Finite Group Databasebeta

← #44559846

Subgroups — #44559846

Order 32128.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548743C212yesmaximal, complemented
2C2×#22548712C211yesmaximal, characteristic, complemented
3C4×#16860810C211yesmaximal, characteristic, complemented
4#30631888C214yesmaximal, complemented
5#30631883C214yesmaximal, complemented
6#30631693C212yesmaximal, complemented
7#423×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×D1004—22no—
10C22×D1004C2212yes—
11#22548743—28no—
12C2×#16860799C2214yescomplemented
13C2×D2008C2214yescomplemented
14#22548712—21no—
15#22548712—21no—
16#22548712C2214yes—
17C4×Dic502C2212yescomplemented
18C4×D1004C2212yescomplemented
19C4×C22⋊D251—22no—
20#22548674C2211yescharacteristic
21#22548672—21no—
22C42×D251—21no—
23#22548637C2218yescomplemented
24C2×M16×C251C2212yes—
25C2×C42×C251C2211yescharacteristic
26C4⋊C8×C251C2214yescomplemented
27C23×D251—22no—
28#16860810—22no—
29#16860810D414yescomplemented
30C2×Dic502C2311yescharacteristic
31C2×D1004—24no—
32C2×D1004—24no—
33C2×D1004—24no—
34C2×D1004—22no—
35C2×D1004—24no—
36C2×D1004—48no—
37C2×D1004C2314yes—
38C2×D1004C2311yescharacteristic
39C2×C22⋊D251—21no—
40C2×C4×D251—21no—
41C2×C4×D251—22no—
42#16860799—28no—
43D2008—28no—
44C4⋊Dic251C2312yes—
45Dic251⋊C4—22no—
46C22⋊Dic251—21no—
47(C2×D251)⋊C4—22no—
48C4×Dic251—21no—
49C4×Dic251—22no—
50C22×C4×C251D412yes—
51C22×C4×C251C2311yescharacteristic
52M16×C251—24no—
53C2×C8×C251C2314yes—
54C42×C251D411yescharacteristic
55C42×C251D411yescharacteristic
56C42×C251C2312yes—
57C22×D251—21no—
58C22×D251—24no—
59C22×D251—48no—
60C22×D251—44no—
61Dic502—21no—
62Dic502C2×D412yescomplemented
63D1004—24no—
64D1004—21no—
65D1004—24no—
66D1004—48no—
67D1004—44no—
68D1004—48no—
69D1004C2×D412yescomplemented
70C22⋊D251—24no—
71C2×Dic251—21no—
72C2×Dic251—22no—
73C4×D251—24no—
74C4×D251—44no—
75C23×C251C2×D411yescharacteristic
76C2×C4×C251—24no—
77C2×C4×C251C2×D412yes—
78C2×C4×C251C2×D412yes—
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D412yes—
81C2×C4×C251#4112yes—
82C2×C4×C251C2411yescharacteristic
83C8×C251—24no—
84C22×C251—22no—
85C22×C251#12112yes—
86C22×C251C22×D411yescharacteristic
87C2×D251—22no—
88C2×D251—42no—
89C2×D251—44no—
90C2×D251—48no—
91C2×D251—88no—
92C4×C251—22no—
93C4×C251#12112yes—
94C4×C251AGL(1,ℤ/8)14yescomplemented
95C4×C251C22×D411yescharacteristic
96C4×C251C2×#4111yescharacteristic
97Dic251—22no—
98Dic251—42no—
99C2×C251—22no—
100C2×C251C2×#12111yescharacteristic
101C2×C251C2×AGL(1,ℤ/8)12yes—
102D251—42no—
103D251—84no—
104C251#303811yescharacteristic, complemented, Sylow
105#3038—2511nomaximal, Sylow
106C2×AGL(1,ℤ/8)—2512no—
107C2×#127—2511no—
108C4×#41—2511no—
109#460—2514no—
110#459—2514no—
