Finite Group Databasebeta

← #11734114

Subgroups — #11734114

Order 896.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×D4×D7C211yesmaximal, characteristic, complemented
2C2×#6502C212yesmaximal, complemented
3C2×#6500C212yesmaximal, complemented
4C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
5C7×#392C211yesmaximal, characteristic, complemented
6#89994C211yesmaximal, characteristic, complemented
7#89732C211yesmaximal, characteristic, complemented
8#89729C212yesmaximal, complemented
9#89727C212yesmaximal, complemented
10#89444C211yesmaximal, characteristic, complemented
11#89439C211yesmaximal, characteristic, complemented
12C24×D7C2212yes—
13C22×C7×D4C2211yescharacteristic
14C22×C7×D4C2212yescomplemented
15C22×C22⋊D7C2211yescharacteristic
16C22×C22⋊D7C2212yes—
17C22×C22⋊D7C2212yescomplemented
18C23×Dic7C2211yescharacteristic, complemented
19C2×D4×D7—28no—
20C2×D4×D7—24no—
21C22×D28C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic, complemented
24C2×C7×C22⋊C4C2212yescomplemented
25C2×C22⋊Dic7C2212yescomplemented
26C2×C22⋊Dic7C2212yescomplemented
27#6502—28no—
28#6500—28no—
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2212yes—
31C2×(C2×D7)⋊C4C2212yescomplemented
32C2×(C2×D7)⋊C4C2212yescomplemented
33C2×(C2×D7)⋊C4C2212yescomplemented
34C2×Dic7⋊C4C2211yescharacteristic, complemented
35C2×Dic7⋊C4C2212yescomplemented
36C2×C4×Dic7C2211yescharacteristic, complemented
37(C22⋊D7)⋊C4—24no—
38C7×#95C2211yescharacteristic, complemented
39C7⋊#95C2211yescharacteristic, complemented
40#2408—71nomaximal, Sylow
41C24×C7C2312yes—
42C24×C7C2311yescharacteristic
43C23×D7—24no—
44C23×D7—24no—
45C23×D7—24no—
46C23×D7—24no—
47C23×D7—24no—
48C23×D7—42no—
49C23×D7—42no—
50C23×D7C2312yes—
51C23×D7C2311yescharacteristic
52C2×C7×D4—24no—
53C2×C7×D4—24no—
54C2×C7×D4—24no—
55C2×C7×D4—22no—
56C2×C7×D4—42no—
57C22×C4×C7C2312yes—
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C2×C22⋊D7—22no—
61C2×C22⋊D7—22no—
62C2×C22⋊D7—24no—
63C2×C22⋊D7—24no—
64C2×C22⋊D7—24no—
65C2×C22⋊D7—24no—
66C2×C22⋊D7—42no—
67C2×C22⋊D7D418yescomplemented
68C2×C22⋊D7D414yescomplemented
69C22×Dic7—22no—
70C22×Dic7—24no—
71C22×Dic7C2312yes—
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75D4×D7—232no—
76C2×D28—24no—
77C2×D28—22no—
78C2×C4×D7—24no—
79C2×C4×D7D414yescomplemented
80C7×C22⋊C4—22no—
81C7×C22⋊C4—42no—
82C22⋊Dic7—24no—
83C22⋊Dic7—24no—
84(C2×D7)⋊C4—22no—
85(C2×D7)⋊C4—24no—
86(C2×D7)⋊C4—24no—
87(C2×D7)⋊C4—24no—
88(C2×D7)⋊C4—42no—
89Dic7⋊C4—22no—
90Dic7⋊C4—24no—
91C4×Dic7—22no—
92C23×D4—71no—
93C2×#121—72no—
94C2×#120—72no—
95C2×C4×D4—71no—
96#394—71no—
97#394—71no—
98#392—71no—
99#392D711yescharacteristic, complemented
100#390—72no—
101#380—71no—
102#379—72no—
103C23×C7—22no—
104C23×C7—21no—
105C23×C7—22no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—21no—
109C23×C7—22no—
110C23×C7—22no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7C2×D414yes—
114C23×C7C2×D412yescomplemented
