Finite Group Databasebeta

← #44560483

Subgroups — #44560483

Order 32128.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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×C4×Dic502C211yesmaximal, characteristic, complemented
2#30631998C212yesmaximal, complemented
3#30631979C212yesmaximal, complemented
4#30631884C212yesmaximal, complemented
5#30631825C212yesmaximal, complemented
6#30631787C214yesmaximal, complemented
7#30631744C211yesmaximal, characteristic, complemented
8#532×C251C211yesmaximal, characteristic
9C22×Dic502C2211yescharacteristic
10C2×#16860798C2214yes—
11C4×Dic502—21no—
12C4×Dic502—21no—
13C4×Dic502C2212yes—
14C4×Dic502C2212yes—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×Dic251⋊C4—21no—
17C2×C4×Dic251—21no—
18#22548648C2214yes—
19C4⋊(C251⋊C8)C2212yes—
20#22548638C2214yes—
21#22548636C2214yes—
22C22⋊(C251⋊C8)C2212yes—
23#22548626C2212yes—
24C2×#41×C251—21no—
25C4⋊Q8×C251C2211yescharacteristic
26#127×C251C2211yescharacteristic
27#124×C251—21no—
28#121×C251—21no—
29#121×C251C2212yes—
30C4×D4×C251—21no—
31C4×D4×C251C2212yes—
32C2×C42×C251C2211yescharacteristic
33C2×Dic502—24no—
34C2×Dic502—21no—
35C2×Dic502C2312yes—
36C22×Dic251—21no—
37C22×Dic251—21no—
38#16860798—28no—
39C2×C251⋊C8C2314yes—
40C4⋊Dic251—21no—
41C4⋊Dic251C2312yes—
42Dic251⋊C4—22no—
43Dic251⋊C4—22no—
44C4×Dic251—22no—
45C4×Dic251—22no—
46#41×C251—44no—
47C2×Q8×C251—21no—
48C2×D4×C251—22no—
49C2×D4×C251—21no—
50C2×D4×C251—21no—
51C2×D4×C251C2312yes—
52C22×C4×C251—22no—
53C22×C4×C251D411yescharacteristic
54C22×C4×C251D411yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C4⋊C4×C251—21no—
57C4⋊C4×C251C2312yes—
58C22⋊C4×C251—22no—
59C22⋊C4×C251—22no—
60C42×C251D411yescharacteristic
61C42×C251D411yescharacteristic
62C42×C251C2311yescharacteristic
63C42×C251C2311yescharacteristic
64Dic502—24no—
65Dic502—22no—
66Dic502#4114yescomplemented
67C2×Dic251—24no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—42no—
71C251⋊C8—24no—
72C23×C251—22no—
73C23×C251C2×D411yescharacteristic
74Q8×C251—42no—
75D4×C251—24no—
76D4×C251—42no—
77D4×C251—44no—
78D4×C251—42no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—44no—
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2×D411yescharacteristic
89C2×C4×C251C2411yescharacteristic
90C22×C251—21no—
91C22×C251—21no—
92C22×C251—42no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251AGL(1,ℤ/8)12yes—
96C22×C251C22×D411yescharacteristic
97C4×C251—21no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—21no—
101C4×C251—42no—
102C4×C251#13712yes—
103C4×C251C2×#4112yes—
104Dic251—24no—
105Dic251—42no—
106C2×C251—22no—
107C2×C251—82no—
108C2×C251#52511yescharacteristic
109C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C251C2×#13711yescharacteristic
111C251#311211yescharacteristic, complemented, Sylow
112#3112—2511nomaximal, Sylow
113#532D25111yescharacteristic
114C2×C4×Q8—2511no—
115#480—2512no—
116#477—2512no—
117#460—2512no—
118#448—2512no—
119#440—2514no—
120#432—2511no—
121C2×#41—21no—
122C22×Q8—2511no—
123C2×SD16—2514no—
124C4⋊Q8C2×D25111yescharacteristic
125#127C2×D25111yescharacteristic
126#124—21no—
127#121—21no—
128#121C2×D25112yes—
129C4×Q8—5021no—
130C4×Q8—5021no—
131C4×Q8—2512no—
132C4×Q8—2512no—
133C4×D4—21no—
134C4×D4C2×D25112yes—
135C2×C4⋊C4—5021no—
136C2×C4⋊C4—2511no—
137C2×C42—5021no—
138C2×C42C2×D25111yescharacteristic
139#107—2514no—
140C4⋊C8—2512no—
141#103—2514no—
142#102—2514no—
143C22⋊C8—2512no—
144#97—2512no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—5024no—
148C2×Q8—5021no—
149C2×Q8—2512no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D25112yes—
154C22×C4—22no—
155C22×C4—5021no—
156C22×C4—5021no—
157C22×C4C22⋊D25111yescharacteristic
158C22×C4C22⋊D25111yescharacteristic
159C22×C4C22×D25111yescharacteristic
160SD16—5028no—
161C2×C8—2514no—
162C4⋊C4—21no—
163C4⋊C4—5022no—
164C4⋊C4—5022no—
165C4⋊C4—5021no—
166C4⋊C4—2512no—
167C4⋊C4C22×D25112yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—5022no—
171C42—5022no—
172C42C22⋊D25111yescharacteristic
173C42C22⋊D25111yescharacteristic
174C42C22×D25111yescharacteristic
175C42C22×D25111yescharacteristic
176C23—22no—
177C23C2×C22⋊D25111yescharacteristic
178Q8—42no—
179Q8—5024no—
180Q8—5022no—
181Q8—2514no—
182D4—24no—
183D4—42no—
184D4—44no—
185D4—42no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—10042no—
192C2×C4—5022no—
193C2×C4—5024no—
194C2×C4—5022no—
195C2×C4C2×C22⋊D25111yescharacteristic
196C2×C4C2×C22⋊D25111yescharacteristic
197C2×C4C2×C22⋊D25111yescharacteristic
198C2×C4C2×C22⋊D25111yescharacteristic
199C2×C4C2×C22⋊D25111yescharacteristic
200C2×C4C23×D25111yescharacteristic
201C8—5024no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22#2254874112yes—
208C22C22×C22⋊D25111yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—10042no—
215C4—5024no—
216C4#2254874512yes—
217C4#41×D25112yes—
218C2—22no—
219C2—82no—
220C2#3063219411yescharacteristic
221C2C2×#2254874111yescharacteristic
222C2C2×#2254874511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting, prime radical#532×C251C2
socleC22×C251C22×D4
hypercenter#532D251
hypocenterC251#3112
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic502C2semidirect
2. #30631998C2semidirect
3. #30631979C2semidirect
4. #30631884C2semidirect
5. #30631825C2semidirect
6. #30631787C2semidirect
7. #30631744C2semidirect
66. Dic502#41semidirect
111. C251#3112semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic5022
#306319982
#306319792
#306318842
#306318252
#306317872
#306317442
#532×C2512
#3112251