Finite Group Databasebeta

← #11731298

Subgroups — #11731298

Order 896.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D7C211yesmaximal, characteristic
2C2×#6446C212yesmaximal, complemented
3C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
4C7×#379C211yesmaximal, characteristic, complemented
5#90019C211yesmaximal, characteristic, complemented
6#89733C212yesmaximal, complemented
7#89731C212yesmaximal, complemented
8#89728C212yesmaximal, complemented
9#89727C211yesmaximal, characteristic, complemented
10#89722C211yesmaximal, characteristic, complemented
11#89718C211yesmaximal, characteristic, complemented
12C25×C7C2211yescharacteristic
13C24×D7C2211yescharacteristic
14C22×C22⋊D7—24no—
15C22×C22⋊D7—28no—
16C22×C22⋊D7C2212yes—
17C22×C22⋊D7C2212yes—
18C23×Dic7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic, complemented
20C22×C4×D7C2211yescharacteristic, complemented
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2212yescomplemented
24C2×C7×C22⋊C4C2212yescomplemented
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2212yescomplemented
28C2×C22⋊Dic7C2212yescomplemented
29C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
31C2×(C2×D7)⋊C4C2212yescomplemented
32C2×(C2×D7)⋊C4C2212yescomplemented
33C2×C4⋊Dic7C2212yescomplemented
34C2×Dic7⋊C4C2212yescomplemented
35#6446—28no—
36D7×C22⋊C4—24no—
37C7⋊#95C2212yescomplemented
38C7⋊#95C2212yescomplemented
39#2405—71nomaximal, Sylow
40C24×C7—24no—
41C24×C7—22no—
42C24×C7—22no—
43C24×C7—22no—
44C24×C7—42no—
45C24×C7C2312yes—
46C24×C7C2311yescharacteristic
47C23×D7—22no—
48C23×D7—22no—
49C23×D7—24no—
50C23×D7—42no—
51C23×D7D414yes—
52C23×D7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2312yes—
56C22×C4×C7C2311yescharacteristic
57C2×C22⋊D7—232no—
58C2×C22⋊D7—24no—
59C2×C22⋊D7—24no—
60C2×C22⋊D7—24no—
61C2×C22⋊D7—24no—
62C2×C22⋊D7—24no—
63C2×C22⋊D7D418yes—
64C22×Dic7—22no—
65C22×Dic7—24no—
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2312yes—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2312yes—
70C2×C4×D7—22no—
71C2×C4×D7—22no—
72C2×C4×D7—24no—
73C2×C4×D7—42no—
74C7×C22⋊C4—22no—
75C7×C22⋊C4—24no—
76C7×C22⋊C4—22no—
77C7×C22⋊C4—42no—
78C22⋊Dic7—22no—
79C22⋊Dic7—24no—
80C22⋊Dic7—22no—
81C22⋊Dic7—42no—
82(C2×D7)⋊C4—22no—
83(C2×D7)⋊C4—22no—
84(C2×D7)⋊C4—24no—
85(C2×D7)⋊C4—42no—
86C4⋊Dic7—24no—
87Dic7⋊C4—24no—
88C23×D4—71no—
89C2×#123—72no—
90C22×C22⋊C4—71no—
91#394—72no—
92#394—72no—
93#386—72no—
94#385—71no—
95#385—71no—
96#379—71no—
97#379—71no—
98#379D711yescharacteristic, complemented
99C23×C7—22no—
100C23×C7—22no—
101C23×C7—28no—
102C23×C7—22no—
103C23×C7—22no—
104C23×C7—22no—
105C23×C7—22no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—48no—
109C23×C7—48no—
110C23×C7—44no—
111C23×C7—44no—
112C23×C7C2×D414yes—
113C23×C7C2×D412yes—
114C23×C7#4112yes—
115C23×C7#4112yes—
116C23×C7C2411yescharacteristic
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—22no—
120C22×D7—28no—
121C22×D7—22no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—22no—
125C22×D7—28no—
126C22×D7—44no—
127C22×D7—48no—
128C22×D7—48no—
129C22×D7C2×D412yes—
130C22×D7C2×D414yes—
131C22×D7C2×D412yes—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C22⋊D7—216no—
140C22⋊D7—216no—
141C2×Dic7—28no—
142C2×Dic7—22no—
143C2×Dic7—22no—
144C2×Dic7—22no—
145C2×Dic7—22no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—22no—
149C2×Dic7—22no—
150C2×Dic7—22no—
151C2×Dic7C2×D414yes—
152C4×D7—44no—
153C4×D7—44no—
154C25—71no—
155C25C2×D711yescharacteristic
156C22×D4—72no—
157C22×D4—72no—
158C22×D4—148no—
159C22×D4—144no—
160C23×C4—71no—
161C23×C4—71no—
162C23×C4—71no—
163#123—148no—
164C2×C4⋊C4—72no—
165C2×C4⋊C4—72no—
166C2×C22⋊C4—71no—
167C2×C22⋊C4—72no—
168C2×C22⋊C4—71no—
169C2×C22⋊C4—71no—
170C2×C22⋊C4—71no—
171C2×C22⋊C4—72no—
172C2×C22⋊C4—72no—
173C2×C22⋊C4—72no—
174C2×C22⋊C4—144no—
175C2×C22⋊C4C2×D711yescharacteristic
176C2×C22⋊C4C2×D711yescharacteristic
177C2×C22⋊C4C2×D712yescomplemented
178C2×C22⋊C4C2×D712yescomplemented
179#95—72no—
180#95—72no—
181C22×C7—22no—
182C22×C7—22no—
183C22×C7—24no—
184C22×C7—24no—
185C22×C7—24no—
186C22×C7—24no—
187C22×C7—24no—
