Finite Group Databasebeta

← #44548987

Subgroups — #44548987

Order 32128.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D251C211yesmaximal, characteristic
2#30631742C212yesmaximal, complemented
3#30631427C211yesmaximal, characteristic, complemented
4#30631426C212yesmaximal, complemented
5#329×C251C211yesmaximal, characteristic, complemented
6C2×Q8×D251C411yescharacteristic
7C2×D4×D251C411yescharacteristic
8#22548704C411yescharacteristic
9#22548703C2211yescharacteristic
10#22548702C411yescharacteristic
11#22548701—22no—
12C22⋊C4×D251—22no—
13C42×D251C2211yescharacteristic
14Dic251⋊C8—22no—
15(C2×D251)⋊C8—22no—
16#22548626C2212yescomplemented
17#124×C251C2211yescharacteristic
18#97×C251C2212yescomplemented
19C23×D251—21no—
20C2×Dic502—21no—
21C2×Dic502C2×C411yescharacteristic
22Q8×D251—42no—
23C2×D1004C2×C411yescharacteristic
24C2×C22⋊D251—21no—
25D4×D251—42no—
26C2×C4×D251—21no—
27C2×C4×D251C2×C411yescharacteristic
28C2×C4×D251D412yes—
29C2×C251⋊C8—22no—
30C22⋊Dic251—22no—
31(C2×D251)⋊C4—22no—
32(C2×D251)⋊C4—22no—
33C4×Dic251C2×C411yescharacteristic
34C4×Dic251D412yes—
35C2×Q8×C251C2×C411yescharacteristic
36C2×D4×C251C2×C411yescharacteristic
37C2×C8×C251—22no—
38C22⋊C4×C251—22no—
39C42×C251C2311yescharacteristic
40C22×D251—21no—
41C22×D251—44no—
42C22×D251—42no—
43C22×D251C22⋊C411yescharacteristic
44Dic502—42no—
45Dic502—41no—
46D1004—41no—
47C22⋊D251—42no—
48C2×Dic251—21no—
49C2×Dic251C22⋊C412yes—
50C2×Dic251C22⋊C411yescharacteristic
51C4×D251—22no—
52C4×D251—42no—
53C4×D251—42no—
54C251⋊C8—42no—
55C23×C251—21no—
56Q8×C251—41no—
57D4×C251—41no—
58C2×C4×C251—21no—
59C2×C4×C251C22×C411yescharacteristic
60C2×C4×C251C2×D412yes—
61C8×C251—42no—
62C22×C251—41no—
63C22×C251—42no—
64C22×C251C2×C22⋊C411yescharacteristic
65C2×D251—22no—
66C2×D251—41no—
67C2×D251—42no—
68C2×D251—84no—
69C2×D251#10012yes—
70C4×C251—21no—
71C4×C251—22no—
72C4×C251—41no—
73Dic251—21no—
74Dic251—22no—
75Dic251—41no—
76C2×C251—81no—
77C2×C251C2×#10011yescharacteristic
78C2×C251#42112yes—
79D251—22no—
80D251—81no—
81C251#153311yescharacteristic, complemented, Sylow
82#1533—2511nomaximal, Sylow
83C2×#124—2511no—
84#432—2512no—
85#329—2511no—
86#329—2512no—
87#329D25111yescharacteristic, complemented
88C22×Q8—2511no—
89C22×D4—2511no—
90#124—5022no—
91#124—2511no—
92#124—2511no—
93#124—2511no—
94#124C2×D25111yescharacteristic
95C2×C22⋊C4—5022no—
96C2×C42—2511no—
97C4⋊C8—5022no—
98C22⋊C8—5022no—
99#97—2512no—
100#97C2×D25112yescomplemented
101C24—5021no—
102C2×Q8—10042no—
103C2×Q8—5021no—
104C2×Q8—2511no—
105C2×Q8C4×D25111yescharacteristic
106C2×D4—10042no—
107C2×D4—5021no—
108C2×D4—2511no—
109C2×D4C4×D25111yescharacteristic
110C22×C4—5021no—
111C22×C4—2512no—
112C22×C4—2511no—
113C2×C8—22no—
114C2×C8—5022no—
115C22⋊C4—22no—
116C22⋊C4—5022no—
117C22⋊C4—5022no—
118C22⋊C4—5022no—
119C42—2511no—
120C42—2512no—
121C42C22×D25111yescharacteristic
122C23—21no—
123C23—10044no—
124C23—10042no—
125C23—5021no—
126C23—2511no—
127Q8—41no—
128Q8—10041no—
129Q8—10042no—
130D4—41no—
131D4—10041no—
132D4—10042no—
133C2×C4—21no—
134C2×C4—10042no—
135C2×C4—10042no—
136C2×C4—5022no—
137C2×C4—5021no—
138C2×C4—2512no—
139C2×C4—2511no—
140C2×C4C2×C4×D25111yescharacteristic
141C2×C4D4×D25112yes—
142C8—42no—
143C8—10042no—
144C22—42no—
145C22—41no—
146C22—20084no—
147C22—10042no—
148C22—10041no—
149C22—5022no—
150C22—2512no—
151C22C22⋊C4×D25111yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—10041no—
156C4—5022no—
157C4—5021no—
158C2—81no—
159C2—20081no—
160C2—5022no—
161C2#100×D25111yescharacteristic
162C2#3063170212yes—
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
Fitting#329×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC2×D4×D251C4
cosocleC42×C251C23
hypercenter#329D251
hypocenterC251#1533
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. #30631742C2semidirect
3. #30631427C2semidirect
4. #30631426C2semidirect
5. #329×C251C2semidirect
16. #22548626C22semidirect
18. #97×C251C22semidirect
81. C251#1533semidirect
87. #329D251semidirect
100. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#124×D2512
#306317422
#306314272
#306314262
#329×C2512
#1533251