Finite Group Databasebeta

← #11738990

Subgroups — #11738990

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90322C211yesmaximal, characteristic, complemented
2#90223C211yesmaximal, characteristic, complemented
3C7×#485C211yesmaximal, characteristic, complemented
4#89820C211yesmaximal, characteristic, complemented
5#89774C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic, complemented
7#89644C211yesmaximal, characteristic, complemented
8#89642C211yesmaximal, characteristic, complemented
9#89636C211yesmaximal, characteristic, complemented
10#89590C211yesmaximal, characteristic, complemented
11#89583C211yesmaximal, characteristic, complemented
12#89576C211yesmaximal, characteristic, complemented
13#89564C211yesmaximal, characteristic, complemented
14#89506C211yesmaximal, characteristic, complemented
15(C22⋊D7)⋊C8C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#117C2211yescharacteristic
20#6507C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22#6493C2211yescharacteristic
23C4×C22⋊D7C2211yescharacteristic
24C22×C7⋊C8C2211yescharacteristic
25C2×D56C2211yescharacteristic
26C2×#1132C2211yescharacteristic
27C2×C8×D7C2211yescharacteristic
28#6460C2211yescharacteristic
29#6458C2211yescharacteristic
30#6457—21no—
31#6456C2211yescharacteristic
32#6456C2211yescharacteristic
33#6455C2211yescharacteristic
34D7×C4⋊C4—21no—
35#6451C2211yescharacteristic
36#6448—21no—
37#6447—21no—
38#6447—21no—
39#6446—21no—
40#6438—21no—
41#6435—21no—
42C7×#107C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×C22⋊C8C2211yescharacteristic
47#6409C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58#6382C2211yescharacteristic
59#3303—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7D411yescharacteristic
65#1161—22no—
66#1161—41no—
67#1157—22no—
68#1157—41no—
69C2×D28—21no—
70C2×D28—21no—
71C2×D28C2311yescharacteristic
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C2×Dic14—21no—
79C2×Dic14C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C42×C7—21no—
89C22⋊Dic7D411yescharacteristic
90#1143—41no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4D411yescharacteristic
98C4⋊Dic7—21no—
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109D56—22no—
110#1132—41no—
111C8×D7—41no—
112#555—71no—
113#547—71no—
114#487—71no—
115#485—71no—
116#485D711yescharacteristic, complemented
117#476—71no—
118#465—71no—
119#465—71no—
120#461—71no—
121#449—71no—
122#449—71no—
123C4×D8—71no—
124C8×D4—71no—
125#416—71no—
126#416—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
129C22×D7—21no—
130C22×D7C2×D411yescharacteristic
131C7×Q8—21no—
132C7×Q8—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×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C22⋊D7—22no—
142C22⋊D7—42no—
143C22⋊D7—81no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149D28—21no—
150D28—21no—
151D28—21no—
152D28—42no—
153D28—81no—
154D28#4112yes—
155C4×D7—21no—
156C4×D7—21no—
157C4×D7—42no—
158C4×D7—42no—
159C4×D7—42no—
160C4×D7—42no—
161Dic14—21no—
162Dic14—21no—
163Dic14—81no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C2×#41—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×D8—71no—
173C22×C8—71no—
174C22×C8—71no—
175C4⋊Q8—71no—
176C4⋊Q8—71no—
177#124—141no—
178#124—141no—
179#123—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8C2×D711yescharacteristic
183#121—71no—
184#121—71no—
185#121—141no—
186#121—141no—
187C4×D4—71no—
188C4×D4—71no—
189C4×D4—71no—
190#117—71no—
191#117—141no—
192#117C2×D711yescharacteristic
193C2×C4⋊C4—141no—
194#107C2×D711yescharacteristic
195#106—71no—
196#106C2×D711yescharacteristic
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208C22⋊C8—71no—
209C22⋊C8C2×D711yescharacteristic
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7C22×D411yescharacteristic
215C2×D7—21no—
216C2×D7—22no—
217C2×D7—21no—
218C2×D7—21no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C4×C7—21no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7C2×#4111yescharacteristic
230C4×C7H(5,𝔽2)11yescharacteristic
231Dic7—21no—
232Dic7—21no—
233Dic7—41no—
234Dic7—41no—
235Dic7#13512yes—
236#41—142no—
237#41—142no—
238#41—281no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8—141no—
242C2×Q8C22×D711yescharacteristic
243C2×D4—71no—
244C2×D4—71no—
245C2×D4—71no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4—141no—
250C22×C4—71no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4C22×D711yescharacteristic
256SD16—281no—
257SD16—281no—
258D8—142no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—281no—
262C2×C8—281no—
263C2×C8C22×D711yescharacteristic
264C2×C8C22×D711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C4⋊C4C22×D711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—71no—
278C22⋊C4—71no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C42—21no—
286C42—71no—
287C42—141no—
288C2×C7—41no—
289C2×C7#54611yescharacteristic
290C2×C7C2×#13511yescharacteristic
291C2×C7#57711yescharacteristic
292D7—41no—
293D7—42no—
294D7—81no—
295C23—71no—
296C23—141no—
297C23—141no—
298C23D4×D711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—141no—
302Q8—141no—
303Q8—561no—
304D4—72no—
305D4—142no—
306D4—141no—
307D4—141no—
308D4—141no—
309D4—282no—
310D4—282no—
311D4—561no—
312D4—561no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—71no—
320C2×C4—71no—
321C2×C4—71no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—282no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4D4×D711yescharacteristic
331C2×C4C23×D711yescharacteristic
332C8—21no—
333C8—21no—
334C8—141no—
335C8—141no—
336C7#330311yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—141no—
341C22—141no—
342C22—141no—
343C22—142no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22C2×D4×D711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—72no—
356C4—141no—
357C4—141no—
358C4—281no—
359C4—281no—
360C4D7×#4111yescharacteristic
361C4#655311yescharacteristic
362C2—41no—
363C2—281no—
364C2—282no—
365C2—561no—
366C2#9034511yescharacteristic
367C2D7×#13511yescharacteristic
368C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
hypercenter#485D7
hypocenterC7#3303
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #90322C2semidirect
2. #90223C2semidirect
3. C7×#485C2semidirect
4. #89820C2semidirect
5. #89774C2semidirect
6. #89660C2semidirect
7. #89644C2semidirect
8. #89642C2semidirect
9. #89636C2semidirect
10. #89590C2semidirect
11. #89583C2semidirect
12. #89576C2semidirect
13. #89564C2semidirect
14. #89506C2semidirect
15. (C22⋊D7)⋊C8C2semidirect
116. #485D7semidirect
336. C7#3303semidirect

Maximal subgroups

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

subgroupindex
#903222
#902232
C7×#4852
#898202
#897742
#896602
#896442
#896422
#896362
#895902
#895832
#895762
#895642
#895062
(C22⋊D7)⋊C82
#33037