Finite Group Databasebeta

← #1646×C251 #44546233

Subgroups — #1646×C251 #44546233

Order 32128.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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#555×C251C211yesmaximal, characteristic
2#487×C251C211yesmaximal, characteristic, complemented
3#474×C251C211yesmaximal, characteristic, complemented
4#461×C251C211yesmaximal, characteristic, complemented
5#333×C251C211yesmaximal, characteristic, complemented
6D4⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×SD16×C251—21no—
9C4⋊Q8×C251C2211yescharacteristic
10C4⋊Q8×C251C2211yescharacteristic
11#124×C251—22no—
12#121×C251—21no—
13#121×C251—21no—
14C4×D4×C251C2211yescharacteristic
15C4×D4×C251C2211yescharacteristic
16#117×C251—21no—
17#117×C251—21no—
18#107×C251—21no—
19C4⋊C8×C251C2211yescharacteristic
20C4⋊C8×C251C2211yescharacteristic
21#103×C251—22no—
22#103×C251—21no—
23#102×C251—21no—
24C4×C8×C251C2211yescharacteristic
25C2×Q8×C251—21no—
26C2×Q8×C251D412yes—
27C2×D4×C251—22no—
28C2×D4×C251D411yescharacteristic
29C2×D4×C251D411yescharacteristic
30C22×C4×C251—21no—
31C22×C4×C251—21no—
32SD16×C251—42no—
33C2×C8×C251—21no—
34C2×C8×C251—21no—
35C2×C8×C251—21no—
36C4⋊C4×C251—21no—
37C4⋊C4×C251—21no—
38C4⋊C4×C251—21no—
39C4⋊C4×C251—21no—
40C4⋊C4×C251D411yescharacteristic
41C4⋊C4×C251D411yescharacteristic
42C22⋊C4×C251—22no—
43C22⋊C4×C251—22no—
44C22⋊C4×C251—21no—
45C22⋊C4×C251—21no—
46C42×C251—22no—
47C42×C251C2311yescharacteristic
48C23×C251—21no—
49C23×C251—21no—
50Q8×C251—42no—
51Q8×C251—41no—
52Q8×C251—41no—
53D4×C251—21no—
54D4×C251—21no—
55D4×C251—21no—
56D4×C251—21no—
57D4×C251—82no—
58C2×C4×C251—21no—
59C2×C4×C251—21no—
60C2×C4×C251—22no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251—41no—
64C2×C4×C251—41no—
65C2×C4×C251—41no—
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—41no—
70C8×C251—41no—
71C8×C251—41no—
72C22×C251—41no—
73C22×C251—41no—
74C22×C251—41no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251—41no—
78C22×C251#12011yescharacteristic
79C4×C251—21no—
80C4×C251—41no—
81C4×C251—41no—
82C4×C251—42no—
83C4×C251—81no—
84C4×C251#13511yescharacteristic
85C4×C251#13511yescharacteristic
86C4×C251AGL(1,ℤ/8)11yescharacteristic
87C4×C251#13711yescharacteristic
88C2×C251—81no—
89C2×C251—81no—
90C2×C251#44911yescharacteristic
91C2×C251#45211yescharacteristic
92C2×C251#45511yescharacteristic
93C251#164611yescharacteristic, direct product factor, Sylow
94#1646C25111yesmaximal, characteristic, direct product factor, Sylow
95#555C2×C25111yescharacteristic
96#487C2×C25111yescharacteristic, complemented
97#474C2×C25111yescharacteristic, complemented
98#461C2×C25111yescharacteristic, complemented
99#333C2×C25111yescharacteristic, complemented
100D4⋊C8C2×C25111yescharacteristic, complemented
101D4⋊C8C2×C25111yescharacteristic, complemented
102C2×SD16—21no—
103C4⋊Q8C22×C25111yescharacteristic
104C4⋊Q8C22×C25111yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C22×C25111yescharacteristic
109C4×D4C22×C25111yescharacteristic
110#117—21no—
111#117—21no—
112#107—21no—
113C4⋊C8C22×C25111yescharacteristic
114C4⋊C8C22×C25111yescharacteristic
115#103—22no—
116#103—21no—
117#102—21no—
118C4×C8C22×C25111yescharacteristic
119C2×Q8—21no—
120C2×Q8D4×C25112yes—
121C2×D4—22no—
122C2×D4D4×C25111yescharacteristic
123C2×D4D4×C25111yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—42no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4D4×C25111yescharacteristic
135C4⋊C4D4×C25111yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C23×C25111yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4C2×D4×C25111yescharacteristic
161C2×C4C2×D4×C25111yescharacteristic
162C2×C4C2×D4×C25111yescharacteristic
163C8—41no—
164C8—41no—
165C8—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#120×C25111yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—81no—
178C4#135×C25111yescharacteristic
179C4#135×C25111yescharacteristic
180C4AGL(1,ℤ/8)×C25111yescharacteristic
181C4#137×C25111yescharacteristic
182C2—81no—
183C2—81no—
184C2#449×C25111yescharacteristic
185C2#452×C25111yescharacteristic
186C2#455×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radical#555×C251C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #487×C251C2semidirect
3. #474×C251C2semidirect
4. #461×C251C2semidirect
5. #333×C251C2semidirect
6. D4⋊C8×C251C2semidirect
7. D4⋊C8×C251C2semidirect
93. C251#1646direct (N × H)
94. #1646C251direct (N × H)
96. #487C2×C251semidirect
97. #474C2×C251semidirect
98. #461C2×C251semidirect
99. #333C2×C251semidirect
100. D4⋊C8C2×C251semidirect
101. D4⋊C8C2×C251semidirect

Maximal subgroups

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

subgroupindex
#555×C2512
#487×C2512
#474×C2512
#461×C2512
#333×C2512
D4⋊C8×C2512
#1646251