Finite Group Databasebeta

← #11735682

Subgroups — #11735682

Order 896.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6460C211yesmaximal, characteristic
2C2×#6453C211yesmaximal, characteristic, complemented
3C7×#400C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic
5#89750C211yesmaximal, characteristic, complemented
6#89447C211yesmaximal, characteristic
7#89442C211yesmaximal, characteristic
8#89442C211yesmaximal, characteristic
9#89440C211yesmaximal, characteristic
10#89439C211yesmaximal, characteristic
11#89433C211yesmaximal, characteristic, complemented
12#89432C211yesmaximal, characteristic, complemented
13#89428C211yesmaximal, characteristic, complemented
14#89426C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37#6460—24no—
38#6453—24no—
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7×#95C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2690—71nomaximal, Sylow
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
68C22×Dic7C2311yescharacteristic
69C2×C4×D7—22no—
70C2×C4×D7—22no—
71C2×C4×D7—41no—
72C2×Dic14—22no—
73C2×Dic14—22no—
74C2×Dic14—41no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77(C2×D7)⋊C4—22no—
78(C2×D7)⋊C4—22no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—41no—
83C4⋊Dic7—22no—
84C4⋊Dic7—22no—
85C4⋊Dic7—22no—
86C4⋊Dic7—41no—
87C4⋊Dic7D414yes—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—41no—
92C4×Dic7—22no—
93C4×Dic7—41no—
94C2×#125—71no—
95C2×C22⋊Q8—71no—
96#400—71no—
97#400D711yescharacteristic, complemented
98#398—71no—
99#397—71no—
100#397—71no—
101#395—71no—
102#391—71no—
103#388—71no—
104#388—71no—
105#387—71no—
106#387—71no—
107#385—71no—
108#380—71no—
109C23×C7C2411yescharacteristic
110C22×D7—21no—
111C22×D7—21no—
112C22×D7—21no—
113C22×D7—21no—
114C22×D7—21no—
115C22×D7—21no—
116C22×D7#4112yes—
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—21no—
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×C7C2×D412yes—
137C2×C4×C7#4112yes—
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—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7C2×D412yes—
158C2×Dic7C2×D412yes—
159C4×D7—44no—
160Dic14—44no—
161C22×Q8—71no—
162C23×C4—71no—
163#125—144no—
164C22⋊Q8—144no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4C2×D711yescharacteristic
175C2×C4⋊C4C2×D711yescharacteristic
176C2×C22⋊C4—71no—
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—71no—
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C42—71no—
183C2×C42—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95C2×D711yescharacteristic
194#95C2×D711yescharacteristic
195#95C2×D711yescharacteristic
196#95C2×D711yescharacteristic
197#95C2×D711yescharacteristic
198C22×C7C22×D411yescharacteristic
199C22×C7C2×#4111yescharacteristic
200C22×C7C2×#4111yescharacteristic
201C22×C7H(5,𝔽2)11yescharacteristic
202C22×C72−1+411yescharacteristic
203C22×C72−1+411yescharacteristic
204C22×C72−1+411yescharacteristic
205C2×D7—22no—
206C2×D7—22no—
207C2×D7—22no—
208C2×D7—41no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C4×C7—22no—
213C4×C7—22no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219Dic7—22no—
220Dic7—22no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226C24—71no—
227C2×Q8—142no—
228C2×Q8—142no—
229C2×Q8—281no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—142no—
238C22×C4—142no—
239C22×C4—281no—
240C22×C4C22×D711yescharacteristic
241C22×C4C22×D711yescharacteristic
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244C22×C4C22×D711yescharacteristic
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—74no—
250C4⋊C4—142no—
251C4⋊C4—142no—
252C4⋊C4—142no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—142no—
256C4⋊C4—281no—
257C4⋊C4—281no—
258C22⋊C4—142no—
259C22⋊C4—142no—
260C22⋊C4—142no—
261C22⋊C4—142no—
262C22⋊C4—142no—
263C22⋊C4—281no—
264C42—142no—
265C42—281no—
266C2×C7#53611yescharacteristic
267C2×C7#53911yescharacteristic
268C2×C7#54111yescharacteristic
269C2×C7#54611yescharacteristic
270C2×C7#54811yescharacteristic
271C2×C7#55211yescharacteristic
272C2×C7#56311yescharacteristic
273D7—42no—
274C23—72no—
275C23—141no—
276C23—141no—
277C23—141no—
278C23—141no—
279C23—141no—
280C23—141no—
281C23C23×D711yescharacteristic
282Q8—284no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—72no—
303C2×C4—72no—
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—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—284no—
324C2×C4D4×D712yes—
325C2×C4#116112yes—
326C7#269011yescharacteristic, complemented, Sylow
327C22—142no—
328C22—142no—
329C22—142no—
330C22—281no—
331C22—281no—
332C22—281no—
333C22—281no—
334C22C2×D4×D711yescharacteristic
335C22C2×#116111yescharacteristic
336C22D7×#4111yescharacteristic
337C22#655311yescharacteristic
338C22#655411yescharacteristic
339C22#655411yescharacteristic
340C22#655411yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—142no—
349C4—142no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4—281no—
354C4—281no—
355C2—282no—
356C2#9032111yescharacteristic
357C2#9033911yescharacteristic
358C2#9036811yescharacteristic
359C2#9038511yescharacteristic
360C2#9038811yescharacteristic
361C2#9039811yescharacteristic
362C2#9040311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2690
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#6453C2semidirect
3. C7×#400C2semidirect
5. #89750C2semidirect
11. #89433C2semidirect
12. #89432C2semidirect
13. #89428C2semidirect
14. #89426C2semidirect
15. #89418C2semidirect
97. #400D7semidirect
326. C7#2690semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64532
C7×#4002
#897652
#897502
#894472
#894422
#894402
#894392
#894332
#894322
#894282
#894262
#894182
#26907