115C23×C7C2411yescharacteristic
116C22×D7—22no—
117C22×D7—22no—
118C22×D7—216no—
119C22×D7—22no—
120C22×D7—22no—
121C22×D7—22no—
122C22×D7—22no—
123C22×D7—22no—
124C22×D7—22no—
125C22×D7—48no—
126C22×D7—48no—
127C22×D7—48no—
128C22×D7—48no—
129C22×D7—48no—
130C22×D7—48no—
131C22×D7C2×D414yes—
132C22×D7C2×D412yes—
133C22×D7C2×D412yes—
134C22×D7#4112yes—
135C7×D4—28no—
136C7×D4—48no—
137C2×C4×C7—22no—
138C2×C4×C7—21no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7C2×D412yes—
144C22⋊D7—28no—
145C22⋊D7—216no—
146C22⋊D7—48no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—22no—
151C2×Dic7—22no—
152C2×Dic7—22no—
153C2×Dic7—22no—
154C2×Dic7—44no—
155C2×Dic7C2×D412yes—
156C2×Dic7C2×D414yes—
157C2×Dic7C2×D412yescomplemented
158D28—28no—
159C4×D7—28no—
160C25—72no—
161C22×D4—71no—
162C22×D4—72no—
163C22×D4—72no—
164C22×D4—71no—
165C22×D4—144no—
166C22×D4—148no—
167C22×D4C2×D711yescharacteristic
168C22×D4C2×D712yescomplemented
169C23×C4—71no—
170C23×C4—71no—
171#121—148no—
172#120—148no—
173C4×D4—144no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—72no—
176C2×C22⋊C4—72no—
177C2×C22⋊C4—72no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—72no—
180C2×C22⋊C4—72no—
181C2×C22⋊C4—72no—
182C2×C22⋊C4—72no—
183C2×C22⋊C4C2×D711yescharacteristic, complemented
184C2×C22⋊C4C2×D712yescomplemented
185C2×C42—71no—
186#95—71no—
187#95C2×D711yescharacteristic, complemented
188C22×C7—24no—
189C22×C7—22no—
190C22×C7—24no—
191C22×C7—24no—
192C22×C7—24no—
193C22×C7—41no—
194C22×C7—42no—
195C22×C7—42no—
196C22×C7—41no—
197C22×C7—42no—
198C22×C7—42no—
199C22×C7—42no—
200C22×C7C22×D411yescharacteristic
201C22×C7C22×D412yes—
202C22×C7C22×D411yescharacteristic
203C22×C7C2×#4111yescharacteristic
204C22×C7H(5,𝔽2)12yes—
205C2×D7—22no—
206C2×D7—22no—
207C2×D7—24no—
208C2×D7—24no—
209C2×D7—24no—
210C2×D7—24no—
211C2×D7—24no—
212C2×D7—432no—
213C2×D7—42no—
214C2×D7—42no—
215C2×D7—42no—
216C2×D7—42no—
217C2×D7#12114yescomplemented
218C4×C7—22no—
219C4×C7—41no—
220C4×C7—42no—
221Dic7—22no—
222Dic7—41no—
223Dic7—42no—
224Dic7#12014yescomplemented
225C24—71no—
226C24—72no—
227C24—144no—
228C24—144no—
229C24—144no—
230C24—144no—
231C24—144no—
232C24—282no—
233C24—282no—
234C24C22×D712yes—
235C24C22×D711yescharacteristic
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—78no—
242C2×D4—74no—
243C2×D4—144no—
244C2×D4—144no—
245C2×D4—144no—
246C2×D4—1432no—
247C2×D4—142no—
248C2×D4—142no—
249C2×D4—144no—
250C2×D4—144no—
251C2×D4—142no—
252C2×D4—282no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—74no—
256C22×C4—72no—
257C22×C4—71no—
258C22×C4—144no—
259C22×C4—142no—
260C22×C4—144no—
261C22×C4C22×D712yes—
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C4⋊C4—144no—
265C4⋊C4—142no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—144no—
269C22⋊C4—142no—
270C22⋊C4—144no—
271C22⋊C4—144no—
272C22⋊C4—144no—
273C22⋊C4—144no—
274C22⋊C4—282no—
275C42—142no—
276C2×C7—44no—
277C2×C7—42no—
