Finite Group Databasebeta

← #44549304

Subgroups — #44549304

Order 32128.

266 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632318C211yesmaximal, characteristic, complemented
2#30631838C212yesmaximal
3#30631423C212yesmaximal, complemented
4#30631413C211yesmaximal, characteristic, complemented
5#394×C251C211yesmaximal, characteristic
6C2×#16860798—22no—
7#22548714—22no—
8#22548706—22no—
9#22548694—21no—
10#22548694C2212yes—
11#22548674—21no—
12#22548636—22no—
13C22⋊(C251⋊C8)C2212yes—
14C22×D4×C251C2211yescharacteristic
15C2×C4⋊C4×C251C2211yescharacteristic
16C2×C22⋊C4×C251—21no—
17C2×C22⋊C4×C251—21no—
18#95×C251C2211yescharacteristic
19C2×Dic502—21no—
20C2×Dic502D412yes—
21#16860798—44no—
22C2×C251⋊C8—22no—
23C4⋊Dic251—21no—
24C4⋊Dic251—21no—
25C4⋊Dic251—21no—
26C4⋊Dic251D412yes—
27Dic251⋊C4—22no—
28Dic251⋊C4—22no—
29C22⋊Dic251—22no—
30C4×Dic251—22no—
31C24×C251—21no—
32C2×D4×C251—22no—
33C2×D4×C251—42no—
34C22×C4×C251—21no—
35C22×C4×C251D411yescharacteristic
36C22×C4×C251D411yescharacteristic
37C22×C4×C251C2311yescharacteristic
38C4⋊C4×C251—22no—
39C22⋊C4×C251—42no—
40C22⋊C4×C251—42no—
41Dic502—42no—
42Dic502—81no—
43C2×Dic251—22no—
44C2×Dic251—22no—
45C251⋊C8—42no—
46C23×C251—22no—
47C23×C251—41no—
48C23×C251—41no—
49C23×C251—41no—
50C23×C251—41no—
51C23×C251—41no—
52C23×C251—41no—
53C23×C251C2×D411yescharacteristic
54D4×C251—24no—
55D4×C251—42no—
56C2×C4×C251—21no—
57C2×C4×C251—21no—
58C2×C4×C251—21no—
59C2×C4×C251—41no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251—41no—
64C2×C4×C251SD1612yescomplemented
65C2×C4×C251C2×D412yes—
66C22×C251—21no—
67C22×C251—21no—
68C22×C251—42no—
69C22×C251—42no—
70C22×C251—42no—
71C22×C251—81no—
72C22×C251—81no—
73C22×C251—81no—
74C22×C251—81no—
75C22×C251#12011yescharacteristic
76C22×C251C2×SD1611yescharacteristic
77C22×C251#13711yescharacteristic
78C4×C251—22no—
79C4×C251—22no—
80C4×C251—41no—
81C4×C251—81no—
82Dic251—42no—
83Dic251—42no—
84C2×C251—21no—
85C2×C251—21no—
86C2×C251—82no—
87C2×C251#44811yescharacteristic
88C2×C251#45411yescharacteristic
89C2×C251B(4,𝔽2)11yescharacteristic
90C251#159911yescharacteristic, complemented, Sylow
91#1599—2511nomaximal, Sylow
92#544—2511no—
93#450—2512no—
94#394D25111yescharacteristic
95#328—2512no—
96#324—2511no—
97C22×D4C2×D25111yescharacteristic
98C2×SD16—5022no—
99C4⋊Q8—5022no—
100#125—5022no—
101C22⋊Q8—5021no—
102C22⋊Q8—2512no—
103#117—5021no—
104C2×C4⋊C4C2×D25111yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#102—5022no—
108C22⋊C8—2512no—
109#95C2×D25111yescharacteristic
110C24—21no—
111C2×Q8—5021no—
112C2×Q8—2512no—
113C2×D4—22no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4C22⋊D25111yescharacteristic
117C22×C4C22⋊D25111yescharacteristic
118C22×C4C22×D25111yescharacteristic
119SD16—10044no—
120C2×C8—5022no—
121C4⋊C4—22no—
122C4⋊C4—5021no—
123C4⋊C4—5021no—
124C4⋊C4—5022no—
125C4⋊C4—5021no—
126C4⋊C4—5022no—
127C4⋊C4—2512no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—5022no—
131C42—5022no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C2×C22⋊D25111yescharacteristic
140Q8—20081no—
141Q8—10042no—
142D4—24no—
143D4—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—5022no—
153C2×C4—5022no—
154C2×C4#1686079812yescomplemented
155C2×C4D4×D25112yes—
156C8—10042no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#2254868311yescharacteristic
167C22C2×#1686079811yescharacteristic
168C22#2254874511yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—81no—
173C4—10042no—
174C4—10042no—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#3063182511yescharacteristic
179C2#3063185911yescharacteristic
180C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#394×C251C2
socleC22×C251#120
prime radicalC22×D4×C251C22
hypercenter#394D251
hypocenterC251#1599
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #30632318C2semidirect
3. #30631423C2semidirect
4. #30631413C2semidirect
64. C2×C4×C251SD16semidirect
90. C251#1599semidirect
154. C2×C4#16860798semidirect

Maximal subgroups

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

subgroupindex
#306323182
#306318382
#306314232
#306314132
#394×C2512
#1599251