Finite Group Databasebeta

← #44559891

Subgroups — #44559891

Order 32128.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22548745C211yesmaximal, characteristic
2C22×#16860800C211yesmaximal, characteristic, complemented
3#30632164C211yesmaximal, characteristic, complemented
4#30631906C214yesmaximal, complemented
5#30631894C214yesmaximal, complemented
6#30631714C211yesmaximal, characteristic, complemented
7#30631696C211yesmaximal, characteristic, complemented
8#30631683C211yesmaximal, characteristic
9#536×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11#22548745—24no—
12C2×#16860800—24no—
13C2×#16860800C2214yes—
14C2×#16860800C2212yes—
15C2×#16860798C2212yes—
16C2×#16860794C2211yescharacteristic
17C22×C251⋊C8C2211yescharacteristic
18C4×Dic502C2214yes—
19#22548680—22no—
20#22548674C2211yescharacteristic
21#22548671—22no—
22C4⋊(C251⋊C8)C2214yes—
23#22548639C2212yes—
24#22548639C2214yes—
25#22548636C2212yes—
26C2×#41×C251C2211yescharacteristic
27C22×Q8×C251C2211yescharacteristic
28C4⋊Q8×C251C2212yes—
29#124×C251C2212yes—
30#123×C251—22no—
31C22⋊Q8×C251—22no—
32#117×C251C2211yescharacteristic
33C2×Dic502—21no—
34C2×Dic502—22no—
35C2×Dic502D414yes—
36C2×Dic502C2312yes—
37C22×Dic251—21no—
38#16860800—24no—
39#16860800—28no—
40#16860798—24no—
41#16860794—22no—
42C2×C251⋊C8—22no—
43C2×C251⋊C8C2312yes—
44C2×C251⋊C8C2312yes—
45C4⋊Dic251C2312yes—
46Dic251⋊C4—24no—
47C22⋊Dic251—21no—
48C4×Dic251—24no—
49C4×Dic251—21no—
50#41×C251—22no—
51#41×C251—22no—
52C2×Q8×C251—21no—
53C2×Q8×C251—22no—
54C2×Q8×C251—22no—
55C2×Q8×C251C2312yes—
56C2×Q8×C251C2311yescharacteristic
57C2×D4×C251—21no—
58C2×D4×C251C2311yescharacteristic
59C22×C4×C251—21no—
60C22×C4×C251—21no—
61C22×C4×C251C2311yescharacteristic
62C4⋊C4×C251—24no—
63C4⋊C4×C251D412yes—
64C22⋊C4×C251—24no—
65C22⋊C4×C251D412yes—
66C42×C251C2312yes—
67Dic502—24no—
68Dic502—22no—
69Dic502C2×D414yes—
70C2×Dic251—24no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C251⋊C8—22no—
74C251⋊C8—22no—
75C23×C251—21no—
76C23×C251C2×D411yescharacteristic
77Q8×C251—22no—
78Q8×C251—21no—
79Q8×C251—22no—
80Q8×C251—21no—
81Q8×C251—42no—
82D4×C251—21no—
83D4×C251—21no—
84D4×C251—42no—
85C2×C4×C251—22no—
86C2×C4×C251—21no—
87C2×C4×C251—42no—
88C2×C4×C251—42no—
89C2×C4×C251C2×D412yes—
90C2×C4×C251C2×D414yes—
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251#4112yes—
93C2×C4×C251C2411yescharacteristic
94C22×C251—22no—
95C22×C251—41no—
96C22×C251—42no—
97C22×C251#12112yes—
98C22×C251C22×D411yescharacteristic
99C4×C251—24no—
100C4×C251—41no—
101C4×C251—42no—
102C4×C251#12112yes—
103C4×C251C22×D411yescharacteristic
104C4×C251C2×#4111yescharacteristic
105Dic251—24no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251—81no—
109C2×C251C2×#12111yescharacteristic
110C2×C251#57812yes—
111C251#304411yescharacteristic, complemented, Sylow
112#3044—2511nomaximal, Sylow
113C2×#137—2511no—
114C22×Q16—2511no—
115#536D25111yescharacteristic
116#519—2511no—
117#464—2514no—
118#462—2514no—
119#424—2511no—
120#419—2511no—
121#416—2511no—
122C2×#41C2×D25111yescharacteristic
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⋊Q8C2×D25112yes—
133#124C2×D25112yes—
134#123—22no—
135C22⋊Q8—22no—
136C4×Q8—5022no—
137C4×Q8—2514no—
138#117—5022no—
139#117—2511no—
140#117C2×D25111yescharacteristic
141C4⋊C8—2514no—
142#103—2512no—
143#103—2514no—
144#102—2512no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—5021no—
151C2×Q8—5022no—
152C2×Q8—2514no—
153C2×Q8—2512no—
154C2×Q8C22×D25112yes—
155C2×Q8C22×D25111yescharacteristic
156C2×D4—21no—
157C2×D4C22×D25111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
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⋊C4—2512no—
172C4⋊C4C22⋊D25112yes—
173C22⋊C4—24no—
174C22⋊C4—5021no—
175C22⋊C4C22⋊D25112yes—
176C42—5024no—
177C42—5021no—
178C42C22×D25112yes—
179C23—21no—
180C23C2×C22⋊D25111yescharacteristic
181Q8—22no—
182Q8—21no—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—5022no—
187Q8—5024no—
188Q8—2514no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5024no—
199C2×C4D4×D25112yes—
200C2×C4C2×C22⋊D25114yes—
201C2×C4C2×C22⋊D25111yescharacteristic
202C2×C4#1686080912yes—
203C2×C4C23×D25111yescharacteristic
204C8—5022no—
205C8—5022no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2254868612yes—
210C22C22×C22⋊D25111yescharacteristic
211C4—24no—
212C4—41no—
213C4—42no—
214C4—10042no—
215C4—5024no—
216C4#2254868612yes—
217C4C2×D4×D25111yescharacteristic
218C4C2×#1686080911yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2254868611yescharacteristic
222C2#3063250212yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#536×C251C2
socleC22×C251C22×D4
prime radicalC2×#41×C251C22
hypercenter#536D251
hypocenterC251#3044
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#16860800C2semidirect
3. #30632164C2semidirect
4. #30631906C2semidirect
5. #30631894C2semidirect
6. #30631714C2semidirect
7. #30631696C2semidirect
111. C251#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
C22×#168608002
#306321642
#306319062
#306318942
#306317142
#306316962
#306316832
#536×C2512
#3044251