Finite Group Databasebeta

← #24918572

Subgroups — #24918572

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128322C211yesmaximal, characteristic, complemented
2#19128093C211yesmaximal, characteristic, complemented
3#19128083C211yesmaximal, characteristic, complemented
4#19128060C211yesmaximal, characteristic, complemented
5#19128051C211yesmaximal, characteristic, complemented
6#19127920C211yesmaximal, characteristic, complemented
7#19127913C211yesmaximal, characteristic, complemented
8#19127908C211yesmaximal, characteristic, complemented
9#19127903C211yesmaximal, characteristic, complemented
10#19127863C211yesmaximal, characteristic, complemented
11#19127857C211yesmaximal, characteristic, complemented
12#19127846C211yesmaximal, characteristic
13#19127840C211yesmaximal, characteristic, complemented
14#19127762C211yesmaximal, characteristic, complemented
15#536×C83C211yesmaximal, characteristic, complemented
16C2×#13050580C2211yescharacteristic
17C2×#13050580C2211yescharacteristic
18C2×#13050580C2211yescharacteristic
19C2×#13050578C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21C2×#13050577C2211yescharacteristic
22C2×#13050577C2211yescharacteristic
23C2×#13050575C2211yescharacteristic
24#14581792C2211yescharacteristic
25#14581787—21no—
26#14581780C2211yescharacteristic
27#14581778—21no—
28#14581774—21no—
29#14581771C2211yescharacteristic
30#14581770—21no—
31#14581766C2211yescharacteristic
32C4×Dic166C2211yescharacteristic
33C4×D332C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35C4⋊(C83⋊C8)C2211yescharacteristic
36#14581716C2211yescharacteristic
37#14581716C2211yescharacteristic
38#14581716C2211yescharacteristic
39#14581716C2211yescharacteristic
40#14581716C2211yescharacteristic
41#14581716C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44C22⋊(C83⋊C8)C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46#14581703C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48C2×#41×C83C2211yescharacteristic
49C22×Q8×C83C2211yescharacteristic
50C4⋊Q8×C83C2211yescharacteristic
51C4⋊Q8×C83C2211yescharacteristic
52#124×C83C2211yescharacteristic
53#124×C83C2211yescharacteristic
54#123×C83—21no—
55#123×C83—21no—
56C22⋊Q8×C83—21no—
57C22⋊Q8×C83—21no—
58#117×C83C2211yescharacteristic
59C2×Dic166—21no—
60C2×Dic166C2311yescharacteristic
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C4×D83—21no—
64#13050580—41no—
65#13050580—41no—
66#13050580—41no—
67#13050578—41no—
68#13050577—41no—
69#13050577—41no—
70#13050577—41no—
71#13050575—41no—
72C2×C83⋊C8C2311yescharacteristic
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C4⋊Dic83—21no—
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81C22⋊Dic83—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86C4×Dic83—21no—
87#41×C83—41no—
88#41×C83—41no—
89C2×Q8×C83—21no—
90C2×Q8×C83—41no—
91C2×Q8×C83—41no—
92C2×Q8×C83C2311yescharacteristic
93C2×Q8×C83C2311yescharacteristic
94C2×Q8×C83C2311yescharacteristic
95C2×D4×C83—21no—
96C2×D4×C83C2311yescharacteristic
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83C2311yescharacteristic
100C4⋊C4×C83—21no—
101C4⋊C4×C83—21no—
102C4⋊C4×C83—21no—
103C4⋊C4×C83—21no—
104C4⋊C4×C83D411yescharacteristic
105C4⋊C4×C83D411yescharacteristic
106C22⋊C4×C83—21no—
107C22⋊C4×C83—21no—
108C22⋊C4×C83—21no—
109C22⋊C4×C83—21no—
110C22⋊C4×C83D411yescharacteristic
111C22⋊C4×C83D411yescharacteristic
112C42×C83C2311yescharacteristic
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C22×D83—21no—
117Q8×C83—21no—
118Q8×C83—21no—
119Q8×C83—21no—
120Q8×C83—21no—
121Q8×C83—21no—
122Q8×C83—21no—
123Q8×C83—42no—
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—42no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2411yescharacteristic
140C22⋊D83—81no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144D332—21no—
145D332—21no—
146C4×D83—81no—
147Dic166—21no—
148Dic166—21no—
149Dic166—81no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83C22×D411yescharacteristic
161C2×D83—41no—
162C2×D83—41no—
163C2×D83—41no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83H(5,𝔽2)11yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177C2×C83—41no—
