Finite Group Databasebeta

← #1610×C251 #44546197

Subgroups — #1610×C251 #44546197

Order 32128.

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

normal subgroups only

This group has 254 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#537×C251C211yesmaximal, characteristic
2#480×C251C212yesmaximal, complemented
3#398×C251C211yesmaximal, characteristic, complemented
4#327×C251C212yesmaximal, complemented
5#324×C251C211yesmaximal, characteristic, complemented
6C2×#41×C251—21no—
7C22⋊Q8×C251—22no—
8#121×C251—22no—
9#121×C251—21no—
10#121×C251C2212yes—
11C2×C4⋊C4×C251—21no—
12C2×C4⋊C4×C251C2211yescharacteristic
13C2×C4⋊C4×C251C2211yescharacteristic
14#107×C251—22no—
15#102×C251—22no—
16C22⋊C8×C251C2212yes—
17#95×C251—21no—
18#95×C251C2211yescharacteristic
19#41×C251—42no—
20#41×C251—42no—
21C2×Q8×C251—21no—
22C2×D4×C251—21no—
23C2×D4×C251—22no—
24C2×D4×C251—21no—
25C2×D4×C251D412yes—
26C22×C4×C251—22no—
27C22×C4×C251—21no—
28C22×C4×C251—21no—
29C22×C4×C251D411yescharacteristic
30C22×C4×C251D411yescharacteristic
31C22×C4×C251C2311yescharacteristic
32C2×C8×C251—22no—
33C4⋊C4×C251—21no—
34C4⋊C4×C251—22no—
35C4⋊C4×C251—22no—
36C4⋊C4×C251—42no—
37C4⋊C4×C251D412yes—
38C22⋊C4×C251—22no—
39C22⋊C4×C251—22no—
40C23×C251—22no—
41C23×C251C2×D411yescharacteristic
42Q8×C251—42no—
43D4×C251—42no—
44D4×C251—42no—
45D4×C251—42no—
46D4×C251—42no—
47D4×C251—81no—
48C2×C4×C251—21no—
49C2×C4×C251—21no—
50C2×C4×C251—22no—
51C2×C4×C251—21no—
52C2×C4×C251—22no—
53C2×C4×C251—41no—
54C2×C4×C251—44no—
55C2×C4×C251—41no—
56C2×C4×C251—41no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251—41no—
60C2×C4×C251—41no—
61C2×C4×C251D812yes—
62C2×C4×C251C2×D412yes—
63C8×C251—42no—
64C22×C251—21no—
65C22×C251—21no—
66C22×C251—42no—
67C22×C251—42no—
68C22×C251—42no—
69C22×C251#12011yescharacteristic
70C22×C251C2×D811yescharacteristic
71C22×C251AGL(1,ℤ/8)11yescharacteristic
72C4×C251—22no—
73C4×C251—22no—
74C4×C251—42no—
75C4×C251—42no—
76C4×C251—41no—
77C4×C251—81no—
78C2×C251—21no—
79C2×C251—21no—
80C2×C251—82no—
81C2×C251#44711yescharacteristic
82C2×C251#45611yescharacteristic
83C2×C251#45811yescharacteristic
84C251#161011yescharacteristic, direct product factor, Sylow
85#1610C25111yesmaximal, characteristic, direct product factor, Sylow
86#537C2×C25111yescharacteristic
87#480C2×C25112yescomplemented
88#398C2×C25111yescharacteristic, complemented
89#327C2×C25112yescomplemented
90#324C2×C25111yescharacteristic, complemented
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C22×C25112yes—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C22×C25111yescharacteristic
98C2×C4⋊C4C22×C25111yescharacteristic
99#107—22no—
100#102—22no—
101C22⋊C8C22×C25112yes—
102#95—21no—
103#95C22×C25111yescharacteristic
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4D4×C25112yes—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4D4×C25111yescharacteristic
115C22×C4D4×C25111yescharacteristic
116C22×C4C23×C25111yescharacteristic
117C2×C8—22no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—42no—
122C4⋊C4D4×C25112yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×D4×C25111yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—44no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4D8×C25112yes—
147C2×C4C2×D4×C25112yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#120×C25111yescharacteristic
155C22C2×D8×C25111yescharacteristic
156C22AGL(1,ℤ/8)×C25111yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—42no—
161C4—41no—
162C4—81no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#447×C25111yescharacteristic
167C2#456×C25111yescharacteristic
168C2#458×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC22×C4C23×C251
prime radical#537×C251C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C22×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #480×C251C2semidirect
3. #398×C251C2semidirect
4. #327×C251C2semidirect
5. #324×C251C2semidirect
84. C251#1610direct (N × H)
85. #1610C251direct (N × H)
87. #480C2×C251semidirect
88. #398C2×C251semidirect
89. #327C2×C251semidirect
90. #324C2×C251semidirect

Maximal subgroups

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

subgroupindex
#537×C2512
#480×C2512
#398×C2512
#327×C2512
#324×C2512
#1610251