Finite Group Databasebeta

← #24913348

Subgroups — #24913348

Order 10624.

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

normal subgroups only

This group has 620 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×C4×C22⋊D83C211yesmaximal, characteristic, complemented
2#19127639C211yesmaximal, characteristic, complemented
3#19127636C211yesmaximal, characteristic, complemented
4#19127615C211yesmaximal, characteristic, complemented
5#19127615C212yesmaximal, complemented
6#19127609C212yesmaximal, complemented
7#19127609C211yesmaximal, characteristic, complemented
8#19127589C212yesmaximal, complemented
9#19127577C212yesmaximal, complemented
10#19127564C211yesmaximal, characteristic, complemented
11C2×#121×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C4×D83C2211yescharacteristic
15C4×C22⋊D83—24no—
16C2×Dic83⋊C4C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C22×D4×C83C2212yes—
32C22×D4×C83C2211yescharacteristic
33C23×C4×C83C2211yescharacteristic
34#121×C83—24no—
35C2×C4⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—22no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—22no—
42C2×C22⋊D83—42no—
43C2×C22⋊D83—42no—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C2×C4×D83—22no—
50C2×C4×D83—42no—
51Dic83⋊C4—22no—
52Dic83⋊C4—42no—
53C22⋊Dic83—22no—
54C22⋊Dic83—24no—
55C22⋊Dic83—42no—
56C22⋊Dic83—42no—
57C22⋊Dic83—42no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—42no—
60C4×Dic83—22no—
61C24×C83C2312yes—
62C24×C83C2311yescharacteristic
63C2×D4×C83—24no—
64C2×D4×C83—22no—
65C2×D4×C83—42no—
66C2×D4×C83—42no—
67C2×D4×C83—42no—
68C22×C4×C83—42no—
69C22×C4×C83D414yescomplemented
70C22×C4×C83C2312yes—
71C22×C4×C83C2311yescharacteristic
72C22×C4×C83C2311yescharacteristic
73C4⋊C4×C83—22no—
74C22⋊C4×C83—24no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83C2×D412yes—
87C23×C83C2411yescharacteristic
88C22×D83—21no—
89C22×D83—21no—
90C22×D83—22no—
91C22×D83—22no—
92C22×D83#4112yes—
93D4×C83—44no—
94D4×C83—48no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—44no—
101C2×C4×C83C2×D412yes—
102C2×C4×C83C2×D412yes—
103C22⋊D83—28no—
104C22⋊D83—48no—
105C2×Dic83—21no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83—21no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—22no—
115C2×Dic83—22no—
116C2×Dic83—22no—
117C2×Dic83#4112yes—
118C4×D83—44no—
119C22×C83—22no—
120C22×C83—22no—
121C22×C83—22no—
122C22×C83—42no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—42no—
130C22×C83—42no—
131C22×C83C22×D411yescharacteristic
132C22×C83C2×#4111yescharacteristic
133C22×C83C2×#4111yescharacteristic
134C22×C83H(5,𝔽2)12yes—
135C22×C83H(5,𝔽2)12yes—
136C2×D83—22no—
137C2×D83—24no—
138C2×D83—41no—
139C2×D83—41no—
140C2×D83—42no—
141C4×C83—22no—
142C4×C83—22no—
143C4×C83—42no—
144Dic83—22no—
145Dic83—41no—
146Dic83—41no—
147Dic83—42no—
148Dic83—42no—
149C2×C83—42no—
150C2×C83—82no—
151C2×C83#52511yescharacteristic
152C2×C83#53411yescharacteristic
153C2×C83#53511yescharacteristic
154C2×C83#53812yes—
155C2×C83#54012yes—
156D83—42no—
157#2635—831nomaximal, Sylow
158C83#263511yescharacteristic, complemented, Sylow
159C2×#121D8311yescharacteristic, complemented
160C2×C4×D4—831no—
161#400—831no—
162#399—831no—
163#394—831no—
164#394—832no—
165#392—831no—
166#392—832no—
167#388—832no—
168#386—832no—
169#381—831no—
170C22×D4—832no—
171C22×D4—831no—
172C22×D4C2×D8312yes—
173C22×D4C2×D8311yescharacteristic
174C23×C4—831no—
175C23×C4C2×D8311yescharacteristic
176#121—24no—
177C4×D4—1664no—
178C2×C4⋊C4—832no—
179C2×C4⋊C4—831no—
180C2×C4⋊C4C2×D8311yescharacteristic
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4—832no—
185C2×C22⋊C4—831no—
186C2×C22⋊C4—832no—
187C2×C22⋊C4—831no—
188C2×C22⋊C4C2×D8312yes—
189C2×C42—831no—
190#95—832no—
191#95—831no—
192#95—831no—
193#95—832no—
194#95—832no—
195C24—831no—
196C24C22×D8312yes—
197C24C22×D8311yescharacteristic
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—3322no—
204C2×D4—3322no—
205C2×D4—1662no—
206C2×D4—1664no—
207C2×D4—1662no—
208C2×D4—1662no—
209C22×C4—42no—
210C22×C4—3322no—
211C22×C4—831no—
212C22×C4—832no—
213C22×C4—832no—
214C22×C4—831no—
215C22×C4—831no—
216C22×C4—1662no—
217C22×C4C22⋊D8314yescomplemented
218C22×C4C22×D8312yes—
219C22×C4C22×D8311yescharacteristic
220C22×C4C22×D8311yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—3322no—
223C4⋊C4—1662no—
224C22⋊C4—24no—
225C22⋊C4—3322no—
226C22⋊C4—3322no—
227C22⋊C4—3322no—
228C22⋊C4—3322no—
229C22⋊C4—1664no—
230C22⋊C4—1662no—
231C22⋊C4—1662no—
232C42—1662no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—832no—
245C23—1661no—
246C23—1661no—
247C23—1662no—
248C23—1662no—
249C23C2×C22⋊D8312yes—
250C23C23×D8311yescharacteristic
251D4—44no—
252D4—48no—
253D4—3328no—
254D4—1668no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—44no—
261C2×C4—3324no—
262C2×C4—832no—
263C2×C4—1661no—
264C2×C4—1661no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—1661no—
274C2×C4—1661no—
275C2×C4C2×C22⋊D8312yes—
276C2×C4C2×C22⋊D8312yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—3321no—
290C22—3321no—
291C22—3322no—
292C22—1662no—
293C22—1664no—
294C22C22×C22⋊D8311yescharacteristic
295C22#41×D8311yescharacteristic
296C22#41×D8311yescharacteristic
297C22#1458183712yes—
298C22#1458183712yes—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—3321no—
303C4—3321no—
304C4—3322no—
305C4—3322no—
306C4—1662no—
307C2—42no—
308C2—82no—
309C2—3322no—
310C2#1912820811yescharacteristic
311C2#1912826011yescharacteristic
312C2#1912826811yescharacteristic
313C2#1912828312yes—
314C2#1912830112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2635
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D83C2semidirect
2. #19127639C2semidirect
3. #19127636C2semidirect
4. #19127615C2semidirect
5. #19127615C2semidirect
6. #19127609C2semidirect
7. #19127609C2semidirect
8. #19127589C2semidirect
9. #19127577C2semidirect
10. #19127564C2semidirect
11. C2×#121×C83C2semidirect
69. C22×C4×C83D4semidirect
158. C83#2635semidirect
159. C2×#121D83semidirect
217. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D832
#191276392
#191276362
#191276152
#191276092
#191275892
#191275772
#191275642
C2×#121×C832
#263583