Finite Group Databasebeta

← #24912461

Subgroups — #24912461

Order 10624.

676 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581788C211yesmaximal, characteristic
2C2×#14581778C212yesmaximal, complemented
3C2×#14581757C211yesmaximal, characteristic, complemented
4#19127619C211yesmaximal, characteristic, complemented
5#19127618C211yesmaximal, characteristic, complemented
6#19127615C211yesmaximal, characteristic, complemented
7#19127601C212yesmaximal, complemented
8#19127601C212yesmaximal, complemented
9#19127601C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×D332C2211yescharacteristic
14C22×C22⋊D83C2212yes—
15C22×C22⋊D83C2212yes—
16#14581788—24no—
17#14581778—28no—
18#14581757—24no—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×C4×Dic83C2212yes—
29C2×C4×Dic83C2211yescharacteristic
30C2×C4×Dic83C2211yescharacteristic
31C2×C4×Dic83C2211yescharacteristic
32C83⋊#95C2211yescharacteristic
33C22×D4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36C2×C22⋊C4×C83C2212yes—
37#95×C83C2211yescharacteristic
38C23×D83C2312yes—
39C2×Dic166—22no—
40C2×Dic166—22no—
41C2×Dic166D414yescomplemented
42C2×D332—22no—
43C2×D332—42no—
44C2×C22⋊D83—24no—
45C2×C22⋊D83—24no—
46C2×C22⋊D83—24no—
47C2×C22⋊D83—24no—
48C2×C22⋊D83—24no—
49C2×C22⋊D83—24no—
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2311yescharacteristic
53C22×Dic83C2311yescharacteristic
54C22×Dic83C2311yescharacteristic
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—24no—
58C22⋊Dic83—24no—
59C22⋊Dic83—42no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—24no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64C4×Dic83—24no—
65C4×Dic83—22no—
66C4×Dic83—22no—
67C4×Dic83D414yes—
68C24×C83C2312yes—
69C2×D4×C83—22no—
70C2×D4×C83—42no—
71C22×C4×C83C2312yes—
72C22×C4×C83C2311yescharacteristic
73C22×C4×C83C2311yescharacteristic
74C22×C4×C83C2311yescharacteristic
75C4⋊C4×C83—22no—
76C22⋊C4×C83—24no—
77C22⋊C4×C83—42no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83C2411yescharacteristic
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83—22no—
92C22×D83—22no—
93D4×C83—44no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83C2×D412yes—
103C2×C4×C83C2×D412yes—
104C22⋊D83—48no—
105C22⋊D83—48no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83C2×D412yes—
114C2×Dic83C2×D412yes—
115C2×Dic83C2×D412yes—
116C2×Dic83C2×D412yes—
117C2×Dic83#4112yes—
118C2×Dic83#4114yes—
119D332—44no—
120Dic166—28no—
121C22×C83—42no—
122C22×C83—42no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83C22×D411yescharacteristic
129C22×C83C22×D411yescharacteristic
130C22×C83C2×#4111yescharacteristic
131C22×C83C2×#4112yes—
132C22×C83H(5,𝔽2)12yes—
133C2×D83—42no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—42no—
139C2×D83—42no—
140C4×C83—22no—
141C4×C83—22no—
142C4×C83—41no—
143C4×C83—42no—
144Dic83—22no—
145Dic83—41no—
146Dic83—42no—
147Dic83#12114yes—
148Dic83#12414yescomplemented
149C2×C83—82no—
150C2×C83C2×#12111yescharacteristic
151C2×C83C2×#12411yescharacteristic
152C2×C83#53511yescharacteristic
153C2×C83#55012yes—
154C2×C83#55912yes—
155D83—82no—
156#2548—831nomaximal, Sylow
157C83#254811yescharacteristic, complemented, Sylow
158C2×#127—831no—
159C2×#124—832no—
160C2×C4×Q8—831no—
