Finite Group Databasebeta

← #44548955

Subgroups — #44548955

Order 32128.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#30632236C211yesmaximal, characteristic
2#30631639C212yesmaximal, complemented
3#30631427C211yesmaximal, characteristic, complemented
4#30631426C212yesmaximal, complemented
5#329×C251C211yesmaximal, characteristic, complemented
6C2×#16860811C411yescharacteristic
7C2×#16860809C411yescharacteristic
8C2×#16860793—22no—
9#22548713C411yescharacteristic
10#22548711C411yescharacteristic
11#22548703C2211yescharacteristic
12#22548688—22no—
13(C22⋊D251)⋊C4—22no—
14C42×D251C2211yescharacteristic
15#22548626C2212yes—
16C8×Dic251—22no—
17#124×C251C2211yescharacteristic
18#97×C251C2212yes—
19#16860811—42no—
20#16860809—42no—
21C2×Dic502C2×C411yescharacteristic
22C2×D1004—21no—
23C2×D1004C2×C411yescharacteristic
24C2×C22⋊D251—21no—
25C2×C22⋊D251—22no—
26C22×Dic251—21no—
27C2×C4×D251—21no—
28C2×C4×D251C2×C411yescharacteristic
29C2×C4×D251D412yes—
30#16860793—28no—
31C2×C251⋊C8—22no—
32Dic251⋊C4—22no—
33(C2×D251)⋊C4—22no—
34C4×Dic251C2×C411yescharacteristic
35C4×Dic251D412yes—
36C2×Q8×C251C2×C411yescharacteristic
37C2×D4×C251C2×C411yescharacteristic
38C2×C8×C251—22no—
39C22⋊C4×C251—22no—
40C42×C251C2311yescharacteristic
41C22×D251—21no—
42C22×D251C22⋊C411yescharacteristic
43Dic502—41no—
44D1004—41no—
45D1004—42no—
46C22⋊D251—42no—
47C22⋊D251—44no—
48C2×Dic251—21no—
49C2×Dic251—42no—
50C2×Dic251C22⋊C411yescharacteristic
51C2×Dic251C22⋊C412yes—
52C4×D251—22no—
53C4×D251—24no—
54C4×D251—42no—
55C251⋊C8—24no—
56C23×C251—21no—
57Q8×C251—41no—
58D4×C251—41no—
59C2×C4×C251—21no—
60C2×C4×C251C22×C411yescharacteristic
61C2×C4×C251C2×D412yes—
62C8×C251—24no—
63C22×C251—41no—
64C22×C251—42no—
65C22×C251C2×C22⋊C411yescharacteristic
66C2×D251—21no—
67C2×D251—22no—
68C2×D251—41no—
69C2×D251—42no—
70C4×C251—21no—
71C4×C251—22no—
72C4×C251—41no—
73Dic251—41no—
74Dic251#10012yes—
75Dic251#10414yes—
76C2×C251—81no—
77C2×C251C2×#10011yescharacteristic
78C2×C251C2×#10412yes—
79D251—41no—
80D251—81no—
81C251#152511yescharacteristic, complemented, Sylow
82#1525—2511nomaximal, Sylow
83#532—2511no—
84C4×M16—2512no—
85#329—2511no—
86#329—2512no—
87#329D25111yescharacteristic, complemented
88C2×#41—2511no—
89C2×#41—2511no—
90C2×M16—5022no—
91C4⋊Q8—2511no—
92#127—2511no—
93#124—2511no—
94#124C2×D25111yescharacteristic
95#121—5022no—
96C4×D4—5022no—
97C2×C42—2511no—
98#97—2512no—
99#97C2×D25112yes—
100C4×C8—5022no—
101#41—10042no—
102#41—10042no—
103C2×Q8—2511no—
104C2×Q8C4×D25111yescharacteristic
105C2×D4—5021no—
106C2×D4—5021no—
107C2×D4—5022no—
108C2×D4—2511no—
109C2×D4C4×D25111yescharacteristic
110C22×C4—5021no—
111C22×C4—5021no—
112C22×C4—2511no—
113C22×C4—2512no—
114M16—5028no—
115C2×C8—22no—
116C2×C8—5022no—
117C4⋊C4—5022no—
118C22⋊C4—22no—
119C22⋊C4—5022no—
120C42—2511no—
121C42—2512no—
122C42C22×D25111yescharacteristic
123C23—21no—
124C23—5021no—
125C23—2511no—
126Q8—41no—
127Q8—10041no—
128D4—41no—
129D4—10041no—
130D4—10042no—
131D4—10042no—
132D4—10044no—
133C2×C4—21no—
134C2×C4—10042no—
135C2×C4—10042no—
136C2×C4—5024no—
137C2×C4—5022no—
138C2×C4—5021no—
139C2×C4—2512no—
140C2×C4—2511no—
141C2×C4C2×C4×D25111yescharacteristic
142C2×C4D4×D25112yes—
143C8—24no—
144C8—5024no—
145C22—42no—
146C22—41no—
147C22—10042no—
148C22—10041no—
149C22—5022no—
150C22—5021no—
151C22C22⋊C4×D25111yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—10041no—
156C4—2514no—
157C4—2512no—
158C2—81no—
159C2—20081no—
160C2—10041no—
161C2#100×D25111yescharacteristic
162C2#104×D25112yes—
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
Fitting#329×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632236C2
cosocleC42×C251C23
hypercenter#329D251
hypocenterC251#1525
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #30631639C2semidirect
3. #30631427C2semidirect
4. #30631426C2semidirect
5. #329×C251C2semidirect
81. C251#1525semidirect
87. #329D251semidirect

Maximal subgroups

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

subgroupindex
#306322362
#306316392
#306314272
#306314262
#329×C2512
#1525251