
55C22×C4×C7C2312yes—
56C22×C4×C7C2311yescharacteristic
57C2×C22⋊D7—24no—
58C2×C22⋊D7—24no—
59C2×C22⋊D7—42no—
60C22×Dic7—22no—
61C22×Dic7—42no—
62C22×Dic7—42no—
63C22×Dic7D414yescomplemented
64C22×Dic7C2312yes—
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C7×C4⋊C4—24no—
70C7×C22⋊C4—22no—
71C7×C22⋊C4—24no—
72C22⋊Dic7—22no—
73C22⋊Dic7—22no—
74C22⋊Dic7—22no—
75(C2×D7)⋊C4—24no—
76(C2×D7)⋊C4—24no—
77(C2×D7)⋊C4—24no—
78(C2×D7)⋊C4—24no—
79(C2×D7)⋊C4—42no—
80C4⋊Dic7—22no—
81Dic7⋊C4—42no—
82C4×Dic7—22no—
83C2×#123D711yescharacteristic, complemented
84C2×#120—71no—
85C2×C4×D4—71no—
86C22×C22⋊C4—71no—
87#396—72no—
88#394—72no—
89#388—72no—
90#386—72no—
91#381—71no—
92#380—71no—
93#379—71no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—21no—
97C23×C7—22no—
98C23×C7—21no—
99C23×C7—22no—
100C23×C7C2×D412yes—
101C23×C7C2×D412yes—
102C23×C7C2×D412yes—
103C23×C7#4112yes—
104C23×C7C2411yescharacteristic
105C22×D7—28no—
106C22×D7—22no—
107C22×D7—22no—
108C22×D7—22no—
109C22×D7—22no—
110C22×D7—22no—
111C22×D7—22no—
112C22×D7—48no—
113C22×D7—48no—
114C22×D7—44no—
115C22×D7—44no—
116C22×D7#4114yes—
117C7×D4—28no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—44no—
127C2×C4×C7C2×D412yes—
128C22⋊D7—48no—
129C2×Dic7—22no—
130C2×Dic7—22no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—22no—
134C2×Dic7—22no—
135C2×Dic7—22no—
136C2×Dic7—22no—
137C2×Dic7—22no—
138C2×Dic7—44no—
139C2×Dic7—44no—
140C2×Dic7C2×D412yes—
141C2×Dic7C2×D412yes—
142C25—71no—
143C22×D4—72no—
144C22×D4C2×D711yescharacteristic
145C23×C4—71no—
146C23×C4—71no—
147C23×C4C2×D711yescharacteristic
148#123—24no—
149#120—144no—
150C4×D4—144no—
151C2×C4⋊C4—71no—
152C2×C4⋊C4—72no—
153C2×C4⋊C4C2×D712yescomplemented
154C2×C22⋊C4—71no—
155C2×C22⋊C4—71no—
156C2×C22⋊C4—71no—
157C2×C22⋊C4—72no—
158C2×C22⋊C4—72no—
159C2×C22⋊C4—72no—
160C2×C22⋊C4—72no—
161C2×C22⋊C4—72no—
162C2×C22⋊C4—144no—
163C2×C22⋊C4C2×D711yescharacteristic
164C2×C22⋊C4C2×D712yescomplemented
165C2×C42—71no—
166#95—72no—
167#95—72no—
168#95—72no—
169C22×C7—22no—
170C22×C7—22no—
171C22×C7—24no—
172C22×C7—24no—
173C22×C7—22no—
174C22×C7—24no—
175C22×C7—41no—
176C22×C7—41no—
177C22×C7—42no—
178C22×C7#12114yes—
179C22×C7C22×D411yescharacteristic
180C22×C7C22×D411yescharacteristic
181C22×C7C2×#4111yescharacteristic
182C22×C7C2×#4112yes—
183C22×C7H(5,𝔽2)12yes—
184C2×D7—24no—
185C2×D7—24no—
186C2×D7—24no—
187C2×D7—416no—
188C2×D7—42no—
189C2×D7—42no—
190C2×D7—42no—
191C2×D7—42no—
192C4×C7—22no—
193C4×C7—42no—
194C4×C7—42no—
195Dic7—22no—
196Dic7—22no—
197Dic7—41no—
198Dic7—41no—
199Dic7—42no—
200C24—72no—
201C24—144no—
202C24—142no—
203C24—142no—
204C24—142no—
205C24—282no—
206C24C22×D711yescharacteristic
207C24C22×D711yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—144no—
211C2×D4—144no—
212C2×D4—282no—
213C2×D4C22⋊D714yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—74no—
220C22×C4—72no—