161#394—831no—
162#394—831no—
163#394—831no—
164#394D8311yescharacteristic, complemented
165#390—832no—
166#390—832no—
167#390—832no—
168C4×C4⋊C4—831no—
169C22×Q8—831no—
170C22×D4—832no—
171C22×D4—832no—
172C22×D4—831no—
173C22×D4C2×D8311yescharacteristic
174#127—1664no—
175#124—1668no—
176C4×Q8—1664no—
177C2×C4⋊C4—831no—
178C2×C4⋊C4—831no—
179C2×C4⋊C4—832no—
180C2×C4⋊C4C2×D8311yescharacteristic
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4—832no—
185C2×C22⋊C4—832no—
186C2×C22⋊C4—832no—
187C2×C22⋊C4C2×D8312yes—
188C2×C22⋊C4C2×D8312yes—
189C2×C42—831no—
190C2×C42—831no—
191C2×C42—831no—
192C2×C42—832no—
193#95—831no—
194#95C2×D8311yescharacteristic
195C24—832no—
196C24C22×D8312yes—
197C2×Q8—834no—
198C2×Q8—1662no—
199C2×Q8—1662no—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4—3322no—
203C2×D4—1664no—
204C2×D4—1664no—
205C2×D4—1664no—
206C2×D4—1664no—
207C2×D4—1664no—
208C2×D4—1664no—
209C2×D4—1662no—
210C22×C4—831no—
211C22×C4—831no—
212C22×C4—831no—
213C22×C4—832no—
214C22×C4—831no—
215C22×C4C22×D8312yes—
216C22×C4C22×D8311yescharacteristic
217C22×C4C22×D8311yescharacteristic
218C22×C4C22×D8311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—1662no—
221C4⋊C4—1664no—
222C4⋊C4—1662no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—3322no—
226C22⋊C4—3322no—
227C22⋊C4—3322no—
228C22⋊C4—1664no—
229C22⋊C4—1664no—
230C22⋊C4—1664no—
231C42—834no—
232C42—1662no—
233C42—1662no—
234C42—1664no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—1662no—
243C23—1662no—
244C23—1662no—
245C23—1662no—
246C23—1662no—
247C23—1662no—
248C23—1662no—
249C23C23×D8311yescharacteristic
250Q8—1668no—
251D4—44no—
252D4—3324no—
253D4—3328no—
254D4—3328no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—832no—
264C2×C4—832no—
265C2×C4—832no—
266C2×C4—834no—
267C2×C4—832no—
268C2×C4—832no—
269C2×C4—1662no—
270C2×C4—1661no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4—1661no—
275C2×C4—1661no—
276C2×C4D4×D8312yes—
277C2×C4D4×D8312yes—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—3322no—
286C22—3322no—
287C22—3322no—
288C22—3322no—
289C22—3322no—
290C22—3322no—
291C22—3322no—
292C22C2×D4×D8311yescharacteristic
293C22C2×D4×D8311yescharacteristic
294C22#41×D8312yes—
295C22#41×D8311yescharacteristic
296C22#1458183712yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—41no—
301C4—3322no—
302C4—3321no—
303C4—834no—
304C4—834no—
305C4—1662no—
306C2—82no—
307C2—6642no—
308C2#121×D8311yescharacteristic
309C2#124×D8311yescharacteristic
310C2#1912826611yescharacteristic
311C2#1912836912yes—
312C2#1912843512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC2×#14581788C2
hypercenter#394D83
hypocenterC83#2548
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14581778C2semidirect
3. C2×#14581757C2semidirect
4. #19127619C2semidirect
5. #19127618C2semidirect
6. #19127615C2semidirect
7. #19127601C2semidirect
8. #19127601C2semidirect
9. #19127601C2semidirect
10. #19127552C2semidirect
11. #394×C83C2semidirect
41. C2×Dic166D4semidirect
148. Dic83#124semidirect
157. C83#2548semidirect
164. #394D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817882
C2×#145817782
C2×#145817572
#191276192
#191276182
#191276152
#191276012
#191275522
#394×C832
#254883