Finite Group Databasebeta

← #24916993

Subgroups — #24916993

Order 10624.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581823C211yesmaximal, characteristic
2C22×#13050580C211yesmaximal, characteristic, complemented
3#19128277C211yesmaximal, characteristic
4#19127923C214yesmaximal, complemented
5#19127911C214yesmaximal, complemented
6#19127728C211yesmaximal, characteristic, complemented
7#19127709C211yesmaximal, characteristic, complemented
8#19127695C211yesmaximal, characteristic
9#519×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12#14581823—24no—
13C2×#13050580—24no—
14C2×#13050580C2212yes—
15C2×#13050580C2214yes—
16C2×#13050577C2212yes—
17C2×#13050574C2211yescharacteristic
18C22×C83⋊C8C2211yescharacteristic
19#14581792C2212yes—
20#14581780C2212yes—
21#14581776—22no—
22#14581772—22no—
23C4⋊(C83⋊C8)C2214yes—
24#14581715C2212yes—
25#14581715C2214yes—
26#14581712C2212yes—
27C22×Q8×C83C2211yescharacteristic
28C4×Q8×C83—22no—
29C4×Q8×C83C2214yes—
30#117×C83—22no—
31#117×C83C2211yescharacteristic
32#117×C83C2211yescharacteristic
33#13050590—22no—
34#13050590—22no—
35C2×Dic166—22no—
36C2×Dic166—22no—
37C2×Dic166—21no—
38C2×Dic166C2312yes—
39C2×Dic166C2311yescharacteristic
40C2×D332C2311yescharacteristic
41C2×C22⋊D83—21no—
42C22×Dic83—21no—
43C2×C4×D83—21no—
44#13050580—28no—
45#13050580—24no—
46#13050577—24no—
47#13050574—22no—
48C2×C83⋊C8—22no—
49C2×C83⋊C8C2312yes—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—24no—
52(C2×D83)⋊C4—24no—
53C2×Q8×C83—21no—
54C2×Q8×C83—22no—
55C2×Q8×C83D414yes—
56C2×Q8×C83C2312yes—
57C22×C4×C83—21no—
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—24no—
60C4⋊C4×C83D412yes—
61C4⋊C4×C83C2312yes—
62C22⋊C4×C83—21no—
63C22⋊C4×C83D412yes—
64C42×C83—24no—
65C42×C83—21no—
66C42×C83C2312yes—
67C23×C83C2×D411yescharacteristic
68C22×D83—21no—
69Q8×C83—24no—
70Q8×C83—22no—
71Q8×C83C2×D414yes—
72C2×C4×C83—24no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83C2×D414yes—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D412yes—
78C2×C4×C83#4112yes—
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—42no—
81C2×Dic83—22no—
82C2×Dic83—21no—
83C2×Dic83—42no—
84D332—21no—
85D332—21no—
86C4×D83—42no—
87Dic166—22no—
88Dic166—21no—
89Dic166—22no—
90Dic166—21no—
91Dic166—42no—
92C83⋊C8—22no—
93C83⋊C8—22no—
94C22×C83—22no—
95C22×C83#12112yes—
96C22×C83C22×D411yescharacteristic
97C2×D83—41no—
98C2×D83—42no—
99C4×C83—24no—
100C4×C83—24no—
101C4×C83—42no—
102C4×C83#12112yes—
103C4×C83C22×D411yescharacteristic
104C4×C83C2×#4111yescharacteristic
105Dic83—42no—
106Dic83—41no—
107C2×C83—22no—
108C2×C83C2×#12111yescharacteristic
109C2×C83#57812yes—
110D83—81no—
111#3044—831nomaximal, Sylow
112C83#304411yescharacteristic, complemented, Sylow
113C2×#137—831no—
114C22×Q16—831no—
115#536—831no—
116#519D8311yescharacteristic, complemented
117#464—834no—
118#462—834no—
119#424—831no—
120#419—831no—
121#416—831no—
122C2×#41—831no—
123C22×Q8—831no—
124C22×Q8C2×D8311yescharacteristic
125#137—1664no—
126C2×Q16—834no—
127C2×Q16—832no—
128C2×Q16—1664no—
129C2×SD16—832no—
130C2×M16—831no—
131C22×C8—831no—
132C4⋊Q8—832no—
133#124—832no—
134#123—1662no—
135C22⋊Q8—1662no—
136C4×Q8—22no—
137C4×Q8C2×D8314yes—
138#117—22no—
139#117C2×D8311yescharacteristic
140#117C2×D8311yescharacteristic
141C4⋊C8—834no—
142#103—832no—
143#103—834no—
144#102—832no—
145#41—1662no—
146#41—1662no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—832no—
150C2×Q8—831no—
151C2×Q8—1662no—
152C2×Q8—1662no—
153C2×Q8—1661no—
154C2×Q8D33214yes—
155C2×Q8C22×D8312yes—
156C2×D4—831no—
157C2×D4—1661no—
158C22×C4—21no—
159C22×C4—1661no—
160C22×C4—1661no—
161C22×C4C22×D8311yescharacteristic
162Q16—1668no—
163Q16—1664no—
164SD16—1664no—
165M16—1662no—
166C2×C8—832no—
167C2×C8—832no—
168C2×C8—1662no—
169C4⋊C4—24no—
170C4⋊C4—1664no—
171C4⋊C4C22⋊D8312yes—
172C4⋊C4C22×D8312yes—
173C22⋊C4—21no—
174C22⋊C4—1664no—
175C22⋊C4C22⋊D8312yes—
176C42—24no—
177C42—21no—
178C42C22×D8312yes—
179C23—1661no—
180C23C2×C22⋊D8311yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—3322no—
184Q8—1662no—
185Q8—1661no—
186Q8—1662no—
187Q8—1661no—
188Q8C2×D33214yes—
189D4—3322no—
190D4—1661no—
191D4—1661no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3322no—
196C2×C4—3322no—
197C2×C4—1662no—
198C2×C4—1661no—
199C2×C4C2×C22⋊D8314yes—
200C2×C4C2×C22⋊D8311yescharacteristic
201C2×C4C2×D33212yes—
202C2×C4#1305059012yes—
203C2×C4C23×D8311yescharacteristic
204C8—1662no—
205C8—1662no—
206C22—22no—
207C22—3321no—
208C22—3322no—
209C22#1458176612yes—
210C22C22×C22⋊D8311yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—3321no—
215C4—3322no—
216C4#1458176612yes—
217C4C22×D33211yescharacteristic
218C4C2×#1305059011yescharacteristic
219C2—22no—
220C2—6641no—
221C2C2×#1458176611yescharacteristic
222C2#1912851612yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#519×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenter#519D83
hypocenterC83#3044
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#13050580C2semidirect
4. #19127923C2semidirect
5. #19127911C2semidirect
6. #19127728C2semidirect
7. #19127709C2semidirect
9. #519×C83C2semidirect
112. C83#3044semidirect
116. #519D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C22×#130505802
#191282772
#191279232
#191279112
#191277282
#191277092
#191276952
#519×C832
#304483