Finite Group Databasebeta

← #24910076

Subgroups — #24910076

Order 10624.

816 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×(C2×D83)⋊C4C211yesmaximal, characteristic
2C2×C22⋊C4×D83C212yesmaximal
3#19127575C214yesmaximal, complemented
4#19127572C214yesmaximal, complemented
5#19127561C211yesmaximal, characteristic, complemented
6#19127559C212yesmaximal, complemented
7#380×C83C211yesmaximal, characteristic, complemented
8C24×D83C2211yescharacteristic
9C23×Dic83C2211yescharacteristic
10C22×C4×D83C2214yes—
11C2×C4⋊Dic83C2212yescomplemented
12C2×Dic83⋊C4C2214yescomplemented
13C2×C22⋊Dic83C2212yes—
14C2×(C2×D83)⋊C4C418yescomplemented
15C2×(C2×D83)⋊C4C2214yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C22⋊C4×D83—28no—
19C83⋊#95C2214yescomplemented
20C83⋊#95C2212yescomplemented
21C23×C4×C83C2211yescharacteristic
22C2×C4⋊C4×C83C2212yescomplemented
23C2×C22⋊C4×C83C2212yes—
24#95×C83C2212yescomplemented
25C23×D83—42no—
26C23×D83—42no—
27C23×D83C2×C414yes—
28C23×D83D414yes—
29C23×D83Q814yes—
30C23×D83C2312yes—
31C22×Dic83—42no—
32C22×Dic83C2×C414yescomplemented
33C22×Dic83C2314yes—
34C22×Dic83C2312yes—
35C2×C4×D83—28no—
36C2×C4×D83—44no—
37C2×C4×D83—44no—
38C4⋊Dic83—42no—
39Dic83⋊C4—44no—
40C22⋊Dic83—24no—
41(C2×D83)⋊C4—28no—
42(C2×D83)⋊C4C2×C418yescomplemented
43(C2×D83)⋊C4C2×C418yescomplemented
44C24×C83C2311yescharacteristic
45C22×C4×C83—42no—
46C22×C4×C83C2×C414yescomplemented
47C22×C4×C83C2314yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C4⋊C4×C83—42no—
51C22⋊C4×C83—24no—
52C23×C83—22no—
53C23×C83—22no—
54C23×C83C22×C412yes—
55C23×C83C2×D412yes—
56C23×C83C2×Q812yes—
57C23×C83C2411yescharacteristic
58C22×D83—24no—
59C22×D83—24no—
60C22×D83—44no—
61C22×D83—44no—
62C22×D83—48no—
63C22×D83—48no—
64C22×D83C4⋊C4116yescomplemented
65C22×D83C22×C414yes—
66C22×D83C2×D414yes—
67C22×D83C2×Q814yes—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—24no—
71C2×C4×C83—24no—
72C2×C4×C83—24no—
73C2×C4×C83—44no—
74C2×C4×C83C22×C412yes—
75C2×C4×C83C22×C412yes—
76C2×Dic83—24no—
77C2×Dic83—24no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—44no—
81C2×Dic83C22×C414yes—
82C4×D83—416no—
83C22×C83—22no—
84C22×C83—22no—
85C22×C83—24no—
86C22×C83—24no—
87C22×C83C2×C4⋊C414yes—
88C22×C83C23×C411yescharacteristic
89C22×C83C22×D411yescharacteristic
90C22×C83C22×Q811yescharacteristic
91C22×C83H(5,𝔽2)12yes—
92C22×C83H(5,𝔽2)12yes—
93C2×D83—24no—
94C2×D83—24no—
95C2×D83—28no—
96C2×D83—28no—
97C2×D83—416no—
98C2×D83C2×C4⋊C418yes—
99C4×C83—22no—
100C4×C83—22no—
101C4×C83—44no—
102Dic83—24no—
103Dic83—44no—
104C2×C83—24no—
105C2×C83C22×C4⋊C411yescharacteristic
106C2×C83#51811yescharacteristic
107C2×C83#51811yescharacteristic
108C2×C83#53412yes—
109C2×C83#54312yes—
110D83—28no—
111#2315—831nomaximal, Sylow
112C83#231511yescharacteristic, complemented, Sylow
113C22×C22⋊C4—831no—
114C22×C22⋊C4—832no—
115#385—834no—
116#385—834no—
117#380—831no—
118#380—832no—
