Finite Group Databasebeta

← #44550409

Subgroups — #44550409

Order 32128.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#22548722C211yesmaximal, characteristic
2#30631723C212yesmaximal, complemented
3#30631661C211yesmaximal, characteristic
4#30631443C212yesmaximal, complemented
5#410×C251C211yesmaximal, characteristic
6#22548722—22no—
7#22548722—22no—
8C2×#16860794C412yescomplemented
9C2×#16860794C2211yescharacteristic
10C22×C251⋊C8C412yescomplemented
11C22×C251⋊C8C2211yescharacteristic
12#22548674C2212yes—
13#22548648—22no—
14#22548646—22no—
15#22548630—22no—
16C2×#41×C251C2211yescharacteristic
17#117×C251C2212yes—
18#99×C251—22no—
19#16860794—24no—
20#16860794—22no—
21C2×C251⋊C8—22no—
22C2×C251⋊C8—22no—
23C2×C251⋊C8C2×C412yes—
24C2×C251⋊C8C2×C411yescharacteristic
25C2×C251⋊C8C2×C411yescharacteristic
26C2×C251⋊C8D412yes—
27C2×C251⋊C8D412yes—
28C4⋊Dic251—22no—
29C22⋊Dic251—22no—
30C4×Dic251—22no—
31#41×C251—22no—
32#41×C251—22no—
33C2×Q8×C251Q811yescharacteristic
34C2×D4×C251D412yes—
35C2×D4×C251Q811yescharacteristic
36C22×C4×C251C2×C412yes—
37C22×C4×C251C2311yescharacteristic
38C4⋊C4×C251—22no—
39C22⋊C4×C251—22no—
40C42×C251—22no—
41C2×Dic251—42no—
42C251⋊C8—22no—
43C251⋊C8C22⋊C411yescharacteristic
44C251⋊C8C22⋊C411yescharacteristic
45C251⋊C8C22⋊C412yes—
46C23×C251C4⋊C412yes—
47C23×C251C2×D411yescharacteristic
48Q8×C251—22no—
49D4×C251—24no—
50D4×C251—22no—
51C2×C4×C251—22no—
52C2×C4×C251—22no—
53C2×C4×C251—42no—
54C2×C4×C251C4⋊C412yes—
55C2×C4×C251C22×C411yescharacteristic
56C2×C4×C251C2×D412yes—
57C2×C4×C251C2×Q811yescharacteristic
58C2×C4×C251#4112yes—
59C22×C251—22no—
60C22×C251—41no—
61C22×C251—42no—
62C22×C251C2×C4⋊C411yescharacteristic
63C22×C251#12112yes—
64C4×C251—22no—
65C4×C251—82no—
66C4×C251C2×C22⋊C411yescharacteristic
67C4×C251#11711yescharacteristic
68C4×C251C22⋊Q812yes—
69Dic251—82no—
70C2×C251—22no—
71C2×C251—21no—
72C2×C251—42no—
73C2×C251#38011yescharacteristic
74C251#181211yescharacteristic, complemented, Sylow
75#1812—2511nomaximal, Sylow
76C2×#131—2511no—
77#427—2512no—
78#410—2511no—
79#410D25111yescharacteristic
80H(3,ℤ/4)—2512no—
81C2×#41C2×D25111yescharacteristic
82#131—5022no—
83#131—5022no—
84C2×M16—2512no—
85C2×M16—2511no—
86C22×C8—2512no—
87C22×C8—2511no—
88#117—2512no—
89#117C2×D25112yes—
90#107—5022no—
91#106—5022no—
92#99—22no—
93#99—5022no—
94#41—22no—
95#41—22no—
96C2×Q8Dic50211yescharacteristic
97C2×D4C22⋊D25112yes—
98C2×D4Dic50211yescharacteristic
99C22×C4C4×D25112yes—
100C22×C4C22×D25111yescharacteristic
101M16—5022no—
102M16—5024no—
103C2×C8—5022no—
104C2×C8—5022no—
105C2×C8—2512no—
106C2×C8—2511no—
107C2×C8—2511no—
108C2×C8—2512no—
109C2×C8—2512no—
110C4⋊C4—22no—
111C4⋊C4—5022no—
112C22⋊C4—22no—
113C22⋊C4—5022no—
114C42—22no—
115C42—5022no—
116C23Dic251⋊C412yes—
117C23C2×C22⋊D25111yescharacteristic
118Q8—22no—
119D4—24no—
120D4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—42no—
124C2×C4—10042no—
125C2×C4Dic251⋊C412yes—
126C2×C4C2×C4×D25111yescharacteristic
127C2×C4D4×D25112yes—
128C2×C4C2×Dic50211yescharacteristic
129C2×C4#1686080912yes—
130C8—5022no—
131C8—2511no—
132C8—2511no—
133C8—2512no—
134C22—22no—
135C22—41no—
136C22—42no—
137C22C2×Dic251⋊C411yescharacteristic
138C22#2254868712yes—
139C4—22no—
140C4—82no—
141C4—20082no—
142C4C22⋊C4×D25111yescharacteristic
143C4#2254867111yescharacteristic
144C4#2254869212yes—
145C2—22no—
146C2—21no—
147C2—42no—
148C2#3063154511yescharacteristic
subgroupG/H
centerC4#22548671
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#410×C251C2
socleC2×C251#380
prime radicalC2×#41×C251C22
cosocleC22×C4×C251C23
hypercenter#410D251
hypocenterC251#1812
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #30631723C2semidirect
4. #30631443C2semidirect
8. C2×#16860794C4semidirect
10. C22×C251⋊C8C4semidirect
74. C251#1812semidirect

Maximal subgroups

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

subgroupindex
C2×#225487222
#306317232
#306316612
#306314432
#410×C2512
#1812251