Finite Group Databasebeta

← #24913355

Subgroups — #24913355

Order 10624.

620 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14581763C211yesmaximal, characteristic, complemented
2C2×#14581754C211yesmaximal, characteristic, complemented
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic, complemented
5#19127617C212yesmaximal, complemented
6#19127610C212yesmaximal, complemented
7#19127609C211yesmaximal, characteristic, complemented
8#19127594C212yesmaximal, complemented
9#19127577C212yesmaximal, complemented
10#19127563C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C22⋊D83C2212yes—
15C22×C4×D83C2211yescharacteristic
16C22×C4×D83C2211yescharacteristic
17#14581763—24no—
18#14581754—24no—
19C2×C4⋊Dic83C2211yescharacteristic
20C2×C4⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C23×D83C2311yescharacteristic
39C2×D332—22no—
40C2×D332—22no—
41C2×D332—22no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—24no—
44C2×C22⋊D83—42no—
45C2×C22⋊D83—42no—
46C2×C22⋊D83—42no—
47C2×C22⋊D83—42no—
48C22×Dic83C2312yes—
49C22×Dic83C2312yes—
50C22×Dic83C2311yescharacteristic
51C22×Dic83C2311yescharacteristic
52C2×C4×D83—22no—
53C2×C4×D83—42no—
54C2×C4×D83—42no—
55C2×C4×D83D414yescomplemented
56C4⋊Dic83—22no—
57C4⋊Dic83—42no—
58C22⋊Dic83—24no—
59C22⋊Dic83—42no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64C4×Dic83—22no—
65C24×C83C2312yes—
66C2×D4×C83—22no—
67C2×D4×C83—42no—
68C22×C4×C83C2311yescharacteristic
69C22×C4×C83C2312yes—
70C22×C4×C83C2311yescharacteristic
71C22×C4×C83C2311yescharacteristic
72C4⋊C4×C83—22no—
73C22⋊C4×C83—24no—
74C22⋊C4×C83—42no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83C2411yescharacteristic
83C22×D83—21no—
84C22×D83—21no—
85C22×D83—21no—
86C22×D83—21no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83C2×D412yes—
92C22×D83#4112yes—
93D4×C83—44no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83C2×D412yes—
103C2×C4×C83#4112yes—
104C22⋊D83—48no—
105C22⋊D83—48no—
106C2×Dic83—22no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—22no—
115C2×Dic83C2×D412yes—
116D332—28no—
117C4×D83—44no—
118C4×D83—44no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—42no—
122C22×C83—42no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83C22×D411yescharacteristic
127C22×C83C2×#4111yescharacteristic
128C22×C83C2×#4111yescharacteristic
129C22×C83H(5,𝔽2)12yes—
130C22×C83H(5,𝔽2)12yes—
131C2×D83—22no—
132C2×D83—22no—
133C2×D83—22no—
134C2×D83—22no—
135C2×D83—24no—
136C2×D83—41no—
137C2×D83—41no—
138C2×D83—42no—
139C2×D83—42no—
140C2×D83—42no—
141C4×C83—22no—
142C4×C83—22no—
143C4×C83—41no—
144C4×C83—42no—
145Dic83—22no—
146Dic83—41no—
147Dic83—42no—
148Dic83—42no—
149C2×C83—82no—
150C2×C83#52511yescharacteristic
151C2×C83#53411yescharacteristic
152C2×C83#53511yescharacteristic
153C2×C83#53812yes—
154C2×C83#54012yes—
155D83—42no—
156D83—42no—
157#2635—831nomaximal, Sylow
158C83#263511yescharacteristic, complemented, Sylow
159C2×#121—831no—
160C2×C4×D4—831no—
161#400—831no—
162#399—831no—
163#394—832no—
164#394D8311yescharacteristic, complemented
165#392—831no—
166#392—832no—
167#388—832no—
168#386—832no—
169#381—831no—
170C22×D4—832no—
171C22×D4—832no—
172C22×D4—831no—
173C22×D4C2×D8311yescharacteristic
174C23×C4—831no—
175C23×C4—831no—
176#121—1664no—
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—831no—
187C2×C22⋊C4C2×D8312yes—
188C2×C22⋊C4C2×D8312yes—
189C2×C42—831no—
190#95—832no—
191#95—831no—
192#95—832no—
193#95—832no—
194#95C2×D8311yescharacteristic
195C24—831no—
196C24—831no—
197C24C22×D8312yes—
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—3322no—
201C2×D4—3322no—
202C2×D4—3322no—
203C2×D4—3322no—
204C2×D4—1662no—
205C2×D4—1662no—
206C2×D4—1664no—
207C2×D4—1664no—
208C2×D4—1662no—
209C22×C4—3322no—
210C22×C4—3322no—
211C22×C4—834no—
212C22×C4—831no—
213C22×C4—832no—
214C22×C4—832no—
215C22×C4—831no—
216C22×C4—1662no—
217C22×C4C22×D8311yescharacteristic
218C22×C4C22×D8312yes—
219C22×C4C22×D8311yescharacteristic
220C22×C4C22×D8311yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—3322no—
223C4⋊C4—1662no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—3322no—
227C22⋊C4—3322no—
228C22⋊C4—3322no—
229C22⋊C4—1662no—
230C22⋊C4—1664no—
231C22⋊C4—1662no—
232C42—1662no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—832no—
241C23—832no—
242C23—1661no—
243C23—1661no—
244C23—1661no—
245C23—1661no—
246C23—1662no—
247C23—1662no—
248C23—1662no—
249C23—1662no—
250C23C23×D8311yescharacteristic
251D4—44no—
252D4—3328no—
253D4—3328no—
254D4—1668no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—3324no—
264C2×C4—3324no—
265C2×C4—832no—
266C2×C4—1661no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4—1662no—
275C2×C4D4×D8312yes—
276C2×C4#1305058912yes—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—3321no—
285C22—3321no—
286C22—3322no—
287C22—3322no—
288C22—3322no—
289C22—1662no—
290C22—1662no—
291C22—1662no—
292C22—1662no—
293C22—1664no—
294C22C2×D4×D8311yescharacteristic
295C22#41×D8311yescharacteristic
296C22C2×#1305058911yescharacteristic
297C22#1458183712yes—
298C22#1458183812yes—
299C4—22no—
300C4—22no—
301C4—41no—
302C4—42no—
303C4—3322no—
304C4—3321no—
305C4—3322no—
306C4—1662no—
307C2—82no—
308C2—3322no—
309C2—3322no—
310C2#1912821111yescharacteristic
311C2#1912826111yescharacteristic
312C2#1912826911yescharacteristic
313C2#1912828712yes—
314C2#1912830412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
hypercenter#394D83
hypocenterC83#2635
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581763C2semidirect
2. C2×#14581754C2semidirect
3. #19127640C2semidirect
4. #19127638C2semidirect
5. #19127617C2semidirect
6. #19127610C2semidirect
7. #19127609C2semidirect
8. #19127594C2semidirect
9. #19127577C2semidirect
10. #19127563C2semidirect
11. #394×C83C2semidirect
55. C2×C4×D83D4semidirect
158. C83#2635semidirect
164. #394D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817632
C2×#145817542
#191276402
#191276382
#191276172
#191276102
#191276092
#191275942
#191275772
#191275632
#394×C832
#263583