Finite Group Databasebeta

← #44550913

Subgroups — #44550913

Order 32128.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#30632454C211yesmaximal, characteristic, complemented
2#30631671C211yesmaximal, characteristic, complemented
3C2×#22548632C212yesmaximal, complemented
4#30631457C212yesmaximal, complemented
5#535×C251C211yesmaximal, characteristic
6#22548722—24no—
7C2×#16860794—22no—
8C2×#16860794C2212yes—
9C2×#16860794C2212yes—
10#22548634—21no—
11#22548632—24no—
12#22548632—21no—
13C2×#41×C251C2211yescharacteristic
14C22×D4×C251C2211yescharacteristic
15#127×C251—21no—
16#124×C251—21no—
17#121×C251—22no—
18#120×C251C414yes—
19#117×C251C2211yescharacteristic
20#16860794—24no—
21#16860794—22no—
22#16860794—22no—
23#16860794—22no—
24#16860794D414yescomplemented
25C2×C251⋊C8—24no—
26C2×C251⋊C8—22no—
27C2×C251⋊C8—22no—
28C24×C251C2×C412yes—
29#41×C251—22no—
30#41×C251—22no—
31C2×Q8×C251D411yescharacteristic
32C2×D4×C251—24no—
33C2×D4×C251—21no—
34C2×D4×C251—21no—
35C2×D4×C251—22no—
36C2×D4×C251—22no—
37C2×D4×C251C2×C412yes—
38C2×D4×C251D411yescharacteristic
39C22×C4×C251D412yes—
40C22×C4×C251C2311yescharacteristic
41C4⋊C4×C251—21no—
42C22⋊C4×C251—24no—
43C22⋊C4×C251C2×C412yes—
44C42×C251—21no—
45C251⋊C8—24no—
46C251⋊C8—42no—
47C23×C251—22no—
48C23×C251—24no—
49C23×C251—82no—
50C23×C251C22⋊C412yes—
51C23×C251C22×C411yescharacteristic
52Q8×C251—22no—
53D4×C251—24no—
54D4×C251—22no—
55D4×C251—41no—
56D4×C251—42no—
57D4×C251—41no—
58D4×C251—84no—
59C2×C4×C251—22no—
60C2×C4×C251—22no—
61C2×C4×C251—22no—
62C2×C4×C251C22⋊C412yes—
63C2×C4×C251C2×D412yes—
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251#4112yes—
67C22×C251—22no—
68C22×C251—41no—
69C22×C251—42no—
70C22×C251—44no—
71C22×C251—42no—
72C22×C251—84no—
73C22×C251C2×C22⋊C411yescharacteristic
74C22×C251C4×D412yes—
75C4×C251—22no—
76C4×C251—42no—
77C4×C251#12011yescharacteristic
78C4×C251#12112yes—
79C4×C251#12311yescharacteristic
80C2×C251—21no—
81C2×C251—22no—
82C2×C251—42no—
83C2×C251—82no—
84C2×C251#38611yescharacteristic
85C251#191411yescharacteristic, complemented, Sylow
86#1914—2511nomaximal, Sylow
87#568—2511no—
88#535D25111yescharacteristic
89#413—2511no—
90C2×#100—2512no—
91#344—2512no—
92C2×#41C2×D25111yescharacteristic
93C22×D4C2×D25111yescharacteristic
94#131—5024no—
95C2×M16—5022no—
96C2×M16—2512no—
97C2×M16—2512no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic25114yes—
102#117C2×D25111yescharacteristic
103#101—5021no—
104#100—5021no—
105#100—5024no—
106C24C2×Dic25112yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D25111yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic25112yes—
116C2×D4C22⋊D25111yescharacteristic
117C22×C4C22⋊D25112yes—
118C22×C4C22×D25111yescharacteristic
119M16—5022no—
120M16—5022no—
121M16—5024no—
122M16—5022no—
123M16—2514no—
124C2×C8—5022no—
125C2×C8—5024no—
126C2×C8—5022no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic25112yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic25112yes—
135C23C22×Dic25111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic25112yes—
147C2×C4D4×D25112yes—
148C2×C4C2×C22⋊D25111yescharacteristic
149C2×C4C2×C22⋊D25111yescharacteristic
150C2×C4#1686080912yes—
151C8—10042no—
152C8—5024no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic25111yescharacteristic
160C22D4×Dic25112yes—
161C4—22no—
162C4—42no—
163C4#2254868311yescharacteristic
164C4#2254868712yes—
165C4#2254869611yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#3063156711yescharacteristic
subgroupG/H
centerC2#30631567
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radical#535×C251C2
socleC2×C251#386
cosocleC22×C4×C251C23
hypercenter#535D251
hypocenterC251#1914
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #30632454C2semidirect
2. #30631671C2semidirect
3. C2×#22548632C2semidirect
4. #30631457C2semidirect
24. #16860794D4semidirect
85. C251#1914semidirect

Maximal subgroups

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

subgroupindex
#306324542
#306316712
C2×#225486322
#306314572
#535×C2512
#1914251