Finite Group Databasebeta

← #44550902

Subgroups — #44550902

Order 32128.

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

normal subgroups only

This group has 314 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
1#30632261C211yesmaximal, characteristic
2#30631705C212yesmaximal
3C2×#22548634C211yesmaximal, characteristic, complemented
4#30631443C212yesmaximal, complemented
5#519×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7C2×#16860794C2212yes—
8#22548713C412yescomplemented
9#22548704C412yescomplemented
10#22548698—22no—
11#22548691—22no—
12#22548674C2211yescharacteristic
13#22548643—24no—
14#22548634—22no—
15C22×Q8×C251C2211yescharacteristic
16C4×Q8×C251—24no—
17#117×C251—22no—
18#117×C251C2212yes—
19#16860810—22no—
20#16860810—22no—
21C2×Dic502C2×C411yescharacteristic
22C2×D1004C2×C411yescharacteristic
23C2×C22⋊D251—21no—
24C2×C4×D251—21no—
25#16860794—22no—
26#16860794—22no—
27C2×C251⋊C8—22no—
28C4⋊Dic251—21no—
29Dic251⋊C4—24no—
30C22⋊Dic251—21no—
31(C2×D251)⋊C4—24no—
32C4×Dic251C2×C412yes—
33C2×Q8×C251—22no—
34C2×Q8×C251—22no—
35C2×Q8×C251C2×C412yes—
36C2×Q8×C251D412yes—
37C22×C4×C251D412yes—
38C22×C4×C251C2311yescharacteristic
39C4⋊C4×C251—24no—
40C4⋊C4×C251—22no—
41C22⋊C4×C251—22no—
42C42×C251—24no—
43C42×C251D414yes—
44C22×D251—21no—
45Dic502—22no—
46D1004—22no—
47C22⋊D251—41no—
48C22⋊D251—41no—
49C2×Dic251—22no—
50C2×Dic251—21no—
51C4×D251—42no—
52C251⋊C8—42no—
53C23×C251C2×D411yescharacteristic
54Q8×C251—24no—
55Q8×C251—24no—
56C2×C4×C251—22no—
57C2×C4×C251—22no—
58C2×C4×C251—24no—
59C2×C4×C251C22⋊C414yes—
60C2×C4×C251C22×C411yescharacteristic
61C2×C4×C251C2×D412yes—
62C2×C4×C251C2×D411yescharacteristic
63C2×C4×C251#4112yes—
64C22×C251—41no—
65C22×C251C2×C22⋊C411yescharacteristic
66C22×C251#12011yescharacteristic
67C22×C251#12311yescharacteristic
68C2×D251—42no—
69C2×D251—41no—
70C4×C251—24no—
71C4×C251—44no—
72C4×C251C4×D412yes—
73C4×C251#12112yes—
74Dic251—42no—
75Dic251—41no—
76C2×C251—21no—
77C2×C251—21no—
78C2×C251—21no—
79C2×C251#38611yescharacteristic
80D251—81no—
81C251#191211yescharacteristic, complemented, Sylow
82#1912—2511nomaximal, Sylow
83#536—2511no—
84#519D25111yescharacteristic, complemented
85#421—2512no—
86C2×#101—2511no—
87H(3,ℤ/4)—2512no—
88C2×#41—2511no—
89C22×Q8C2×D25111yescharacteristic
90C2×M16—2512no—
91C4⋊Q8—2512no—
92#124—2512no—
93#123—5022no—
94C22⋊Q8—5022no—
95C4×Q8—24no—
96#117—22no—
97#117—2511no—
98#117C2×D25112yes—
99#104—5024no—
100#101—5022no—
101#41—5022no—
102#41—5022no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—2511no—
106C2×Q8C4×D25112yes—
107C2×Q8D100412yes—
108C2×D4—5021no—
109C2×D4—2511no—
110C22×C4—5021no—
111C22×C4C22⋊D25112yes—
112C22×C4C22×D25111yescharacteristic
113M16—5022no—
114M16—5022no—
115C2×C8—5022no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—5024no—
119C4⋊C4—5021no—
120C22⋊C4—22no—
121C22⋊C4—5021no—
122C22⋊C4—5024no—
123C42—24no—
124C42—2512no—
125C42C22⋊D25114yes—
126C23—5021no—
127C23C2×C22⋊D25111yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—5022no—
131D4—10041no—
132D4—10041no—
133D4—5022no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—10042no—
138C2×C4—5022no—
139C2×C4—5021no—
140C2×C4(C2×D251)⋊C414yes—
141C2×C4C2×C4×D25111yescharacteristic
142C2×C4C2×C22⋊D25112yes—
143C2×C4C2×D100411yescharacteristic
144C2×C4#1686081012yes—
145C8—10042no—
146C22—41no—
147C22—10042no—
148C22—10041no—
149C22C2×(C2×D251)⋊C411yescharacteristic
150C22#2254868411yescharacteristic
151C22#2254869811yescharacteristic
152C4—24no—
153C4—44no—
154C4—10042no—
155C4—10041no—
156C4C4×C22⋊D25112yes—
157C4#2254868912yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—20081no—
162C2#3063156611yescharacteristic
subgroupG/H
centerC2#30631566
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#519×C251C2
socleC2×C251#386
prime radicalC2×#16860810C22
cosocleC22×C4×C251C23
hypercenter#519D251
hypocenterC251#1912
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. C2×#22548634C2semidirect
4. #30631443C2semidirect
5. #519×C251C2semidirect
8. #22548713C4semidirect
9. #22548704C4semidirect
81. C251#1912semidirect
84. #519D251semidirect

Maximal subgroups

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

subgroupindex
#306322612
#306317052
C2×#225486342
#306314432
#519×C2512
#1912251