Finite Group Databasebeta

← #24915721

Subgroups — #24915721

Order 10624.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127640C211yesmaximal, characteristic, complemented
2#19127639C211yesmaximal, characteristic, complemented
3#19127636C211yesmaximal, characteristic
4#19127635C211yesmaximal, characteristic, complemented
5#19127635C211yesmaximal, characteristic, complemented
6#19127631C211yesmaximal, characteristic, complemented
7#19127623C211yesmaximal, characteristic
8#19127623C211yesmaximal, characteristic
9#19127622C211yesmaximal, characteristic, complemented
10#19127621C211yesmaximal, characteristic, complemented
11#19127614C211yesmaximal, characteristic
12#19127563C211yesmaximal, characteristic
13#19127560C211yesmaximal, characteristic
14#19127558C211yesmaximal, characteristic
15#398×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2211yescharacteristic
20C2×C4⋊Dic83C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C83⋊#95C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C2×C4⋊C4×C83C2211yescharacteristic
45C2×C4⋊C4×C83C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48#95×C83C2211yescharacteristic
49#95×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51C23×D83C2311yescharacteristic
52C2×Dic166—41no—
53C2×Dic166—41no—
54C2×Dic166—41no—
55C22×Dic83C2311yescharacteristic
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C2×C4×D83—41no—
63C2×C4×D83—41no—
64C2×C4×D83Q814yescomplemented
65C4⋊Dic83—41no—
66C4⋊Dic83—41no—
67C4⋊Dic83—41no—
68Dic83⋊C4—41no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72Dic83⋊C4—41no—
73Dic83⋊C4—41no—
74(C2×D83)⋊C4—41no—
75(C2×D83)⋊C4—41no—
76(C2×D83)⋊C4—41no—
77(C2×D83)⋊C4—41no—
78(C2×D83)⋊C4—41no—
79(C2×D83)⋊C4—41no—
80C22×C4×C83C2311yescharacteristic
81C22×C4×C83C2311yescharacteristic
82C22×C4×C83C2311yescharacteristic
83C22×C4×C83C2311yescharacteristic
84C22×C4×C83C2311yescharacteristic
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2311yescharacteristic
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C23×C83C2411yescharacteristic
92C22×D83—21no—
93C22×D83—21no—
94C22×D83—21no—
95C22×D83—21no—
96C22×D83—21no—
97C22×D83—21no—
98C22×D83C2×Q812yes—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
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×C83C2×Q812yes—
120C2×Dic83—21no—
121C2×Dic83—21no—
122C2×Dic83—21no—
123C2×Dic83—21no—
124C2×Dic83—21no—
125C2×Dic83—21no—
126C2×Dic83—21no—
127C2×Dic83—21no—
128C2×Dic83—21no—
129C2×Dic83—21no—
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×Dic83C2×Q812yes—
141C4×D83—44no—
142Dic166—44no—
143C22×C83C22×Q811yescharacteristic
144C22×C83H(5,𝔽2)11yescharacteristic
145C22×C83H(5,𝔽2)11yescharacteristic
146C22×C83H(5,𝔽2)11yescharacteristic
147C22×C83H(5,𝔽2)11yescharacteristic
148C22×C832−1+411yescharacteristic
149C22×C832−1+411yescharacteristic
150C2×D83—22no—
151C2×D83—22no—
152C2×D83—22no—
153C2×D83—41no—
154C2×D83—41no—
155C2×D83—41no—
156C2×D83—41no—
157C4×C83—22no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164Dic83—22no—
165Dic83—41no—
166Dic83—41no—
167Dic83—41no—
168Dic83—41no—
169Dic83—41no—
170Dic83—41no—
171C2×C83#54311yescharacteristic
172C2×C83#54411yescharacteristic
173C2×C83#54411yescharacteristic
174C2×C83#56011yescharacteristic
175C2×C83#56211yescharacteristic
176C2×C83#56211yescharacteristic
177C2×C83#56311yescharacteristic
178D83—42no—
179#2841—831nomaximal, Sylow
180C83#284111yescharacteristic, complemented, Sylow
181#400—831no—
182#400—831no—
183#399—831no—
184#398—831no—
185#398—831no—
186#398—831no—
187#398D8311yescharacteristic, complemented
188#396—831no—
189#396—831no—
190#395—831no—
191#395—831no—
