Finite Group Databasebeta

← #44559895

Subgroups — #44559895

Order 32128.

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

normal subgroups only

This group has 458 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×#22548746C211yesmaximal, characteristic
2C22×#16860800C211yesmaximal, characteristic, complemented
3#30632263C211yesmaximal, characteristic
4#30631909C214yesmaximal, complemented
5#30631897C214yesmaximal, complemented
6#30631714C211yesmaximal, characteristic, complemented
7#30631695C211yesmaximal, characteristic, complemented
8#30631681C211yesmaximal, characteristic
9#519×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12#22548746—24no—
13C2×#16860800—24no—
14C2×#16860800C2214yes—
15C2×#16860800C2212yes—
16C2×#16860797C2212yes—
17C2×#16860794C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic
19#22548715C2212yes—
20#22548703C2212yes—
21#22548699—22no—
22#22548695—22no—
23C4⋊(C251⋊C8)C2214yes—
24#22548638C2212yes—
25#22548638C2214yes—
26#22548635C2212yes—
27C22×Q8×C251C2211yescharacteristic
28C4×Q8×C251—22no—
29C4×Q8×C251C2214yes—
30#117×C251—22no—
31#117×C251C2211yescharacteristic
32#117×C251C2211yescharacteristic
33#16860810—22no—
34#16860810—22no—
35C2×Dic502—21no—
36C2×Dic502—22no—
37C2×Dic502—22no—
38C2×Dic502C2312yes—
39C2×Dic502C2311yescharacteristic
40C2×D1004C2311yescharacteristic
41C2×C22⋊D251—21no—
42C22×Dic251—21no—
43C2×C4×D251—21no—
44#16860800—24no—
45#16860800—28no—
46#16860797—24no—
47#16860794—22no—
48C2×C251⋊C8—22no—
49C2×C251⋊C8C2312yes—
50C2×C251⋊C8C2312yes—
51C4⋊Dic251—24no—
52(C2×D251)⋊C4—24no—
53C2×Q8×C251—21no—
54C2×Q8×C251—22no—
55C2×Q8×C251D414yes—
56C2×Q8×C251C2312yes—
57C22×C4×C251—21no—
58C22×C4×C251C2311yescharacteristic
59C4⋊C4×C251—24no—
60C4⋊C4×C251D412yes—
61C4⋊C4×C251C2312yes—
62C22⋊C4×C251—21no—
63C22⋊C4×C251D412yes—
64C42×C251—24no—
65C42×C251—21no—
66C42×C251C2312yes—
67C22×D251—21no—
68Dic502—22no—
69Dic502—21no—
70Dic502—22no—
71Dic502—21no—
72Dic502—42no—
73D1004—21no—
74D1004—21no—
75C22⋊D251—42no—
76C2×Dic251—22no—
77C2×Dic251—21no—
78C2×Dic251—42no—
79C4×D251—42no—
80C251⋊C8—22no—
81C251⋊C8—22no—
82C23×C251C2×D411yescharacteristic
83Q8×C251—24no—
84Q8×C251—22no—
85Q8×C251C2×D414yes—
86C2×C4×C251—24no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251C2×D414yes—
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2×D412yes—
92C2×C4×C251#4112yes—
93C2×C4×C251C2411yescharacteristic
94C22×C251—22no—
95C22×C251#12112yes—
96C22×C251C22×D411yescharacteristic
97C2×D251—41no—
98C2×D251—42no—
99C4×C251—24no—
100C4×C251—24no—
101C4×C251—42no—
102C4×C251#12112yes—
103C4×C251C22×D411yescharacteristic
104C4×C251C2×#4111yescharacteristic
105Dic251—41no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251C2×#12111yescharacteristic
109C2×C251#57812yes—
110D251—81no—
111C251#304411yescharacteristic, complemented, Sylow
112#3044—2511nomaximal, Sylow
113C2×#137—2511no—
114C22×Q16—2511no—
115#536—2511no—
116#519D25111yescharacteristic, complemented
117#464—2514no—
118#462—2514no—
119#424—2511no—
120#419—2511no—
121#416—2511no—
122C2×#41—2511no—
123C22×Q8—2511no—
124C22×Q8C2×D25111yescharacteristic
125#137—5024no—
126C2×Q16—5024no—
127C2×Q16—2514no—
128C2×Q16—2512no—
129C2×SD16—2512no—
130C2×M16—2511no—
131C22×C8—2511no—
132C4⋊Q8—2512no—
133#124—2512no—
134#123—5022no—
135C22⋊Q8—5022no—
136C4×Q8—22no—
137C4×Q8C2×D25114yes—
138#117—22no—
139#117C2×D25111yescharacteristic
140#117C2×D25111yescharacteristic
141C4⋊C8—2514no—
142#103—2512no—
143#103—2514no—
144#102—2512no—
145#41—5022no—
146#41—5022no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—5022no—
150C2×Q8—5022no—
151C2×Q8—5021no—
152C2×Q8—2512no—
153C2×Q8—2511no—
154C2×Q8D100414yes—
155C2×Q8C22×D25112yes—
156C2×D4—5021no—
157C2×D4—2511no—
158C22×C4—21no—
159C22×C4—5021no—
160C22×C4—5021no—
161C22×C4C22×D25111yescharacteristic
162Q16—5028no—
163Q16—5024no—
164SD16—5024no—
165M16—5022no—
166C2×C8—5022no—
167C2×C8—2512no—
168C2×C8—2512no—
169C4⋊C4—24no—
170C4⋊C4—5024no—
171C4⋊C4C22⋊D25112yes—
172C4⋊C4C22×D25112yes—
173C22⋊C4—21no—
174C22⋊C4—5024no—
175C22⋊C4C22⋊D25112yes—
176C42—24no—
177C42—21no—
178C42C22×D25112yes—
179C23—5021no—
180C23C2×C22⋊D25111yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—10042no—
184Q8—5022no—
185Q8—5021no—
186Q8—5022no—
187Q8—5021no—
188Q8C2×D100414yes—
189D4—10042no—
190D4—5021no—
191D4—5021no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—10042no—
196C2×C4—10042no—
197C2×C4—5022no—
198C2×C4—5021no—
199C2×C4C2×C22⋊D25114yes—
200C2×C4C2×C22⋊D25111yescharacteristic
201C2×C4C2×D100412yes—
202C2×C4#1686081012yes—
203C2×C4C23×D25111yescharacteristic
204C8—5022no—
205C8—5022no—
206C22—22no—
207C22—10042no—
208C22—10041no—
209C22#2254868912yes—
210C22C22×C22⋊D25111yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—10041no—
215C4—10042no—
216C4#2254868912yes—
217C4C22×D100411yescharacteristic
218C4C2×#1686081011yescharacteristic
219C2—22no—
220C2—20081no—
221C2C2×#2254868911yescharacteristic
222C2#3063250212yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#519×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860810C22
hypercenter#519D251
hypocenterC251#3044
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#16860800C2semidirect
4. #30631909C2semidirect
5. #30631897C2semidirect
6. #30631714C2semidirect
7. #30631695C2semidirect
9. #519×C251C2semidirect
111. C251#3044semidirect
116. #519D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C22×#168608002
#306322632
#306319092
#306318972
#306317142
#306316952
#306316812
#519×C2512
#3044251