Finite Group Databasebeta

← #44550844

Subgroups — #44550844

Order 32128.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#22548743C211yesmaximal, characteristic
2#30632453C211yesmaximal, characteristic, complemented
3#30631705C211yesmaximal, characteristic, complemented
4#30631691C211yesmaximal, characteristic, complemented
5C2×#22548631C211yesmaximal, characteristic, complemented
6C251⋊#338C211yesmaximal, characteristic, complemented
7#344×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×D1004C2211yescharacteristic
10#22548743—22no—
11#22548743—22no—
12C2×#16860799C412yescomplemented
13C2×D2008C412yescomplemented
14#22548723—22no—
15#22548721—22no—
16C2×#16860794C2211yescharacteristic
17C2×#16860794C2211yescharacteristic
18M16×D251—22no—
19#22548643—22no—
20#22548635—22no—
21#22548631—22no—
22C2×M16×C251C2211yescharacteristic
23C2×M16×C251C2211yescharacteristic
24#117×C251C2211yescharacteristic
25C23×D251—21no—
26#16860810—21no—
27#16860810—21no—
28#16860810D412yes—
29#16860810D412yes—
30C2×Dic502C2×C411yescharacteristic
31C2×D1004—21no—
32C2×D1004—21no—
33C2×D1004—21no—
34C2×D1004—21no—
35C2×D1004—21no—
36C2×D1004C2×C412yes—
37C2×D1004C2×C411yescharacteristic
38C2×C22⋊D251—21no—
39C2×C4×D251—21no—
40#16860799—24no—
41D2008—24no—
42#16860794—21no—
43#16860794—21no—
44#16860794—21no—
45#16860794—21no—
46#16860793—24no—
47C2×C251⋊C8—21no—
48C2×C251⋊C8D412yes—
49C8×D251—24no—
50C22×C4×C251C2311yescharacteristic
51M16×C251—21no—
52M16×C251—21no—
53M16×C251—21no—
54M16×C251D412yes—
55C2×C8×C251—21no—
56C2×C8×C251C2×C412yes—
57C4⋊C4×C251—21no—
58C22⋊C4×C251—21no—
59C42×C251—21no—
60C22×D251—21no—
61C22×D251—22no—
62C22×D251—41no—
63C22×D251—41no—
64C22×D251—82no—
65Dic502—21no—
66Dic502C22⋊C412yes—
67D1004—22no—
68D1004—21no—
69D1004—22no—
70D1004—41no—
71D1004—41no—
72D1004C22⋊C412yes—
73C22⋊D251—22no—
74C22⋊D251—22no—
75C2×Dic251—21no—
76C4×D251—22no—
77C4×D251—22no—
78C251⋊C8—22no—
79C251⋊C8—41no—
80C23×C251#4111yescharacteristic
81C2×C4×C251—41no—
82C2×C4×C251C22×C411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251#4111yescharacteristic
88C8×C251—22no—
89C8×C251—22no—
90C22×C251—41no—
91C22×C251C4×D411yescharacteristic
92C22×C251#12111yescharacteristic
93C22×C251#12311yescharacteristic
94C2×D251—22no—
95C2×D251—42no—
96C2×D251—81no—
97C2×D251—82no—
98C2×D251—82no—
99C2×D251—82no—
100C4×C251—81no—
101C4×C251C2×C22⋊C411yescharacteristic
102C4×C251C4×D411yescharacteristic
103C4×C251#12011yescharacteristic
104C4×C251#12111yescharacteristic
105Dic251—22no—
106C2×C251—21no—
107C2×C251—21no—
108C2×C251—21no—
109C2×C251#38611yescharacteristic
110D251—42no—
111D251—82no—
112C251#190311yescharacteristic, complemented, Sylow
113#1903—2511nomaximal, Sylow
114C2×AGL(1,ℤ/8)—2511no—
115#568—2511no—
116#421—2511no—
117#418—2511no—
118C2×#100—2511no—
119#344D25111yescharacteristic, complemented
120#338—2511no—
121C2×#41—2511no—
122C22×D4—2511no—
123AGL(1,ℤ/8)—5022no—
124AGL(1,ℤ/8)—5022no—
125C2×SD16—2512no—
126C2×D8—2512no—
127#131—5022no—
128#131—5022no—
129C2×M16—5022no—
130C2×M16—2511no—
131C2×M16—2511no—
132C2×M16C2×D25111yescharacteristic
133C2×M16C2×D25111yescharacteristic
134#117C2×D25111yescharacteristic
135#104—5022no—
136#102—5022no—
137#100—5022no—
138C24—5021no—
139#41—5021no—
140#41—5021no—
141#41—2512no—
142#41—2512no—
143C2×Q8—2511no—
144C2×D4—5021no—
145C2×D4—5021no—
146C2×D4—5021no—
147C2×D4—5021no—
148C2×D4—5021no—
149C2×D4—5021no—
150C2×D4—2512no—
151C2×D4—2511no—
152C22×C4—5021no—
153C22×C4C22×D25111yescharacteristic
154SD16—5024no—
155D8—5024no—
156M16—21no—
157M16—21no—
158M16—21no—
159M16—5024no—
160M16—5021no—
161M16—5021no—
162M16—5021no—
163M16—5021no—
164M16D100412yes—
165C2×C8—21no—
166C2×C8—5024no—
167C2×C8—5021no—
168C2×C8—2512no—
169C2×C8C4×D25112yes—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—20082no—
174C23—10041no—
175C23—10041no—
176C23—5022no—
177C23—5021no—
178C23#1686081011yescharacteristic
179Q8—5021no—
180Q8—2512no—
181D4—10041no—
182D4—10041no—
183D4—5022no—
184D4—5022no—
185D4—5022no—
186D4—5022no—
187D4—5021no—
188D4—2512no—
189C2×C4—41no—
190C2×C4—5021no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4C2×C4×D25111yescharacteristic
194C2×C4D4×D25111yescharacteristic
195C2×C4D4×D25111yescharacteristic
196C2×C4D4×D25111yescharacteristic
197C2×C4C2×D100411yescharacteristic
198C2×C4#1686081111yescharacteristic
199C8—22no—
200C8—22no—
201C8—10041no—
202C8—5022no—
203C22—41no—
204C22—20082no—
205C22—20082no—
206C22—20082no—
207C22—20081no—
208C22—10042no—
209C22—5022no—
210C22C4×D100411yescharacteristic
211C22#2254868811yescharacteristic
212C22#2254870011yescharacteristic
213C4—81no—
214C4—5022no—
215C4C22⋊C4×D25111yescharacteristic
216C4#2254867711yescharacteristic
217C4#2254868511yescharacteristic
218C4#2254869011yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—20082no—
223C2—10042no—
224C2#3063156511yescharacteristic
subgroupG/H
centerC2#30631565
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#344×C251C2
socleC2×C251#386
prime radicalC2×#22548743C2
cosocleC22×C4×C251C23
hypercenter#344D251
hypocenterC251#1903
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #30632453C2semidirect
3. #30631705C2semidirect
4. #30631691C2semidirect
5. C2×#22548631C2semidirect
6. C251⋊#338C2semidirect
7. #344×C251C2semidirect
12. C2×#16860799C4semidirect
13. C2×D2008C4semidirect
112. C251#1903semidirect
119. #344D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
#306324532
#306317052
#306316912
C2×#225486312
C251⋊#3382
#344×C2512
#1903251