119#380D8311yescharacteristic, complemented
120C25—831no—
121C23×C4—831no—
122C23×C4—834no—
123C23×C4C2×D8311yescharacteristic
124C2×C4⋊C4—834no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4C2×D8312yescomplemented
127C2×C22⋊C4—838no—
128C2×C22⋊C4—832no—
129C2×C22⋊C4—832no—
130C2×C22⋊C4—832no—
131C2×C22⋊C4—834no—
132C2×C22⋊C4—1668no—
133C2×C22⋊C4C2×D8312yes—
134#95—834no—
135#95—832no—
136#95C2×D8312yescomplemented
137C24—3322no—
138C24—3322no—
139C24—834no—
140C24—834no—
141C24—834no—
142C24—832no—
143C24C22×D8311yescharacteristic
144C22×C4—42no—
145C22×C4—3324no—
146C22×C4—3324no—
147C22×C4—3322no—
148C22×C4—834no—
149C22×C4—832no—
150C22×C4—834no—
151C22×C4—1668no—
152C22×C4C4×D8314yescomplemented
153C22×C4C22×D8314yes—
154C22×C4C22×D8311yescharacteristic
155C22×C4C22×D8311yescharacteristic
156C4⋊C4—42no—
157C4⋊C4—3324no—
158C4⋊C4—3322no—
159C22⋊C4—24no—
160C22⋊C4—838no—
161C22⋊C4—838no—
162C22⋊C4—1668no—
163C22⋊C4—1664no—
164C23—22no—
165C23—22no—
166C23—3328no—
167C23—3324no—
168C23—3324no—
169C23—3328no—
170C23—8316no—
171C23—834no—
172C23—834no—
173C23—834no—
174C23—1664no—
175C23—1664no—
176C23C2×C4×D8312yes—
177C23D4×D8312yes—
178C23Q8×D8312yes—
179C23C23×D8311yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—44no—
186C2×C4—33216no—
187C2×C4—3324no—
188C2×C4—834no—
189C2×C4—1664no—
190C2×C4—1664no—
191C2×C4—1664no—
192C2×C4—1664no—
193C2×C4C2×C4×D8312yes—
194C2×C4C2×C4×D8312yes—
195C22—24no—
196C22—22no—
197C22—22no—
198C22—24no—
199C22—33216no—
200C22—838no—
201C22—1668no—
202C22—1664no—
203C22—1664no—
204C22—1668no—
205C22C4⋊C4×D8314yes—
206C22C22×C4×D8311yescharacteristic
207C22C2×D4×D8311yescharacteristic
208C22C2×Q8×D8311yescharacteristic
209C22#1458183712yes—
210C22#1458183812yes—
211C4—22no—
212C4—22no—
213C4—44no—
214C4—3324no—
215C4—1664no—
216C2—24no—
217C2—1668no—
218C2C2×C4⋊C4×D8311yescharacteristic
219C2#1912817411yescharacteristic
220C2#1912817511yescharacteristic
221C2#1912825912yes—
222C2#1912832612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutatorC22×C83C23×C4
Fitting#380×C83C2
socle, cosocleC23×C83C24
prime radicalC24×D83C22
hypercenter#380D83
hypocenterC83#2315
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #19127575C2semidirect
4. #19127572C2semidirect
5. #19127561C2semidirect
6. #19127559C2semidirect
7. #380×C83C2semidirect
11. C2×C4⋊Dic83C22semidirect
12. C2×Dic83⋊C4C22semidirect
14. C2×(C2×D83)⋊C4C4semidirect
19. C83⋊#95C22semidirect
20. C83⋊#95C22semidirect
22. C2×C4⋊C4×C83C22semidirect
24. #95×C83C22semidirect
32. C22×Dic83C2×C4semidirect
42. (C2×D83)⋊C4C2×C4semidirect
43. (C2×D83)⋊C4C2×C4semidirect
46. C22×C4×C83C2×C4semidirect
64. C22×D83C4⋊C4semidirect
112. C83#2315semidirect
119. #380D83semidirect
126. C2×C4⋊C4C2×D83semidirect
136. #95C2×D83semidirect
152. C22×C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D83)⋊C42
C2×C22⋊C4×D832
#191275752
#191275722
#191275612
#191275592
#380×C832
#231583