Finite Group Databasebeta

← #24911254

Subgroups — #24911254

Order 10624.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581778C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic, complemented
4#19127636C211yesmaximal, characteristic, complemented
5#19127630C211yesmaximal, characteristic, complemented
6#19127605C211yesmaximal, characteristic, complemented
7#19127601C211yesmaximal, characteristic
8#19127601C211yesmaximal, characteristic
9#19127598C211yesmaximal, characteristic, complemented
10#19127595C211yesmaximal, characteristic, complemented
11#19127593C211yesmaximal, characteristic, complemented
12#19127590C211yesmaximal, characteristic, complemented
13#19127589C211yesmaximal, characteristic, complemented
14#19127552C211yesmaximal, characteristic, complemented
15#397×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18#14581778—24no—
19#14581757—24no—
20C2×C4⋊Dic83C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×C4×Dic83C2211yescharacteristic
37C2×C4×Dic83C2211yescharacteristic
38C2×C4×Dic83C2211yescharacteristic
39C2×C4×Dic83C2211yescharacteristic
40C2×C4×Dic83C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×Dic166—22no—
55C2×Dic166—22no—
56C2×Dic166D414yes—
57C2×C22⋊D83—22no—
58C2×C22⋊D83—22no—
59C2×C22⋊D83—22no—
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C4⋊Dic83—41no—
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70Dic83⋊C4—22no—
71Dic83⋊C4—22no—
72Dic83⋊C4—22no—
73Dic83⋊C4—41no—
74C22⋊Dic83—22no—
75C22⋊Dic83—22no—
76C22⋊Dic83—22no—
77(C2×D83)⋊C4—22no—
78(C2×D83)⋊C4—22no—
79(C2×D83)⋊C4—22no—
80(C2×D83)⋊C4—22no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—41no—
83C4×Dic83—22no—
84C4×Dic83—22no—
85C4×Dic83—22no—
86C4×Dic83—22no—
87C4×Dic83—22no—
88C24×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C4⋊C4×C83—22no—
96C22⋊C4×C83—22no—
97C22⋊C4×C83—41no—
98C22⋊C4×C83—41no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83C2411yescharacteristic
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83—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×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83C2×D412yes—
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×Dic83C2×D412yes—
149C2×Dic83C2×D412yes—
150C2×Dic83#4112yes—
151C2×Dic83#4112yes—
152C2×Dic83#4112yes—
153C2×Dic83#4112yes—
154Dic166—28no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83C22×D411yescharacteristic
163C22×C83C2×#4111yescharacteristic
164C22×C83C2×#4111yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83H(5,𝔽2)11yescharacteristic
168C22×C832−1+411yescharacteristic
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C4×C83—22no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182Dic83—22no—
183Dic83—22no—
184Dic83—22no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188Dic83#12314yes—
189C2×C83—81no—
190C2×C83C2×#12311yescharacteristic
191C2×C83#52911yescharacteristic
192C2×C83#54211yescharacteristic
193C2×C83#54811yescharacteristic
194C2×C83#55011yescharacteristic
195C2×C83#55611yescharacteristic
196C2×C83#55911yescharacteristic
197D83—81no—
198#2448—831nomaximal, Sylow
199C83#244811yescharacteristic, complemented, Sylow
200C2×#124—831no—
201C2×C4×Q8—831no—
202#399—831no—
203#399—831no—
204#397—831no—
205#397D8311yescharacteristic, complemented
206#391—831no—
207#390—831no—
208#390—831no—
209#389—831no—
210#388—831no—
211#388—831no—
212#388—831no—
213#388—831no—
214C4×C4⋊C4—831no—
215C22×Q8—831no—
216C22×D4—831no—
217#124—1664no—
218C4×Q8—1664no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4—831no—
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4C2×D8311yescharacteristic
227C2×C22⋊C4—831no—
228C2×C22⋊C4—831no—
229C2×C22⋊C4—831no—
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⋊C4C2×D8311yescharacteristic
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4C2×D8311yescharacteristic
239C2×C42—831no—
240C2×C42—831no—
241C2×C42—831no—
242C2×C42—831no—
243C2×C42—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95—831no—
249#95C2×D8311yescharacteristic
250#95C2×D8311yescharacteristic
251#95C2×D8311yescharacteristic
252C24—831no—
253C24C22×D8311yescharacteristic
254C2×Q8—834no—
255C2×Q8—1662no—
256C2×Q8—1662no—
257C2×D4—1662no—
258C2×D4—1662no—
259C2×D4—1662no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4C22×D8311yescharacteristic
268C22×C4C22×D8311yescharacteristic
269C22×C4C22×D8311yescharacteristic
270C22×C4C22×D8311yescharacteristic
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—3321no—
275C4⋊C4—3321no—
276C4⋊C4—1662no—
277C4⋊C4—1662no—
278C4⋊C4—1662no—
279C4⋊C4—1662no—
280C4⋊C4—1662no—
281C22⋊C4—22no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—3321no—
285C22⋊C4—3321no—
286C22⋊C4—1662no—
287C22⋊C4—1662no—
288C22⋊C4—1662no—
289C22⋊C4—1662no—
290C22⋊C4—1662no—
291C22⋊C4—1662no—
292C22⋊C4—1662no—
293C42—1662no—
294C42—1662no—
295C42—1662no—
296C42—1662no—
297C42—1662no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—1661no—
306C23—1661no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23C23×D8311yescharacteristic
313Q8—1668no—
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—21no—
331C2×C4—21no—
332C2×C4—832no—
333C2×C4—832no—
334C2×C4—832no—
335C2×C4—832no—
336C2×C4—832no—
337C2×C4—832no—
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—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22C2×D4×D8311yescharacteristic
369C22#41×D8311yescharacteristic
370C22#41×D8311yescharacteristic
371C22#41×D8311yescharacteristic
372C22#41×D8311yescharacteristic
373C22#1458183711yescharacteristic
374C22#1458183911yescharacteristic
375C4—22no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3321no—
382C4—3321no—
383C4—3321no—
384C4—834no—
385C4—1662no—
386C4—1662no—
387C4—1662no—
388C2—81no—
389C2—6641no—
390C2#123×D8311yescharacteristic
391C2#1912823411yescharacteristic
392C2#1912831911yescharacteristic
393C2#1912835911yescharacteristic
394C2#1912836911yescharacteristic
395C2#1912841811yescharacteristic
396C2#1912843511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#397D83
hypocenterC83#2448
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#14581757C2semidirect
3. #19127637C2semidirect
4. #19127636C2semidirect
5. #19127630C2semidirect
6. #19127605C2semidirect
9. #19127598C2semidirect
10. #19127595C2semidirect
11. #19127593C2semidirect
12. #19127590C2semidirect
13. #19127589C2semidirect
14. #19127552C2semidirect
15. #397×C83C2semidirect
199. C83#2448semidirect
205. #397D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
C2×#145817572
#191276372
#191276362
#191276302
#191276052
#191276012
#191275982
#191275952
#191275932
#191275902
#191275892
#191275522
#397×C832
#244883