Finite Group Databasebeta

← #44549324

Subgroups — #44549324

Order 32128.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632313C211yesmaximal, characteristic
2#30631845C212yesmaximal, complemented
3#30631599C211yesmaximal, characteristic
4#30631425C212yesmaximal, complemented
5#323×C251C211yesmaximal, characteristic
6C22×Dic502C2211yescharacteristic
7C2×Dic1004—22no—
8#22548694—21no—
9#22548694C2212yes—
10#22548692—22no—
11#22548692—22no—
12C2×C22⋊Dic251—21no—
13C2×C22⋊Dic251—21no—
14#22548640—22no—
15C251⋊#95—21no—
16C251⋊#95C2211yescharacteristic
17C2×C22⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yes—
19C2×Dic502—21no—
20C2×Dic502—22no—
21C2×Dic502—42no—
22C2×Dic502D412yes—
23C22×Dic251—21no—
24C22×Dic251—21no—
25C22×Dic251—21no—
26C22×Dic251—21no—
27Dic1004—44no—
28C4⋊Dic251—21no—
29C4⋊Dic251D412yes—
30Dic251⋊C4—22no—
31Dic251⋊C4—22no—
32C22⋊Dic251—22no—
33C22⋊Dic251—22no—
34C22⋊Dic251—42no—
35C24×C251D411yescharacteristic
36C22×C4×C251D411yescharacteristic
37C22×C4×C251C2311yescharacteristic
38C2×C8×C251—22no—
39C22⋊C4×C251—22no—
40Dic502—24no—
41Dic502—42no—
42Dic502—42no—
43Dic502—81no—
44C2×Dic251—22no—
45C2×Dic251—22no—
46C2×Dic251—22no—
47C2×Dic251—41no—
48C2×Dic251—41no—
49C2×Dic251—41no—
50C2×Dic251—41no—
51C2×Dic251—41no—
52C2×Dic251—81no—
53C2×Dic251—81no—
54C23×C251—21no—
55C23×C251—21no—
56C23×C251—41no—
57C23×C251—41no—
58C23×C251Q1612yescomplemented
59C23×C251C2×D411yescharacteristic
60C2×C4×C251—21no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251C2×D412yes—
64C8×C251—42no—
65C22×C251—21no—
66C22×C251—21no—
67C22×C251—22no—
68C22×C251—42no—
69C22×C251—41no—
70C22×C251—41no—
71C22×C251—81no—
72C22×C251#12011yescharacteristic
73C22×C251C2×Q1611yescharacteristic
74C22×C251#13711yescharacteristic
75C4×C251—22no—
76C4×C251—41no—
77Dic251—42no—
78Dic251—42no—
79Dic251—42no—
80Dic251—81no—
81C2×C251—21no—
82C2×C251—21no—
83C2×C251—42no—
84C2×C251#45111yescharacteristic
85C2×C251#45511yescharacteristic
86C2×C251B(4,𝔽2)11yescharacteristic
87C251#160211yescharacteristic, complemented, Sylow
88#1602—2511nomaximal, Sylow
89#543—2511no—
90#451—2512no—
91#393—2511no—
92#328—2512no—
93#323D25111yescharacteristic
94C22×Q8—2511no—
95C2×Q16—5022no—
96C22⋊Q8—5022no—
97C22⋊Q8—5022no—
98C22⋊Q8—5021no—
99C22⋊Q8—2512no—
100C2×C22⋊C4—5021no—
101C2×C22⋊C4—5021no—
102C2×C22⋊C4C2×D25111yescharacteristic
103#103—5022no—
104C22⋊C8C2×D25112yes—
105#95—5021no—
106#95—2511no—
107C24D100411yescharacteristic
108C2×Q8—10042no—
109C2×Q8—5021no—
110C2×Q8—5022no—
111C2×Q8—2512no—
112C22×C4—5021no—
113C22×C4—5021no—
114C22×C4—5021no—
115C22×C4—5021no—
116C22×C4D100411yescharacteristic
117C22×C4C22×D25111yescharacteristic
118Q16—10044no—
119C2×C8—22no—
120C4⋊C4—5022no—
121C4⋊C4—5021no—
122C4⋊C4—5022no—
123C4⋊C4—2512no—
124C22⋊C4—22no—
125C22⋊C4—10042no—
126C22⋊C4—5022no—
127C22⋊C4—5022no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic100412yescomplemented
133C23C2×D100411yescharacteristic
134Q8—20081no—
135Q8—10042no—
136Q8—10042no—
137Q8—5024no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—20081no—
142C2×C4—20081no—
143C2×C4—10041no—
144C2×C4—10041no—
145C2×C4—10041no—
146C2×C4—10041no—
147C2×C4—10041no—
148C2×C4—5022no—
149C2×C4—5022no—
150C2×C4—5022no—
151C2×C4D4×D25112yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2254868511yescharacteristic
161C22C2×Dic100411yescharacteristic
162C22#2254874811yescharacteristic
163C4—22no—
164C4—41no—
165C4—20081no—
166C4—10042no—
167C4—10042no—
168C4—10042no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#3063184511yescharacteristic
173C2#3063186711yescharacteristic
174C2#3063187511yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251#120
prime radicalC24×C251D4
hypercenter#323D251
hypocenterC251#1602
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #30631845C2semidirect
4. #30631425C2semidirect
58. C23×C251Q16semidirect
87. C251#1602semidirect
132. C23Dic1004semidirect

Maximal subgroups

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

subgroupindex
#306323132
#306318452
#306315992
#306314252
#323×C2512
#1602251