Finite Group Databasebeta

← #24907935

Subgroups — #24907935

Order 10624.

304 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 79 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×#14581825C211yesmaximal, characteristic
2#19128467C211yesmaximal, characteristic
3#19127719C211yesmaximal, characteristic
4#19127709C211yesmaximal, characteristic, complemented
5C2×#14581710C211yesmaximal, characteristic, complemented
6C83⋊#338C211yesmaximal, characteristic, complemented
7#344×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C22×Dic166C2211yescharacteristic
10#14581825—22no—
11#14581825—22no—
12C2×Dic332C412yescomplemented
13C2×#13050579C412yescomplemented
14#14581800—22no—
15#14581798—22no—
16C2×#13050574C2211yescharacteristic
17C2×#13050574C2211yescharacteristic
18M16×D83—22no—
19#14581720—22no—
20#14581715—22no—
21#14581710—22no—
22C2×M16×C83C2211yescharacteristic
23C2×M16×C83C2211yescharacteristic
24#117×C83C2211yescharacteristic
25#13050590—21no—
26#13050590—21no—
27#13050590D412yes—
28#13050590D412yes—
29C2×Dic166—21no—
30C2×Dic166—21no—
31C2×Dic166—21no—
32C2×Dic166—21no—
33C2×Dic166—21no—
34C2×Dic166C2×C412yes—
35C2×Dic166C2×C411yescharacteristic
36C2×D332C2×C411yescharacteristic
37C2×C22⋊D83—21no—
38C22×Dic83—21no—
39C2×C4×D83—21no—
40Dic332—24no—
41#13050579—24no—
42#13050574—21no—
43#13050574—21no—
44#13050574—21no—
45#13050574—21no—
46#13050573—24no—
47C2×C83⋊C8—21no—
48C2×C83⋊C8D412yes—
49C8×D83—24no—
50C22×C4×C83C2311yescharacteristic
51M16×C83—21no—
52M16×C83—21no—
53M16×C83—21no—
54M16×C83D412yes—
55C2×C8×C83—21no—
56C2×C8×C83C2×C412yes—
57C4⋊C4×C83—21no—
58C22⋊C4×C83—21no—
59C42×C83—21no—
60C23×C83#4111yescharacteristic
61C22×D83—21no—
62C2×C4×C83—41no—
63C2×C4×C83C22×C411yescharacteristic
64C2×C4×C83C2×D411yescharacteristic
65C2×C4×C83C2×D411yescharacteristic
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83C2×D411yescharacteristic
68C2×C4×C83#4111yescharacteristic
69C22⋊D83—22no—
70C22⋊D83—22no—
71C2×Dic83—21no—
72C2×Dic83—22no—
73C2×Dic83—41no—
74C2×Dic83—41no—
75D332—21no—
76D332C22⋊C412yes—
77C4×D83—22no—
78C4×D83—22no—
79Dic166—21no—
80Dic166—22no—
81Dic166—22no—
82Dic166—41no—
83Dic166—41no—
84Dic166C22⋊C412yes—
85C8×C83—22no—
86C8×C83—22no—
87C83⋊C8—22no—
88C83⋊C8—41no—
89C22×C83—41no—
90C22×C83C4×D411yescharacteristic
91C22×C83#12111yescharacteristic
92C22×C83#12311yescharacteristic
93C2×D83—22no—
94C2×D83—81no—
95C4×C83—81no—
96C4×C83C2×C22⋊C411yescharacteristic
97C4×C83C4×D411yescharacteristic
98C4×C83#12011yescharacteristic
99C4×C83#12111yescharacteristic
100Dic83—22no—
101Dic83—42no—
102C2×C83—21no—
103C2×C83—21no—
104C2×C83—21no—
105C2×C83#38611yescharacteristic
106D83—42no—
107#1902—831nomaximal, Sylow
108C83#190211yescharacteristic, complemented, Sylow
109C2×#137—831no—
110#568—831no—
111#421—831no—
112#419—831no—
113C2×#101—831no—
114#344D8311yescharacteristic, complemented
115#338—831no—
116C2×#41—831no—
117C22×Q8—831no—
118#137—1662no—
119#137—1662no—
120C2×Q16—832no—
121C2×SD16—832no—
122#131—1662no—
123#131—1662no—
124C2×M16—831no—
125C2×M16—831no—
126C2×M16—1662no—
127C2×M16C2×D8311yescharacteristic
128C2×M16C2×D8311yescharacteristic
129#117C2×D8311yescharacteristic
130#104—1662no—
131#103—1662no—
132#101—1662no—
133#41—832no—
134#41—832no—
135#41—1661no—
136#41—1661no—
137C2×Q8—831no—
138C2×Q8—832no—
139C2×Q8—1661no—
140C2×Q8—1661no—
141C2×Q8—1661no—
142C2×Q8—1661no—
143C2×Q8—1661no—
144C2×D4—831no—
145C2×D4—1661no—
146C22×C4—1661no—
147C22×C4—1661no—
148C22×C4C22×D8311yescharacteristic
149Q16—1664no—
150SD16—1664no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—1664no—
155M16—1661no—
156M16—1661no—
157M16—1661no—
158M16—1661no—
159M16D33212yes—
160C2×C8—21no—
161C2×C8—832no—
162C2×C8—1664no—
163C2×C8—1661no—
164C2×C8C4×D8312yes—
165C4⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C23—1661no—
169C23#1305059011yescharacteristic
170Q8—3321no—
171Q8—3321no—
172Q8—832no—
173Q8—1662no—
174Q8—1662no—
175Q8—1661no—
176D4—832no—
177D4—1662no—
178D4—1662no—
179D4—1661no—
180C2×C4—41no—
181C2×C4—3321no—
182C2×C4—3321no—
183C2×C4—1662no—
184C2×C4—1661no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4C2×C4×D8311yescharacteristic
188C2×C4D4×D8311yescharacteristic
189C2×C4D4×D8311yescharacteristic
190C2×C4D4×D8311yescharacteristic
191C2×C4C2×D33211yescharacteristic
192C2×C4#1305059111yescharacteristic
193C8—22no—
194C8—22no—
195C8—3321no—
196C8—1662no—
197C22—41no—
198C22—6641no—
199C22—1662no—
200C22C4×D33211yescharacteristic
201C22#1458176511yescharacteristic
202C22#1458177711yescharacteristic
203C4—81no—
204C4—3322no—
205C4—1662no—
206C4C22⋊C4×D8311yescharacteristic
207C4#1458175411yescharacteristic
208C4#1458176211yescharacteristic
209C4#1458176711yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—3322no—
214C2#1912757911yescharacteristic
subgroupG/H
centerC2#19127579
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#344×C83C2
socleC2×C83#386
prime radicalC2×#13050590C22
cosocleC22×C4×C83C23
hypercenter#344D83
hypocenterC83#1902
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #19127709C2semidirect
5. C2×#14581710C2semidirect
6. C83⋊#338C2semidirect
7. #344×C83C2semidirect
12. C2×Dic332C4semidirect
13. C2×#13050579C4semidirect
108. C83#1902semidirect
114. #344D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818252
#191284672
#191277192
#191277092
C2×#145817102
C83⋊#3382
#344×C832
#190283