Finite Group Databasebeta

← #44549067

Subgroups — #44549067

Order 32128.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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#30632235C211yesmaximal, characteristic
2#30631743C211yesmaximal, characteristic, complemented
3#30631739C211yesmaximal, characteristic, complemented
4#30631434C211yesmaximal, characteristic, complemented
5#30631432C211yesmaximal, characteristic, complemented
6#30631431C211yesmaximal, characteristic, complemented
7#331×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9#22548715C2211yescharacteristic
10#22548702—21no—
11#22548686—21no—
12#22548686C412yes—
13D4×Dic251—21no—
14D4×Dic251C412yes—
15C42×D251C2211yescharacteristic
16C4⋊(C251⋊C8)C2211yescharacteristic
17C4⋊(C251⋊C8)C2211yescharacteristic
18(C2×D251)⋊C8—21no—
19(C2×D251)⋊C8—21no—
20#22548625—21no—
21C8×Dic251—21no—
22#127×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4⋊C8×C251C2211yescharacteristic
25#16860809—42no—
26#16860809—42no—
27C2×Dic502—21no—
28C2×C22⋊D251—22no—
29C22×Dic251—22no—
30C2×C4×D251C2×C411yescharacteristic
31C2×C4×D251D411yescharacteristic
32C2×C4×D251D411yescharacteristic
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251C2×C412yes—
37C22⋊Dic251—22no—
38C22⋊Dic251—22no—
39C4×Dic251C2×C411yescharacteristic
40C4×Dic251D411yescharacteristic
41C4×Dic251D411yescharacteristic
42C2×D4×C251—21no—
43C2×D4×C251—21no—
44C2×D4×C251C2×C412yes—
45C2×C8×C251—21no—
46C2×C8×C251—21no—
47C42×C251C2311yescharacteristic
48C22×D251C22⋊C411yescharacteristic
49Dic502—41no—
50Dic502—41no—
51C22⋊D251—44no—
52C2×Dic251—22no—
53C2×Dic251—44no—
54C2×Dic251C22⋊C411yescharacteristic
55C2×Dic251C22⋊C411yescharacteristic
56C2×Dic251C22⋊C411yescharacteristic
57C4×D251—21no—
58C4×D251—21no—
59C4×D251—21no—
60C4×D251—21no—
61C4×D251—41no—
62C251⋊C8—41no—
63C251⋊C8—41no—
64C23×C251—22no—
65D4×C251—42no—
66D4×C251—41no—
67D4×C251—41no—
68D4×C251—41no—
69D4×C251—41no—
70C2×C4×C251C22×C411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—41no—
74C8×C251—41no—
75C22×C251—42no—
76C22×C251—42no—
77C22×C251—42no—
78C22×C251C2×C22⋊C411yescharacteristic
79C2×D251—21no—
80C2×D251—21no—
81C2×D251—21no—
82C4×C251—21no—
83C4×C251#13511yescharacteristic
84C4×C251#13511yescharacteristic
85C4×C251AGL(1,ℤ/8)11yescharacteristic
86C4×C251AGL(1,ℤ/8)11yescharacteristic
87Dic251—21no—
88Dic251—21no—
89Dic251—21no—
90Dic251—42no—
91C2×C251—82no—
92C2×C251#41311yescharacteristic
93C2×C251#41611yescharacteristic
94C2×C251#41811yescharacteristic
95D251—41no—
96C251#155311yescharacteristic, complemented, Sylow
97#1553—2511nomaximal, Sylow
98#532—2511no—
99#432—2511no—
100#431—2511no—
101#332—2511no—
102#332—2511no—
103#331—2511no—
104#331D25111yescharacteristic, complemented
105C2×#41—5021no—
106C4⋊Q8—2511no—
107#127C2×D25111yescharacteristic
108#124—5021no—
109#121—5021no—
110#121—2512no—
111C4×D4—5021no—
112C4×D4—2512no—
113C2×C42—2511no—
114C4⋊C8—2511no—
115C4⋊C8—2511no—
116C4⋊C8C2×D25111yescharacteristic
117C4⋊C8C2×D25111yescharacteristic
118C22⋊C8—5021no—
119C22⋊C8—5021no—
120#97—5021no—
121C4×C8—5021no—
122#41—10042no—
123#41—10042no—
124C2×Q8—5021no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—5022no—
128C2×D4C4×D25112yes—
129C22×C4—5022no—
130C22×C4—2511no—
131C22×C4—2511no—
132C22×C4—2511no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—5021no—
136C2×C8—5021no—
137C4⋊C4—5021no—
138C4⋊C4—2512no—
139C22⋊C4—5022no—
140C22⋊C4—5022no—
141C42—2511no—
142C42—2511no—
143C42—2511no—
144C42C22×D25111yescharacteristic
145C23—22no—
146C23—2511no—
147Q8—10041no—
148Q8—10041no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—10044no—
155C2×C4—10041no—
156C2×C4—10044no—
157C2×C4—5022no—
158C2×C4—5021no—
159C2×C4—5021no—
160C2×C4—5021no—
161C2×C4—5021no—
162C2×C4—2511no—
163C2×C4—2511no—
164C2×C4—2511no—
165C2×C4C2×C4×D25111yescharacteristic
166C2×C4D4×D25111yescharacteristic
167C2×C4D4×D25111yescharacteristic
168C8—41no—
169C8—41no—
170C8—10041no—
171C8—10041no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—5021no—
176C22—5021no—
177C22—5021no—
178C22C22⋊C4×D25111yescharacteristic
179C4—21no—
180C4—10042no—
181C4—5021no—
182C4—5021no—
183C4—5021no—
184C4#2254873411yescharacteristic
185C4#2254873611yescharacteristic
186C4#2254873911yescharacteristic
187C4#2254874211yescharacteristic
188C2—82no—
189C2—10041no—
190C2#3063166811yescharacteristic
191C2#3063168011yescharacteristic
192C2#3063169011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
Fitting#331×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632235C2
cosocleC42×C251C23
hypercenter#331D251
hypocenterC251#1553
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #30631743C2semidirect
3. #30631739C2semidirect
4. #30631434C2semidirect
5. #30631432C2semidirect
6. #30631431C2semidirect
7. #331×C251C2semidirect
96. C251#1553semidirect
104. #331D251semidirect

Maximal subgroups

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

subgroupindex
#306322352
#306317432
#306317392
#306314342
#306314322
#306314312
#331×C2512
#1553251