Finite Group Databasebeta

← #24907679

Subgroups — #24907679

Order 10624.

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

normal subgroups only

This group has 272 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
1#19128178C211yesmaximal, characteristic
2#19127744C211yesmaximal, characteristic
3#19127743C211yesmaximal, characteristic
4C2×#14581710C211yesmaximal, characteristic
5#19127607C211yesmaximal, characteristic
6#19127461C211yesmaximal, characteristic
7#339×C83C211yesmaximal, characteristic
8C22×Dic166C2211yescharacteristic
9C2×#13050574C2211yescharacteristic
10C4×Dic166—21no—
11C4×Dic166—21no—
12#14581757—21no—
13#14581757—21no—
14#14581751C2211yescharacteristic
15#14581748—22no—
16C2×C4⋊Dic83C2211yescharacteristic
17#14581726—21no—
18#14581725—21no—
19#14581724—21no—
20#14581723—21no—
21#14581710C414yescomplemented
22C83⋊#95—22no—
23C2×M16×C83C2211yescharacteristic
24#117×C83C2211yescharacteristic
25C2×C42×C83C2211yescharacteristic
26C2×Dic166—21no—
27C2×Dic166—21no—
28C2×Dic166—21no—
29C2×Dic166—21no—
30C2×Dic166C2×C412yes—
31C2×Dic166Q812yes—
32C22×Dic83—21no—
33C22×Dic83D412yes—
34#13050574—21no—
35#13050574C2×C412yes—
36C2×C83⋊C8—21no—
37C4⋊Dic83—21no—
38C4⋊Dic83—21no—
39C4⋊Dic83D411yescharacteristic
40C4⋊Dic83D411yescharacteristic
41Dic83⋊C4—22no—
42Dic83⋊C4—22no—
43C22⋊Dic83—21no—
44C4×Dic83—22no—
45C4×Dic83—22no—
46C4×Dic83—21no—
47C22×C4×C83—21no—
48C22×C4×C83C2311yescharacteristic
49M16×C83—21no—
50M16×C83C2×C412yes—
51C2×C8×C83—21no—
52C4⋊C4×C83D411yescharacteristic
53C4⋊C4×C83D411yescharacteristic
54C22⋊C4×C83—21no—
55C42×C83—21no—
56C42×C83—21no—
57C42×C83—21no—
58C23×C83C2×D411yescharacteristic
59C2×C4×C83—22no—
60C2×C4×C83—21no—
61C2×C4×C83—21no—
62C2×C4×C83—41no—
63C2×C4×C83—41no—
64C2×C4×C83C22×C411yescharacteristic
65C2×C4×C83C2×D411yescharacteristic
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83C2×Q811yescharacteristic
68C2×C4×C83#4111yescharacteristic
69C2×C4×C83#4111yescharacteristic
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—21no—
73C2×Dic83—21no—
74C2×Dic83—42no—
75C2×Dic83—41no—
76C2×Dic83C4⋊C414yes—
77Dic166—22no—
78Dic166—22no—
79Dic166—22no—
80Dic166—22no—
81C8×C83—22no—
82C83⋊C8—22no—
83C22×C83—21no—
84C22×C83—21no—
85C22×C83C2×C4⋊C411yescharacteristic
86C22×C83#12011yescharacteristic
87C22×C83#12311yescharacteristic
88C4×C83—42no—
89C4×C83—41no—
90C4×C83—41no—
91C4×C83C4×D411yescharacteristic
92C4×C83C4×D411yescharacteristic
93C4×C83C22⋊Q811yescharacteristic
94C4×C83C22⋊Q811yescharacteristic
95Dic83—24no—
96Dic83—41no—
97Dic83—41no—
98Dic83—81no—
99C2×C83—21no—
100C2×C83—21no—
101C2×C83#38511yescharacteristic
102C2×C83#45411yescharacteristic
103C2×C83#45611yescharacteristic
104#1857—831nomaximal, Sylow
105C83#185711yescharacteristic, complemented, Sylow
106#519—831no—
107#428—831no—
108#428—831no—
109C2×#101—831no—
110#391—831no—
111#339—831no—
112#339D8311yescharacteristic
113C22×Q8—831no—
114C2×M16—831no—
115C2×M16C2×D8311yescharacteristic
116C4×Q8—1661no—
117C4×Q8—1661no—
118C4×Q8—1661no—
119C4×Q8—1661no—
120#117—831no—
121#117—1662no—
122#117C2×D8311yescharacteristic
123C2×C4⋊C4—831no—
124C2×C42C2×D8311yescharacteristic
125#107—1661no—
126#107—1661no—
127#106—1661no—
128#106—1661no—
129#101—834no—
130#95—1662no—
131C2×Q8—832no—
132C2×Q8—832no—
133C2×Q8—1661no—
134C2×Q8—1661no—
135C2×Q8—1661no—
136C2×Q8—1661no—
137C22×C4—21no—
138C22×C4—832no—
139C22×C4—1661no—
140C22×C4C22×D8311yescharacteristic
141M16—21no—
142M16—832no—
143M16—1661no—
144M16C4×D8312yes—
145C2×C8—21no—
146C2×C8—1661no—
147C4⋊C4—831no—
148C4⋊C4—831no—
149C4⋊C4—1662no—
150C4⋊C4—1662no—
151C4⋊C4—1661no—
152C4⋊C4—1661no—
153C4⋊C4D33211yescharacteristic
154C4⋊C4D33211yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1661no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—1662no—
161C42—1662no—
162C42—1661no—
163C23D4×D8311yescharacteristic
164Q8—1662no—
165Q8—1662no—
166Q8—1662no—
167Q8—1662no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—3322no—
174C2×C4—3321no—
175C2×C4—834no—
176C2×C4—1661no—
177C2×C4—1661no—
178C2×C4—1662no—
179C2×C4—1662no—
180C2×C4C2×C4×D8311yescharacteristic
181C2×C4D4×D8311yescharacteristic
182C2×C4C2×D33211yescharacteristic
183C2×C4Q8×D8311yescharacteristic
184C2×C4#1305058911yescharacteristic
185C2×C4#1305059011yescharacteristic
186C8—22no—
187C8—1662no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D8311yescharacteristic
191C22#1458176211yescharacteristic
192C22#1458177411yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—3321no—
197C4—3321no—
198C4—6641no—
199C4—1664no—
200C4(C22⋊D83)⋊C411yescharacteristic
201C4C4×D33211yescharacteristic
202C4#1458177011yescharacteristic
203C4#1458177211yescharacteristic
204C2—21no—
205C2—21no—
206C2#1912757511yescharacteristic
207C2#1912787511yescharacteristic
208C2#1912788511yescharacteristic
subgroupG/H
centerC22#14581762
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#339×C83C2
socleC22×C83#120
prime radicalC23×C83C2×D4
cosocleC22×C4×C83C23
hypercenter#339D83
hypocenterC83#1857
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
21. #14581710C4semidirect
105. C83#1857semidirect

Maximal subgroups

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

subgroupindex
#191281782
#191277442
#191277432
C2×#145817102
#191276072
#191274612
#339×C832
#185783