Finite Group Databasebeta

← #44548694

Subgroups — #44548694

Order 32128.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×Dic251⋊C4C211yesmaximal, characteristic
2#30631651C211yesmaximal, characteristic
3#30631412C214yesmaximal, complemented
4#407×C251C211yesmaximal, characteristic
5C23×Dic251C411yescharacteristic
6C23×Dic251C411yescharacteristic
7C2×#16860794—21no—
8C2×Dic251⋊C4—21no—
9C2×Dic251⋊C4—21no—
10C2×Dic251⋊C4—21no—
11C2×Dic251⋊C4—21no—
12C2×Dic251⋊C4C412yes—
13C2×Dic251⋊C4C2212yes—
14C22⋊(C251⋊C8)—21no—
15C22⋊(C251⋊C8)C2212yes—
16C23×C4×C251C2211yescharacteristic
17C2×M16×C251—21no—
18C22⋊C8×C251—21no—
19C22⋊C8×C251C2212yes—
20C22×Dic251—24no—
21C22×Dic251—21no—
22C22×Dic251—21no—
23C22×Dic251—21no—
24C22×Dic251—21no—
25C22×Dic251—42no—
26C22×Dic251C2×C412yes—
27C22×Dic251C2×C412yes—
28#16860794—42no—
29C2×C251⋊C8—22no—
30Dic251⋊C4—22no—
31Dic251⋊C4—22no—
32Dic251⋊C4—22no—
33Dic251⋊C4—22no—
34C24×C251C2×C411yescharacteristic
35C22×C4×C251—42no—
36C22×C4×C251C2×C411yescharacteristic
37C22×C4×C251D411yescharacteristic
38C22×C4×C251D411yescharacteristic
39C22×C4×C251D411yescharacteristic
40C22×C4×C251D411yescharacteristic
41C22×C4×C251C2311yescharacteristic
42M16×C251—42no—
43C2×C8×C251—22no—
44C2×Dic251—24no—
45C2×Dic251—24no—
46C2×Dic251—22no—
47C2×Dic251—22no—
48C2×Dic251—22no—
49C2×Dic251—42no—
50C2×Dic251—42no—
51C2×Dic251—42no—
52C2×Dic251M1614yescomplemented
53C251⋊C8—42no—
54C23×C251—22no—
55C23×C251—22no—
56C23×C251—22no—
57C23×C251C22⋊C411yescharacteristic
58C23×C251C22⋊C411yescharacteristic
59C23×C251C22×C411yescharacteristic
60C2×C4×C251—22no—
61C2×C4×C251—22no—
62C2×C4×C251—41no—
63C2×C4×C251—41no—
64C2×C4×C251—41no—
65C2×C4×C251—41no—
66C2×C4×C251C22⋊C411yescharacteristic
67C2×C4×C251C22⋊C411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C8×C251—42no—
71C22×C251—28no—
72C22×C251—21no—
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—21no—
76C22×C251—21no—
77C22×C251—21no—
78C22×C251#9912yes—
79C22×C251#10112yes—
80C22×C251C2×C22⋊C411yescharacteristic
81C22×C251C2×M1612yes—
82C4×C251—21no—
83C4×C251—21no—
84C4×C251—21no—
85C4×C251—21no—
86Dic251—24no—
87Dic251—42no—
88C2×C251—22no—
89C2×C251—22no—
90C2×C251—22no—
91C2×C251#40711yescharacteristic
92C2×C251C2×#9911yescharacteristic
93C2×C251C2×#10111yescharacteristic
94C251#146311yescharacteristic, complemented, Sylow
95#1463—2511nomaximal, Sylow
96C22×C4⋊C4—2511no—
97#407—2511no—
98#407D25111yescharacteristic
99#324—2514no—
100C23×C4—2511no—
101C23×C4—2511no—
102C23×C4C2×D25111yescharacteristic
103C2×M16—21no—
104C2×M16—5021no—
105C2×C4⋊C4—5021no—
106C2×C4⋊C4—5021no—
107C2×C4⋊C4—5021no—
108C2×C4⋊C4—5021no—
109C2×C4⋊C4—2512no—
110C2×C4⋊C4—2512no—
111C22⋊C8—21no—
112C22⋊C8—5021no—
113C22⋊C8—2512no—
114C22⋊C8C2×D25112yes—
115C24C4×D25111yescharacteristic
116C22×C4—42no—
117C22×C4—10042no—
118C22×C4—5021no—
119C22×C4—5021no—
120C22×C4—5021no—
121C22×C4—5021no—
122C22×C4—5024no—
123C22×C4—2512no—
124C22×C4—2512no—
125C22×C4C4×D25111yescharacteristic
126C22×C4C22⋊D25111yescharacteristic
127C22×C4C22⋊D25111yescharacteristic
128C22×C4D100411yescharacteristic
129C22×C4D100411yescharacteristic
130C22×C4C22×D25111yescharacteristic
131M16—42no—
132M16—10042no—
133C2×C8—22no—
134C2×C8—5022no—
135C4⋊C4—5022no—
136C4⋊C4—5022no—
137C4⋊C4—5022no—
138C4⋊C4—5022no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23(C2×D251)⋊C411yescharacteristic
143C23(C2×D251)⋊C411yescharacteristic
144C23C2×C4×D25111yescharacteristic
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—10042no—
152C2×C4—10042no—
153C2×C4—10042no—
154C2×C4—5022no—
155C2×C4—5024no—
156C2×C4—5024no—
157C2×C4—5022no—
158C2×C4—5022no—
159C2×C4—2514no—
160C2×C4(C2×D251)⋊C411yescharacteristic
161C2×C4(C2×D251)⋊C411yescharacteristic
162C2×C4C2×C22⋊D25111yescharacteristic
163C2×C4C2×D100411yescharacteristic
164C8—42no—
165C8—10042no—
166C22—28no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22#2254862912yes—
174C22#2254863312yes—
175C22C2×(C2×D251)⋊C411yescharacteristic
176C22M16×D25112yes—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—10042no—
182C4—5024no—
183C2—22no—
184C2—22no—
185C2—22no—
186C2#3063165011yescharacteristic
187C2C2×#2254862911yescharacteristic
188C2C2×#2254863311yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC24×C251C2×C4
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1463
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #30631412C2semidirect
52. C2×Dic251M16semidirect
94. C251#1463semidirect

Maximal subgroups

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

subgroupindex
C22×Dic251⋊C42
#306316512
#306314122
#407×C2512
#1463251