Finite Group Databasebeta

← #24907960

Subgroups — #24907960

Order 10624.

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

normal subgroups only

This group has 310 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×#14581822C211yesmaximal, characteristic
2#19128178C211yesmaximal, characteristic, complemented
3#19127718C211yesmaximal, characteristic
4#19127709C211yesmaximal, characteristic, complemented
5#19127471C211yesmaximal, characteristic, complemented
6#19127457C211yesmaximal, characteristic, complemented
7#410×C83C211yesmaximal, characteristic
8C22×Dic166C2211yescharacteristic
9#14581822—22no—
10#14581822—22no—
11C2×#13050580C412yescomplemented
12C2×#13050578C412yescomplemented
13C2×#13050574C2211yescharacteristic
14C2×#13050574C2211yescharacteristic
15C4×Dic166—22no—
16#14581757—22no—
17#14581751C2211yescharacteristic
18#14581748—22no—
19#14581719—22no—
20#14581715—22no—
21C2×#41×C83C2211yescharacteristic
22#117×C83C2211yescharacteristic
23#117×C83C2211yescharacteristic
24#99×C83—22no—
25C2×Dic166—21no—
26C2×Dic166—21no—
27C2×Dic166—21no—
28C2×Dic166C2×C412yes—
29C2×Dic166D412yes—
30C2×Dic166D412yes—
31C22×Dic83—21no—
32#13050580—24no—
33#13050578—24no—
34#13050574—21no—
35#13050574—21no—
36#13050574—21no—
37#13050574—21no—
38C2×C83⋊C8—21no—
39C2×C83⋊C8C2×C412yes—
40C4⋊Dic83—21no—
41Dic83⋊C4—24no—
42C22⋊Dic83—21no—
43C4×Dic83—24no—
44C4×Dic83D412yes—
45#41×C83—21no—
46#41×C83—21no—
47#41×C83—21no—
48#41×C83—21no—
49C2×Q8×C83C2×C411yescharacteristic
50C2×D4×C83—21no—
51C2×D4×C83C2×C411yescharacteristic
52C22×C4×C83—21no—
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—21no—
55C4⋊C4×C83—21no—
56C22⋊C4×C83—21no—
57C22⋊C4×C83D412yes—
58C42×C83—21no—
59C42×C83—21no—
60C23×C83—21no—
61C23×C83C2×D411yescharacteristic
62Q8×C83—21no—
63Q8×C83—21no—
64D4×C83—21no—
65D4×C83—21no—
66D4×C83—41no—
67D4×C83—41no—
68C2×C4×C83—21no—
69C2×C4×C83—22no—
70C2×C4×C83—41no—
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×C83#4111yescharacteristic
78C2×C4×C83#4111yescharacteristic
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83Dic166—22no—
84Dic166—22no—
85Dic166—22no—
86Dic166C22⋊C414yes—
87C83⋊C8—22no—
88C83⋊C8—41no—
89C22×C83—41no—
90C22×C83—41no—
91C22×C83—81no—
92C22×C83C4×D411yescharacteristic
93C22×C83#12011yescharacteristic
94C22×C83#12111yescharacteristic
95C4×C83—42no—
96C4×C83—41no—
97C4×C83—81no—
98C4×C83C2×C22⋊C411yescharacteristic
99C4×C83C4×D411yescharacteristic
100C4×C83#12111yescharacteristic
101C4×C83#12311yescharacteristic
102Dic83—24no—
103Dic83—42no—
104C2×C83—21no—
105C2×C83—21no—
106C2×C83—21no—
107C2×C83—81no—
108C2×C83#38611yescharacteristic
109#1905—831nomaximal, Sylow
110C83#190511yescharacteristic, complemented, Sylow
111C2×#137—831no—
112#519—831no—
113#421—831no—
114#419—831no—
115#410D8311yescharacteristic
116#344—831no—
117H(3,ℤ/4)—831no—
118C2×#41C2×D8311yescharacteristic
119C22×Q8—831no—
120#137—1662no—
121#137—1662no—
122C2×Q16—832no—
123C2×SD16—832no—
124C2×M16—831no—
125C2×M16—831no—
126C4×Q8—1662no—
127C4×Q8—1662no—
128#117—831no—
129#117—1662no—
130#117C2×D8311yescharacteristic
131#117C2×D8311yescharacteristic
132#104—1662no—
133#103—1662no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—832no—
140C2×Q8—832no—
141C2×Q8—832no—
142C2×Q8—1661no—
143C2×Q8—1661no—
144C2×Q8—1661no—
145C2×Q8C4×D8311yescharacteristic
146C2×D4—21no—
147C2×D4C4×D8311yescharacteristic
148C22×C4—21no—
149C22×C4—1661no—
150C22×C4C22×D8311yescharacteristic
151Q16—1664no—
152SD16—1664no—
153M16—1661no—
154M16—1661no—
155M16—1661no—
156M16—1661no—
157C2×C8—832no—
158C2×C8—1661no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1664no—
162C4⋊C4—1661no—
163C22⋊C4—21no—
164C22⋊C4—1661no—
165C22⋊C4C22⋊D8312yes—
166C42—21no—
167C42—21no—
168C42—832no—
169C42—1664no—
170C23—21no—
171C23C2×C22⋊D8311yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—834no—
175Q8—1662no—
176Q8—1662no—
177Q8—1662no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4C2×C4×D8311yescharacteristic
192C2×C4D4×D8311yescharacteristic
193C2×C4D4×D8311yescharacteristic
194C2×C4D4×D8311yescharacteristic
195C2×C4#1305058911yescharacteristic
196C2×C4#1305059011yescharacteristic
197C8—3321no—
198C8—1662no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D8311yescharacteristic
203C22#1458176011yescharacteristic
204C22#1458176311yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—3322no—
209C4—1664no—
210C4C22⋊C4×D8311yescharacteristic
211C4(C22⋊D83)⋊C411yescharacteristic
212C4#1458176511yescharacteristic
213C4#1458177411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#1912757711yescharacteristic
subgroupG/H
centerC2#19127577
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#410×C83C2
socleC2×C83#386
prime radicalC2×#41×C83C22
cosocleC22×C4×C83C23
hypercenter#410D83
hypocenterC83#1905
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #19128178C2semidirect
4. #19127709C2semidirect
5. #19127471C2semidirect
6. #19127457C2semidirect
11. C2×#13050580C4semidirect
12. C2×#13050578C4semidirect
110. C83#1905semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
#191281782
#191277182
#191277092
#191274712
#191274572
#410×C832
#190583