Finite Group Databasebeta

← #24918785

Subgroups — #24918785

Order 10624.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#19128249C211yesmaximal, characteristic, complemented
2#19128233C211yesmaximal, characteristic, complemented
3#19127999C212yesmaximal, complemented
4#19127997C211yesmaximal, characteristic, complemented
5#19127991C211yesmaximal, characteristic, complemented
6#19127934C212yesmaximal, complemented
7#19127929C212yesmaximal, complemented
8#19127914C212yesmaximal, complemented
9#19127898C211yesmaximal, characteristic, complemented
10C4⋊C8×D83C211yesmaximal, characteristic, complemented
11#459×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C2×#13050578C2212yes—
14C2×C8×D83C2212yes—
15#14581792C2211yescharacteristic
16#14581785—21no—
17#14581783C2211yescharacteristic
18#14581779—21no—
19#14581779C2211yescharacteristic
20#14581774—21no—
21#14581772C2211yescharacteristic
22#14581763—21no—
23#14581763C2212yes—
24#14581763C2211yescharacteristic
25C4×Dic166C2211yescharacteristic
26D4×Dic83—21no—
27D4×Dic83C2212yes—
28C4×C22⋊D83—21no—
29#14581751—21no—
30C42×D83C2211yescharacteristic
31#14581726C2212yes—
32#14581723C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581717C2212yes—
36#14581713C2212yes—
37#14581713C2212yes—
38C2×D8×C83C2212yes—
39#127×C83C2211yescharacteristic
40C4×D4×C83C2211yescharacteristic
41C4⋊C8×C83C2211yescharacteristic
42#102×C83C2212yes—
43#13050589—44no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×C22⋊D83—22no—
47C2×C22⋊D83—21no—
48C22×Dic83—22no—
49C2×C4×D83D411yescharacteristic
50C2×C4×D83D411yescharacteristic
51C2×C4×D83C2311yescharacteristic
52#13050578—42no—
53C2×C83⋊C8C2312yes—
54C8×D83—42no—
55C4⋊Dic83—21no—
56C4⋊Dic83—21no—
57C4⋊Dic83—21no—
58C4⋊Dic83C2311yescharacteristic
59C4⋊Dic83C2312yes—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62C22⋊Dic83—22no—
63C22⋊Dic83—22no—
64C22⋊Dic83—21no—
65C22⋊Dic83—21no—
66C22⋊Dic83—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×D4×C83—21no—
73C2×D4×C83—21no—
74C2×D4×C83C2311yescharacteristic
75C2×D4×C83C2312yes—
76C22×C4×C83—21no—
77D8×C83—42no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C23×C83—22no—
84C22×D83C2×D411yescharacteristic
85D4×C83—21no—
86D4×C83—22no—
87D4×C83—21no—
88D4×C83—22no—
89D4×C83—42no—
90D4×C83—42no—
91C2×C4×C83—21no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—42no—
97C22⋊D83—44no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83—44no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105C4×D83—22no—
106C4×D83—22no—
107C4×D83—21no—
108C4×D83—21no—
109Dic166—21no—
110Dic166—21no—
111Dic166—42no—
112C8×C83—22no—
113C83⋊C8—22no—
114C22×C83—41no—
115C22×C83—42no—
116C22×C83—41no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—41no—
120C22×C83C22×D411yescharacteristic
121C2×D83—21no—
122C2×D83—21no—
123C2×D83AGL(1,ℤ/8)12yes—
124C4×C83—21no—
125C4×C83—41no—
126C4×C83#13512yes—
127C4×C83H(5,𝔽2)11yescharacteristic
128C4×C832−1+411yescharacteristic
129Dic83—21no—
130Dic83—21no—
131Dic83—21no—
132Dic83—41no—
133Dic83—42no—
134Dic83—41no—
135C2×C83—81no—
136C2×C83—82no—
137C2×C83#53711yescharacteristic
138C2×C83C2×#13511yescharacteristic
139C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
140D83—22no—
