Finite Group Databasebeta

← #11732302

Subgroups — #11732302

Order 896.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#386C211yesmaximal, characteristic
2#89993C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89984C211yesmaximal, characteristic, complemented
5#89982C211yesmaximal, characteristic, complemented
6#89722C211yesmaximal, characteristic, complemented
7#89721C211yesmaximal, characteristic, complemented
8#89720C211yesmaximal, characteristic, complemented
9#89720C211yesmaximal, characteristic, complemented
10#89720C211yesmaximal, characteristic, complemented
11#89718C211yesmaximal, characteristic, complemented
12#89433C211yesmaximal, characteristic, complemented
13#89432C211yesmaximal, characteristic, complemented
14#89427C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7×#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2780—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C24×C7C2311yescharacteristic
54C2×C7×D4—41no—
55C2×C7×D4—41no—
56C2×C7×D4Q814yes—
57C22×C4×C7—22no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×Dic7—41no—
66C22×Dic7—41no—
67C22×Dic7—41no—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C7×C22⋊C4—22no—
77C7×C22⋊C4—41no—
78C7×C22⋊C4—41no—
79C22⋊Dic7—22no—
80C22⋊Dic7—22no—
81C22⋊Dic7—41no—
82C22⋊Dic7—41no—
83C22⋊Dic7—41no—
84C22⋊Dic7—41no—
85C22⋊Dic7—41no—
86C4⋊Dic7—41no—
87C4⋊Dic7—41no—
88C4⋊Dic7—41no—
89C4⋊Dic7—41no—
90C4⋊Dic7—41no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96Dic7⋊C4—41no—
97C4×Dic7—41no—
98#400—71no—
99#399—71no—
100#398—71no—
101#396—71no—
102#396—71no—
103#396—71no—
104#390—71no—
105#389—71no—
106#387—71no—
107#386—71no—
108#386D711yescharacteristic
109#385—71no—
110#385—71no—
111#385—71no—
112#380—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7C2×Q812yes—
126C23×C7C2×Q812yes—
127C23×C7C2411yescharacteristic
128C7×D4—44no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—44no—
143C2×C4×C7C2×Q812yes—
144C2×C4×C7#4112yes—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—44no—
169C2×Dic7#4112yes—
170C22×D4C2×D711yescharacteristic
171C23×C4—71no—
172C23×C4C2×D711yescharacteristic
173C2×C4⋊C4—71no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C22⋊C4—71no—
187C2×C22⋊C4—71no—
188C2×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4—71no—
191C2×C22⋊C4C2×D711yescharacteristic
192C2×C22⋊C4C2×D711yescharacteristic
193C2×C22⋊C4C2×D711yescharacteristic
194C2×C42—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95C2×D711yescharacteristic
204#95C2×D711yescharacteristic
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—22no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×Q811yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7C2×#4111yescharacteristic
222C22×C7H(5,𝔽2)11yescharacteristic
223C22×C7H(5,𝔽2)11yescharacteristic
224C22×C7H(5,𝔽2)11yescharacteristic
225C22×C7H(5,𝔽2)11yescharacteristic
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231Dic7—22no—
232Dic7—41no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239C24C22×D711yescharacteristic
240C24C22×D711yescharacteristic
241C2×D4—41no—
242C2×D4—41no—
243C2×D4Dic1414yes—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—71no—
248C22×C4—71no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—281no—
256C22×C4—281no—
257C22×C4—281no—
258C22×C4C22×D711yescharacteristic
259C22×C4C22×D711yescharacteristic
260C22×C4C22×D711yescharacteristic
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C4⋊C4—142no—
264C4⋊C4—142no—
265C4⋊C4—281no—
266C4⋊C4—281no—
267C4⋊C4—281no—
268C4⋊C4—281no—
269C4⋊C4—281no—
270C4⋊C4—281no—
271C4⋊C4—281no—
272C4⋊C4—281no—
273C4⋊C4—281no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—142no—
278C22⋊C4—142no—
279C22⋊C4—281no—
280C22⋊C4—281no—
281C22⋊C4—281no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C42—281no—
285C2×C7—42no—
286C2×C7—42no—
287C2×C7#53811yescharacteristic
288C2×C7#54011yescharacteristic
289C2×C7#54311yescharacteristic
290C2×C7#55311yescharacteristic
291C2×C7#55411yescharacteristic
292C2×C7#55411yescharacteristic
293C2×C7#56011yescharacteristic
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23C2×Dic1412yes—
307C23C2×Dic1412yes—
308C23C23×D711yescharacteristic
309D4—44no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—72no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—284no—
349C2×C4C2×Dic1412yes—
350C2×C4#115912yes—
351C7#278011yescharacteristic, complemented, Sylow
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22C22×Dic1411yescharacteristic
367C22C2×#115911yescharacteristic
368C22#654811yescharacteristic
369C22#654811yescharacteristic
370C22#654811yescharacteristic
371C22D7×#4111yescharacteristic
372C22#655311yescharacteristic
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—142no—
379C4—281no—
380C4—281no—
381C4—281no—
382C4—281no—
383C4—281no—
384C4—281no—
385C4—281no—
386C2—42no—
387C2—42no—
388C2#9017411yescharacteristic
389C2#9023011yescharacteristic
390C2#9023411yescharacteristic
391C2#9028411yescharacteristic
392C2#9030511yescharacteristic
393C2#9031111yescharacteristic
394C2#9035411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2780
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #89984C2semidirect
5. #89982C2semidirect
6. #89722C2semidirect
7. #89721C2semidirect
8. #89720C2semidirect
9. #89720C2semidirect
10. #89720C2semidirect
11. #89718C2semidirect
12. #89433C2semidirect
13. #89432C2semidirect
14. #89427C2semidirect
15. #89419C2semidirect
351. C7#2780semidirect

Maximal subgroups

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

subgroupindex
C7×#3862
#899932
#899922
#899842
#899822
#897222
#897212
#897202
#897182
#894332
#894322
#894272
#894192
#27807