Finite Group Databasebeta

← #11731474

Subgroups — #11731474

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×C4×C22⋊D7C211yesmaximal, characteristic, complemented
2C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
3C2×#6449C211yesmaximal, characteristic, complemented
4C2×#6445C211yesmaximal, characteristic
5C7×#380C211yesmaximal, characteristic, complemented
6#89766C212yesmaximal, complemented
7#89763C212yesmaximal, complemented
8#89727C211yesmaximal, characteristic, complemented
9#89715C211yesmaximal, characteristic, complemented
10#89442C212yesmaximal, complemented
11#89437C212yesmaximal, complemented
12C24×D7C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C23×Dic7C2211yescharacteristic
16C22×D28C2212yes—
17C22×C4×D7C2211yescharacteristic, complemented
18C2×C7×C4⋊C4C2212yescomplemented
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic7C2211yescharacteristic
22C4×C22⋊D7—24no—
23C2×(C2×D7)⋊C4—24no—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×(C2×D7)⋊C4C2212yescomplemented
30C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
31C2×C4⋊Dic7C2212yescomplemented
32C2×Dic7⋊C4C2211yescharacteristic, complemented
33C2×C4×Dic7C2211yescharacteristic, complemented
34#6449—24no—
35#6445—24no—
36C7×#95C2212yescomplemented
37C7⋊#95C2212yescomplemented
38C7⋊#95C2212yescomplemented
39#2534—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7—22no—
42C23×D7—24no—
43C23×D7—22no—
44C23×D7—22no—
45C23×D7—42no—
46C23×D7C2311yescharacteristic
47C23×D7C2312yes—
48C22×C4×C7—42no—
49C22×C4×C7D414yescomplemented
50C22×C4×C7C2312yes—
51C22×C4×C7C2312yes—
52C22×C4×C7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C2×C22⋊D7—22no—
55C2×C22⋊D7—22no—
56C2×C22⋊D7D414yes—
57C22×Dic7—22no—
58C22×Dic7—22no—
59C22×Dic7—22no—
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2312yes—
63C22×Dic7C2311yescharacteristic
64C2×D28—24no—
65C2×D28—24no—
66C2×D28—42no—
67C2×C4×D7—22no—
68C2×C4×D7—42no—
69C7×C4⋊C4—42no—
70C7×C22⋊C4—22no—
71C7×C22⋊C4—22no—
72C22⋊Dic7—22no—
73(C2×D7)⋊C4—24no—
74(C2×D7)⋊C4—24no—
75(C2×D7)⋊C4—24no—
76(C2×D7)⋊C4—24no—
77(C2×D7)⋊C4—22no—
78(C2×D7)⋊C4—24no—
79(C2×D7)⋊C4—42no—
80C4⋊Dic7—24no—
81Dic7⋊C4—22no—
82C4×Dic7—22no—
83C2×#123—71no—
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#380D711yescharacteristic, complemented
93#379—71no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7C2×D412yes—
97C23×C7C2×D412yes—
98C23×C7#4112yes—
99C23×C7C2411yescharacteristic
100C22×D7—28no—
101C22×D7—22no—
102C22×D7—21no—
103C22×D7—22no—
104C22×D7—22no—
105C22×D7—21no—
106C22×D7—22no—
107C22×D7—22no—
108C22×D7—22no—
109C22×D7—22no—
110C22×D7—22no—
111C22×D7—48no—
112C22×D7—48no—
113C22×D7—44no—
114C22×D7—44no—
115C22×D7C2×D412yes—
116C22×D7#4114yes—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—44no—
126C2×C4×C7C2×D412yes—
127C2×C4×C7C2×D412yes—
128C22⋊D7—28no—
129C2×Dic7—21no—
130C2×Dic7—21no—
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×Dic7C2×D412yes—
140D28—48no—
141C4×D7—44no—
142C25—71no—
143C22×D4—71no—
144C22×D4—72no—
145C23×C4—71no—
146C23×C4—71no—
147C23×C4C2×D711yescharacteristic
148#123—144no—
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—72no—
156C2×C22⋊C4—72no—
157C2×C22⋊C4—71no—
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×D711yescharacteristic
165C2×C42—71no—
166#95—72no—
167#95—72no—
168#95C2×D712yescomplemented
169C22×C7—22no—
170C22×C7—22no—
171C22×C7—24no—
172C22×C7—24no—
173C22×C7#12114yes—
174C22×C7C22×D411yescharacteristic
175C22×C7C22×D411yescharacteristic
176C22×C7C2×#4111yescharacteristic
177C22×C7C2×#4112yes—
178C22×C7H(5,𝔽2)12yes—
179C2×D7—22no—
