Finite Group Databasebeta

← #11740340

Subgroups — #11740340

Order 896.

814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 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
3C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5#89992C211yesmaximal, characteristic, complemented
6#89983C211yesmaximal, characteristic, complemented
7#89765C212yesmaximal, complemented
8#89759C212yesmaximal, complemented
9#89733C212yesmaximal, complemented
10#89728C212yesmaximal, complemented
11#89722C211yesmaximal, characteristic, complemented
12C24×D7C2211yescharacteristic
13C22×C7×D4C2211yescharacteristic
14C23×C4×C7C2211yescharacteristic
15C22×C22⋊D7C2211yescharacteristic
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2212yes—
19C7×#123—24no—
20C2×C7×C4⋊C4C2212yescomplemented
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2212yescomplemented
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2212yescomplemented
26#6500—24no—
27C2×(C2×D7)⋊C4—24no—
28C2×(C2×D7)⋊C4C2212yes—
29C2×(C2×D7)⋊C4C2212yes—
30C2×(C2×D7)⋊C4C2212yes—
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×C4⋊Dic7C2212yescomplemented
34D7×C22⋊C4—24no—
35C7⋊#95C2212yescomplemented
36C7⋊#95C2212yescomplemented
37C7⋊#95C2212yescomplemented
38C7⋊#95C2212yescomplemented
39#2412—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C24×C7C2311yescharacteristic
42C23×D7—22no—
43C23×D7—22no—
44C23×D7—24no—
45C23×D7—42no—
46C23×D7D414yes—
47C23×D7C2311yescharacteristic
48C23×D7C2311yescharacteristic
49C2×C7×D4—22no—
50C2×C7×D4—22no—
51C2×C7×D4—22no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7C2312yes—
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2312yes—
58C2×C22⋊D7—22no—
59C2×C22⋊D7—22no—
60C2×C22⋊D7—24no—
61C2×C22⋊D7—42no—
62C22×Dic7—22no—
63C22×Dic7—42no—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2312yes—
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2312yes—
68C2×C4×D7—24no—
69C2×C4×D7—24no—
70C2×C4×D7—42no—
71C7×C4⋊C4—24no—
72C7×C22⋊C4—24no—
73C7×C22⋊C4D414yescomplemented
74C22⋊Dic7—22no—
75C22⋊Dic7—22no—
76C22⋊Dic7—42no—
77(C2×D7)⋊C4—24no—
78(C2×D7)⋊C4—24no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—24no—
82C4⋊Dic7—42no—
83C2×#123D711yescharacteristic, complemented
84C2×#120—71no—
85C22×C22⋊C4—71no—
86C22×C22⋊C4—71no—
87#397—72no—
88#394—72no—
89#386—71no—
90#386—72no—
91#385—71no—
92#381—71no—
93#380—72no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—21no—
97C23×C7—21no—
98C23×C7—22no—
99C23×C7—21no—
100C23×C7—22no—
101C23×C7C2×D412yes—
102C23×C7C2×D412yes—
103C23×C7#4112yes—
104C23×C7C2411yescharacteristic
105C22×D7—21no—
106C22×D7—21no—
107C22×D7—22no—
108C22×D7—28no—
109C22×D7—21no—
110C22×D7—21no—
111C22×D7—22no—
112C22×D7—22no—
113C22×D7—28no—
114C22×D7—44no—
115C22×D7—48no—
116C22×D7—48no—
117C22×D7C2×D412yes—
118C22×D7C2×D412yes—
119C22×D7#4114yes—
120C7×D4—44no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—44no—
130C2×C4×C7C2×D414yes—
131C22⋊D7—44no—
132C22⋊D7—44no—
133C2×Dic7—22no—
134C2×Dic7—22no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—22no—
138C2×Dic7—22no—
139C2×Dic7—22no—
140C2×Dic7—22no—
141C2×Dic7—22no—
142C2×Dic7—22no—
143C2×Dic7—44no—
144C4×D7—48no—
145C25—71no—
146C22×D4—71no—
147C22×D4—71no—
148C22×D4C2×D711yescharacteristic
149C23×C4—71no—
150C23×C4—72no—
151C23×C4C2×D711yescharacteristic
152#123—24no—
153#120—144no—
154C2×C4⋊C4—72no—
155C2×C4⋊C4C2×D712yescomplemented
156C2×C22⋊C4—71no—
157C2×C22⋊C4—71no—
158C2×C22⋊C4—72no—
159C2×C22⋊C4—72no—
160C2×C22⋊C4—72no—
161C2×C22⋊C4—71no—
162C2×C22⋊C4—71no—
163C2×C22⋊C4—72no—
164C2×C22⋊C4—144no—
165C2×C22⋊C4—144no—
166C2×C22⋊C4C2×D711yescharacteristic
167C2×C22⋊C4C2×D712yescomplemented
168#95—72no—
169#95—72no—
170#95—72no—
171#95—72no—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—24no—
175C22×C7—24no—
