Finite Group Databasebeta

← #24907421

Subgroups — #24907421

Order 10624.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14581840C211yesmaximal, characteristic
2C2×#14581720C212yesmaximal, complemented
3#19127700C212yesmaximal, complemented
4C2×#14581711C211yesmaximal, characteristic, complemented
5#391×C83C211yesmaximal, characteristic, complemented
6#14581840—24no—
7#14581840—22no—
8#14581840C414yescomplemented
9C2×#13050591—22no—
10C2×#13050590—22no—
11C2×#13050590C2212yes—
12C2×Q8×D83—22no—
13C2×#13050574C2212yes—
14#14581720—24no—
15#14581715—22no—
16#14581712—22no—
17#14581711—22no—
18C22×Q8×C83C2211yescharacteristic
19C2×C4⋊C4×C83C2211yescharacteristic
20C2×C42×C83C2211yescharacteristic
21#95×C83—22no—
22#13050591—28no—
23#13050591—28no—
24#13050590—28no—
25#13050590—28no—
26#13050590—22no—
27#13050590—22no—
28#13050590C2×C414yes—
29#13050590D414yes—
30C2×Dic166—22no—
31C2×Dic166D412yes—
32Q8×D83—28no—
33Q8×D83—28no—
34C2×D332—22no—
35C2×D332D412yes—
36C2×C22⋊D83—22no—
37C2×C4×D83—24no—
38C2×C4×D83—22no—
39#13050574—22no—
40#13050574—22no—
41C2×C83⋊C8—22no—
42C2×Q8×C83—22no—
43C2×Q8×C83—22no—
44C2×Q8×C83C2×C412yes—
45C2×Q8×C83D412yes—
46C22×C4×C83—21no—
47C22×C4×C83—21no—
48C22×C4×C83D412yes—
49C22×C4×C83C2311yescharacteristic
50C4⋊C4×C83—22no—
51C42×C83—22no—
52C23×C83C2×D411yescharacteristic
53C22×D83—22no—
54Q8×C83—24no—
55Q8×C83—24no—
56C2×C4×C83—22no—
57C2×C4×C83—22no—
58C2×C4×C83—22no—
59C2×C4×C83—41no—
60C2×C4×C83—41no—
61C2×C4×C83—42no—
62C2×C4×C83—41no—
63C2×C4×C83C22⋊C414yes—
64C2×C4×C83C22×C411yescharacteristic
65C2×C4×C83C2×D412yes—
66C2×C4×C83C2×D412yes—
67C2×C4×C83C2×D411yescharacteristic
68C22⋊D83—24no—
69C22⋊D83—24no—
70C2×Dic83—22no—
71D332—28no—
72D332—22no—
73D332C22⋊C414yes—
74C4×D83—216no—
75C4×D83—24no—
76C4×D83—24no—
77Dic166—28no—
78Dic166—22no—
79Dic166C22⋊C414yes—
80C83⋊C8—42no—
81C22×C83—21no—
82C22×C83—21no—
83C22×C83C2×C22⋊C411yescharacteristic
84C22×C83#12011yescharacteristic
85C22×C83#12011yescharacteristic
86C2×D83—24no—
87C2×D83—42no—
88C2×D83—42no—
89C4×C83—24no—
90C4×C83—42no—
91C4×C83—81no—
92C4×C83C2×C22⋊C412yes—
93C4×C83#12012yes—
94Dic83—24no—
95C2×C83—21no—
96C2×C83—21no—
97C2×C83#37911yescharacteristic
98C2×C83#45411yescharacteristic
99C2×C83#45611yescharacteristic
100D83—44no—
101#1793—831nomaximal, Sylow
102C83#179311yescharacteristic, complemented, Sylow
103C2×2−1+4—831no—
104C2×#104—832no—
105#417—832no—
106C2×#101—831no—
107#391D8311yescharacteristic, complemented
1082−1+4—834no—
1092−1+4—1664no—
1102−1+4—1662no—
111C2×#41—832no—
112C2×#41—1662no—
113C2×#41—1662no—
114C22×Q8—1662no—
115C22×Q8C2×D8311yescharacteristic
116C2×M16—832no—
117C2×C4⋊C4C2×D8311yescharacteristic
118C2×C42C2×D8311yescharacteristic
119#104—1664no—
120#103—1662no—
121#102—1662no—
122#101—1662no—
123#95—22no—
124#41—834no—
125#41—834no—
126#41—1668no—
127#41—1668no—
128#41—1668no—
129#41—1668no—
130#41—1662no—
131#41—1662no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—832no—
135C2×Q8—1662no—
136C2×Q8—1668no—
137C2×Q8—1668no—
138C2×Q8C4×D8312yes—
139C2×Q8D33212yes—
140C2×D4—832no—
141C2×D4—1662no—
142C2×D4—1662no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1664no—
146C22×C4—1662no—
147C22×C4C22⋊D8312yes—
148C22×C4C22×D8311yescharacteristic
149M16—1662no—
150M16—1662no—
151C2×C8—1662no—
152C4⋊C4—22no—
153C42—22no—
154C23—1662no—
155C23C2×C22⋊D8311yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—834no—
159Q8—1668no—
160Q8—1662no—
161D4—834no—
162D4—1668no—
163D4—1664no—
164D4—1664no—
165D4—1662no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—16616no—
174C2×C4—1662no—
175C2×C4—1664no—
176C2×C4—1664no—
177C2×C4(C2×D83)⋊C414yes—
178C2×C4C2×C4×D8311yescharacteristic
179C2×C4D4×D8312yes—
180C2×C4D4×D8312yes—
181C2×C4C2×D33211yescharacteristic
182C8—3322no—
183C22—21no—
184C22—21no—
185C22—3322no—
186C22—3322no—
187C22—1664no—
188C22C2×(C2×D83)⋊C411yescharacteristic
189C22#1458176011yescharacteristic
190C22#1458176011yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—1664no—
195C4C22⋊C4×D8312yes—
196C4#1458176212yes—
197C2—21no—
198C2—21no—
199C2—3324no—
200C2#1912755611yescharacteristic
201C2#1912787311yescharacteristic
202C2#1912788311yescharacteristic
subgroupG/H
centerC22#14581760
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#391×C83C2
socleC22×C83#120
prime radicalC2×#14581840C2
cosocleC22×C4×C83C23
hypercenter#391D83
hypocenterC83#1793
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#14581720C2semidirect
3. #19127700C2semidirect
4. C2×#14581711C2semidirect
5. #391×C83C2semidirect
8. #14581840C4semidirect
102. C83#1793semidirect
107. #391D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818402
C2×#145817202
#191277002
C2×#145817112
#391×C832
#179383