Finite Group Databasebeta

← #44549641

Subgroups — #44549641

Order 32128.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632415C211yesmaximal, characteristic
2#30632083C211yesmaximal, characteristic
3#30631894C211yesmaximal, characteristic
4#30631434C211yesmaximal, characteristic
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic
6#30631417C211yesmaximal, characteristic
7#475×C251C211yesmaximal, characteristic
8C2×#16860800—21no—
9#22548715C2211yescharacteristic
10#22548713—21no—
11#22548707—21no—
12#22548705—22no—
13C4×Dic502—21no—
14C4×Dic502C2211yescharacteristic
15#22548680—21no—
16#22548646—22no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548639—21no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×Q8×C251C2211yescharacteristic
22#106×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502—21no—
26C2×Dic502D411yescharacteristic
27#16860800—42no—
28C2×C251⋊C8—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—21no—
31C4⋊Dic251D411yescharacteristic
32C4⋊Dic251D412yes—
33Dic251⋊C4—22no—
34Dic251⋊C4—22no—
35Dic251⋊C4—22no—
36Dic251⋊C4—21no—
37C4×Dic251—22no—
38C4×Dic251—21no—
39C2×Q8×C251—21no—
40C2×Q8×C251D411yescharacteristic
41C2×C8×C251—21no—
42C4⋊C4×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C42×C251—21no—
47C42×C251C2311yescharacteristic
48Dic502—21no—
49Dic502—41no—
50Dic502—41no—
51Dic502SD1612yes—
52C2×Dic251—21no—
53C2×Dic251—21no—
54C2×Dic251—22no—
55C251⋊C8—22no—
56C251⋊C8—41no—
57Q8×C251—21no—
58Q8×C251—21no—
59Q8×C251—41no—
60Q8×C251—41no—
61C2×C4×C251—21no—
62C2×C4×C251—21no—
63C2×C4×C251—41no—
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C8×C251—41no—
68C22×C251#12011yescharacteristic
69C4×C251—21no—
70C4×C251—41no—
71C4×C251—41no—
72C4×C251—81no—
73C4×C251C2×SD1611yescharacteristic
74C4×C251#13511yescharacteristic
75C4×C251#13711yescharacteristic
76C4×C251#13711yescharacteristic
77Dic251—22no—
78Dic251—42no—
79Dic251—41no—
80C2×C251#44811yescharacteristic
81C2×C251#45211yescharacteristic
82C2×C251#45611yescharacteristic
83C251#165311yescharacteristic, complemented, Sylow
84#1653—2511nomaximal, Sylow
85#557—2511no—
86#498—2511no—
87#475D25111yescharacteristic
88#462—2511no—
89#332—2511no—
90Q8⋊C8—2511no—
91Q8⋊C8—2511no—
92C2×Q16—5021no—
93C4⋊Q8—5021no—
94C4⋊Q8—2511no—
95C4⋊Q8C2×D25111yescharacteristic
96#125—5021no—
97#125—5022no—
98C4×Q8—5021no—
99C4×Q8—5021no—
100C4×Q8—2511no—
101C4×Q8C2×D25111yescharacteristic
102#106—21no—
103#106—5022no—
104C4⋊C8—2511no—
105C4⋊C8C2×D25111yescharacteristic
106#103—21no—
107#103—5021no—
108C4×C8—2511no—
109C2×Q8—21no—
110C2×Q8—5021no—
111C2×Q8—2511no—
112C2×Q8C22⋊D25111yescharacteristic
113Q16—10042no—
114C2×C8—21no—
115C2×C8—5021no—
116C2×C8—5021no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—5022no—
121C4⋊C4—5022no—
122C4⋊C4—5022no—
123C4⋊C4—5021no—
124C4⋊C4—5021no—
125C4⋊C4—2511no—
126C4⋊C4—2512no—
127C4⋊C4C22⋊D25111yescharacteristic
128C42—21no—
129C42—5022no—
130C42—5021no—
131C42C22×D25111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—41no—
135Q8—41no—
136Q8—10041no—
137Q8—10041no—
138Q8—5021no—
139Q8—2512no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—5021no—
144C2×C4—5022no—
145C2×C4—5021no—
146C2×C4D4×D25111yescharacteristic
147C2×C4D4×D25111yescharacteristic
148C2×C4C2×C22⋊D25111yescharacteristic
149C8—41no—
150C8—10041no—
151C8—5022no—
152C22#2254868311yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—81no—
157C4—10042no—
158C4—10041no—
159C4—5022no—
160C4SD16×D25111yescharacteristic
161C4#2254873511yescharacteristic
162C4#2254874411yescharacteristic
163C4#2254874611yescharacteristic
164C2#3063182411yescharacteristic
165C2#3063184711yescharacteristic
166C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#475×C251C2
socle, prime radicalC22×C251#120
hypercenter#475D251
hypocenterC251#1653
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
83. C251#1653semidirect

Maximal subgroups

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

subgroupindex
#306324152
#306320832
#306318942
#306314342
Q8⋊(C251⋊C8)2
#306314172
#475×C2512
#1653251