Finite Group Databasebeta

← #24908807

Subgroups — #24908807

Order 10624.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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#19128329C211yesmaximal, characteristic
2#19127744C211yesmaximal, characteristic
3#19127743C211yesmaximal, characteristic
4#19127571C211yesmaximal, characteristic
5#19127469C211yesmaximal, characteristic
6#19127461C211yesmaximal, characteristic
7#339×C83C211yesmaximal, characteristic
8C2×#13050574C2211yescharacteristic
9#14581792—21no—
10#14581791—21no—
11#14581784—21no—
12#14581783—21no—
13#14581769—22no—
14#14581751C2211yescharacteristic
15C2×C4⋊Dic83—22no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2211yescharacteristic
18#14581726—21no—
19#14581725—21no—
20#14581724—21no—
21#14581723—21no—
22C2×M16×C83C2211yescharacteristic
23#117×C83C2211yescharacteristic
24C2×C42×C83C2211yescharacteristic
25C2×Dic166—22no—
26C22×Dic83—21no—
27C22×Dic83D412yes—
28#13050574—21no—
29#13050574Q812yescomplemented
30C2×C83⋊C8—21no—
31C4⋊Dic83—22no—
32C4⋊Dic83—21no—
33C4⋊Dic83—21no—
34C4⋊Dic83—21no—
35C4⋊Dic83—21no—
36C4⋊Dic83—44no—
37C4⋊Dic83D411yescharacteristic
38C4⋊Dic83D411yescharacteristic
39Dic83⋊C4—22no—
40Dic83⋊C4—22no—
41C22⋊Dic83—21no—
42C4×Dic83—21no—
43C22×C4×C83—21no—
44C22×C4×C83C2311yescharacteristic
45M16×C83—21no—
46M16×C83Q812yescomplemented
47C2×C8×C83—21no—
48C4⋊C4×C83D411yescharacteristic
49C4⋊C4×C83D411yescharacteristic
50C22⋊C4×C83—21no—
51C42×C83—21no—
52C42×C83—21no—
53C42×C83Q812yescomplemented
54C23×C83C2×D411yescharacteristic
55C2×C4×C83—22no—
56C2×C4×C83—22no—
57C2×C4×C83—22no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83C2×D411yescharacteristic
61C2×C4×C83C2×D411yescharacteristic
62C2×C4×C83C2×Q811yescharacteristic
63C2×C4×C83C2×Q811yescharacteristic
64C2×C4×C83C2×Q811yescharacteristic
65C2×C4×C83#4111yescharacteristic
66C2×Dic83—21no—
67C2×Dic83—21no—
68C2×Dic83—22no—
69C2×Dic83—42no—
70C2×Dic83—42no—
71C2×Dic83—41no—
72Dic166—82no—
73C8×C83—22no—
74C83⋊C8—22no—
75C22×C83—21no—
76C22×C83—21no—
77C22×C83#12011yescharacteristic
78C22×C83C22⋊Q811yescharacteristic
79C22×C83C4⋊Q811yescharacteristic
80C4×C83—24no—
81C4×C83—41no—
82C4×C83—41no—
83C4×C83C22⋊Q811yescharacteristic
84C4×C83C22⋊Q811yescharacteristic
85C4×C83C4⋊Q811yescharacteristic
86C4×C83C4⋊Q811yescharacteristic
87Dic83—41no—
88Dic83—42no—
89Dic83—41no—
90Dic83—81no—
91C2×C83—21no—
92C2×C83—21no—
93C2×C83#39511yescharacteristic
94C2×C83#45311yescharacteristic
95C2×C83#45611yescharacteristic
96#2060—831nomaximal, Sylow
97C83#206011yescharacteristic, complemented, Sylow
98#544—831no—
99#428—831no—
100#428—831no—
101#384—831no—
102#343—831no—
103#339—831no—
104#339D8311yescharacteristic
105C2×M16—831no—
106C2×M16C2×D8311yescharacteristic
107C4⋊Q8—1661no—
108C4⋊Q8—1661no—
109#125—1661no—
110#125—1661no—
111C22⋊Q8—1662no—
112#117—831no—
113#117C2×D8311yescharacteristic
114C2×C4⋊C4—831no—
115C2×C4⋊C4—831no—
116C2×C4⋊C4—1662no—
117C2×C42C2×D8311yescharacteristic
118#107—1661no—
119#107—1661no—
120#106—1661no—
121#106—1661no—
122C2×Q8—1662no—
123C22×C4—21no—
124C22×C4—832no—
125C22×C4—1661no—
126C22×C4C22×D8311yescharacteristic
127M16—21no—
128M16—832no—
129M16—1661no—
130M16Dic16612yescomplemented
131C2×C8—21no—
132C2×C8—1661no—
133C4⋊C4—3324no—
134C4⋊C4—831no—
135C4⋊C4—831no—
136C4⋊C4—1662no—
137C4⋊C4—1662no—
138C4⋊C4—1662no—
139C4⋊C4—1661no—
140C4⋊C4—1661no—
141C4⋊C4—1661no—
142C4⋊C4—1661no—
143C4⋊C4D33211yescharacteristic
144C4⋊C4D33211yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—1661no—
147C42—21no—
148C42—21no—
149C42—1661no—
150C42Dic16612yescomplemented
151C23D4×D8311yescharacteristic
152Q8—6642no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—3322no—
159C2×C4—3322no—
160C2×C4—3321no—
161C2×C4—1661no—
162C2×C4—1661no—
163C2×C4—1662no—
164C2×C4D4×D8311yescharacteristic
165C2×C4C2×D33211yescharacteristic
166C2×C4Q8×D8311yescharacteristic
167C2×C4C2×Dic16611yescharacteristic
168C2×C4C2×Dic16611yescharacteristic
169C2×C4#1305058911yescharacteristic
170C8—22no—
171C8—1662no—
172C22—21no—
173C22—21no—
174C22#1458176211yescharacteristic
175C22#1458176911yescharacteristic
176C22#1458179111yescharacteristic
177C4—24no—
178C4—41no—
179C4—41no—
180C4—3322no—
181C4—3321no—
182C4—3321no—
183C4—6641no—
184C4#1458176911yescharacteristic
185C4#1458177211yescharacteristic
186C4#1458179111yescharacteristic
187C4#1458179211yescharacteristic
188C2—21no—
189C2—21no—
190C2#1912762211yescharacteristic
191C2#1912787011yescharacteristic
192C2#1912788511yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#339×C83C2
socleC22×C83#120
prime radicalC23×C83C2×D4
hypercenter#339D83
hypocenterC83#2060
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
29. #13050574Q8semidirect
46. M16×C83Q8semidirect
53. C42×C83Q8semidirect
97. C83#2060semidirect
130. M16Dic166semidirect
150. C42Dic166semidirect

Maximal subgroups

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

subgroupindex
#191283292
#191277442
#191277432
#191275712
#191274692
#191274612
#339×C832
#206083