Finite Group Databasebeta

← #24918094

Subgroups — #24918094

Order 10624.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128312C211yesmaximal, characteristic, complemented
2#19128307C211yesmaximal, characteristic
3#19128100C211yesmaximal, characteristic, complemented
4#19128075C211yesmaximal, characteristic, complemented
5#19128066C211yesmaximal, characteristic, complemented
6#19127950C212yesmaximal, complemented
7#19127933C212yesmaximal, complemented
8#19127804C212yesmaximal, complemented
9#19127786C212yesmaximal, complemented
10#19127658C211yesmaximal, characteristic, complemented
11#491×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2212yes—
13C2×#13050577C2212yes—
14C2×#13050573C2211yescharacteristic
15C2×C8×D83C2211yescharacteristic
16#14581789C2211yescharacteristic
17#14581787—22no—
18#14581784C2211yescharacteristic
19#14581782—22no—
20#14581777—22no—
21#14581768C2212yes—
22#14581767—22no—
23#14581767C2212yes—
24Q8×Dic83C2212yes—
25#14581754C2212yes—
26#14581749C2211yescharacteristic
27#14581725C2212yes—
28#14581723C2212yes—
29#14581714C2212yes—
30#14581714C2212yes—
31#14581702C2211yescharacteristic
32C4×C83⋊C8C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C4⋊Q8×C83C2211yescharacteristic
35#125×C83C2211yescharacteristic
36#103×C83C2212yes—
37#103×C83C2212yes—
38#97×C83C2211yescharacteristic
39C2×D332—22no—
40C2×D332—22no—
41C2×D332C2312yes—
42C2×C4×D83—22no—
43C2×C4×D83C2311yescharacteristic
44#13050577—24no—
45#13050577—24no—
46#13050573—22no—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2311yescharacteristic
49C8×D83—22no—
50C4⋊Dic83—22no—
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4D412yes—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4D412yes—
59C4×Dic83—22no—
60C4×Dic83C2311yescharacteristic
61C2×Q8×C83C2312yes—
62C2×C8×C83C2311yescharacteristic
63C2×C8×C83C2311yescharacteristic
64C4⋊C4×C83—22no—
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C42×C83C2311yescharacteristic
68C22×D83—22no—
69C22×D83C2×D411yescharacteristic
70Q8×C83—22no—
71Q8×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83C2×D412yes—
75C2×C4×C83C2411yescharacteristic
76C2×Dic83—22no—
77C2×Dic83C2×D412yes—
78C2×Dic83C2×D411yescharacteristic
79D332—22no—
80D332—22no—
81D332—82no—
82D332—82no—
83C4×D83—21no—
84C4×D83—21no—
85C4×D83—82no—
86C8×C83—21no—
87C8×C83—21no—
88C83⋊C8#4112yes—
89C83⋊C8#4112yes—
90C22×C83C22×D411yescharacteristic
91C2×D83—21no—
92C2×D83—22no—
93C2×D83—42no—
94C2×D83—42no—
95C2×D83—42no—
96C4×C83—22no—
97C4×C83—42no—
98C4×C83—42no—
99C4×C83C2×#4111yescharacteristic
100C4×C83C2×#4111yescharacteristic
101Dic83—21no—
102Dic83—22no—
103Dic83—42no—
104C2×C83#53211yescharacteristic
105C2×C83#57712yes—
106D83—41no—
107D83—82no—
108#3174—831nomaximal, Sylow
109C83#317411yescharacteristic, complemented, Sylow
110#541—831no—
111#540—831no—
112#499—831no—
113#492—831no—
114#491D8311yescharacteristic, complemented
115#490—831no—
116#468—832no—
117#465—832no—
118#440—832no—
119C4×SD16—832no—
120#405—831no—
121C2×SD16—832no—
122C2×SD16—832no—
123C2×M16—831no—
124C22×C8—831no—
125C4⋊Q8C2×D8311yescharacteristic
126#127—831no—
127#126—1662no—
128#125—831no—
129#125—1662no—
130#125C2×D8311yescharacteristic
131#123—1662no—
132C22⋊Q8—832no—
133#121—832no—
134#121—1662no—
135C4×Q8—832no—
136C4×D4—832no—
137#117—831no—
138#107—832no—
139#106—832no—
140#103C2×D8312yes—
141#103C2×D8312yes—
142#102—832no—
143#102—832no—
144#97—831no—
145#97C2×D8311yescharacteristic
146C4×C8—831no—
147C4×C8—831no—
148C2×Q8C22×D8312yes—
149C2×D4—832no—
150C2×D4—1662no—
151C2×D4—1662no—
152C22×C4—831no—
153C22×C4—1662no—
154SD16—1664no—
155SD16—1664no—
156M16—1662no—
157C2×C8—831no—
158C2×C8—831no—
159C2×C8—1662no—
160C2×C8C22×D8311yescharacteristic
161C2×C8C22×D8311yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—832no—
165C4⋊C4—832no—
166C4⋊C4—1662no—
167C4⋊C4—1662no—
168C4⋊C4—1662no—
169C4⋊C4C22×D8312yes—
170C22⋊C4—832no—
171C22⋊C4—1662no—
172C22⋊C4—1662no—
173C22⋊C4—1662no—
174C42—831no—
175C42—1662no—
176C42C22×D8311yescharacteristic
177C23—831no—
178C23—1662no—
179Q8—22no—
180Q8—22no—
181D4—6642no—
182D4—6642no—
183D4—1662no—
184D4—1662no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—832no—
188C2×C4—831no—
189C2×C4—6642no—
190C2×C4—1662no—
191C2×C4—1661no—
192C2×C4—1661no—
193C2×C4D4×D8312yes—
194C2×C4C23×D8311yescharacteristic
195C8—21no—
196C8—21no—
197C8—832no—
198C8—832no—
199C22—3322no—
200C22—3322no—
201C22—3322no—
202C22—1661no—
203C22—1662no—
204C22C2×D4×D8311yescharacteristic
205C4—22no—
206C4—42no—
207C4—42no—
208C4—3322no—
209C4—1661no—
210C4—1662no—
211C4#41×D8311yescharacteristic
212C4#41×D8311yescharacteristic
213C2—3321no—
214C2—6642no—
215C2#1912825011yescharacteristic
216C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#491×C83C2
socleC22×C83C22×D4
prime radical#19128307C2
hypercenter#491D83
hypocenterC83#3174
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #19128312C2semidirect
3. #19128100C2semidirect
4. #19128075C2semidirect
5. #19128066C2semidirect
6. #19127950C2semidirect
7. #19127933C2semidirect
8. #19127804C2semidirect
9. #19127786C2semidirect
10. #19127658C2semidirect
11. #491×C83C2semidirect
109. C83#3174semidirect
114. #491D83semidirect

Maximal subgroups

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

subgroupindex
#191283122
#191283072
#191281002
#191280752
#191280662
#191279502
#191279332
#191278042
#191277862
#191276582
#491×C832
#317483