192#393—831no—
193#381—831no—
194#380—831no—
195#380—831no—
196C22×Q8—831no—
197C23×C4—831no—
198C2×C4⋊C4—831no—
199C2×C4⋊C4—831no—
200C2×C4⋊C4—831no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—831no—
204C2×C4⋊C4—831no—
205C2×C4⋊C4—831no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4C2×D8311yescharacteristic
208C2×C4⋊C4C2×D8311yescharacteristic
209C2×C4⋊C4C2×D8311yescharacteristic
210C2×C4⋊C4C2×D8311yescharacteristic
211C2×C22⋊C4—831no—
212C2×C22⋊C4—831no—
213C2×C22⋊C4—831no—
214C2×C22⋊C4—831no—
215C2×C22⋊C4—831no—
216C2×C22⋊C4—831no—
217#95—831no—
218#95—831no—
219#95—831no—
220#95—831no—
221#95—831no—
222#95—831no—
223#95—831no—
224#95—831no—
225#95—831no—
226#95—831no—
227#95—831no—
228#95C2×D8311yescharacteristic
229#95C2×D8311yescharacteristic
230#95C2×D8311yescharacteristic
231C24—831no—
232C2×Q8—3321no—
233C2×Q8—3321no—
234C2×Q8—3321no—
235C22×C4—3321no—
236C22×C4—3321no—
237C22×C4—834no—
238C22×C4—831no—
239C22×C4—831no—
240C22×C4—831no—
241C22×C4—831no—
242C22×C4—831no—
243C22×C4—831no—
244C22×C4—831no—
245C22×C4C22×D8311yescharacteristic
246C22×C4C22×D8311yescharacteristic
247C22×C4C22×D8311yescharacteristic
248C22×C4C22×D8311yescharacteristic
249C22×C4C22×D8311yescharacteristic
250C22×C4C22×D8311yescharacteristic
251C22×C4C22×D8311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—3321no—
257C4⋊C4—3321no—
258C4⋊C4—3321no—
259C4⋊C4—3321no—
260C4⋊C4—3321no—
261C4⋊C4—3321no—
262C4⋊C4—3321no—
263C4⋊C4—3321no—
264C4⋊C4—3321no—
265C22⋊C4—3321no—
266C22⋊C4—3321no—
267C22⋊C4—3321no—
268C22⋊C4—3321no—
269C22⋊C4—3321no—
270C22⋊C4—3321no—
271C23—832no—
272C23—1661no—
273C23—1661no—
274C23—1661no—
275C23—1661no—
276C23—1661no—
277C23—1661no—
278C23C23×D8311yescharacteristic
279Q8—3324no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
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—21no—
299C2×C4—21no—
300C2×C4—3324no—
301C2×C4—832no—
302C2×C4—1661no—
303C2×C4—1661no—
304C2×C4—1661no—
305C2×C4—1661no—
306C2×C4—1661no—
307C2×C4—1661no—
308C2×C4—1661no—
309C2×C4—1661no—
310C2×C4—1661no—
311C2×C4—1661no—
312C2×C4—1661no—
313C2×C4—1661no—
314C2×C4—1661no—
315C2×C4—1661no—
316C2×C4—1661no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4—1661no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4Q8×D8312yes—
323C22—3321no—
324C22—3321no—
325C22—3321no—
326C22—3321no—
327C22—1662no—
328C22—1662no—
329C22—1662no—
330C22C2×Q8×D8311yescharacteristic
331C22#1458183711yescharacteristic
332C22#1458183711yescharacteristic
333C22#1458183811yescharacteristic
334C22#1458183811yescharacteristic
335C22#1458183911yescharacteristic
336C22#1458184011yescharacteristic
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3321no—
345C4—3321no—
346C4—3321no—
347C4—3321no—
348C4—3321no—
349C4—3321no—
350C4—1662no—
351C2—3322no—
352C2#1912832611yescharacteristic
353C2#1912832811yescharacteristic
354C2#1912833111yescharacteristic
355C2#1912844111yescharacteristic
356C2#1912844611yescharacteristic
357C2#1912844711yescharacteristic
358C2#1912845511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2841
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #19127640C2semidirect
2. #19127639C2semidirect
4. #19127635C2semidirect
5. #19127635C2semidirect
6. #19127631C2semidirect
9. #19127622C2semidirect
10. #19127621C2semidirect
15. #398×C83C2semidirect
64. C2×C4×D83Q8semidirect
180. C83#2841semidirect
187. #398D83semidirect

Maximal subgroups

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

subgroupindex
#191276402
#191276392
#191276362
#191276352
#191276312
#191276232
#191276222
#191276212
#191276142
#191275632
#191275602
#191275582
#398×C832
#284183