Finite Group Databasebeta

← #24912470

Subgroups — #24912470

Order 10624.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581770C211yesmaximal, characteristic
2C2×#14581770C211yesmaximal, characteristic
3#19127635C211yesmaximal, characteristic, complemented
4#19127622C211yesmaximal, characteristic, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127605C211yesmaximal, characteristic, complemented
8#19127597C211yesmaximal, characteristic, complemented
9#19127596C211yesmaximal, characteristic, complemented
10#19127575C211yesmaximal, characteristic
11#19127572C211yesmaximal, characteristic
12#19127558C211yesmaximal, characteristic
13#19127548C211yesmaximal, characteristic
14#95×D83C211yesmaximal, characteristic
15#398×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581770—24no—
22#14581770—24no—
23C2×C4⋊Dic83C2211yescharacteristic
24C2×C4⋊Dic83C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×Dic83⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
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×C4⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×Dic166—22no—
55C2×Dic166—22no—
56C2×Dic166—22no—
57C2×Dic166—22no—
58C2×Dic166—41no—
59C2×Dic166—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—22no—
70C2×C4×D83—22no—
71C2×C4×D83—41no—
72C2×C4×D83—41no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C2×C4×D83Q814yescomplemented
76C4⋊Dic83—22no—
77C4⋊Dic83—41no—
78Dic83⋊C4—22no—
79Dic83⋊C4—22no—
80Dic83⋊C4—22no—
81Dic83⋊C4—22no—
82Dic83⋊C4—22no—
83Dic83⋊C4—41no—
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4D414yes—
87(C2×D83)⋊C4Q814yescomplemented
88C4×Dic83—41no—
89C4×Dic83—41no—
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22×C4×C83C2311yescharacteristic
97C4⋊C4×C83—22no—
98C4⋊C4×C83—22no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—41no—
101C23×C83C2411yescharacteristic
102C22×D83—21no—
103C22×D83—21no—
104C22×D83—21no—
105C22×D83—21no—
106C22×D83C2×D412yes—
107C22×D83C2×Q812yes—
108C22×D83C2×Q812yes—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
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×C83C2×D412yes—
128C2×C4×C83C2×Q812yes—
129C2×C4×C83C2×Q812yes—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
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×Dic83C2×D412yes—
147C2×Dic83C2×Q812yes—
148C2×Dic83C2×Q812yes—
149C2×Dic83#4112yes—
150C2×Dic83#4112yes—
151C4×D83—44no—
152C4×D83—44no—
153C4×D83—44no—
154Dic166—44no—
155Dic166—44no—
156C22×C83C22×D411yescharacteristic
157C22×C83C22×Q811yescharacteristic
158C22×C83C22×Q811yescharacteristic
159C22×C83C2×#4111yescharacteristic
160C22×C83C2×#4111yescharacteristic
161C22×C83H(5,𝔽2)11yescharacteristic
162C22×C83H(5,𝔽2)11yescharacteristic
163C2×D83—22no—
164C2×D83—22no—
165C2×D83—22no—
166C2×D83—22no—
167C2×D83—22no—
168C2×D83—22no—
169C2×D83C4⋊Q814yes—
170C4×C83—22no—
171C4×C83—22no—
172C4×C83—22no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83—41no—
177Dic83—22no—
178Dic83—22no—
179Dic83—22no—
180Dic83—22no—
181Dic83—22no—
182Dic83—41no—
183Dic83—41no—
184C2×C83C2×C4⋊Q811yescharacteristic
185C2×C83#53311yescharacteristic
186C2×C83#54311yescharacteristic
187C2×C83#54611yescharacteristic
188C2×C83#54611yescharacteristic
189C2×C83#55411yescharacteristic
190C2×C83#55411yescharacteristic
191D83—24no—
192#2549—831nomaximal, Sylow
193C83#254911yescharacteristic, complemented, Sylow
194C2×C22⋊Q8—831no—
195C2×C22⋊Q8—831no—
196#398—831no—
197#398D8311yescharacteristic, complemented
198#395—831no—
199#395—831no—
200#391—831no—
201#391—831no—
