Finite Group Databasebeta

← #11731362

Subgroups — #11731362

Order 896.

440 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#380C211yesmaximal, characteristic
2#89982C211yesmaximal, characteristic
3C2×C7⋊#95C211yesmaximal, characteristic
4#89758C211yesmaximal, characteristic, complemented
5#89754C211yesmaximal, characteristic, complemented
6#89753C211yesmaximal, characteristic, complemented
7#89750C211yesmaximal, characteristic, complemented
8#89722C211yesmaximal, characteristic
9#89715C211yesmaximal, characteristic
10#89715C211yesmaximal, characteristic
11C22⋊C4×Dic7C211yesmaximal, characteristic
12C7⋊#401C211yesmaximal, characteristic, complemented
13#89432C211yesmaximal, characteristic, complemented
14#89426C211yesmaximal, characteristic, complemented
15#89423C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C7×#95C2211yescharacteristic
32C7×#95C2211yescharacteristic
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2211yescharacteristic
37C7⋊#95C2211yescharacteristic
38C7⋊#95C2211yescharacteristic
39C7⋊#95C2211yescharacteristic
40C7⋊#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2493—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—41no—
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7—22no—
63C22×Dic7—22no—
64C22×Dic7—22no—
65C22×Dic7—22no—
66C22×Dic7—22no—
67C22×Dic7—41no—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C7×C4⋊C4—22no—
77C7×C4⋊C4—41no—
78C7×C22⋊C4—22no—
79C7×C22⋊C4—22no—
80C22⋊Dic7—22no—
81C22⋊Dic7—22no—
82C4⋊Dic7—22no—
83C4⋊Dic7—41no—
84Dic7⋊C4—22no—
85Dic7⋊C4—41no—
86C4×Dic7—22no—
87C4×Dic7—41no—
88#401—71no—
89#400—71no—
90#400—71no—
91#400—71no—
92#387—71no—
93#387—71no—
94#387—71no—
95#385—71no—
96#385—71no—
97#383—71no—
98#381—71no—
99#381—71no—
100#380D711yescharacteristic
101C4×C22⋊C4—71no—
102C2×#95—71no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7#4112yes—
108C23×C7#4112yes—
109C23×C7#4112yes—
110C23×C7C2411yescharacteristic
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—44no—
129C2×C4×C7#4112yes—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
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—44no—
154C2×Dic7—44no—
155C2×Dic7#4112yes—
156C23×C4—71no—
157C23×C4—71no—
158C23×C4C2×D711yescharacteristic
159C2×C4⋊C4—71no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4C2×D711yescharacteristic
164C2×C4⋊C4C2×D711yescharacteristic
165C2×C22⋊C4—71no—
166C2×C22⋊C4—71no—
167C2×C22⋊C4C2×D711yescharacteristic
168C2×C22⋊C4C2×D711yescharacteristic
169C2×C42—71no—
170C2×C42—71no—
171#95—71no—
172#95—71no—
173#95—71no—
174#95—71no—
175#95—71no—
176#95—71no—
177#95—71no—
178#95—71no—
179#95—71no—
180#95—71no—
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95C2×D711yescharacteristic
190#95C2×D711yescharacteristic
191C22×C7—22no—
192C22×C7—22no—
193C22×C7—22no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7#12614yes—
198C22×C7C2×#4111yescharacteristic
199C22×C7C2×#4111yescharacteristic
200C22×C7C2×#4111yescharacteristic
201C22×C7C2×#4111yescharacteristic
202C22×C7C2×#4111yescharacteristic
203C22×C7H(5,𝔽2)11yescharacteristic
204C22×C72−1+411yescharacteristic
205C4×C7—22no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211Dic7—22no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219C24C22×D711yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—142no—
232C22×C4—142no—
233C22×C4—142no—
234C22×C4—142no—
235C22×C4—142no—
236C22×C4—281no—
237C22×C4C22×D711yescharacteristic
238C22×C4C22×D711yescharacteristic
239C22×C4C22×D711yescharacteristic
240C22×C4C22×D711yescharacteristic
241C22×C4C22×D711yescharacteristic
242C22×C4C22×D711yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—142no—
246C4⋊C4—142no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—142no—
252C22⋊C4—142no—
253C42—142no—
254C42—281no—
255C2×C7—24no—
256C2×C7C2×#12611yescharacteristic
257C2×C7#52911yescharacteristic
258C2×C7#53911yescharacteristic
259C2×C7#55111yescharacteristic
260C2×C7#55111yescharacteristic
261C2×C7#55211yescharacteristic
262C2×C7#55211yescharacteristic
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#115712yes—
268C23#115912yes—
269C23#116112yes—
270C23C23×D711yescharacteristic
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—44no—
289C2×C4—72no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—284no—
314C2×C4—284no—
315C2×C4#115912yes—
316C7#249311yescharacteristic, complemented, Sylow
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22#646114yes—
324C22C2×#115711yescharacteristic
325C22C2×#115911yescharacteristic
326C22C2×#115911yescharacteristic
327C22#654811yescharacteristic
328C22C2×#116111yescharacteristic
329C22D7×#4111yescharacteristic
330C22#655411yescharacteristic
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—142no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4—281no—
343C4—281no—
344C4—281no—
345C2—24no—
346C2#9018611yescharacteristic
347C2C2×#646111yescharacteristic
348C2#9022211yescharacteristic
349C2#9029111yescharacteristic
350C2#9029311yescharacteristic
351C2#9032811yescharacteristic
352C2#9033311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#380C2
prime radicalC24×C7C23
hypercenter#380D7
hypocenterC7#2493
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
4. #89758C2semidirect
5. #89754C2semidirect
6. #89753C2semidirect
7. #89750C2semidirect
12. C7⋊#401C2semidirect
13. #89432C2semidirect
14. #89426C2semidirect
15. #89423C2semidirect
316. C7#2493semidirect

Maximal subgroups

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

subgroupindex
C7×#3802
#899822
C2×C7⋊#952
#897582
#897542
#897532
#897502
#897222
#897152
C22⋊C4×Dic72
C7⋊#4012
#894322
#894262
#894232
#24937