Finite Group Databasebeta

← #24912713

Subgroups — #24912713

Order 10624.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#19127618C211yesmaximal, characteristic
2#19127614C212yesmaximal, complemented
3#19127606C211yesmaximal, characteristic, complemented
4#19127591C212yesmaximal, complemented
5#19127579C212yesmaximal
6#19127568C212yesmaximal, complemented
7#19127563C212yesmaximal, complemented
8#19127548C212yesmaximal, complemented
9#400×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12C22×C4×D83C2212yes—
13C2×(C2×D83)⋊C4C2212yes—
14C2×(C2×D83)⋊C4C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×C4×Dic83C2212yes—
19C2×C4×Dic83C2211yescharacteristic
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28#95×C83C2211yescharacteristic
29#95×C83C2212yes—
30#95×C83C2212yes—
31C23×D83C2312yes—
32C2×Dic166—22no—
33C2×Dic166—41no—
34C2×Dic166—41no—
35C2×D332—24no—
36C2×D332—41no—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C2×C4×D83—24no—
41C2×C4×D83—24no—
42C2×C4×D83—42no—
43(C2×D83)⋊C4—24no—
44(C2×D83)⋊C4—24no—
45(C2×D83)⋊C4—24no—
46(C2×D83)⋊C4—42no—
47(C2×D83)⋊C4—42no—
48C4×Dic83—24no—
49C4×Dic83—41no—
50C22×C4×C83C2312yes—
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—41no—
56C4⋊C4×C83—41no—
57C23×C83C2411yescharacteristic
58C22×D83—22no—
59C22×D83—22no—
60C22×D83—22no—
61C22×D83—22no—
62C22×D83—22no—
63C22×D83—22no—
64C22×D83#4114yes—
65C2×C4×C83—21no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83#4112yes—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83#4114yes—
87D332—44no—
88C4×D83—48no—
89Dic166—44no—
90C22×C83C2×#4112yes—
91C22×C83C2×#4112yes—
92C22×C83C2×#4111yescharacteristic
93C22×C83H(5,𝔽2)11yescharacteristic
94C22×C83H(5,𝔽2)11yescharacteristic
95C2×D83—24no—
96C2×D83—24no—
97C2×D83—24no—
98C2×D83—42no—
99C2×D83—42no—
100C2×D83—42no—
101C2×D83—42no—
102C4×C83—22no—
103C4×C83—42no—
104C4×C83—41no—
105C4×C83—41no—
106C4×C83—42no—
107Dic83—24no—
108Dic83—42no—
109Dic83—42no—
110C2×C83#52912yes—
111C2×C83#53811yescharacteristic
112C2×C83#55112yes—
113C2×C83#55111yescharacteristic
114C2×C83#55511yescharacteristic
115D83—44no—
116#2572—831nomaximal, Sylow
117C83#257211yescharacteristic, complemented, Sylow
118#400D8311yescharacteristic, complemented
119#394—831no—
120#393—832no—
121#391—831no—
122#388—832no—
123#386—832no—
124#383—832no—
125#381—832no—
126C4×C22⋊C4—832no—
127C22×Q8—831no—
128C22×D4—831no—
129C23×C4—832no—
130C2×C4⋊C4C2×D8311yescharacteristic
131C2×C4⋊C4C2×D8311yescharacteristic
132C2×C22⋊C4—832no—
133C2×C22⋊C4—832no—
134C2×C22⋊C4—832no—
135C2×C22⋊C4—832no—
136C2×C22⋊C4—832no—
137C2×C42—831no—
138C2×C42—832no—
139#95—832no—
140#95—832no—
141#95—832no—
142#95—831no—
143#95—832no—
144#95—832no—
145#95C2×D8312yes—
146#95C2×D8311yescharacteristic
147#95C2×D8312yes—
148C24—832no—
149C2×Q8—3321no—
150C2×Q8—3321no—
151C2×Q8—1662no—
152C2×D4—3321no—
153C2×D4—1664no—
154C22×C4—3322no—
155C22×C4—832no—
156C22×C4—832no—
157C22×C4—832no—
158C22×C4—1664no—
159C22×C4—1664no—
160C22×C4C22×D8311yescharacteristic
161C22×C4C22×D8312yes—
162C22×C4C22×D8311yescharacteristic
163C22×C4C22×D8312yes—
164C22×C4C22×D8311yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—41no—
167C22⋊C4—3322no—
168C22⋊C4—3322no—
169C22⋊C4—1664no—
170C22⋊C4—1664no—
171C22⋊C4—1664no—
172C42—3321no—
173C42—1664no—
174C23—834no—
175C23—1662no—
176C23—1662no—
177C23—1662no—
178C23—1662no—
179C23—1662no—
180C23—1662no—
181C23C23×D8311yescharacteristic
182Q8—3324no—
183D4—3324no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—3328no—
197C2×C4—834no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4#1305059112yes—
207C22—3322no—
208C22—3322no—
209C22—3322no—
210C22—3322no—
211C22—1664no—
212C22—1664no—
213C22—1664no—
214C22#41×D8312yes—
215C22#41×D8312yes—
216C22C2×#1305059111yescharacteristic
217C22#1458183711yescharacteristic
218C22#1458183811yescharacteristic
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—42no—
224C4—3322no—
225C4—3322no—
226C4—1664no—
227C2—3324no—
228C2#1912823612yes—
229C2#1912829111yescharacteristic
230C2#1912837612yes—
231C2#1912837911yescharacteristic
232C2#1912841311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC22×D332C22
hypercenter#400D83
hypocenterC83#2572
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #19127614C2semidirect
3. #19127606C2semidirect
4. #19127591C2semidirect
6. #19127568C2semidirect
7. #19127563C2semidirect
8. #19127548C2semidirect
9. #400×C83C2semidirect
117. C83#2572semidirect
118. #400D83semidirect

Maximal subgroups

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

subgroupindex
#191276182
#191276142
#191276062
#191275912
#191275792
#191275682
#191275632
#191275482
#400×C832
#257283