Finite Group Databasebeta

← #24907975

Subgroups — #24907975

Order 10624.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×#14581800C211yesmaximal, characteristic
3#19127703C211yesmaximal, characteristic
4C2×#14581716C211yesmaximal, characteristic, complemented
5#19127469C211yesmaximal, characteristic, complemented
6#19127465C211yesmaximal, characteristic, complemented
7C2×#101×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9#14581823—22no—
10#14581823C414yescomplemented
11C2×#13050580—21no—
12C2×#13050577—21no—
13#14581800—22no—
14#14581800—22no—
15C2×#13050574C2211yescharacteristic
16C2×#13050573—21no—
17C22×C83⋊C8C2211yescharacteristic
18C2×C8×D83—21no—
19C2×C4⋊Dic83C2211yescharacteristic
20#14581717—21no—
21#14581716—21no—
22#14581716—21no—
23#14581714—21no—
24C22×Q8×C83C2211yescharacteristic
25C2×M16×C83C2211yescharacteristic
26C2×M16×C83C2211yescharacteristic
27#101×C83—22no—
28#13050590—21no—
29#13050590—21no—
30#13050590C2×C412yes—
31#13050590D412yes—
32C2×Dic166D411yescharacteristic
33C2×D332D411yescharacteristic
34C2×C22⋊D83—21no—
35C22×Dic83—21no—
36C2×C4×D83—21no—
37#13050580—24no—
38#13050577—24no—
39#13050574—21no—
40#13050574C2×C412yes—
41#13050573—24no—
42C2×C83⋊C8—21no—
43C2×C83⋊C8—21no—
44C2×C83⋊C8—21no—
45C2×C83⋊C8D411yescharacteristic
46C2×C83⋊C8D411yescharacteristic
47C8×D83—24no—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C2×Q8×C83—21no—
51C2×Q8×C83—21no—
52C2×Q8×C83—21no—
53C2×Q8×C83—21no—
54C2×Q8×C83—21no—
55C2×Q8×C83C2×C412yes—
56C22×C4×C83—21no—
57C22×C4×C83C2311yescharacteristic
58M16×C83—21no—
59M16×C83—21no—
60M16×C83—21no—
61M16×C83D412yes—
62C2×C8×C83—21no—
63C2×C8×C83—21no—
64C23×C83#4111yescharacteristic
65C22×D83—21no—
66Q8×C83—22no—
67Q8×C83—22no—
68Q8×C83—41no—
69Q8×C83—41no—
70C2×C4×C83—22no—
71C2×C4×C83—41no—
72C2×C4×C83—41no—
73C2×C4×C83C22×C411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83#4111yescharacteristic
79C22⋊D83—22no—
80C22⋊D83—22no—
81C2×Dic83—21no—
82C2×Dic83—41no—
83C2×Dic83—81no—
84D332—21no—
85D332C22⋊C412yes—
86C4×D83—22no—
87C4×D83—22no—
88Dic166—21no—
89Dic166C22⋊C412yes—
90C8×C83—22no—
91C8×C83—41no—
92C83⋊C8—22no—
93C83⋊C8—21no—
94C83⋊C8—21no—
95C22×C83—21no—
96C22×C83—21no—
97C22×C83C4×D411yescharacteristic
98C22×C83#12111yescharacteristic
99C22×C83#12311yescharacteristic
100C2×D83—22no—
101C2×D83—41no—
102C2×D83—41no—
103C4×C83—42no—
104C4×C83C2×C22⋊C411yescharacteristic
105C4×C83C4×D411yescharacteristic
106C4×C83#12011yescharacteristic
107C4×C83#12111yescharacteristic
108Dic83—22no—
109Dic83—81no—
110C2×C83—21no—
111C2×C83—21no—
112C2×C83#38611yescharacteristic
113C2×C83#47111yescharacteristic
114C2×C83#47311yescharacteristic
115D83—42no—
116#1908—831nomaximal, Sylow
117C83#190811yescharacteristic, complemented, Sylow
118C2×#137—831no—
119C2×#131—831no—
120#417—831no—
121C2×#103—831no—
122C2×#101D8311yescharacteristic, complemented
123#343—831no—
124#341—831no—
125C2×#41—831no—
126C22×Q8C2×D8311yescharacteristic
127#137—834no—
128#137—1662no—
129C2×Q16—1661no—
130C2×SD16—1661no—
131#131—1662no—
132#131—1662no—
133C2×M16—831no—
134C2×M16—1661no—
135C2×M16C2×D8311yescharacteristic
136C2×M16C2×D8311yescharacteristic
137C22×C8—831no—
138C22×C8—1661no—
139C2×C4⋊C4—831no—
140#103—1661no—
141#103—1661no—
142#103—1661no—
143#102—1661no—
144#101—22no—
145#41—832no—
146#41—832no—
147#41—1661no—
148#41—1661no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—831no—
155C2×Q8C4×D8312yes—
156C2×D4—831no—
157C2×D4—1661no—
158C22×C4—21no—
159C22×C4—1661no—
160C22×C4—1661no—
161C22×C4C22×D8311yescharacteristic
162Q16—1664no—
163SD16—1664no—
164M16—21no—
165M16—21no—
166M16—21no—
167M16—832no—
168M16—1664no—
169M16—1661no—
170M16C22⋊D8312yes—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—831no—
174C2×C8—831no—
175C2×C8—1664no—
176C2×C8—1661no—
177C2×C8—1661no—
178C2×C8—1661no—
179C4⋊C4—1661no—
180C4⋊C4—1661no—
181C23—1661no—
182C23#1305059011yescharacteristic
183Q8—22no—
184Q8—22no—
185Q8—41no—
186Q8—41no—
187Q8—832no—
188Q8—1661no—
189D4—832no—
190D4—1662no—
191D4—1662no—
192D4—1661no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—3321no—
197C2×C4—6641no—
198C2×C4—1661no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4C2×C4×D8311yescharacteristic
202C2×C4D4×D8311yescharacteristic
203C2×C4D4×D8311yescharacteristic
204C2×C4D4×D8311yescharacteristic
205C2×C4C2×C22⋊D8311yescharacteristic
206C2×C4#1305058911yescharacteristic
207C8—22no—
208C8—41no—
209C8—1661no—
210C8—1661no—
211C8—1662no—
212C22—21no—
213C22—21no—
214C22—3321no—
215C22—3321no—
216C22—1662no—
217C22C4×C22⋊D8311yescharacteristic
218C22#1458176511yescharacteristic
219C22#1458177411yescharacteristic
220C4—42no—
221C4—6641no—
222C4—1662no—
223C4C22⋊C4×D8311yescharacteristic
224C4(C22⋊D83)⋊C411yescharacteristic
225C4#1458176011yescharacteristic
226C4#1458176311yescharacteristic
227C2—21no—
228C2—21no—
229C2—3322no—
230C2#1912757711yescharacteristic
231C2#1912796711yescharacteristic
232C2#1912797511yescharacteristic
subgroupG/H
centerC22#14581765
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×#101×C83C2
socleC22×C83#121
prime radicalC2×#13050590C22
cosocleC22×C4×C83C23
hypercenterC2×#101D83
hypocenterC83#1908
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
4. C2×#14581716C2semidirect
5. #19127469C2semidirect
6. #19127465C2semidirect
7. C2×#101×C83C2semidirect
10. #14581823C4semidirect
117. C83#1908semidirect
122. C2×#101D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C2×#145818002
#191277032
C2×#145817162
#191274692
#191274652
C2×#101×C832
#190883