Finite Group Databasebeta

← #11733652

Subgroups — #11733652

Order 896.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6457C211yesmaximal, characteristic
2C7×#389C211yesmaximal, characteristic, complemented
3#89762C211yesmaximal, characteristic, complemented
4#89761C211yesmaximal, characteristic, complemented
5#89761C211yesmaximal, characteristic, complemented
6#89760C211yesmaximal, characteristic
7#89759C211yesmaximal, characteristic, complemented
8#89753C211yesmaximal, characteristic, complemented
9#89744C211yesmaximal, characteristic, complemented
10#89712C211yesmaximal, characteristic, complemented
11#89712C211yesmaximal, characteristic, complemented
12#89442C211yesmaximal, characteristic, complemented
13#89439C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic
15#89424C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C42×C7C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41#6457C418yescomplemented
42C7×#95C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52#2376—71nomaximal, Sylow
53C23×D7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C2×D28—41no—
69C2×D28—41no—
70C2×D28C2×C414yes—
71C2×C4×D7—41no—
72C2×C4×D7—41no—
73C2×C4×D7—41no—
74C2×C4×D7—41no—
75C2×C4×D7—41no—
76C2×C4×D7C2×C414yescomplemented
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4C2×C414yescomplemented
81C42×C7—41no—
82(C2×D7)⋊C4—41no—
83(C2×D7)⋊C4—41no—
84(C2×D7)⋊C4—41no—
85(C2×D7)⋊C4—41no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89(C2×D7)⋊C4C2×C414yes—
90(C2×D7)⋊C4C2×C414yes—
91(C2×D7)⋊C4C2×C414yescomplemented
92C4⋊Dic7—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96Dic7⋊C4C2×C414yescomplemented
97C4×Dic7—41no—
98C4×Dic7—41no—
99C2×#123—71no—
100#390—71no—
101#389—71no—
102#389D711yescharacteristic, complemented
103#388—71no—
104#388—71no—
105#388—71no—
106#388—71no—
107#387—71no—
108#387—71no—
109#386—71no—
110#385—71no—
111#385—71no—
112#380—71no—
113#380—71no—
114C23×C7C2411yescharacteristic
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7C22×C412yes—
128C22×D7C22×C412yes—
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—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7C22×C412yes—
148C2×C4×C7C22×C412yes—
149C2×C4×C7C22×C412yes—
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×Dic7C22×C412yes—
167C2×Dic7C22×C412yes—
168D28—44no—
169C4×D7—44no—
170C4×D7—44no—
171C22×D4—71no—
172C23×C4—71no—
173C23×C4—71no—
174#123—78no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4C2×D711yescharacteristic
181C2×C4⋊C4C2×D711yescharacteristic
182C2×C4⋊C4C2×D711yescharacteristic
183C2×C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C22⋊C4—71no—
194C2×C42—71no—
195C2×C42—71no—
196C2×C42C2×D711yescharacteristic
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95C2×D711yescharacteristic
206#95C2×D711yescharacteristic
207C22×C7C23×C411yescharacteristic
208C22×C7H(5,𝔽2)11yescharacteristic
209C22×C7H(5,𝔽2)11yescharacteristic
210C22×C7H(5,𝔽2)11yescharacteristic
211C22×C7H(5,𝔽2)11yescharacteristic
212C22×C72−1+411yescharacteristic
213C22×C72−1+411yescharacteristic
214C2×D7—22no—
215C2×D7—22no—
216C2×D7—22no—
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—22no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—22no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235Dic7—22no—
236Dic7—22no—
237Dic7—41no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241C24—71no—
242C24—71no—
243C2×D4—74no—
244C2×D4—281no—
245C2×D4—281no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—74no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—281no—
254C22×C4—281no—
255C22×C4—281no—
256C22×C4—281no—
257C22×C4—281no—
258C22×C4C22×D711yescharacteristic
259C22×C4C22×D711yescharacteristic
260C22×C4C22×D711yescharacteristic
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C22×C4C22×D711yescharacteristic
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—74no—
269C4⋊C4—281no—
270C4⋊C4—281no—
271C4⋊C4—281no—
272C4⋊C4—281no—
273C4⋊C4C4×D714yescomplemented
274C22⋊C4—74no—
275C22⋊C4—74no—
276C22⋊C4—74no—
277C22⋊C4—281no—
278C22⋊C4—281no—
279C22⋊C4—281no—
280C22⋊C4—281no—
281C22⋊C4—281no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C42—41no—
285C42—281no—
286C42—281no—
287C2×C7#51811yescharacteristic
288C2×C7#52011yescharacteristic
289C2×C7#52011yescharacteristic
290C2×C7#56011yescharacteristic
291C2×C7#56211yescharacteristic
292C2×C7#56211yescharacteristic
293C2×C7#56311yescharacteristic
294D7—42no—
295D7—42no—
296C23—72no—
297C23—72no—
298C23—141no—
299C23—141no—
300C23—141no—
301C23—141no—
302C23—141no—
303C23—141no—
304C23—141no—
305C23—141no—
306C23—141no—
307C23—141no—
308C23—141no—
309C23—141no—
310C23C23×D711yescharacteristic
311D4—284no—
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—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—72no—
331C2×C4—72no—
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×C4—284no—
350C2×C4C2×C4×D712yes—
351C2×C4C2×C4×D712yes—
352C2×C4C2×C4×D712yes—
353C7#237611yescharacteristic, complemented, Sylow
354C22—142no—
355C22—142no—
356C22—142no—
357C22—142no—
358C22—142no—
359C22—142no—
360C22—281no—
361C22—281no—
362C22—281no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—281no—
368C22C22×C4×D711yescharacteristic
369C22#654811yescharacteristic
370C22#654811yescharacteristic
371C22#655111yescharacteristic
372C22#655111yescharacteristic
373C22#655311yescharacteristic
374C22#655311yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—142no—
383C4—142no—
384C4—281no—
385C4—281no—
386C4—281no—
387C4—281no—
388C2—282no—
389C2—282no—
390C2#9019511yescharacteristic
391C2#9023611yescharacteristic
392C2#9026511yescharacteristic
393C2#9031111yescharacteristic
394C2#9033811yescharacteristic
395C2#9039111yescharacteristic
396C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#389C2
socle, cosocleC23×C7C24
prime radicalC22×D28C22
hypercenter#389D7
hypocenterC7#2376
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C7×#389C2semidirect
3. #89762C2semidirect
4. #89761C2semidirect
5. #89761C2semidirect
7. #89759C2semidirect
8. #89753C2semidirect
9. #89744C2semidirect
10. #89712C2semidirect
11. #89712C2semidirect
12. #89442C2semidirect
13. #89439C2semidirect
15. #89424C2semidirect
41. #6457C4semidirect
76. C2×C4×D7C2×C4semidirect
80. C7×C4⋊C4C2×C4semidirect
91. (C2×D7)⋊C4C2×C4semidirect
96. Dic7⋊C4C2×C4semidirect
102. #389D7semidirect
273. C4⋊C4C4×D7semidirect
353. C7#2376semidirect

Maximal subgroups

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

subgroupindex
C2×#64572
C7×#3892
#897622
#897612
#897602
#897592
#897532
#897442
#897122
#894422
#894392
#894372
#894242
#23767