Finite Group Databasebeta

← #24913591

Subgroups — #24913591

Order 10624.

598 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581766C211yesmaximal, characteristic, complemented
2#19127631C211yesmaximal, characteristic, complemented
3#19127626C211yesmaximal, characteristic, complemented
4#19127618C211yesmaximal, characteristic, complemented
5#19127616C212yesmaximal, complemented
6#19127610C211yesmaximal, characteristic, complemented
7#19127609C212yesmaximal, complemented
8#19127601C212yesmaximal, complemented
9#19127591C212yesmaximal, complemented
10C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
11C2×#123×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C22⋊D83C2212yes—
15#14581766—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C22×D4×C83C2211yescharacteristic
30C23×C4×C83C2211yescharacteristic
31#123×C83—24no—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2211yescharacteristic
37C23×D83C2312yes—
38C2×D332—22no—
39C2×D332—42no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—24no—
42C2×C22⋊D83—42no—
43C2×C22⋊D83—42no—
44C2×C22⋊D83—42no—
45C2×C22⋊D83—42no—
46C22×Dic83C2312yes—
47C22×Dic83C2312yes—
48C22×Dic83C2312yes—
49C4⋊Dic83—22no—
50Dic83⋊C4—42no—
51C22⋊Dic83—24no—
52C22⋊Dic83—42no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—42no—
57C4×Dic83—42no—
58C24×C83C2311yescharacteristic
59C24×C83C2311yescharacteristic
60C2×D4×C83—22no—
61C2×D4×C83—42no—
62C22×C4×C83—42no—
63C22×C4×C83D414yescomplemented
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—22no—
70C4⋊C4×C83—22no—
71C22⋊C4×C83—22no—
72C22⋊C4×C83—22no—
73C22⋊C4×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—21no—
77C23×C83—21no—
78C23×C83—21no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—21no—
82C23×C83—21no—
83C23×C83C2×D412yes—
84C23×C83C2411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83—22no—
92D4×C83—44no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—44no—
103C2×C4×C83C2×D412yes—
104C2×C4×C83C2×D412yes—
105C22⋊D83—48no—
106C22⋊D83—48no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—22no—
115C2×Dic83#4114yes—
116D332—44no—
117C22×C83—22no—
118C22×C83—22no—
119C22×C83—22no—
120C22×C83—42no—
121C22×C83—42no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83C22×D411yescharacteristic
128C22×C83C2×#4112yes—
129C22×C83H(5,𝔽2)12yes—
130C22×C83H(5,𝔽2)11yescharacteristic
131C22×C83H(5,𝔽2)11yescharacteristic
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—42no—
139C4×C83—22no—
140C4×C83—22no—
141C4×C83—41no—
142C4×C83—41no—
143C4×C83—41no—
144Dic83—24no—
145Dic83—42no—
146Dic83—42no—
147C2×C83—42no—
148C2×C83—81no—
149C2×C83#53211yescharacteristic
150C2×C83#53411yescharacteristic
151C2×C83#53511yescharacteristic
152C2×C83#53812yes—
153C2×C83#54012yes—
154D83—82no—
155#2661—831nomaximal, Sylow
156C83#266111yescharacteristic, complemented, Sylow
157C2×#123D8311yescharacteristic, complemented
158C2×#121—831no—
159#398—831no—
160#397—831no—
161#394—831no—
162#394—832no—
163#392—832no—
164#392—831no—
165#390—832no—
166#388—832no—
167C4×C22⋊C4—831no—
168C22×D4—832no—
169C22×D4—832no—
170C22×D4—831no—
171C22×D4C2×D8311yescharacteristic
172C23×C4C2×D8311yescharacteristic
173#123—24no—
174#121—1664no—
175C2×C4⋊C4—832no—
176C2×C4⋊C4—831no—
177C2×C4⋊C4C2×D8311yescharacteristic
178C2×C4⋊C4C2×D8311yescharacteristic
179C2×C22⋊C4—832no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4—832no—
185C2×C22⋊C4C2×D8311yescharacteristic
186C2×C22⋊C4C2×D8311yescharacteristic
187C2×C22⋊C4C2×D8311yescharacteristic
188C2×C42—832no—
189#95—831no—
190#95—831no—
191#95—832no—
192#95—832no—
193C24—832no—
194C24C22×D8311yescharacteristic
195C24C22×D8311yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—3322no—
199C2×D4—3322no—
200C2×D4—3322no—
201C2×D4—3322no—
202C2×D4—3322no—
203C2×D4—1664no—
204C2×D4—1664no—
205C2×D4—1662no—
206C22×C4—42no—
207C22×C4—832no—
208C22×C4—832no—
209C22×C4—832no—
210C22×C4C22⋊D8314yescomplemented
211C22×C4C22×D8311yescharacteristic
212C22×C4C22×D8311yescharacteristic
213C22×C4C22×D8311yescharacteristic
214C22×C4C22×D8311yescharacteristic
215C22×C4C22×D8311yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—3322no—
219C4⋊C4—1662no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—3322no—
224C22⋊C4—3322no—
225C22⋊C4—3322no—
226C22⋊C4—1664no—
227C22⋊C4—1664no—
228C22⋊C4—1664no—
229C42—3322no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—1662no—
240C23—1662no—
241C23—1662no—
242C23—1662no—
243C23—1662no—
244C23—1662no—
245C23—1662no—
246C23C2×C22⋊D8312yes—
247C23C23×D8311yescharacteristic
248D4—44no—
249D4—3324no—
250D4—3328no—
251D4—3328no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—834no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4C2×C22⋊D8312yes—
272C2×C4C2×C22⋊D8312yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—3322no—
284C22—3322no—
285C22—3322no—
286C22—3322no—
287C22—3322no—
288C22—3322no—
289C22—3322no—
290C22C22×C22⋊D8311yescharacteristic
291C22#41×D8312yes—
292C22#1458183712yes—
293C22#1458183811yescharacteristic
294C22#1458183811yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—3322no—
301C4—3322no—
302C4—1664no—
303C2—42no—
304C2—81no—
305C2—6642no—
306C2#1912824711yescharacteristic
307C2#1912826011yescharacteristic
308C2#1912826511yescharacteristic
309C2#1912828712yes—
310C2#1912830312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
hypercenterC2×#123D83
hypocenterC83#2661
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14581766C2semidirect
2. #19127631C2semidirect
3. #19127626C2semidirect
4. #19127618C2semidirect
5. #19127616C2semidirect
6. #19127610C2semidirect
7. #19127609C2semidirect
8. #19127601C2semidirect
9. #19127591C2semidirect
10. C4×C22⋊Dic83C2semidirect
11. C2×#123×C83C2semidirect
63. C22×C4×C83D4semidirect
156. C83#2661semidirect
157. C2×#123D83semidirect
210. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
#191276312
#191276262
#191276182
#191276162
#191276102
#191276092
#191276012
#191275912
C4×C22⋊Dic832
C2×#123×C832
#266183