111#423D25111yescharacteristic, complemented
112#418—2512no—
113C2×#41—2511no—
114C22×D4—5022no—
115C22×D4—2512no—
116AGL(1,ℤ/8)—5028no—
117C2×SD16—2514no—
118C2×D8—2514no—
119C2×M16C2×D25112yes—
120#127—5021no—
121#127—5021no—
122#127—2514no—
123C4×Q8—2512no—
124C4×D4—5022no—
125C4×D4—2512no—
126#117—5021no—
127#117—2511no—
128C2×C42—5021no—
129C2×C42C2×D25111yescharacteristic
130C4⋊C8C2×D25114yescomplemented
131#102—2518no—
132C24—5022no—
133#41—5022no—
134#41—2514no—
135C2×Q8—2511no—
136C2×D4—10048no—
137C2×D4—5021no—
138C2×D4—5024no—
139C2×D4—5022no—
140C2×D4—5024no—
141C2×D4—5024no—
142C2×D4—5024no—
143C2×D4—2514no—
144C2×D4—2511no—
145C22×C4—5022no—
146C22×C4—5021no—
147C22×C4D100412yes—
148C22×C4C22×D25111yescharacteristic
149SD16—5028no—
150D8—5028no—
151M16—24no—
152C2×C8C22×D25114yes—
153C4⋊C4—5022no—
154C4⋊C4—2512no—
155C22⋊C4—5021no—
156C22⋊C4—5022no—
157C42—5021no—
158C42—5022no—
159C42D100411yescharacteristic
160C42D100411yescharacteristic
161C42C22×D25112yes—
162C23—10048no—
163C23—10044no—
164C23—5024no—
165C23—5021no—
166C23C2×D100411yescharacteristic
167Q8—5021no—
168Q8—2512no—
169D4—10048no—
170D4—10048no—
171D4—10044no—
172D4—5024no—
173D4—5024no—
174D4—5024no—
175D4—5021no—
176D4—2512no—
177C2×C4—24no—
178C2×C4—10044no—
179C2×C4—5022no—
180C2×C4—5021no—
181C2×C4—5024no—
182C2×C4D4×D25112yes—
183C2×C4C2×D100412yes—
184C2×C4C2×D100412yes—
185C2×C4C2×D100411yescharacteristic
186C2×C4#1686081112yes—
187C2×C4C23×D25111yescharacteristic
188C8—24no—
189C22—22no—
190C22—20088no—
191C22—10048no—
192C22—10044no—
193C22—10042no—
194C22—5022no—
195C22#2254869012yes—
196C22C22×D100411yescharacteristic
197C4—22no—
198C4—10042no—
199C4—5022no—
200C4#2254869012yes—
201C4#2254874314yescomplemented
202C4C2×D4×D25111yescharacteristic
203C4C2×#1686081111yescharacteristic
204C2—22no—
205C2—20084no—
206C2—10042no—
207C2C2×#2254869011yescharacteristic
208C2C2×#2254874312yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#423×C251C2
socleC22×C251C22×D4
hypercenter#423D251
hypocenterC251#3038
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×#22548712C2semidirect
3. C4×#16860810C2semidirect
4. #30631888C2semidirect
5. #30631883C2semidirect
6. #30631693C2semidirect
7. #423×C251C2semidirect
12. C2×#16860799C22semidirect
13. C2×D2008C22semidirect
17. C4×Dic502C22semidirect
18. C4×D1004C22semidirect
23. #22548637C22semidirect
26. C4⋊C8×C251C22semidirect
29. #16860810D4semidirect
62. Dic502C2×D4semidirect
69. D1004C2×D4semidirect
94. C4×C251AGL(1,ℤ/8)semidirect
104. C251#3038semidirect
111. #423D251semidirect
130. C4⋊C8C2×D251semidirect
201. C4#22548743semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487122
C4×#168608102
#306318882
#306318832
#306316932
#423×C2512
#3038251