180C2×D7—24no—
181C2×D7—24no—
182C2×D7—24no—
183C2×D7—24no—
184C2×D7—416no—
185C2×D7—42no—
186C2×D7—41no—
187C2×D7—41no—
188C2×D7—42no—
189C2×D7—42no—
190C2×D7—42no—
191C2×D7—42no—
192C4×C7—22no—
193C4×C7—22no—
194C4×C7—42no—
195C4×C7—42no—
196Dic7—22no—
197Dic7—42no—
198Dic7—41no—
199Dic7—41no—
200C24—72no—
201C24—71no—
202C24—144no—
203C24—142no—
204C24—142no—
205C24—142no—
206C24—282no—
207C24C22×D711yescharacteristic
208C2×D4—74no—
209C2×D4—144no—
210C2×D4—144no—
211C2×D4—142no—
212C2×D4—142no—
213C2×D4—282no—
214C22×C4—42no—
215C22×C4—71no—
216C22×C4—71no—
217C22×C4—72no—
218C22×C4—71no—
219C22×C4—142no—
220C22×C4—142no—
221C22×C4—142no—
222C22×C4—142no—
223C22×C4—282no—
224C22×C4D2814yescomplemented
225C22×C4C22×D712yes—
226C22×C4C22×D712yes—
227C22×C4C22×D711yescharacteristic
228C22×C4C22×D711yescharacteristic
229C4⋊C4—42no—
230C4⋊C4—144no—
231C4⋊C4—142no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—144no—
235C22⋊C4—144no—
236C22⋊C4—144no—
237C22⋊C4—142no—
238C22⋊C4—144no—
239C22⋊C4—144no—
240C22⋊C4—142no—
241C22⋊C4—282no—
242C42—142no—
243C2×C7—24no—
244C2×C7C2×#12111yescharacteristic
245C2×C7#52511yescharacteristic
246C2×C7#53411yescharacteristic
247C2×C7#54612yes—
248C2×C7#55112yes—
249D7—42no—
250D7—44no—
251C23—22no—
252C23—22no—
253C23—72no—
254C23—74no—
255C23—148no—
256C23—142no—
257C23—141no—
258C23—142no—
259C23—142no—
260C23—142no—
261C23—142no—
262C23—141no—
263C23—142no—
264C23—142no—
265C23—142no—
266C23—288no—
267C23—288no—
268C23—284no—
269C23—284no—
270C23C2×D2812yes—
271C23D4×D712yes—
272C23#116112yes—
273C23C23×D711yescharacteristic
274D4—148no—
275D4—288no—
276C2×C4—22no—
277C2×C4—22no—
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—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—142no—
290C2×C4—141no—
291C2×C4—142no—
292C2×C4—142no—
293C2×C4—142no—
294C2×C4—142no—
295C2×C4—284no—
296C2×C4—284no—
297C2×C4C2×D2812yes—
298C2×C4C2×D2812yes—
299C7#253411yescharacteristic, complemented, Sylow
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—142no—
305C22—144no—
306C22—144no—
307C22—144no—
308C22—144no—
309C22—2816no—
310C22—281no—
311C22—282no—
312C22—282no—
313C22—281no—
314C22—282no—
315C22—282no—
316C22—282no—
317C22#645814yes—
318C22C22×D2811yescharacteristic
319C22C2×D4×D711yescharacteristic
320C22#654812yes—
321C22C2×#116111yescharacteristic
322C22D7×#4112yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—142no—
328C4—281no—
329C4—282no—
330C4—281no—
331C2—24no—
332C2—284no—
333C2—282no—
334C2#9018411yescharacteristic
335C2C2×#645811yescharacteristic
336C2#9021511yescharacteristic
337C2#9030112yes—
338C2#9032812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#380C2
prime radicalC2×#6445C2
hypercenter#380D7
hypocenterC7#2534
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D7C2semidirect
2. C22×(C2×D7)⋊C4C2semidirect
3. C2×#6449C2semidirect
5. C7×#380C2semidirect
6. #89766C2semidirect
7. #89763C2semidirect
8. #89727C2semidirect
9. #89715C2semidirect
10. #89442C2semidirect
11. #89437C2semidirect
17. C22×C4×D7C22semidirect
18. C2×C7×C4⋊C4C22semidirect
29. C2×(C2×D7)⋊C4C22semidirect
30. C2×(C2×D7)⋊C4C22semidirect
31. C2×C4⋊Dic7C22semidirect
32. C2×Dic7⋊C4C22semidirect
33. C2×C4×Dic7C22semidirect
36. C7×#95C22semidirect
37. C7⋊#95C22semidirect
38. C7⋊#95C22semidirect
49. C22×C4×C7D4semidirect
92. #380D7semidirect
153. C2×C4⋊C4C2×D7semidirect
168. #95C2×D7semidirect
224. C22×C4D28semidirect
299. C7#2534semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D72
C22×(C2×D7)⋊C42
C2×#64492
C2×#64452
C7×#3802
#897662
#897632
#897272
#897152
#894422
#894372
#25347