202#389—831no—
203#389—831no—
204#385—831no—
205#385—831no—
206#380—831no—
207C4×C22⋊C4—831no—
208C2×#95—831no—
209C22×Q8—831no—
210C22×Q8—831no—
211C23×C4—831no—
212C23×C4—831no—
213C23×C4—831no—
214C22⋊Q8—1664no—
215C22⋊Q8—1664no—
216C2×C4⋊C4—831no—
217C2×C4⋊C4—831no—
218C2×C4⋊C4—831no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4—831no—
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4C2×D8311yescharacteristic
225C2×C4⋊C4C2×D8311yescharacteristic
226C2×C4⋊C4C2×D8311yescharacteristic
227C2×C4⋊C4C2×D8311yescharacteristic
228C2×C22⋊C4—831no—
229C2×C22⋊C4—831no—
230C2×C22⋊C4—831no—
231C2×C22⋊C4—831no—
232C2×C42—831no—
233C2×C42—831no—
234#95—831no—
235#95—831no—
236#95—831no—
237#95—831no—
238#95—831no—
239#95—831no—
240#95—831no—
241#95—831no—
242#95—831no—
243#95C2×D8311yescharacteristic
244#95C2×D8311yescharacteristic
245#95C2×D8311yescharacteristic
246C24—831no—
247C2×Q8—3321no—
248C2×Q8—3321no—
249C2×Q8—1662no—
250C2×Q8—1662no—
251C2×Q8—1662no—
252C2×Q8—1662no—
253C22×C4—3321no—
254C22×C4—3321no—
255C22×C4—3321no—
256C22×C4—3321no—
257C22×C4—834no—
258C22×C4—831no—
259C22×C4—831no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—1662no—
266C22×C4—1662no—
267C22×C4—1662no—
268C22×C4—1662no—
269C22×C4C22×D8311yescharacteristic
270C22×C4C22×D8311yescharacteristic
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273C22×C4C22×D8311yescharacteristic
274C22×C4C22×D8311yescharacteristic
275C22×C4C22×D8311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—3321no—
281C4⋊C4—3321no—
282C4⋊C4—1662no—
283C4⋊C4—1662no—
284C4⋊C4—1662no—
285C4⋊C4—1662no—
286C4⋊C4—1662no—
287C4⋊C4—1662no—
288C22⋊C4—834no—
289C22⋊C4—834no—
290C22⋊C4—1662no—
291C22⋊C4—1662no—
292C42—3321no—
293C42—3321no—
294C23—832no—
295C23—832no—
296C23—832no—
297C23—1661no—
298C23—1661no—
299C23—1661no—
300C23—1661no—
301C23C23×D8311yescharacteristic
302Q8—3324no—
303Q8—3324no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—3324no—
323C2×C4—3324no—
324C2×C4—3324no—
325C2×C4—832no—
326C2×C4—832no—
327C2×C4—832no—
328C2×C4—832no—
329C2×C4—832no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
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×C4D4×D8312yes—
347C2×C4Q8×D8312yes—
348C2×C4Q8×D8312yes—
349C22—834no—
350C22—1662no—
351C22—1662no—
352C22—1662no—
353C22—1662no—
354C22—1662no—
355C22—1662no—
356C22C2×D4×D8311yescharacteristic
357C22C2×Q8×D8311yescharacteristic
358C22C2×Q8×D8311yescharacteristic
359C22#41×D8311yescharacteristic
360C22#41×D8311yescharacteristic
361C22#1458183711yescharacteristic
362C22#1458183811yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—3321no—
371C4—3321no—
372C4—1662no—
373C4—1662no—
374C4—1662no—
375C4—1662no—
376C4—1662no—
377C2—1664no—
378C2C4⋊Q8×D8311yescharacteristic
379C2#1912825711yescharacteristic
380C2#1912832611yescharacteristic
381C2#1912834311yescharacteristic
382C2#1912834811yescharacteristic
383C2#1912840511yescharacteristic
384C2#1912840711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2549
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #19127635C2semidirect
4. #19127622C2semidirect
5. #19127621C2semidirect
6. #19127606C2semidirect
7. #19127605C2semidirect
8. #19127597C2semidirect
9. #19127596C2semidirect
15. #398×C83C2semidirect
75. C2×C4×D83Q8semidirect
87. (C2×D83)⋊C4Q8semidirect
193. C83#2549semidirect
197. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
#191276352
#191276222
#191276212
#191276062
#191276052
#191275972
#191275962
#191275752
#191275722
#191275582
#191275482
#95×D832
#398×C832
#254983