Finite Group Databasebeta

← #1621×C251 #44546208

Subgroups — #1621×C251 #44546208

Order 32128.

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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#550×C251C211yesmaximal, characteristic, complemented
2#493×C251C211yesmaximal, characteristic, complemented
3#460×C251C211yesmaximal, characteristic, complemented
4#459×C251C211yesmaximal, characteristic, complemented
5#331×C251C211yesmaximal, characteristic, complemented
6Q8⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#41×C251—21no—
9C2×SD16×C251—21no—
10C2×D8×C251—21no—
11C2×D8×C251—22no—
12#127×C251—21no—
13#127×C251C2211yescharacteristic
14#127×C251C2211yescharacteristic
15#121×C251—22no—
16#121×C251—21no—
17C4×Q8×C251C2211yescharacteristic
18C4×D4×C251—21no—
19C4×D4×C251C2211yescharacteristic
20C4⋊C8×C251C2211yescharacteristic
21C4⋊C8×C251C2211yescharacteristic
22#102×C251—21no—
23#102×C251—21no—
24C4×C8×C251C2211yescharacteristic
25#41×C251—42no—
26#41×C251—42no—
27C2×Q8×C251D411yescharacteristic
28C2×D4×C251—22no—
29C2×D4×C251—22no—
30C2×D4×C251—21no—
31C2×D4×C251—21no—
32C2×D4×C251—21no—
33C2×D4×C251—21no—
34C2×D4×C251—21no—
35C2×D4×C251D411yescharacteristic
36C2×D4×C251D412yes—
37C22×C4×C251—22no—
38C22×C4×C251—21no—
39SD16×C251—42no—
40D8×C251—44no—
41D8×C251—42no—
42C2×C8×C251—21no—
43C2×C8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251D411yescharacteristic
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—22no—
49C22⋊C4×C251—21no—
50C42×C251—21no—
51C42×C251C2311yescharacteristic
52C23×C251—21no—
53C23×C251—22no—
54C23×C251—41no—
55Q8×C251—21no—
56Q8×C251D812yescomplemented
57D4×C251—21no—
58D4×C251—21no—
59D4×C251—42no—
60D4×C251—44no—
61D4×C251—42no—
62D4×C251—41no—
63D4×C251—41no—
64D4×C251—41no—
65D4×C251—41no—
66D4×C251—41no—
67D4×C251—41no—
68D4×C251—41no—
69D4×C251—41no—
70D4×C251—82no—
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—21no—
74C2×C4×C251—44no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C8×C251—22no—
81C8×C251—41no—
82C8×C251—41no—
83C22×C251—42no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251—42no—
87C22×C251—42no—
88C22×C251—41no—
89C22×C251—81no—
90C22×C251—81no—
91C22×C251—81no—
92C22×C251#12011yescharacteristic
93C4×C251—22no—
94C4×C251—21no—
95C4×C251—41no—
96C4×C251—41no—
97C4×C251C2×D811yescharacteristic
98C4×C251#13511yescharacteristic
99C4×C251AGL(1,ℤ/8)11yescharacteristic
100C4×C251AGL(1,ℤ/8)11yescharacteristic
101C2×C251—81no—
102C2×C251—82no—
103C2×C251—161no—
104C2×C251#44711yescharacteristic
105C2×C251#44911yescharacteristic
106C2×C251#45311yescharacteristic
107C251#162111yescharacteristic, direct product factor, Sylow
108#1621C25111yesmaximal, characteristic, direct product factor, Sylow
109#550C2×C25111yescharacteristic, complemented
110#493C2×C25111yescharacteristic, complemented
111#460C2×C25111yescharacteristic, complemented
112#459C2×C25111yescharacteristic, complemented
113#331C2×C25111yescharacteristic, complemented
114Q8⋊C8C2×C25111yescharacteristic, complemented
115D4⋊C8C2×C25111yescharacteristic, complemented
116C2×#41—21no—
117C2×SD16—21no—
118C2×D8—21no—
119C2×D8—22no—
120#127—21no—
121#127C22×C25111yescharacteristic
122#127C22×C25111yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C22×C25111yescharacteristic
126C4×D4—21no—
127C4×D4C22×C25111yescharacteristic
128C4⋊C8C22×C25111yescharacteristic
129C4⋊C8C22×C25111yescharacteristic
130#102—21no—
131#102—21no—
132C4×C8C22×C25111yescharacteristic
133#41—42no—
134#41—42no—
135C2×Q8D4×C25111yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4D4×C25111yescharacteristic
144C2×D4D4×C25112yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—42no—
148D8—44no—
149D8—42no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C4⋊C4—21no—
154C4⋊C4D4×C25111yescharacteristic
155C4⋊C4D4×C25111yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C23×C25111yescharacteristic
160C23—21no—
161C23—22no—
162C23—41no—
163Q8—21no—
164Q8D8×C25112yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—42no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—41no—
178D4—82no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×D4×C25111yescharacteristic
186C2×C4C2×D4×C25111yescharacteristic
187C2×C4C2×D4×C25111yescharacteristic
188C8—22no—
189C8—41no—
190C8—41no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22#120×C25111yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4C2×D8×C25111yescharacteristic
206C4#135×C25111yescharacteristic
207C4AGL(1,ℤ/8)×C25111yescharacteristic
208C4AGL(1,ℤ/8)×C25111yescharacteristic
209C2—81no—
210C2—82no—
211C2—161no—
212C2#447×C25111yescharacteristic
213C2#449×C25111yescharacteristic
214C2#453×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #550×C251C2semidirect
2. #493×C251C2semidirect
3. #460×C251C2semidirect
4. #459×C251C2semidirect
5. #331×C251C2semidirect
6. Q8⋊C8×C251C2semidirect
7. D4⋊C8×C251C2semidirect
56. Q8×C251D8semidirect
107. C251#1621direct (N × H)
108. #1621C251direct (N × H)
109. #550C2×C251semidirect
110. #493C2×C251semidirect
111. #460C2×C251semidirect
112. #459C2×C251semidirect
113. #331C2×C251semidirect
114. Q8⋊C8C2×C251semidirect
115. D4⋊C8C2×C251semidirect
164. Q8D8×C251semidirect

Maximal subgroups

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

subgroupindex
#550×C2512
#493×C2512
#460×C2512
#459×C2512
#331×C2512
Q8⋊C8×C2512
D4⋊C8×C2512
#1621251