Finite Group Databasebeta

← #24906403

Subgroups — #24906403

Order 10624.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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#19127851C212yesmaximal
2#19127618C211yesmaximal, characteristic
3#19127438C212yesmaximal, complemented
4#19127427C211yesmaximal, characteristic, complemented
5#544×C83C211yesmaximal, characteristic, complemented
6C22×D332C2211yescharacteristic
7C2×#13050577—22no—
8C2×(C2×D83)⋊C4—21no—
9C2×(C2×D83)⋊C4—21no—
10#14581712—22no—
11C22⋊(C83⋊C8)C2212yes—
12C83⋊#95C2211yescharacteristic
13C4⋊Q8×C83—22no—
14#125×C83—22no—
15C22⋊Q8×C83—21no—
16C22⋊Q8×C83C2212yes—
17#117×C83—21no—
18C2×C4⋊C4×C83C2211yescharacteristic
19C23×D83—21no—
20C2×D332—22no—
21C2×D332—42no—
22C22×Dic83—21no—
23#13050577—44no—
24C2×C83⋊C8—22no—
25(C2×D83)⋊C4—42no—
26(C2×D83)⋊C4—42no—
27C2×Q8×C83—21no—
28C2×Q8×C83D412yes—
29C22×C4×C83D411yescharacteristic
30C22×C4×C83D411yescharacteristic
31C22×C4×C83C2311yescharacteristic
32C4⋊C4×C83—21no—
33C4⋊C4×C83—21no—
34C4⋊C4×C83—21no—
35C4⋊C4×C83—22no—
36C4⋊C4×C83—22no—
37C4⋊C4×C83—22no—
38C4⋊C4×C83D412yes—
39C22⋊C4×C83—22no—
40C42×C83—22no—
41C23×C83C2×D411yescharacteristic
42C22×D83—22no—
43C22×D83—41no—
44C22×D83—41no—
45C22×D83—41no—
46C22×D83—41no—
47C22×D83—41no—
48C22×D83—41no—
49Q8×C83—42no—
50Q8×C83—81no—
51C2×C4×C83—21no—
52C2×C4×C83—21no—
53C2×C4×C83—22no—
54C2×C4×C83—22no—
55C2×C4×C83—21no—
56C2×C4×C83—41no—
57C2×C4×C83—41no—
58C2×C4×C83SD1612yescomplemented
59C2×C4×C83C2×D412yes—
60C2×Dic83—41no—
61C2×Dic83—41no—
62C2×Dic83—41no—
63D332—24no—
64D332—42no—
65C83⋊C8—42no—
66C22×C83—21no—
67C22×C83—21no—
68C22×C83#12011yescharacteristic
69C22×C83C2×SD1611yescharacteristic
70C22×C83#13711yescharacteristic
71C2×D83—42no—
72C2×D83—42no—
73C2×D83—42no—
74C2×D83—81no—
75C2×D83—81no—
76C2×D83—81no—
77C2×D83—81no—
78C4×C83—22no—
79C4×C83—22no—
80C4×C83—42no—
81C4×C83—41no—
82C4×C83—42no—
83Dic83—81no—
84C2×C83—21no—
85C2×C83—21no—
86C2×C83#44811yescharacteristic
87C2×C83#45411yescharacteristic
88C2×C83B(4,𝔽2)11yescharacteristic
89D83—82no—
90#1599—831nomaximal, Sylow
91C83#159911yescharacteristic, complemented, Sylow
92#544D8311yescharacteristic, complemented
93#450—832no—
94#394—831no—
95#328—832no—
96#324—831no—
97C22×D4—831no—
98C2×SD16—1662no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D8312yes—
103#117—21no—
104C2×C4⋊C4C2×D8311yescharacteristic
105C2×C22⋊C4—1661no—
106C2×C22⋊C4—1661no—
107#102—1662no—
108C22⋊C8—832no—
109#95—831no—
110C24—1661no—
111C2×Q8—21no—
112C2×Q8C22⋊D8312yes—
113C2×D4—3322no—
114C2×D4—1662no—
115C22×C4—1661no—
116C22×C4C22⋊D8311yescharacteristic
117C22×C4C22⋊D8311yescharacteristic
118C22×C4C22×D8311yescharacteristic
119SD16—3324no—
120C2×C8—1662no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D8312yes—
128C22⋊C4—22no—
129C22⋊C4—3322no—
130C22⋊C4—3322no—
131C42—22no—
132C23—3321no—
133C23—3321no—
134C23—3321no—
135C23—3321no—
136C23—3321no—
137C23—3321no—
138C23—1662no—
139C23C2×C22⋊D8311yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—3322no—
143D4—1664no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—3321no—
152C2×C4—3321no—
153C2×C4—3321no—
154C2×C4#1305057712yescomplemented
155C2×C4C2×C22⋊D8312yes—
156C8—3322no—
157C22—21no—
158C22—21no—
159C22—3322no—
160C22—3322no—
161C22—3322no—
162C22—6641no—
163C22—6641no—
164C22—6641no—
165C22—6641no—
166C22#1458176111yescharacteristic
167C22C2×#1305057711yescharacteristic
168C22#1458182311yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—6641no—
175C2—21no—
176C2—21no—
177C2—6642no—
178C2#1912784011yescharacteristic
179C2#1912787411yescharacteristic
180C2#1912788811yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#544×C83C2
socleC22×C83#120
prime radicalC22×D332C22
hypercenter#544D83
hypocenterC83#1599
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #19127438C2semidirect
4. #19127427C2semidirect
5. #544×C83C2semidirect
58. C2×C4×C83SD16semidirect
91. C83#1599semidirect
92. #544D83semidirect
154. C2×C4#13050577semidirect

Maximal subgroups

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

subgroupindex
#191278512
#191276182
#191274382
#191274272
#544×C832
#159983