Finite Group Databasebeta

← #24916697

Subgroups — #24916697

Order 10624.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14581823C212yesmaximal, complemented
2#19128212C211yesmaximal, characteristic
3#19127857C214yesmaximal, complemented
4#19127840C214yesmaximal, complemented
5#19127710C212yesmaximal, complemented
6#19127665C211yesmaximal, characteristic, complemented
7C23×Q8×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9#14581823—28no—
10C2×#13050580C2214yescomplemented
11C2×#13050577C2214yescomplemented
12C2×#13050574C2212yes—
13#14581775—21no—
14#14581771C2212yes—
15#14581766C2212yes—
16#14581761—21no—
17C4×C22⋊D83—22no—
18#14581751C2211yescharacteristic
19#14581716C2218yescomplemented
20C22⋊(C83⋊C8)C2214yes—
21C22×Q8×C83—28no—
22C22×Q8×C83—21no—
23C22×Q8×C83—22no—
24C22×Q8×C83C2214yes—
25C22×Q8×C83C2212yes—
26C23×C4×C83—21no—
27C23×C4×C83C2211yescharacteristic
28#13050590—24no—
29C2×Dic166C2311yescharacteristic
30C2×D332C2311yescharacteristic
31C2×C22⋊D83—22no—
32C2×C22⋊D83—21no—
33C2×C4×D83—21no—
34#13050580—28no—
35#13050577—28no—
36#13050574—24no—
37C2×C83⋊C8C2314yes—
38C4⋊Dic83C2312yes—
39Dic83⋊C4—22no—
40C22⋊Dic83—22no—
41C22⋊Dic83—21no—
42(C2×D83)⋊C4—22no—
43C4×Dic83—21no—
44C24×C83D411yescharacteristic
45C2×Q8×C83—216no—
46C2×Q8×C83—28no—
47C2×Q8×C83—216no—
48C2×Q8×C83—24no—
49C2×Q8×C83—24no—
50C2×Q8×C83—22no—
51C2×Q8×C83D418yescomplemented
52C2×Q8×C83C2314yes—
53C22×C4×C83—28no—
54C22×C4×C83—24no—
55C22×C4×C83—22no—
56C22×C4×C83—24no—
57C22×C4×C83D411yescharacteristic
58C22×C4×C83D412yes—
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C23×C83—22no—
62C23×C83—24no—
63C23×C83C2×D411yescharacteristic
64C23×C83C2×D412yes—
65C22×D83—21no—
66Q8×C83—216no—
67Q8×C83—28no—
68Q8×C83—24no—
69Q8×C83C2×D418yes—
70C2×C4×C83—216no—
71C2×C4×C83—28no—
72C2×C4×C83—22no—
73C2×C4×C83—24no—
74C2×C4×C83—24no—
75C2×C4×C83—24no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D414yes—
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—42no—
81C22⋊D83—44no—
82C2×Dic83—22no—
83C2×Dic83—21no—
84D332—22no—
85C4×D83—42no—
86Dic166—22no—
87C83⋊C8—24no—
88C22×C83—28no—
89C22×C83—22no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83#12012yes—
93C22×C83#13714yescomplemented
94C22×C83C22×D411yescharacteristic
95C2×D83—41no—
96C2×D83—42no—
97C4×C83—28no—
98C4×C83—22no—
99C4×C83#12012yes—
100C4×C83C22×D412yes—
101Dic83—41no—
102Dic83—42no—
103C2×C83—22no—
104C2×C83—24no—
105C2×C83C2×#12011yescharacteristic
106C2×C83C2×#13712yes—
107D83—81no—
108#3004—831nomaximal, Sylow
109C83#300411yescharacteristic, complemented, Sylow
110C23×Q8D8311yescharacteristic, complemented
111C2×#137—832no—
112#525—831no—
113#451—834no—
114#448—834no—
115#419—832no—
116#407—831no—
117C2×#41—831no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D8314yes—
122C22×Q8C2×D8312yes—
123C23×C4—21no—
124C23×C4C2×D8311yescharacteristic
125#137—1668no—
126C2×Q16—834no—
127C2×SD16—834no—
128C2×M16—832no—
129#123—1661no—
130C22⋊Q8—832no—
131#121—832no—
132#120—1661no—
133C4×D4—1662no—
134#117—831no—
135#103—838no—
136C22⋊C8—834no—
137C24C22⋊D8311yescharacteristic
138#41—1664no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—831no—
146C2×Q8C22⋊D8318yescomplemented
147C2×Q8C22×D8314yes—
148C2×D4—831no—
149C2×D4—1662no—
150C2×D4—1661no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—1661no—
156C22×C4C22⋊D8312yes—
157C22×C4C22⋊D8311yescharacteristic
158C22×C4C22×D8312yes—
159C22×C4C22×D8311yescharacteristic
160Q16—1668no—
161SD16—1668no—
162M16—1664no—
163C2×C8—834no—
164C4⋊C4—832no—
165C4⋊C4—1662no—
166C22⋊C4—1661no—
167C22⋊C4—1662no—
168C22⋊C4—1662no—
169C42—1661no—
170C23—22no—
171C23—24no—
172C23—1661no—
173C23C2×C22⋊D8311yescharacteristic
174C23C2×C22⋊D8312yes—
175Q8—216no—
176Q8—28no—
177Q8—24no—
178Q8—1662no—
179Q8C2×C22⋊D8318yes—
180D4—3322no—
181D4—3324no—
182D4—1662no—
183C2×C4—216no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—3322no—
190C2×C4—1662no—
191C2×C4—1661no—
192C2×C4C2×C22⋊D8312yes—
193C2×C4C2×C22⋊D8311yescharacteristic
194C2×C4C2×C22⋊D8314yes—
195C2×C4C23×D8311yescharacteristic
196C8—1664no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—3321no—
202C22—3322no—
203C22#1458176112yes—
204C22#1458182314yescomplemented
205C22C22×C22⋊D8311yescharacteristic
206C4—28no—
207C4—22no—
208C4—3321no—
209C4—3322no—
210C4#1458176112yes—
211C4C22×C22⋊D8312yes—
212C2—22no—
213C2—24no—
214C2—6641no—
215C2C2×#1458176111yescharacteristic
216C2C2×#1458182312yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC23×Q8×C83C2
socleC22×C83C22×D4
prime radical#19128212C2
hypercenterC23×Q8D83
hypocenterC83#3004
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581823C2semidirect
3. #19127857C2semidirect
4. #19127840C2semidirect
5. #19127710C2semidirect
6. #19127665C2semidirect
7. C23×Q8×C83C2semidirect
10. C2×#13050580C22semidirect
11. C2×#13050577C22semidirect
19. #14581716C22semidirect
51. C2×Q8×C83D4semidirect
93. C22×C83#137semidirect
109. C83#3004semidirect
110. C23×Q8D83semidirect
146. C2×Q8C22⋊D83semidirect
204. C22#14581823semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
#191282122
#191278572
#191278402
#191277102
#191276652
C23×Q8×C832
#300483