Finite Group Databasebeta

← #1521×C251 #44546108

Subgroups — #1521×C251 #44546108

Order 32128.

218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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#529×C251C211yesmaximal, characteristic
2#431×C251C211yesmaximal, characteristic, complemented
3#383×C251C211yesmaximal, characteristic, complemented
4#328×C251C212yesmaximal, complemented
5#327×C251C212yesmaximal
6#126×C251—21no—
7#125×C251C411yescharacteristic, complemented
8#124×C251C411yescharacteristic, complemented
9#123×C251—21no—
10C22⋊Q8×C251C2211yescharacteristic
11#121×C251C2211yescharacteristic
12C4×Q8×C251C411yescharacteristic, complemented
13C4×D4×C251—21no—
14C4×D4×C251C411yescharacteristic, complemented
15#117×C251—21no—
16C2×C42×C251C2211yescharacteristic
17C4⋊C8×C251—21no—
18C22⋊C8×C251C2212yes—
19C4×C8×C251—21no—
20#95×C251—21no—
21#95×C251—21no—
22#95×C251C2212yes—
23C2×Q8×C251C2×C411yescharacteristic
24C2×D4×C251—21no—
25C2×D4×C251C2×C411yescharacteristic
26C22×C4×C251—21no—
27C22×C4×C251—22no—
28C22×C4×C251D411yescharacteristic
29C22×C4×C251D411yescharacteristic
30C22×C4×C251C2311yescharacteristic
31C2×C8×C251—22no—
32C4⋊C4×C251—21no—
33C4⋊C4×C251—21no—
34C4⋊C4×C251—21no—
35C4⋊C4×C251—21no—
36C4⋊C4×C251C2×C411yescharacteristic
37C4⋊C4×C251C2×C411yescharacteristic
38C22⋊C4×C251—21no—
39C22⋊C4×C251—21no—
40C22⋊C4×C251—21no—
41C22⋊C4×C251—21no—
42C22⋊C4×C251—21no—
43C42×C251—21no—
44C42×C251C2×C411yescharacteristic
45C42×C251C2×C411yescharacteristic
46C42×C251D411yescharacteristic
47C42×C251D411yescharacteristic
48C23×C251—21no—
49C23×C251C2×D411yescharacteristic
50Q8×C251—41no—
51D4×C251—41no—
52D4×C251—42no—
53C2×C4×C251—22no—
54C2×C4×C251—21no—
55C2×C4×C251—21no—
56C2×C4×C251—21no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251—42no—
60C2×C4×C251—42no—
61C2×C4×C251—42no—
62C2×C4×C251—42no—
63C2×C4×C251C22⋊C411yescharacteristic
64C2×C4×C251C22⋊C411yescharacteristic
65C2×C4×C251C22⋊C411yescharacteristic
66C2×C4×C251C22⋊C411yescharacteristic
67C2×C4×C251C22×C411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—42no—
70C22×C251—21no—
71C22×C251—21no—
72C22×C251—41no—
73C22×C251—41no—
74C22×C251—41no—
75C22×C251C2×C22⋊C411yescharacteristic
76C22×C251#13512yes—
77C4×C251—21no—
78C4×C251—21no—
79C4×C251—21no—
80C4×C251—21no—
81C4×C251—21no—
82C4×C251—21no—
83C4×C251—41no—
84C4×C251—41no—
85C4×C251—41no—
86C4×C251—82no—
87C2×C251—22no—
88C2×C251—81no—
89C2×C251#41011yescharacteristic
90C2×C251#41611yescharacteristic
91C2×C251#42111yescharacteristic
92C251#152111yescharacteristic, direct product factor, Sylow
93#1521C25111yesmaximal, characteristic, direct product factor, Sylow
94#529C2×C25111yescharacteristic
95#431C2×C25111yescharacteristic, complemented
96#383C2×C25111yescharacteristic, complemented
97#328C2×C25112yescomplemented
98#327C2×C25112yes—
99#126—21no—
100#125C4×C25111yescharacteristic, complemented
101#124C4×C25111yescharacteristic, complemented
102#123—21no—
103C22⋊Q8C22×C25111yescharacteristic
104#121C22×C25111yescharacteristic
105C4×Q8C4×C25111yescharacteristic, complemented
106C4×D4—21no—
107C4×D4C4×C25111yescharacteristic, complemented
108#117—21no—
109C2×C42C22×C25111yescharacteristic
110C4⋊C8—21no—
111C22⋊C8C22×C25112yes—
112C4×C8—21no—
113#95—21no—
114#95—21no—
115#95C22×C25112yes—
116C2×Q8C2×C4×C25111yescharacteristic
117C2×D4—21no—
118C2×D4C2×C4×C25111yescharacteristic
119C22×C4—21no—
120C22×C4—22no—
121C22×C4D4×C25111yescharacteristic
122C22×C4D4×C25111yescharacteristic
123C22×C4C23×C25111yescharacteristic
124C2×C8—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4C2×C4×C25111yescharacteristic
130C4⋊C4C2×C4×C25111yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C42—21no—
137C42C2×C4×C25111yescharacteristic
138C42C2×C4×C25111yescharacteristic
139C42D4×C25111yescharacteristic
140C42D4×C25111yescharacteristic
141C23—21no—
142C23C2×D4×C25111yescharacteristic
143Q8—41no—
144D4—41no—
145D4—42no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4C22⋊C4×C25111yescharacteristic
157C2×C4C22⋊C4×C25111yescharacteristic
158C2×C4C22⋊C4×C25111yescharacteristic
159C2×C4C22⋊C4×C25111yescharacteristic
160C2×C4C22×C4×C25111yescharacteristic
161C2×C4C2×D4×C25111yescharacteristic
162C8—42no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22C2×C22⋊C4×C25111yescharacteristic
169C22#135×C25112yes—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—82no—
180C2—22no—
181C2—81no—
182C2#410×C25111yescharacteristic
183C2#416×C25111yescharacteristic
184C2#421×C25111yescharacteristic
subgroupG/H
center, socleC22×C251C2×C22⋊C4
commutatorC2×C4C22×C4×C251
Frattini, cosocleC22×C4C23×C251
prime radical#121×C251C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C2×C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #431×C251C2semidirect
3. #383×C251C2semidirect
4. #328×C251C2semidirect
7. #125×C251C4semidirect
8. #124×C251C4semidirect
12. C4×Q8×C251C4semidirect
14. C4×D4×C251C4semidirect
92. C251#1521direct (N × H)
93. #1521C251direct (N × H)
95. #431C2×C251semidirect
96. #383C2×C251semidirect
97. #328C2×C251semidirect
100. #125C4×C251semidirect
101. #124C4×C251semidirect
105. C4×Q8C4×C251semidirect
107. C4×D4C4×C251semidirect

Maximal subgroups

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

subgroupindex
#529×C2512
#431×C2512
#383×C2512
#328×C2512
#327×C2512
#1521251