Finite Group Databasebeta

← #11740244

Subgroups — #11740244

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×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×#6442C211yesmaximal, characteristic
3#90002C211yesmaximal, characteristic, complemented
4#90001C211yesmaximal, characteristic, complemented
5#89983C211yesmaximal, characteristic
6#89982C211yesmaximal, characteristic
7#89758C211yesmaximal, characteristic, complemented
8#89756C211yesmaximal, characteristic, complemented
9#89752C211yesmaximal, characteristic, complemented
10#89750C211yesmaximal, characteristic, complemented
11#89749C211yesmaximal, characteristic, complemented
12#89744C211yesmaximal, characteristic, complemented
13#89724C211yesmaximal, characteristic
14#89724C211yesmaximal, characteristic
15#89723C211yesmaximal, characteristic
16C22×C7×Q8C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C7×C22⋊Q8—24no—
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38#6442—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#2698—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—41no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—41no—
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C22⋊C4—22no—
79C7×C22⋊C4D414yes—
80C22⋊Dic7—22no—
81C22⋊Dic7—22no—
82C22⋊Dic7—22no—
83C22⋊Dic7—41no—
84C4⋊Dic7—22no—
85C4⋊Dic7—41no—
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×C22⋊Q8D711yescharacteristic
96#400—71no—
97#398—71no—
98#398—71no—
99#397—71no—
100#397—71no—
101#395—71no—
102#391—71no—
103#389—71no—
104#388—71no—
105#387—71no—
106#385—71no—
107#382—71no—
108#381—71no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7C2×D412yes—
116C23×C7C2411yescharacteristic
117C7×Q8—44no—
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×C7C2×D412yes—
135C2×C4×C7C2×D412yes—
136C2×C4×C7#4112yes—
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×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7#4112yes—
161C22×Q8C2×D711yescharacteristic
162C23×C4C2×D711yescharacteristic
163#126—144no—
164C22⋊Q8—24no—
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⋊C4C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
181C2×C22⋊C4C2×D711yescharacteristic
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#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198C22×C7—22no—
199C22×C7—22no—
200C22×C7—22no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7C22×D411yescharacteristic
206C22×C7C2×#4111yescharacteristic
207C22×C7C2×#4111yescharacteristic
208C22×C7H(5,𝔽2)11yescharacteristic
209C22×C72−1+411yescharacteristic
210C22×C72−1+411yescharacteristic
211C22×C72−1+411yescharacteristic
212C4×C7—22no—
213C4×C7—22no—
214C4×C7—22no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218Dic7—22no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226C24C22×D711yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
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—22no—
250C4⋊C4—142no—
251C4⋊C4—142no—
252C4⋊C4—142no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—281no—
256C4⋊C4—281no—
257C4⋊C4—281no—
258C22⋊C4—22no—
259C22⋊C4—142no—
260C22⋊C4—142no—
261C22⋊C4—142no—
262C22⋊C4—281no—
263C22⋊C4C22⋊D714yes—
264C42—142no—
265C42—142no—
266C2×C7—42no—
267C2×C7#53611yescharacteristic
268C2×C7#54111yescharacteristic
269C2×C7#54211yescharacteristic
270C2×C7#54611yescharacteristic
271C2×C7#54711yescharacteristic
272C2×C7#55211yescharacteristic
273C2×C7#56311yescharacteristic
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23C2×C22⋊D712yes—
281C23C23×D711yescharacteristic
282Q8—44no—
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—44no—
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—141no—
323C2×C4#115912yes—
324C2×C4C2×C22⋊D712yes—
325C2×C4C2×C22⋊D712yes—
326C7#269811yescharacteristic, complemented, Sylow
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C2×#115911yescharacteristic
335C22#654811yescharacteristic
336C22#655111yescharacteristic
337C22#655111yescharacteristic
338C22D7×#4111yescharacteristic
339C22#655411yescharacteristic
340C22C22×C22⋊D711yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—142no—
348C4—281no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4—281no—
354C4—281no—
355C2—42no—
356C2#9031611yescharacteristic
357C2#9033011yescharacteristic
358C2#9033111yescharacteristic
359C2#9033711yescharacteristic
360C2#9049711yescharacteristic
361C2#9050711yescharacteristic
362C2#9051511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC24×C7C23
hypercenterC2×C22⋊Q8D7
hypocenterC7#2698
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #90002C2semidirect
4. #90001C2semidirect
7. #89758C2semidirect
8. #89756C2semidirect
9. #89752C2semidirect
10. #89750C2semidirect
11. #89749C2semidirect
12. #89744C2semidirect
326. C7#2698semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#64422
#900022
#900012
#899832
#899822
#897582
#897562
#897522
#897502
#897492
#897442
#897242
#897232
#26987