Finite Group Databasebeta

← #24908367

Subgroups — #24908367

Order 10624.

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

normal subgroups only

This group has 296 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
1C22×#13050577C211yesmaximal, characteristic
2C2×#14581727C211yesmaximal, characteristic, complemented
3#19127710C211yesmaximal, characteristic, complemented
4#19127707C211yesmaximal, characteristic
5C2×#14581708C211yesmaximal, characteristic
6#19127659C211yesmaximal, characteristic, complemented
7C2×#101×C83C211yesmaximal, characteristic, complemented
8C22×D332C2211yescharacteristic
9C2×#13050577—21no—
10C2×#13050577—21no—
11C2×#13050577—22no—
12C2×#13050577C414yescomplemented
13C2×#13050574C2211yescharacteristic
14C22×C83⋊C8C2211yescharacteristic
15#14581751C2211yescharacteristic
16#14581727—22no—
17#14581716—22no—
18#14581714—22no—
19#14581708—22no—
20#14581702—21no—
21C8×Dic83—21no—
22C22×Q8×C83C2211yescharacteristic
23C2×M16×C83C2211yescharacteristic
24C2×M16×C83C2211yescharacteristic
25#101×C83—22no—
26C23×D83—21no—
27C2×D332—21no—
28C2×D332—21no—
29C2×D332—21no—
30C2×D332—21no—
31C2×D332—21no—
32C2×D332C2×C412yes—
33#13050577—24no—
34#13050577—24no—
35#13050574—21no—
36#13050574—21no—
37C2×C83⋊C8—21no—
38C2×C83⋊C8C2×C412yes—
39C2×C83⋊C8D411yescharacteristic
40C2×C83⋊C8D411yescharacteristic
41C2×C83⋊C8D412yes—
42C4⋊Dic83D412yes—
43C22⋊Dic83—21no—
44C4×Dic83—21no—
45C2×Q8×C83—21no—
46C2×Q8×C83—21no—
47C2×Q8×C83—21no—
48C2×Q8×C83—21no—
49C2×Q8×C83—21no—
50C2×Q8×C83C2×C412yes—
51C22×C4×C83—21no—
52C22×C4×C83C2311yescharacteristic
53M16×C83—21no—
54M16×C83—21no—
55M16×C83—21no—
56M16×C83D412yes—
57C2×C8×C83—21no—
58C2×C8×C83—21no—
59C23×C83#4111yescharacteristic
60C22×D83—22no—
61C22×D83—41no—
62C22×D83—41no—
63C22×D83—82no—
64Q8×C83—22no—
65Q8×C83—22no—
66Q8×C83—41no—
67Q8×C83—41no—
68C2×C4×C83—22no—
69C2×C4×C83—41no—
70C2×C4×C83—41no—
71C2×C4×C83C22×C411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83#4111yescharacteristic
77C2×Dic83—22no—
78D332—22no—
79D332—22no—
80D332—41no—
81D332—41no—
82C8×C83—22no—
83C8×C83—41no—
84C83⋊C8—41no—
85C83⋊C8C22⋊C412yes—
86C83⋊C8C22⋊C412yes—
87C22×C83—22no—
88C22×C83C4×D411yescharacteristic
89C22×C83#12111yescharacteristic
90C22×C83#12411yescharacteristic
91C2×D83—42no—
92C2×D83—82no—
93C2×D83—82no—
94C2×D83—82no—
95C4×C83—42no—
96C4×C83C2×C22⋊C411yescharacteristic
97C4×C83C4×D411yescharacteristic
98C4×C83#12111yescharacteristic
99C4×C83#12711yescharacteristic
100Dic83—42no—
101C2×C83—21no—
102C2×C83—21no—
103C2×C83#39011yescharacteristic
104C2×C83#47112yes—
105D83—82no—
106#1977—831nomaximal, Sylow
107C83#197711yescharacteristic, complemented, Sylow
108C22×SD16—831no—
109C2×#108—831no—
110#419—831no—
111#418—831no—
112C2×#101D8311yescharacteristic, complemented
113C2×#100—831no—
114#405—831no—
115C22×Q8C2×D8311yescharacteristic
116C22×D4—831no—
117C2×SD16—834no—
118C2×SD16—1661no—
119C2×SD16—1661no—
120C2×SD16—1662no—
121C2×M16—831no—
122C2×M16C2×D8311yescharacteristic
123C2×M16C2×D8311yescharacteristic
124C22×C8—831no—
125#117—831no—
126#108—1662no—
127#103—1662no—
128#102—1662no—
129#101—22no—
130#100—1662no—
131#97—1661no—
132C4×C8—1661no—
133C24—1661no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C4×D8312yes—
140C2×D4—832no—
141C2×D4—1661no—
142C2×D4—1661no—
143C2×D4—1661no—
144C2×D4—1661no—
145C2×D4—1661no—
146C22×C4—21no—
147C22×C4C22×D8311yescharacteristic
148SD16—1664no—
149SD16—1664no—
150M16—21no—
151M16—21no—
152M16—21no—
153M16—1661no—
154M16—1661no—
155M16C22⋊D8312yes—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—832no—
159C2×C8—831no—
160C2×C8—831no—
161C2×C8—832no—
162C2×C8—1661no—
163C4⋊C4—832no—
164C22⋊C4—1661no—
165C42—1661no—
166C23—3321no—
167C23—3321no—
168C23—6642no—
169C23—1662no—
170C23#1305059011yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—3321no—
176D4—3321no—
177D4—1662no—
178D4—1662no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—1662no—
183C2×C4C2×C4×D8311yescharacteristic
184C2×C4D4×D8311yescharacteristic
185C2×C4D4×D8311yescharacteristic
186C2×C4D4×D8311yescharacteristic
187C2×C4C2×C22⋊D8311yescharacteristic
188C2×C4#1305058911yescharacteristic
189C8—22no—
190C8—41no—
191C8—3321no—
192C8—832no—
193C8—832no—
194C22—22no—
195C22—3322no—
196C22—6642no—
197C22—6642no—
198C22—6642no—
199C22C4×C22⋊D8311yescharacteristic
200C22#1458176511yescharacteristic
201C22#1458177811yescharacteristic
202C4—42no—
203C4—3322no—
204C4C22⋊C4×D8311yescharacteristic
205C4(C22⋊D83)⋊C411yescharacteristic
206C4#1458176411yescharacteristic
207C4#1458178811yescharacteristic
208C2—21no—
209C2—21no—
210C2—6642no—
211C2#1912760111yescharacteristic
212C2#1912796712yes—
subgroupG/H
centerC22#14581765
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×#101×C83C2
socleC22×C83#121
prime radicalC22×D332C22
cosocleC22×C4×C83C23
hypercenterC2×#101D83
hypocenterC83#1977
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
2. C2×#14581727C2semidirect
3. #19127710C2semidirect
6. #19127659C2semidirect
7. C2×#101×C83C2semidirect
12. C2×#13050577C4semidirect
107. C83#1977semidirect
112. C2×#101D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505772
C2×#145817272
#191277102
#191277072
C2×#145817082
#191276592
C2×#101×C832
#197783