Finite Group Databasebeta

← #44550331

Subgroups — #44550331

Order 32128.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#30632523C211yesmaximal, characteristic
2#30631705C212yesmaximal, complemented
3#30631681C212yesmaximal, complemented
4#30631654C211yesmaximal, characteristic, complemented
5#410×C251C211yesmaximal, characteristic, complemented
6#22548763C412yescomplemented
7#22548762—22no—
8#22548761—22no—
9#22548760—22no—
10#22548760C412yescomplemented
11C2×#16860810—22no—
12C2×#16860810C2212yes—
13#41×D251—24no—
14C2×#16860794C2211yescharacteristic, complemented
15C22×C251⋊C8C2211yescharacteristic, complemented
16#22548643—24no—
17#22548638—22no—
18#22548635—22no—
19C22⋊(C251⋊C8)—22no—
20C2×#41×C251C2211yescharacteristic
21#117×C251C2212yescomplemented
22#99×C251—22no—
23#16860811—24no—
24#16860810—24no—
25#16860810—24no—
26#16860810—24no—
27#16860810—22no—
28#16860810—22no—
29#16860810C2×C414yes—
30#16860810D414yes—
31#16860809—28no—
32#16860809—24no—
33C2×Dic502—22no—
34C2×Dic502D412yes—
35Q8×D251—24no—
36C2×D1004—22no—
37C2×D1004D412yes—
38C2×C22⋊D251—24no—
39C2×C22⋊D251—22no—
40D4×D251—28no—
41D4×D251—24no—
42C2×C4×D251—24no—
43C2×C4×D251—22no—
44#16860794—22no—
45C2×C251⋊C8—21no—
46C2×C251⋊C8—21no—
47C2×C251⋊C8—22no—
48#41×C251—22no—
49#41×C251—22no—
50C2×Q8×C251C2×C411yescharacteristic
51C2×D4×C251C2×C411yescharacteristic
52C2×D4×C251D412yes—
53C22×C4×C251D412yes—
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—22no—
56C22⋊C4×C251—22no—
57C42×C251—22no—
58C22×D251—24no—
59C22×D251—22no—
60Dic502—24no—
61Dic502—22no—
62Dic502C22⋊C414yes—
63D1004—24no—
64D1004—22no—
65D1004C22⋊C414yes—
66C22⋊D251—28no—
67C22⋊D251—24no—
68C22⋊D251—24no—
69C2×Dic251—24no—
70C2×Dic251—22no—
71C4×D251—28no—
72C4×D251—24no—
73C4×D251—24no—
74C251⋊C8—41no—
75C251⋊C8—41no—
76C23×C251C22⋊C412yes—
77C23×C251C2×D411yescharacteristic
78Q8×C251—22no—
79D4×C251—24no—
80D4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—42no—
84C2×C4×C251C22⋊C412yes—
85C2×C4×C251C22×C411yescharacteristic
86C2×C4×C251C2×D412yes—
87C2×C4×C251C2×D412yes—
88C2×C4×C251C2×D411yescharacteristic
89C22×C251—22no—
90C22×C251—41no—
91C22×C251—42no—
92C22×C251C2×C22⋊C411yescharacteristic
93C22×C251#12012yes—
94C2×D251—24no—
95C2×D251—82no—
96C2×D251—84no—
97C4×C251—22no—
98C4×C251—82no—
99C4×C251C2×C22⋊C412yes—
100C4×C251#12011yescharacteristic
101C4×C251#12011yescharacteristic
102Dic251—24no—
103C2×C251—22no—
104C2×C251—21no—
105C2×C251—42no—
106C2×C251#37911yescharacteristic
107D251—44no—
108C251#179411yescharacteristic, complemented, Sylow
109#1794—2511nomaximal, Sylow
110#585—2511no—
111#421—2512no—
112#416—2512no—
113#410D25111yescharacteristic, complemented
114#408—2511no—
1152−1+4—5022no—
1162−1+4—2512no—
117H(5,𝔽2)—5022no—
118H(5,𝔽2)—5022no—
119H(5,𝔽2)—2512no—
120C2×#41—5024no—
121C2×#41—5022no—
122C2×#41—2512no—
123C2×#41C2×D25111yescharacteristic
124C2×M16—2511no—
125C22×C8—2511no—
126#117C2×D25112yescomplemented
127#104—5024no—
128#103—5022no—
129#102—5022no—
130#99—22no—
131C22⋊C8—5022no—
132#41—22no—
133#41—22no—
134#41—5024no—
135#41—5024no—
136#41—5028no—
137#41—5024no—
138#41—5024no—
139#41—5024no—
140#41—5022no—
141#41—5022no—
142#41—2514no—
143#41—2514no—
144C2×Q8—5022no—
145C2×Q8—5024no—
146C2×Q8—2512no—
147C2×Q8C4×D25111yescharacteristic
148C2×D4—5022no—
149C2×D4—5028no—
150C2×D4—5024no—
151C2×D4—5024no—
152C2×D4—5022no—
153C2×D4—2512no—
154C2×D4C4×D25111yescharacteristic
155C2×D4C22⋊D25112yes—
156C22×C4—5024no—
157C22×C4—5022no—
158C22×C4D100412yes—
159C22×C4C22×D25111yescharacteristic
160M16—5022no—
161C2×C8—5022no—
162C2×C8—5021no—
163C2×C8—5021no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—5024no—
168C23—5022no—
169C23(C2×D251)⋊C412yes—
170C23C2×C22⋊D25111yescharacteristic
171Q8—22no—
172Q8—5024no—
173Q8—5022no—
174Q8—2514no—
175D4—24no—
176D4—22no—
177D4—5024no—
178D4—5028no—
179D4—5024no—
180D4—5024no—
181D4—5022no—
182D4—2514no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—5028no—
187C2×C4—5024no—
188C2×C4—5022no—
189C2×C4—5024no—
190C2×C4—5024no—
191C2×C4(C2×D251)⋊C412yes—
192C2×C4C2×C4×D25111yescharacteristic
193C2×C4D4×D25112yes—
194C2×C4D4×D25112yes—
195C2×C4C2×D100411yescharacteristic
196C8—10041no—
197C8—10041no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—20082no—
202C22—20084no—
203C22—5024no—
204C22C2×(C2×D251)⋊C411yescharacteristic
205C22#2254868312yes—
206C4—22no—
207C4—82no—
208C4—5024no—
209C4C22⋊C4×D25112yes—
210C4#2254868511yescharacteristic
211C4#2254868511yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—10044no—
216C2#3063154211yescharacteristic
subgroupG/H
centerC4#22548685
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#410×C251C2
socleC2×C251#379
prime radical#30632523C2
cosocleC22×C4×C251C23
hypercenter#410D251
hypocenterC251#1794
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. #30631705C2semidirect
3. #30631681C2semidirect
4. #30631654C2semidirect
5. #410×C251C2semidirect
6. #22548763C4semidirect
10. #22548760C4semidirect
14. C2×#16860794C22semidirect
15. C22×C251⋊C8C22semidirect
21. #117×C251C22semidirect
108. C251#1794semidirect
113. #410D251semidirect
126. #117C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306317052
#306316812
#306316542
#410×C2512
#1794251