Finite Group Databasebeta

← #44550314

Subgroups — #44550314

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#30631669C212yesmaximal, complemented
3#30631655C212yesmaximal, complemented
4#30631654C211yesmaximal, characteristic, complemented
5#413×C251C211yesmaximal, characteristic, complemented
6#22548763—21no—
7#22548762—22no—
8#22548761—22no—
9#22548760—21no—
10#22548760C414yes—
11C2×#16860810—22no—
12C2×#16860810C2212yes—
13#41×D251—24no—
14C2×#16860794C2211yescharacteristic, complemented
15C22×C251⋊C8C2211yescharacteristic, complemented
16#22548633—22no—
17#22548631—22no—
18C22⋊(C251⋊C8)—22no—
19(C2×D251)⋊C8—24no—
20C2×#41×C251C2211yescharacteristic
21C2×M16×C251C2212yescomplemented
22#101×C251—21no—
23#100×C251—21no—
24#16860811—24no—
25#16860810—24no—
26#16860810—24no—
27#16860810—24no—
28#16860810—22no—
29#16860810—22no—
30#16860810—22no—
31#16860810—22no—
32#16860809—24no—
33#16860809—28no—
34C2×Dic502—22no—
35C2×Dic502D412yes—
36Q8×D251—24no—
37C2×D1004—22no—
38C2×D1004D412yes—
39C2×C22⋊D251—24no—
40C2×C22⋊D251C2×C414yes—
41D4×D251—24no—
42D4×D251—28no—
43C2×C4×D251—24no—
44C2×C4×D251D414yes—
45#16860794—22no—
46C2×C251⋊C8—22no—
47C2×C251⋊C8—21no—
48C2×C251⋊C8—21no—
49#41×C251—22no—
50#41×C251—22no—
51C2×Q8×C251D411yescharacteristic
52C2×D4×C251C2×C412yes—
53C2×D4×C251D411yescharacteristic
54C22×C4×C251D412yes—
55C22×C4×C251C2311yescharacteristic
56M16×C251—22no—
57M16×C251—22no—
58C2×C8×C251—22no—
59C22×D251—24no—
60C22×D251C22⋊C414yes—
61Dic502—24no—
62Dic502—22no—
63Dic502—22no—
64D1004—24no—
65D1004—22no—
66D1004—22no—
67C22⋊D251—28no—
68C22⋊D251—24no—
69C22⋊D251—24no—
70C2×Dic251—24no—
71C2×Dic251C22⋊C414yes—
72C4×D251—28no—
73C4×D251—24no—
74C4×D251—24no—
75C251⋊C8—41no—
76C251⋊C8—41no—
77C23×C251C22⋊C412yes—
78C23×C251C22×C411yescharacteristic
79Q8×C251—22no—
80D4×C251—24no—
81D4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251C22⋊C412yes—
85C2×C4×C251C2×D412yes—
86C2×C4×C251C2×D412yes—
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2×D411yescharacteristic
89C8×C251—42no—
90C22×C251—22no—
91C22×C251—41no—
92C22×C251—42no—
93C22×C251C2×C22⋊C412yes—
94C22×C251C2×C22⋊C411yescharacteristic
95C2×D251—24no—
96C2×D251—44no—
97C2×D251—84no—
98C4×C251—22no—
99C4×C251#12012yes—
100C4×C251#12011yescharacteristic
101C4×C251#12011yescharacteristic
102Dic251—24no—
103C2×C251—22no—
104C2×C251—21no—
105C2×C251—42no—
106C2×C251#37911yescharacteristic
107D251—44no—
108C251#179111yescharacteristic, complemented, Sylow
109#1791—2511nomaximal, Sylow
110#585—2511no—
111#413—2512no—
112#413D25111yescharacteristic, complemented
113#408—2511no—
114#408—2512no—
1152−1+4—5022no—
1162−1+4—5021no—
117H(5,𝔽2)—5022no—
118H(5,𝔽2)—5021no—
119H(5,𝔽2)—2514no—
120C2×#41—5024no—
121C2×#41—5022no—
122C2×#41—2512no—
123C2×#41C2×D25111yescharacteristic
124C2×M16—2511no—
125C2×M16C2×D25112yescomplemented
126C22×C8—2511no—
127#101—21no—
128#101—5022no—
129#100—21no—
130#100—5022no—
131C22⋊C8—5024no—
132C22⋊C8—5022no—
133#41—22no—
134#41—22no—
135#41—5024no—
136#41—5024no—
137#41—5024no—
138#41—5024no—
139#41—5028no—
140#41—5024no—
141#41—5022no—
142#41—5022no—
143#41—5022no—
144#41—5022no—
145C2×Q8—5022no—
146C2×Q8—5024no—
147C2×Q8—2512no—
148C2×Q8C22⋊D25111yescharacteristic
149C2×D4—5022no—
150C2×D4—5024no—
151C2×D4—5028no—
152C2×D4—5024no—
153C2×D4—2514no—
154C2×D4—2512no—
155C2×D4C4×D25112yes—
156C2×D4C22⋊D25111yescharacteristic
157C22×C4—5024no—
158C22×C4—2514no—
159C22×C4D100412yes—
160C22×C4C22×D25111yescharacteristic
161M16—22no—
162M16—22no—
163M16—5022no—
164C2×C8—22no—
165C2×C8—5022no—
166C2×C8—5021no—
167C2×C8—5021no—
168C23—5024no—
169C23—2514no—
170C23(C2×D251)⋊C412yes—
171C23C2×C4×D25111yescharacteristic
172Q8—22no—
173Q8—5024no—
174Q8—5022no—
175Q8—5022no—
176D4—24no—
177D4—22no—
178D4—5024no—
179D4—5028no—
180D4—5024no—
181D4—5024no—
182D4—5022no—
183D4—5022no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—5028no—
187C2×C4—5024no—
188C2×C4—5024no—
189C2×C4—5024no—
190C2×C4—2514no—
191C2×C4(C2×D251)⋊C412yes—
192C2×C4D4×D25112yes—
193C2×C4D4×D25112yes—
194C2×C4C2×C22⋊D25111yescharacteristic
195C2×C4C2×D100411yescharacteristic
196C8—42no—
197C8—10041no—
198C8—10041no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—20084no—
203C22—10044no—
204C22—5024no—
205C22C22⋊C4×D25112yes—
206C22C2×(C2×D251)⋊C411yescharacteristic
207C4—22no—
208C4—5024no—
209C4#2254868312yes—
210C4#2254868511yescharacteristic
211C4#2254868511yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—10044no—
216C2#3063154211yescharacteristic
subgroupG/H
centerC4#22548685
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#413×C251C2
socleC2×C251#379
prime radical#30632523C2
cosocleC22×C4×C251C23
hypercenter#413D251
hypocenterC251#1791
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #30631669C2semidirect
3. #30631655C2semidirect
4. #30631654C2semidirect
5. #413×C251C2semidirect
14. C2×#16860794C22semidirect
15. C22×C251⋊C8C22semidirect
21. C2×M16×C251C22semidirect
108. C251#1791semidirect
112. #413D251semidirect
125. C2×M16C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306316692
#306316552
#306316542
#413×C2512
#1791251