141#3247—831nomaximal, Sylow
142C83#324711yescharacteristic, complemented, Sylow
143#532—831no—
144#529—831no—
145#478—832no—
146#477—831no—
147#476—831no—
148#466—832no—
149#465—832no—
150#463—832no—
151#460—831no—
152#459D8311yescharacteristic, complemented
153C2×C4⋊C8—831no—
154C2×#41—1661no—
155C2×SD16—832no—
156C2×D8C2×D8312yes—
157C22×C8—832no—
158C4⋊Q8—831no—
159#127C2×D8311yescharacteristic
160#126—1661no—
161#125—831no—
162#124—831no—
163#124—1661no—
164#123—1661no—
165C22⋊Q8—831no—
166#121—831no—
167#121—832no—
168#121—1661no—
169C4×Q8—831no—
170C4×D4—832no—
171C4×D4—1661no—
172C4×D4—1661no—
173C4×D4C2×D8311yescharacteristic
174#117—1661no—
175C2×C42—831no—
176#107—832no—
177#106—832no—
178C4⋊C8—832no—
179C4⋊C8—831no—
180C4⋊C8C2×D8311yescharacteristic
181#103—832no—
182#102—832no—
183#102—832no—
184#102C2×D8312yes—
185#41—3324no—
186C2×Q8—831no—
187C2×Q8—1661no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—1661no—
191C2×D4—1662no—
192C2×D4C22×D8311yescharacteristic
193C2×D4C22×D8312yes—
194C22×C4—21no—
195C22×C4—831no—
196C22×C4—831no—
197C22×C4—831no—
198C22×C4—1662no—
199SD16—3322no—
200D8—42no—
201C2×C8—3322no—
202C2×C8—832no—
203C2×C8C22×D8312yes—
204C4⋊C4—831no—
205C4⋊C4—832no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4—1661no—
211C4⋊C4C22×D8311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—1661no—
214C22⋊C4—1661no—
215C22⋊C4—1661no—
216C22⋊C4—1662no—
217C22⋊C4—1662no—
218C22⋊C4—1661no—
219C42—831no—
220C42—831no—
221C42—831no—
222C42—1661no—
223C42C22×D8311yescharacteristic
224C23—21no—
225C23—22no—
226C23—831no—
227Q8—3322no—
228Q8—1661no—
229Q8—1661no—
230D4—21no—
231D4—22no—
232D4—21no—
233D4—22no—
234D4—42no—
235D4—42no—
236D4—3324no—
237D4—3322no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—3324no—
241C2×C4—831no—
242C2×C4—831no—
243C2×C4—831no—
244C2×C4—1662no—
245C2×C4—1661no—
246C2×C4—1661no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4D4×D8311yescharacteristic
252C2×C4D4×D8311yescharacteristic
253C2×C4C23×D8311yescharacteristic
254C8—22no—
255C8—1662no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—832no—
263C22—1661no—
264C22—1661no—
265C22C2×D4×D8311yescharacteristic
266C4—21no—
267C4—41no—
268C4—3322no—
269C4—3321no—
270C4—3321no—
271C4—1661no—
272C4—1661no—
273C4—1661no—
274C4#1458181112yes—
275C4#1458183711yescharacteristic
276C4#1458183911yescharacteristic
277C2—81no—
278C2—82no—
279C2—1662no—
280C2#1912828011yescharacteristic
281C2C2×#1458181111yescharacteristic
282C2AGL(1,ℤ/8)×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#459×C83C2
socleC22×C83C22×D4
hypercenter#459D83
hypocenterC83#3247
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #19128249C2semidirect
2. #19128233C2semidirect
3. #19127999C2semidirect
4. #19127997C2semidirect
5. #19127991C2semidirect
6. #19127934C2semidirect
7. #19127929C2semidirect
8. #19127914C2semidirect
9. #19127898C2semidirect
10. C4⋊C8×D83C2semidirect
11. #459×C83C2semidirect
142. C83#3247semidirect
152. #459D83semidirect

Maximal subgroups

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

subgroupindex
#191282492
#191282332
#191279992
#191279972
#191279912
#191279342
#191279292
#191279142
#191278982
C4⋊C8×D832
#459×C832
#324783