176C22×C7—41no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—42no—
180C22×C7—42no—
181C22×C7#12114yes—
182C22×C7C22×D411yescharacteristic
183C22×C7C22×D411yescharacteristic
184C22×C7C2×#4111yescharacteristic
185C22×C7C2×#4112yes—
186C22×C7H(5,𝔽2)12yes—
187C2×D7—22no—
188C2×D7—22no—
189C2×D7—24no—
190C2×D7—24no—
191C2×D7—22no—
192C2×D7—22no—
193C2×D7—22no—
194C2×D7—416no—
195C2×D7—42no—
196C2×D7—42no—
197C2×D7#12314yescomplemented
198C4×C7—24no—
199C4×C7—42no—
200C4×C7—41no—
201Dic7—41no—
202Dic7—42no—
203Dic7—41no—
204Dic7—42no—
205C24—71no—
206C24—71no—
207C24—74no—
208C24—144no—
209C24—142no—
210C24—142no—
211C24—282no—
212C24C22×D711yescharacteristic
213C24C22×D711yescharacteristic
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—142no—
218C2×D4—142no—
219C2×D4—144no—
220C2×D4—282no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—71no—
225C22×C4—72no—
226C22×C4—72no—
227C22×C4—71no—
228C22×C4—144no—
229C22×C4—142no—
230C22×C4—144no—
231C22×C4—282no—
232C22×C4—282no—
233C22×C4C22×D712yes—
234C22×C4C22×D711yescharacteristic
235C22×C4C22×D712yes—
236C4⋊C4—24no—
237C4⋊C4—282no—
238C22⋊C4—24no—
239C22⋊C4—144no—
240C22⋊C4—144no—
241C22⋊C4—142no—
242C22⋊C4—142no—
243C22⋊C4—142no—
244C22⋊C4—142no—
245C22⋊C4—144no—
246C22⋊C4—282no—
247C22⋊C4C22⋊D714yescomplemented
248C2×C7—24no—
249C2×C7—81no—
250C2×C7C2×#12111yescharacteristic
251C2×C7C2×#12311yescharacteristic
252C2×C7#53411yescharacteristic
253C2×C7#54612yes—
254C2×C7#55112yes—
255D7—24no—
256D7—42no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—21no—
264C23—72no—
265C23—74no—
266C23—72no—
267C23—148no—
268C23—141no—
269C23—142no—
270C23—141no—
271C23—148no—
272C23—141no—
273C23—142no—
274C23—142no—
275C23—141no—
276C23—284no—
277C23—288no—
278C23—288no—
279C23D4×D712yes—
280C23#115912yes—
281C23C2×C22⋊D712yes—
282C23C23×D711yescharacteristic
283D4—44no—
284D4—284no—
285D4—284no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—142no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—142no—
299C2×C4—142no—
300C2×C4—142no—
301C2×C4—142no—
302C2×C4—142no—
303C2×C4—142no—
304C2×C4—142no—
305C2×C4—288no—
306C2×C4—284no—
307C2×C4C2×C22⋊D714yes—
308C7#241211yescharacteristic, complemented, Sylow
309C22—24no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—74no—
319C22—142no—
320C22—144no—
321C22—142no—
322C22—144no—
323C22—142no—
324C22—142no—
325C22—142no—
326C22—2816no—
327C22—282no—
328C22—282no—
329C22#650114yes—
330C22C2×D4×D711yescharacteristic
331C22C2×#115911yescharacteristic
332C22D7×#4112yes—
333C22#655312yes—
334C22C22×C22⋊D711yescharacteristic
335C4—24no—
336C4—42no—
337C4—41no—
338C4—282no—
339C4—281no—
340C4—282no—
341C4—281no—
342C2—24no—
343C2—81no—
344C2—144no—
345C2—282no—
346C2D7×#12311yescharacteristic
347C2#9034411yescharacteristic
348C2#9034912yes—
349C2C2×#650111yescharacteristic
350C2#9052012yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radicalC2×#6500C2
hypercenterC2×#123D7
hypocenterC7#2412
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. C22×(C2×D7)⋊C4C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. #89992C2semidirect
6. #89983C2semidirect
7. #89765C2semidirect
8. #89759C2semidirect
9. #89733C2semidirect
10. #89728C2semidirect
11. #89722C2semidirect
20. C2×C7×C4⋊C4C22semidirect
22. C2×C7×C22⋊C4C22semidirect
25. C2×C22⋊Dic7C22semidirect
33. C2×C4⋊Dic7C22semidirect
35. C7⋊#95C22semidirect
36. C7⋊#95C22semidirect
37. C7⋊#95C22semidirect
38. C7⋊#95C22semidirect
73. C7×C22⋊C4D4semidirect
83. C2×#123D7semidirect
155. C2×C4⋊C4C2×D7semidirect
167. C2×C22⋊C4C2×D7semidirect
197. C2×D7#123semidirect
247. C22⋊C4C22⋊D7semidirect
308. C7#2412semidirect

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
C22×(C2×D7)⋊C42
C2×D7×C22⋊C42
#899922
#899832
#897652
#897592
#897332
#897282
#897222
#24127