178C2×C83—81no—
179C2×C83#53511yescharacteristic
180C2×C83#57711yescharacteristic
181C2×C83#57811yescharacteristic
182D83—81no—
183#3225—831nomaximal, Sylow
184C83#322511yescharacteristic, complemented, Sylow
185#542—831no—
186#536D8311yescharacteristic, complemented
187#497—831no—
188#495—831no—
189#489—831no—
190#487—831no—
191#464—831no—
192#463—831no—
193#462—831no—
194#461—831no—
195#452—831no—
196#451—831no—
197#449—831no—
198#448—831no—
199#433—831no—
200C2×#41C2×D8311yescharacteristic
201C22×Q8C2×D8311yescharacteristic
202C2×Q16—831no—
203C2×Q16—831no—
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16—831no—
207C2×SD16—831no—
208C2×SD16—831no—
209C2×D8—831no—
210C4⋊Q8—831no—
211C4⋊Q8C2×D8311yescharacteristic
212C4⋊Q8C2×D8311yescharacteristic
213#126—1661no—
214#124—831no—
215#124—1661no—
216#124C2×D8311yescharacteristic
217#124C2×D8311yescharacteristic
218#123—21no—
219#123—21no—
220#123—1661no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—831no—
224C22⋊Q8—1661no—
225#121—831no—
226C4×Q8—831no—
227C4×D4—831no—
228#117C2×D8311yescharacteristic
229C4⋊C8—831no—
230C4⋊C8—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103—831no—
236#103—831no—
237#102—831no—
238#102—831no—
239C22⋊C8—831no—
240C22⋊C8—831no—
241#97—831no—
242C4×C8—831no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—831no—
249C2×Q8—1661no—
250C2×Q8C22×D8311yescharacteristic
251C2×Q8C22×D8311yescharacteristic
252C2×Q8C22×D8311yescharacteristic
253C2×D4—21no—
254C2×D4—831no—
255C2×D4—1661no—
256C2×D4C22×D8311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—1661no—
260C22×C4C22×D8311yescharacteristic
261Q16—3321no—
262Q16—3321no—
263Q16—3321no—
264SD16—3321no—
265SD16—3321no—
266SD16—3321no—
267SD16—3321no—
268D8—3321no—
269C2×C8—831no—
270C2×C8—831no—
271C2×C8—831no—
272C2×C8—831no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—831no—
278C4⋊C4—831no—
279C4⋊C4—1661no—
280C4⋊C4—1661no—
281C4⋊C4—1661no—
282C4⋊C4C22⋊D8311yescharacteristic
283C4⋊C4C22⋊D8311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4C22⋊D8311yescharacteristic
294C22⋊C4C22⋊D8311yescharacteristic
295C42—1661no—
296C42C22×D8311yescharacteristic
297C42C22×D8311yescharacteristic
298C23—21no—
299C23—1661no—
300C23C2×C22⋊D8311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—6641no—
309Q8—1661no—
310Q8—1661no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—6641no—
315D4—1661no—
316D4—1661no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—6641no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4C2×C22⋊D8311yescharacteristic
329C2×C4C2×C22⋊D8311yescharacteristic
330C2×C4C2×C22⋊D8311yescharacteristic
331C2×C4C2×C22⋊D8311yescharacteristic
332C2×C4C2×C22⋊D8311yescharacteristic
333C2×C4C23×D8311yescharacteristic
334C8—1661no—
335C8—1661no—
336C8—1661no—
337C8—1661no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—3321no—
345C22—3321no—
346C22—3321no—
347C22C22×C22⋊D8311yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3321no—
357C4—3321no—
358C4—3321no—
359C4#1458183711yescharacteristic
360C4#1458183711yescharacteristic
361C2—41no—
362C2—81no—
363C2—6641no—
364C2#1912826811yescharacteristic
365C2#1912850911yescharacteristic
366C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#536×C83C2
socleC22×C83C22×D4
prime radical#19127846C2
hypercenter#536D83
hypocenterC83#3225
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #19128322C2semidirect
2. #19128093C2semidirect
3. #19128083C2semidirect
4. #19128060C2semidirect
5. #19128051C2semidirect
6. #19127920C2semidirect
7. #19127913C2semidirect
8. #19127908C2semidirect
9. #19127903C2semidirect
10. #19127863C2semidirect
11. #19127857C2semidirect
13. #19127840C2semidirect
14. #19127762C2semidirect
15. #536×C83C2semidirect
184. C83#3225semidirect
186. #536D83semidirect

Maximal subgroups

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

subgroupindex
#191283222
#191280932
#191280832
#191280602
#191280512
#191279202
#191279132
#191279082
#191279032
#191278632
#191278572
#191278462
#191278402
#191277622
#536×C832
#322583