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—142no—
224C22×C4—282no—
225C22×C4—282no—
226C22×C4C22×D712yes—
227C22×C4C22×D712yes—
228C22×C4C22×D711yescharacteristic
229C4⋊C4—24no—
230C4⋊C4—142no—
231C4⋊C4—282no—
232C22⋊C4—22no—
233C22⋊C4—24no—
234C22⋊C4—144no—
235C22⋊C4—144no—
236C22⋊C4—144no—
237C22⋊C4—142no—
238C22⋊C4—142no—
239C22⋊C4—142no—
240C22⋊C4—144no—
241C22⋊C4—282no—
242C42—142no—
243C2×C7—24no—
244C2×C7—42no—
245C2×C7C2×#12111yescharacteristic
246C2×C7#52511yescharacteristic
247C2×C7#53411yescharacteristic
248C2×C7#54612yes—
249C2×C7#55112yes—
250D7—44no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—74no—
258C23—148no—
259C23—142no—
260C23—142no—
261C23—142no—
262C23—142no—
263C23—142no—
264C23—142no—
265C23—288no—
266C23—288no—
267C23—284no—
268C23—284no—
269C23D4×D712yes—
270C23#115912yes—
271C23C2×C22⋊D712yes—
272C23C2×C22⋊D712yes—
273C23C23×D711yescharacteristic
274D4—28no—
275D4—288no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—44no—
285C2×C4—72no—
286C2×C4—72no—
287C2×C4—142no—
288C2×C4—142no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—142no—
292C2×C4—142no—
293C2×C4—142no—
294C2×C4—142no—
295C2×C4—142no—
296C2×C4—284no—
297C2×C4—284no—
298C2×C4C2×C22⋊D712yes—
299C7#253411yescharacteristic, complemented, Sylow
300C22—24no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—22no—
305C22—24no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—144no—
310C22—144no—
311C22—144no—
312C22—2816no—
313C22—282no—
314C22—282no—
315C22—282no—
316C22—282no—
317C22#650214yes—
318C22C2×D4×D711yescharacteristic
319C22C2×#115911yescharacteristic
320C22D7×#4112yes—
321C22#655312yes—
322C22C22×C22⋊D711yescharacteristic
323C4—22no—
324C4—42no—
325C4—42no—
326C4—142no—
327C4—142no—
328C4—282no—
329C4—281no—
330C4—281no—
331C2—24no—
332C2—42no—
333C2—284no—
334C2#9034411yescharacteristic
335C2#9034711yescharacteristic
336C2#9034912yes—
337C2C2×#650211yescharacteristic
338C2#9052012yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radicalC2×#6500C2
hypercenterC2×#123D7
hypocenterC7#2534
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C7×#123C2semidirect
3. C2×D4×Dic7C2semidirect
4. C22×(C2×D7)⋊C4C2semidirect
5. #89983C2semidirect
6. #89766C2semidirect
7. #89762C2semidirect
8. #89731C2semidirect
9. #89730C2semidirect
10. #89727C2semidirect
11. #89714C2semidirect
17. C23×Dic7C22semidirect
19. C2×C7×C4⋊C4C22semidirect
21. C2×C7×C22⋊C4C22semidirect
24. C2×C22⋊Dic7C22semidirect
33. C2×C4⋊Dic7C22semidirect
34. C2×Dic7⋊C4C22semidirect
35. C2×C4×Dic7C22semidirect
36. C7⋊#95C22semidirect
37. C7⋊#95C22semidirect
38. C7⋊#95C22semidirect
63. C22×Dic7D4semidirect
83. C2×#123D7semidirect
153. C2×C4⋊C4C2×D7semidirect
164. C2×C22⋊C4C2×D7semidirect
299. C7#2534semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1232
C2×#65002
C2×D4×Dic72
C22×(C2×D7)⋊C42
#899832
#897662
#897622
#897312
#897302
#897272
#897142
#25347