278C2×C7C2×#12011yescharacteristic
279C2×C7C2×#12111yescharacteristic
280C2×C7#53511yescharacteristic
281C2×C7D4212yes—
282C2×C7#54612yes—
283D7—24no—
284D7—44no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—21no—
294C23—22no—
295C23—72no—
296C23—72no—
297C23—72no—
298C23—74no—
299C23—1416no—
300C23—142no—
301C23—142no—
302C23—142no—
303C23—142no—
304C23—142no—
305C23—142no—
306C23—142no—
307C23—142no—
308C23—288no—
309C23—288no—
310C23—288no—
311C23—288no—
312C23—288no—
313C23—288no—
314C23D4×D714yes—
315C23D4×D712yescomplemented
316C23C23×D711yescharacteristic
317D4—28no—
318D4—48no—
319D4—148no—
320D4—1416no—
321D4—148no—
322D4—288no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—72no—
330C2×C4—74no—
331C2×C4—72no—
332C2×C4—148no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—142no—
336C2×C4—142no—
337C2×C4—141no—
338C2×C4—142no—
339C2×C4—142no—
340C2×C4—284no—
341C2×C4D4×D712yes—
342C7#240811yescharacteristic, complemented, Sylow
343C22—24no—
344C22—24no—
345C22—24no—
346C22—22no—
347C22—24no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—74no—
356C22—142no—
357C22—142no—
358C22—144no—
359C22—144no—
360C22—144no—
361C22—144no—
362C22—144no—
363C22—2832no—
364C22—282no—
365C22—282no—
366C22—282no—
367C22—282no—
368C22C2×D4×D711yescharacteristic
369C22C2×D4×D712yes—
370C22C2×D4×D711yescharacteristic
371C22#654812yes—
372C22D7×#4111yescharacteristic
373C4—22no—
374C4—41no—
375C4—42no—
376C4—74no—
377C4—142no—
378C4—281no—
379C4—282no—
380C2—44no—
381C2—42no—
382C2—144no—
383C2—284no—
384C2D7×#12011yescharacteristic
385C2#9028112yes—
386C2D7×#12111yescharacteristic
387C2#9030112yes—
388C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2408
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D7C2semidirect
2. C2×#6502C2semidirect
3. C2×#6500C2semidirect
4. C2×(C22⋊D7)⋊C4C2semidirect
5. C7×#392C2semidirect
6. #89994C2semidirect
7. #89732C2semidirect
8. #89729C2semidirect
9. #89727C2semidirect
10. #89444C2semidirect
11. #89439C2semidirect
14. C22×C7×D4C22semidirect
17. C22×C22⋊D7C22semidirect
18. C23×Dic7C22semidirect
23. C2×C7×C22⋊C4C22semidirect
24. C2×C7×C22⋊C4C22semidirect
25. C2×C22⋊Dic7C22semidirect
26. C2×C22⋊Dic7C22semidirect
31. C2×(C2×D7)⋊C4C22semidirect
32. C2×(C2×D7)⋊C4C22semidirect
33. C2×(C2×D7)⋊C4C22semidirect
34. C2×Dic7⋊C4C22semidirect
35. C2×Dic7⋊C4C22semidirect
36. C2×C4×Dic7C22semidirect
38. C7×#95C22semidirect
39. C7⋊#95C22semidirect
67. C2×C22⋊D7D4semidirect
68. C2×C22⋊D7D4semidirect
79. C2×C4×D7D4semidirect
99. #392D7semidirect
114. C23×C7C2×D4semidirect
157. C2×Dic7C2×D4semidirect
168. C22×D4C2×D7semidirect
183. C2×C22⋊C4C2×D7semidirect
184. C2×C22⋊C4C2×D7semidirect
187. #95C2×D7semidirect
217. C2×D7#121semidirect
224. Dic7#120semidirect
315. C23D4×D7semidirect
342. C7#2408semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D72
C2×#65022
C2×#65002
C2×(C22⋊D7)⋊C42
C7×#3922
#899942
#897322
#897292
#897272
#894442
#894392
#24087