Finite Group Databasebeta

← #24912509

Subgroups — #24912509

Order 10624.

524 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581787C211yesmaximal, characteristic, complemented
2C2×#14581776C211yesmaximal, characteristic
3#19127633C211yesmaximal, characteristic, complemented
4#19127627C211yesmaximal, characteristic, complemented
5#19127617C211yesmaximal, characteristic
6#19127600C211yesmaximal, characteristic, complemented
7#19127595C211yesmaximal, characteristic, complemented
8#19127587C211yesmaximal, characteristic, complemented
9#19127577C211yesmaximal, characteristic
10#19127576C211yesmaximal, characteristic, complemented
11#19127566C211yesmaximal, characteristic, complemented
12#19127558C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
14#19127548C211yesmaximal, characteristic, complemented
15#399×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19#14581787—24no—
20#14581776—24no—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×C4⋊Dic83C2211yescharacteristic
23C2×C4⋊Dic83C2211yescharacteristic
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C2×C4×Dic83C2211yescharacteristic
36C2×C4×Dic83C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×C22⋊D83—22no—
55C2×C22⋊D83—22no—
56C2×C22⋊D83—41no—
57C22×Dic83—22no—
58C22×Dic83—22no—
59C22×Dic83—41no—
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C2×C4×D83—22no—
68C2×C4×D83—22no—
69C2×C4×D83—41no—
70C4⋊Dic83—22no—
71C4⋊Dic83—22no—
72C4⋊Dic83—22no—
73C4⋊Dic83—41no—
74C4⋊Dic83—41no—
75Dic83⋊C4—22no—
76C22⋊Dic83—22no—
77C22⋊Dic83—22no—
78(C2×D83)⋊C4—22no—
79(C2×D83)⋊C4—22no—
80(C2×D83)⋊C4—22no—
81(C2×D83)⋊C4—22no—
82(C2×D83)⋊C4D414yes—
83C4×Dic83—22no—
84C4×Dic83—22no—
85C4×Dic83—41no—
86C24×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C4⋊C4×C83—22no—
94C4⋊C4×C83—22no—
95C4⋊C4×C83—22no—
96C22⋊C4×C83—22no—
97C22⋊C4×C83—22no—
98C22⋊C4×C83—22no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83#4112yes—
106C23×C83C2411yescharacteristic
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83C2×D412yes—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83C2×D412yes—
130C2×C4×C83#4112yes—
131C2×C4×C83#4112yes—
132C22⋊D83—44no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83—44no—
153C2×Dic83C2×D412yes—
154C2×Dic83#4112yes—
155C4×D83—44no—
156C22×C83—22no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83C22×D411yescharacteristic
164C22×C83C2×#4111yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83C2×#4111yescharacteristic
168C22×C83H(5,𝔽2)11yescharacteristic
169C22×C832−1+411yescharacteristic
170C2×D83—22no—
171C2×D83—22no—
172C2×D83—22no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C4×C83—22no—
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183Dic83—22no—
184Dic83—22no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190C2×C83—42no—
191C2×C83#52911yescharacteristic
192C2×C83#53211yescharacteristic
193C2×C83#53911yescharacteristic
194C2×C83#54611yescharacteristic
195C2×C83#54711yescharacteristic
196C2×C83#55111yescharacteristic
197C2×C83#55211yescharacteristic
198D83—42no—
199#2553—831nomaximal, Sylow
200C83#255311yescharacteristic, complemented, Sylow
201C2×#126—831no—
202C2×#123—831no—
203#399D8311yescharacteristic, complemented
204#398—831no—
205#397—831no—
206#394—831no—
207#389—831no—
208#388—831no—
209#387—831no—
210#386—831no—
211#385—831no—
212#382—831no—
213#380—831no—
214C4×C22⋊C4—831no—
215C4×C22⋊C4—831no—
216C22×D4—831no—
217C23×C4—831no—
218C23×C4—831no—
219#126—1664no—
220#123—1664no—
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4C2×D8311yescharacteristic
228C2×C4⋊C4C2×D8311yescharacteristic
229C2×C4⋊C4C2×D8311yescharacteristic
230C2×C22⋊C4—831no—
231C2×C22⋊C4—831no—
232C2×C22⋊C4—831no—
233C2×C22⋊C4—831no—
234C2×C22⋊C4—831no—
235C2×C22⋊C4—831no—
236C2×C22⋊C4—831no—
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4C2×D8311yescharacteristic
239C2×C22⋊C4C2×D8311yescharacteristic
240C2×C42—831no—
241C2×C42—831no—
242C2×C42—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95—831no—
249#95—831no—
250#95—831no—
251#95—831no—
252#95C2×D8311yescharacteristic
253C24—831no—
254C24C22×D8311yescharacteristic
255C2×D4—3321no—
256C2×D4—1662no—
257C2×D4—1662no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4—1662no—
268C22×C4—1662no—
269C22×C4—1662no—
270C22×C4—1662no—
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273C22×C4C22×D8311yescharacteristic
274C22×C4C22×D8311yescharacteristic
275C22×C4C22×D8311yescharacteristic
276C22×C4C22×D8311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—3321no—
281C4⋊C4—3321no—
282C4⋊C4—1662no—
283C4⋊C4—1662no—
284C4⋊C4—1662no—
285C4⋊C4—1662no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—834no—
290C22⋊C4—1662no—
291C22⋊C4—1662no—
292C22⋊C4—1662no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C22⋊C4—1662no—
296C42—3321no—
297C42—1662no—
298C42—1662no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—832no—
306C23—1661no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23#1305059112yes—
313C23C23×D8311yescharacteristic
314D4—3324no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—3324no—
331C2×C4—3324no—
332C2×C4—832no—
333C2×C4—832no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4D4×D8312yes—
354C2×C4#1305058912yes—
355C2×C4#1305058912yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—1662no—
368C22—1662no—
369C22—1662no—
370C22C2×D4×D8311yescharacteristic
371C22#41×D8311yescharacteristic
372C22C2×#1305058911yescharacteristic
373C22C2×#1305058911yescharacteristic
374C22C2×#1305059111yescharacteristic
375C22#1458183811yescharacteristic
376C22#1458183911yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3321no—
384C4—3321no—
385C4—3321no—
386C4—3321no—
387C4—3321no—
388C4—1662no—
389C4—1662no—
390C2—42no—
391C2—3322no—
392C2#1912823711yescharacteristic
393C2#1912824911yescharacteristic
394C2#1912829511yescharacteristic
395C2#1912834211yescharacteristic
396C2#1912834911yescharacteristic
397C2#1912837411yescharacteristic
398C2#1912839311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#399D83
hypocenterC83#2553
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14581787C2semidirect
3. #19127633C2semidirect
4. #19127627C2semidirect
6. #19127600C2semidirect
7. #19127595C2semidirect
8. #19127587C2semidirect
10. #19127576C2semidirect
11. #19127566C2semidirect
12. #19127558C2semidirect
13. C22⋊C4×Dic83C2semidirect
14. #19127548C2semidirect
15. #399×C83C2semidirect
200. C83#2553semidirect
203. #399D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817872
C2×#145817762
#191276332
#191276272
#191276172
#191276002
#191275952
#191275872
#191275772
#191275762
#191275662
#191275582
C22⋊C4×Dic832
#191275482
#399×C832
#255383