Finite Group Databasebeta

← #11731417

Subgroups — #11731417

Order 896.

448 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6442C211yesmaximal, characteristic
2C2×#6442C211yesmaximal, characteristic
3C7×#380C211yesmaximal, characteristic
4#89984C211yesmaximal, characteristic
5#89982C211yesmaximal, characteristic
6#89756C211yesmaximal, characteristic, complemented
7#89753C211yesmaximal, characteristic, complemented
8#89752C211yesmaximal, characteristic, complemented
9#89744C211yesmaximal, characteristic, complemented
10#89724C211yesmaximal, characteristic
11#89724C211yesmaximal, characteristic
12#89433C211yesmaximal, characteristic, complemented
13#89429C211yesmaximal, characteristic, complemented
14#89426C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39#6442—24no—
40#6442—24no—
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#2722—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—41no—
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C7×C4⋊C4—22no—
73C7×C4⋊C4—41no—
74C7×C22⋊C4—22no—
75C7×C22⋊C4—22no—
76C22⋊Dic7—22no—
77C22⋊Dic7—22no—
78C22⋊Dic7—41no—
79C22⋊Dic7D414yes—
80C4⋊Dic7—22no—
81C4⋊Dic7—22no—
82C4⋊Dic7—22no—
83C4⋊Dic7—41no—
84Dic7⋊C4—22no—
85Dic7⋊C4—22no—
86Dic7⋊C4—22no—
87Dic7⋊C4—22no—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—41no—
91Dic7⋊C4—41no—
92C4×Dic7—22no—
93C4×Dic7—22no—
94C2×#126—71no—
95C2×#126—71no—
96#398—71no—
97#398—71no—
98#398—71no—
99#398—71no—
100#397—71no—
101#397—71no—
102#389—71no—
103#389—71no—
104#387—71no—
105#387—71no—
106#385—71no—
107#380—71no—
108#380D711yescharacteristic
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7C2×D412yes—
116C23×C7C2411yescharacteristic
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—44no—
134C2×C4×C7#4112yes—
135C2×C4×C7#4112yes—
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—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7C2×D412yes—
159C2×Dic7C2×D412yes—
160C23×C4C2×D711yescharacteristic
161#126—144no—
162#126—144no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
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⋊C4—71no—
175C2×C4⋊C4C2×D711yescharacteristic
176C2×C4⋊C4C2×D711yescharacteristic
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—71no—
180C2×C22⋊C4—71no—
181C2×C22⋊C4C2×D711yescharacteristic
182C2×C22⋊C4C2×D711yescharacteristic
183C2×C42—71no—
184C2×C42—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#95—71no—
194#95—71no—
195#95C2×D711yescharacteristic
196#95C2×D711yescharacteristic
197C22×C7—22no—
198C22×C7—22no—
199C22×C7—22no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7C22×D411yescharacteristic
205C22×C7C2×#4111yescharacteristic
206C22×C7C2×#4111yescharacteristic
207C22×C7H(5,𝔽2)11yescharacteristic
208C22×C72−1+411yescharacteristic
209C22×C72−1+411yescharacteristic
210C22×C72−1+411yescharacteristic
211C4×C7—22no—
212C4×C7—22no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217Dic7—22no—
218Dic7—22no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C24C22×D711yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
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—142no—
248C4⋊C4—142no—
249C4⋊C4—142no—
250C4⋊C4—142no—
251C4⋊C4—142no—
252C4⋊C4—142no—
253C4⋊C4—142no—
254C4⋊C4—281no—
255C4⋊C4—281no—
256C4⋊C4—281no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—74no—
260C22⋊C4—142no—
261C22⋊C4—142no—
262C22⋊C4—281no—
263C42—142no—
264C42—142no—
265C2×C7—42no—
266C2×C7#53711yescharacteristic
267C2×C7#54111yescharacteristic
268C2×C7#54111yescharacteristic
269C2×C7#54711yescharacteristic
270C2×C7#54711yescharacteristic
271C2×C7#55111yescharacteristic
272C2×C7#56311yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23D4×D712yes—
280C23C23×D711yescharacteristic
281C2×C4—21no—
282C2×C4—21no—
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—44no—
298C2×C4—72no—
299C2×C4—72no—
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—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#115712yes—
323C2×C4#115912yes—
324C7#272211yescharacteristic, complemented, Sylow
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C2×#115711yescharacteristic
333C22C2×D4×D711yescharacteristic
334C22C2×#115911yescharacteristic
335C22#654811yescharacteristic
336C22#655111yescharacteristic
337C22#655411yescharacteristic
338C22#655411yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—142no—
346C4—142no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C2—42no—
354C2#9018611yescharacteristic
355C2#9019811yescharacteristic
356C2#9020911yescharacteristic
357C2#9028811yescharacteristic
358C2#9030211yescharacteristic
359C2#9031611yescharacteristic
360C2#9033711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#380C2
prime radicalC24×C7C23
hypercenter#380D7
hypocenterC7#2722
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
6. #89756C2semidirect
7. #89753C2semidirect
8. #89752C2semidirect
9. #89744C2semidirect
12. #89433C2semidirect
13. #89429C2semidirect
14. #89426C2semidirect
15. #89419C2semidirect
324. C7#2722semidirect

Maximal subgroups

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

subgroupindex
C2×#64422
C7×#3802
#899842
#899822
#897562
#897532
#897522
#897442
#897242
#894332
#894292
#894262
#894192
#27227