Finite Group Databasebeta

← #44550845

Subgroups — #44550845

Order 32128.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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×#22548740C211yesmaximal, characteristic
2#30631704C211yesmaximal, characteristic, complemented
3#30631693C211yesmaximal, characteristic, complemented
4C2×#22548631C211yesmaximal, characteristic, complemented
5#30631459C211yesmaximal, characteristic, complemented
6C251⋊#338C211yesmaximal, characteristic, complemented
7#568×C251C211yesmaximal, characteristic, complemented
8C22×D1004C2211yescharacteristic
9#22548740—22no—
10#22548740—22no—
11C2×#16860797C412yescomplemented
12C2×#16860795C412yescomplemented
13C2×#16860794C2211yescharacteristic
14C2×#16860794C2211yescharacteristic
15#22548674C2211yescharacteristic
16#22548642—22no—
17#22548637—22no—
18#22548631—22no—
19C2×#41×C251C2211yescharacteristic
20#131×C251—22no—
21#131×C251—22no—
22C2×M16×C251—22no—
23C2×M16×C251C2211yescharacteristic
24C2×M16×C251C2211yescharacteristic
25C23×D251—21no—
26C2×D1004—21no—
27C2×D1004—21no—
28C2×D1004—21no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004C2×C412yes—
32#16860797—24no—
33#16860795—24no—
34#16860794—21no—
35#16860794—21no—
36#16860794—21no—
37#16860794—21no—
38C2×C251⋊C8—21no—
39C2×C251⋊C8C2×C412yes—
40C4⋊Dic251—21no—
41C22⋊Dic251—21no—
42C4×Dic251—21no—
43#41×C251—21no—
44#41×C251—21no—
45#41×C251D412yes—
46#41×C251D412yes—
47C2×Q8×C251C2×C411yescharacteristic
48C2×D4×C251—21no—
49C2×D4×C251C2×C411yescharacteristic
50C22×C4×C251—21no—
51C22×C4×C251C2311yescharacteristic
52M16×C251—24no—
53M16×C251—21no—
54M16×C251—21no—
55M16×C251—21no—
56M16×C251D412yes—
57C2×C8×C251—24no—
58C2×C8×C251—21no—
59C2×C8×C251D412yes—
60C22×D251—22no—
61C22×D251—41no—
62C22×D251—41no—
63C22×D251—82no—
64D1004—22no—
65D1004—22no—
66D1004—41no—
67D1004—41no—
68C2×Dic251—41no—
69C251⋊C8—22no—
70C251⋊C8—41no—
71C23×C251—21no—
72C23×C251#4111yescharacteristic
73Q8×C251—21no—
74Q8×C251C22⋊C412yes—
75D4×C251—22no—
76D4×C251—22no—
77D4×C251—21no—
78D4×C251C22⋊C412yes—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
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—22no—
91C22×C251—41no—
92C22×C251—81no—
93C22×C251C4×D411yescharacteristic
94C22×C251#12111yescharacteristic
95C22×C251#12311yescharacteristic
96C2×D251—42no—
97C2×D251—82no—
98C2×D251—82no—
99C2×D251—82no—
100C4×C251—22no—
101C4×C251C2×C22⋊C411yescharacteristic
102C4×C251C4×D411yescharacteristic
103C4×C251#12011yescharacteristic
104C4×C251#12111yescharacteristic
105Dic251—81no—
106C2×C251—21no—
107C2×C251—21no—
108C2×C251—21no—
109C2×C251—42no—
110C2×C251#38611yescharacteristic
111D251—82no—
112C251#190311yescharacteristic, complemented, Sylow
113#1903—2511nomaximal, Sylow
114C2×AGL(1,ℤ/8)—2511no—
115#568D25111yescharacteristic, complemented
116#421—2511no—
117#418—2511no—
118C2×#100—2511no—
119#344—2511no—
120#338—2511no—
121C2×#41C2×D25111yescharacteristic
122C22×D4—2511no—
123AGL(1,ℤ/8)—5022no—
124AGL(1,ℤ/8)—5022no—
125C2×SD16—2512no—
126C2×D8—2512no—
127#131—22no—
128#131—22no—
129C2×M16—22no—
130C2×M16—2511no—
131C2×M16—2511no—
132C2×M16C2×D25111yescharacteristic
133C2×M16C2×D25111yescharacteristic
134#117—2511no—
135#104—5022no—
136#102—5022no—
137#100—5022no—
138C24—5021no—
139#41—21no—
140#41—21no—
141#41C22⋊D25112yes—
142#41D100412yes—
143C2×Q8C4×D25111yescharacteristic
144C2×D4—21no—
145C2×D4—5021no—
146C2×D4—5021no—
147C2×D4—5021no—
148C2×D4—5021no—
149C2×D4—5021no—
150C2×D4—2512no—
151C2×D4C4×D25111yescharacteristic
152C22×C4—21no—
153C22×C4C22×D25111yescharacteristic
154SD16—5024no—
155D8—5024no—
156M16—24no—
157M16—21no—
158M16—21no—
159M16—21no—
160M16—5021no—
161M16—5021no—
162M16—5021no—
163M16—5021no—
164M16C22⋊D25112yes—
165C2×C8—24no—
166C2×C8—21no—
167C2×C8—5021no—
168C2×C8—2512no—
169C2×C8C22⋊D25112yes—
170C4⋊C4—5021no—
171C22⋊C4—5021no—
172C42—5021no—
173C23—21no—
174C23—20082no—
175C23—10041no—
176C23—10041no—
177C23—5022no—
178C23#1686081011yescharacteristic
179Q8—21no—
180Q8(C2×D251)⋊C412yes—
181D4—22no—
182D4—22no—
183D4—21no—
184D4—10041no—
185D4—10041no—
186D4—5022no—
187D4—5022no—
188D4(C2×D251)⋊C412yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—10041no—
193C2×C4C2×C4×D25111yescharacteristic
194C2×C4C2×C22⋊D25111yescharacteristic
195C2×C4C2×C22⋊D25111yescharacteristic
196C2×C4C2×C22⋊D25111yescharacteristic
197C2×C4C2×D100411yescharacteristic
198C2×C4#1686081011yescharacteristic
199C8—22no—
200C8—22no—
201C8—10041no—
202C8—5022no—
203C22—22no—
204C22—41no—
205C22—81no—
206C22—20082no—
207C22—20082no—
208C22—20082no—
209C22—10042no—
210C22C4×C22⋊D25111yescharacteristic
211C22#2254868911yescharacteristic
212C22#2254869811yescharacteristic
213C4—22no—
214C4—20081no—
215C4C2×(C2×D251)⋊C411yescharacteristic
216C4C4×C22⋊D25111yescharacteristic
217C4#2254868411yescharacteristic
218C4#2254868911yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—20082no—
224C2#3063156611yescharacteristic
subgroupG/H
centerC2#30631566
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#568×C251C2
socleC2×C251#386
prime radicalC2×#22548740C2
cosocleC22×C4×C251C23
hypercenter#568D251
hypocenterC251#1903
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #30631704C2semidirect
3. #30631693C2semidirect
4. C2×#22548631C2semidirect
5. #30631459C2semidirect
6. C251⋊#338C2semidirect
7. #568×C251C2semidirect
11. C2×#16860797C4semidirect
12. C2×#16860795C4semidirect
112. C251#1903semidirect
115. #568D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
#306317042
#306316932
C2×#225486312
#306314592
C251⋊#3382
#568×C2512
#1903251