188C22×C7—416no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7—42no—
192C22×C7—42no—
193C22×C7#12314yescomplemented
194C22×C7C22×D411yescharacteristic
195C22×C7C22×D412yes—
196C22×C7C2×#4111yescharacteristic
197C22×C7C2×#4111yescharacteristic
198C22×C7H(5,𝔽2)12yes—
199C2×D7—22no—
200C2×D7—22no—
201C2×D7—24no—
202C2×D7—24no—
203C2×D7—22no—
204C2×D7—22no—
205C2×D7—22no—
206C2×D7—416no—
207C2×D7—42no—
208C2×D7—42no—
209C2×D7#12014yes—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—42no—
213Dic7—24no—
214Dic7—41no—
215Dic7—41no—
216Dic7—42no—
217C24—24no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—42no—
222C24—71no—
223C24—71no—
224C24—74no—
225C24—144no—
226C24—142no—
227C24—142no—
228C24—282no—
229C24C22×D712yes—
230C24C22×D711yescharacteristic
231C2×D4—78no—
232C2×D4—144no—
233C2×D4—144no—
234C2×D4—144no—
235C2×D4—144no—
236C2×D4—1432no—
237C2×D4—144no—
238C22×C4—72no—
239C22×C4—71no—
240C22×C4—72no—
241C22×C4—71no—
242C22×C4—142no—
243C22×C4—142no—
244C22×C4—144no—
245C22×C4—144no—
246C22×C4—142no—
247C22×C4—282no—
248C22×C4C22×D711yescharacteristic
249C22×C4C22×D712yes—
250C22×C4C22×D711yescharacteristic
251C4⋊C4—144no—
252C4⋊C4—144no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—142no—
258C22⋊C4—142no—
259C22⋊C4—144no—
260C22⋊C4—142no—
261C22⋊C4—142no—
262C22⋊C4—144no—
263C22⋊C4—282no—
264C22⋊C4—282no—
265C2×C7—24no—
266C2×C7—44no—
267C2×C7C2×#12011yescharacteristic
268C2×C7C2×#12311yescharacteristic
269C2×C7#53411yescharacteristic
270C2×C7#54612yes—
271C2×C7#54612yes—
272D7—24no—
273D7—42no—
274C23—22no—
275C23—22no—
276C23—28no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—44no—
287C23—74no—
288C23—72no—
289C23—72no—
290C23—148no—
291C23—141no—
292C23—141no—
293C23—142no—
294C23—148no—
295C23—141no—
296C23—142no—
297C23—141no—
298C23—142no—
299C23—284no—
300C23—288no—
301C23—288no—
302C23#115712yes—
303C23D4×D714yes—
304C23D4×D712yes—
305C23#115912yes—
306C23C23×D711yescharacteristic
307D4—1416no—
308D4—1416no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—74no—
317C2×C4—148no—
318C2×C4—142no—
319C2×C4—142no—
320C2×C4—142no—
321C2×C4—142no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—142no—
325C2×C4—142no—
326C2×C4—142no—
327C2×C4—284no—
328C2×C4—284no—
329C7#240511yescharacteristic, complemented, Sylow
330C22—24no—
331C22—22no—
332C22—22no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—74no—
343C22—142no—
344C22—144no—
345C22—142no—
346C22—144no—
347C22—142no—
348C22—142no—
349C22—142no—
350C22—2816no—
351C22—282no—
352C22—282no—
353C22#644614yescomplemented
354C22C2×#115711yescharacteristic
355C22C2×D4×D711yescharacteristic
356C22C2×D4×D712yes—
357C22C2×#115911yescharacteristic
358C22#654812yes—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—144no—
363C4—282no—
364C4—281no—
365C4—281no—
366C2—24no—
367C2—44no—
368C2—144no—
369C2—282no—
370C2C2×#644611yescharacteristic
371C2#9018112yes—
372C2D7×#12011yescharacteristic
373C2#9028011yescharacteristic
374C2#9028212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#379C2
prime radicalC23×C22⋊D7C2
hypercenter#379D7
hypocenterC7#2405
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#6446C2semidirect
3. C2×D7×C22⋊C4C2semidirect
4. C7×#379C2semidirect
5. #90019C2semidirect
6. #89733C2semidirect
7. #89731C2semidirect
8. #89728C2semidirect
9. #89727C2semidirect
10. #89722C2semidirect
11. #89718C2semidirect
19. C22×C4×D7C22semidirect
20. C22×C4×D7C22semidirect
23. C2×C7×C22⋊C4C22semidirect
24. C2×C7×C22⋊C4C22semidirect
27. C2×C22⋊Dic7C22semidirect
28. C2×C22⋊Dic7C22semidirect
29. C2×(C2×D7)⋊C4C22semidirect
30. C2×(C2×D7)⋊C4C22semidirect
31. C2×(C2×D7)⋊C4C22semidirect
32. C2×(C2×D7)⋊C4C22semidirect
33. C2×C4⋊Dic7C22semidirect
34. C2×Dic7⋊C4C22semidirect
37. C7⋊#95C22semidirect
38. C7⋊#95C22semidirect
98. #379D7semidirect
177. C2×C22⋊C4C2×D7semidirect
178. C2×C22⋊C4C2×D7semidirect
193. C22×C7#123semidirect
329. C7#2405semidirect
353. C22#6446semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D72
C2×#64462
C2×D7×C22⋊C42
C7×#3792
#900192
#897332
#897312
#897282
#897272
#897222
#897182
#24057