Finite Group Databasebeta

← #24912537

Subgroups — #24912537

Order 10624.

696 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14581778C212yesmaximal, complemented
2C2×#14581765C212yesmaximal, complemented
3#19127621C211yesmaximal, characteristic, complemented
4#19127618C211yesmaximal, characteristic, complemented
5#19127609C212yesmaximal, complemented
6#19127589C212yesmaximal, complemented
7#19127577C212yesmaximal, complemented
8C4×(C2×D83)⋊C4C212yesmaximal, complemented
9#390×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12C22×C22⋊D83C2212yes—
13C22×C22⋊D83C2212yes—
14C22×C4×D83C2212yes—
15#14581778—28no—
16#14581765—28no—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×C4×Dic83C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2212yes—
26C83⋊#95C2211yescharacteristic
27C83⋊#95C2212yes—
28C22×D4×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2211yescharacteristic
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2212yes—
32C2×C42×C83C2211yescharacteristic
33C23×D83C2312yes—
34C2×Dic166—24no—
35C2×Dic166—41no—
36C2×D332—24no—
37C2×D332—41no—
38C2×C22⋊D83—24no—
39C2×C22⋊D83—24no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—24no—
42C2×C22⋊D83—42no—
43C2×C22⋊D83—42no—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47C2×C4×D83—24no—
48C2×C4×D83—24no—
49C2×C4×D83—42no—
50Dic83⋊C4—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—24no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4D418yescomplemented
56C4×Dic83—24no—
57C24×C83C2312yes—
58C2×D4×C83—22no—
59C2×D4×C83—41no—
60C2×D4×C83—41no—
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—41no—
66C22⋊C4×C83—24no—
67C22⋊C4×C83—24no—
68C42×C83—41no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83C2411yescharacteristic
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83C2×D414yes—
84D4×C83—44no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83C2×D414yes—
94C2×C4×C83#4112yes—
95C22⋊D83—48no—
96C22⋊D83—48no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83C2×D414yes—
105C2×Dic83#4114yes—
106D332—44no—
107C4×D83—48no—
108Dic166—44no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83C22×D412yes—
117C22×C83C2×#4112yes—
118C22×C83C2×#4111yescharacteristic
119C22×C83H(5,𝔽2)11yescharacteristic
120C22×C83H(5,𝔽2)11yescharacteristic
121C2×D83—24no—
122C2×D83—24no—
123C2×D83—24no—
124C2×D83—42no—
125C2×D83—42no—
126C2×D83—42no—
127C2×D83—42no—
128C4×C83—22no—
129C4×C83—24no—
130C4×C83—41no—
131C4×C83—41no—
132Dic83—24no—
133Dic83—24no—
134Dic83—42no—
135C2×C83—82no—
136C2×C83#53212yes—
137C2×C83#53811yescharacteristic
138C2×C83D4211yescharacteristic
139C2×C83#54612yes—
140C2×C83#55511yescharacteristic
141D83—44no—
142#2555—831nomaximal, Sylow
143C83#255511yescharacteristic, complemented, Sylow
144C2×#124—832no—
145C2×#121—832no—
146#395—831no—
147#394—831no—
148#392—832no—
149#390D8311yescharacteristic, complemented
150#388—832no—
151#386—832no—
152C4×C22⋊C4—832no—
153C22×Q8—831no—
154C22×D4—832no—
155C22×D4—832no—
156C22×D4—831no—
157C22×D4C2×D8311yescharacteristic
158C23×C4—832no—
159#124—1668no—
160#121—1668no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4C2×D8311yescharacteristic
163C2×C22⋊C4—832no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4C2×D8312yes—
169C2×C22⋊C4C2×D8312yes—
170C2×C42—832no—
171C2×C42C2×D8311yescharacteristic
172#95—832no—
173#95—832no—
174#95—831no—
175#95—831no—
176C24—832no—
177C24C22×D8312yes—
178C2×Q8—3321no—
179C2×Q8—1664no—
180C2×D4—22no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—3321no—
184C2×D4—3322no—
185C2×D4—3322no—
186C2×D4—1664no—
187C2×D4—1664no—
188C2×D4—1664no—
189C2×D4—1664no—
190C2×D4—1664no—
191C22×C4—3322no—
192C22×C4—832no—
193C22×C4—832no—
194C22×C4—832no—
195C22×C4—1664no—
196C22×C4—1664no—
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8311yescharacteristic
199C22×C4C22×D8312yes—
200C22×C4C22×D8311yescharacteristic
201C4⋊C4—41no—
202C4⋊C4—1664no—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4—838no—
206C22⋊C4—1664no—
207C22⋊C4—1664no—
208C22⋊C4—1664no—
209C22⋊C4—1664no—
210C42—41no—
211C42—1664no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—834no—
220C23—1662no—
221C23—1662no—
222C23—1662no—
223C23—1662no—
224C23—1662no—
225C23—1662no—
226C23C23×D8311yescharacteristic
227Q8—3324no—
228D4—44no—
229D4—3324no—
230D4—3328no—
231D4—3328no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—3328no—
241C2×C4—834no—
242C2×C4—834no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4D4×D8314yes—
251C2×C4#1305059012yes—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—3322no—
260C22—3322no—
261C22—3322no—
262C22—3322no—
263C22—1664no—
264C22—1664no—
265C22—1664no—
266C22C2×D4×D8312yes—
267C22#41×D8312yes—
268C22C2×#1305059011yescharacteristic
269C22#1458183711yescharacteristic
270C22#1458183711yescharacteristic
271C4—24no—
272C4—22no—
273C4—41no—
274C4—41no—
275C4—3322no—
276C4—1664no—
277C4—1664no—
278C2—82no—
279C2—3324no—
280C2#1912824812yes—
281C2#1912828911yescharacteristic
282C2#1912833611yescharacteristic
283C2#1912834012yes—
284C2#1912841311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
hypercenter#390D83
hypocenterC83#2555
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14581778C2semidirect
2. C2×#14581765C2semidirect
3. #19127621C2semidirect
4. #19127618C2semidirect
5. #19127609C2semidirect
6. #19127589C2semidirect
7. #19127577C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. #390×C83C2semidirect
55. (C2×D83)⋊C4D4semidirect
143. C83#2555semidirect
149. #390D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
C2×#145817652
#191276212
#191276182
#191276092
#191275892
#191275772
C4×(C2×D83)⋊C42
#390×C832
#255583