Finite Group Databasebeta

← #44560019

Subgroups — #44560019

Order 32128.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#16860800C211yesmaximal, characteristic, complemented
2C2×#22548723C211yesmaximal, characteristic
3#30632263C211yesmaximal, characteristic
4#30632261C211yesmaximal, characteristic
5#30631944C214yesmaximal, complemented
6#30631927C214yesmaximal, complemented
7#30631723C211yesmaximal, characteristic, complemented
8#30631697C211yesmaximal, characteristic, complemented
9#419×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12C2×#16860800—24no—
13C2×#16860800C2212yes—
14C2×#16860800C2212yes—
15#22548723—24no—
16C2×#16860793C2212yes—
17C22×C251⋊C8C2211yescharacteristic
18C2×C8×D251C2212yes—
19#22548715—21no—
20#22548713—21no—
21#22548704—21no—
22#22548703—21no—
23#22548699—22no—
24#22548698—22no—
25#22548695C2214yes—
26#22548691C2214yes—
27#22548674C2211yescharacteristic
28#22548648C2212yes—
29#22548646C2212yes—
30#22548640C2214yes—
31C22×Q8×C251C2211yescharacteristic
32C2×M16×C251C2211yescharacteristic
33#117×C251C2211yescharacteristic
34#103×C251C2214yes—
35#16860810—22no—
36#16860810—22no—
37C2×Dic502—21no—
38C2×Dic502—22no—
39C2×Dic502—22no—
40C2×Dic502D411yescharacteristic
41C2×Dic502C2312yes—
42C2×D1004D411yescharacteristic
43C2×C22⋊D251D412yes—
44C22×Dic251—21no—
45C2×C4×D251C2312yes—
46#16860800—24no—
47#16860800—24no—
48#16860793—24no—
49C2×C251⋊C8D414yes—
50C2×C251⋊C8C2311yescharacteristic
51C2×C251⋊C8C2311yescharacteristic
52C8×D251—24no—
53C4⋊Dic251—24no—
54C4⋊Dic251C2312yes—
55Dic251⋊C4—24no—
56C22⋊Dic251—21no—
57(C2×D251)⋊C4—24no—
58(C2×D251)⋊C4—24no—
59C4×Dic251—21no—
60C2×Q8×C251—21no—
61C2×Q8×C251—22no—
62C2×Q8×C251—22no—
63C2×Q8×C251C2312yes—
64C22×C4×C251—21no—
65C22×C4×C251C2311yescharacteristic
66M16×C251—22no—
67C2×C8×C251C2312yes—
68C4⋊C4×C251C2312yes—
69C22⋊C4×C251—21no—
70C42×C251—21no—
71C22×D251C2×D412yes—
72Dic502—22no—
73Dic502—22no—
74Dic502—22no—
75Dic502—42no—
76D1004—22no—
77C22⋊D251—24no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—42no—
81C2×Dic251C2×D412yes—
82C4×D251—22no—
83C4×D251—22no—
84C251⋊C8C2×D412yes—
85C251⋊C8C2×D412yes—
86C23×C251C2×D411yescharacteristic
87Q8×C251—22no—
88Q8×C251—22no—
89Q8×C251—42no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—42no—
93C2×C4×C251C2×D411yescharacteristic
94C2×C4×C251C2×D412yes—
95C2×C4×C251#4112yes—
96C2×C4×C251C2411yescharacteristic
97C8×C251—22no—
98C22×C251—22no—
99C22×C251#12112yes—
100C22×C251C22×D411yescharacteristic
101C2×D251—22no—
102C2×D251—24no—
103C4×C251—42no—
104C4×C251—42no—
105C4×C251#12112yes—
106C4×C251C22×D411yescharacteristic
107C4×C251C2×#4111yescharacteristic
108Dic251—22no—
109Dic251—42no—
110Dic251—42no—
111C2×C251—22no—
112C2×C251C2×#12111yescharacteristic
113C2×C251#57812yes—
114D251—42no—
115C251#305911yescharacteristic, complemented, Sylow
116#3059—2511nomaximal, Sylow
117C22×Q16—2511no—
118C2×#131—2511no—
119#536—2511no—
120#536—2511no—
121#470—2514no—
122#467—2514no—
123#427—2511no—
124#419—2511no—
125#419D25111yescharacteristic, complemented
126C2×#41—2511no—
127C22×Q8—2511no—
128C22×Q8C2×D25111yescharacteristic
129C2×Q16—5024no—
130C2×Q16—2512no—
131C2×Q16—2512no—
132#131—5024no—
133C2×M16—2512no—
134C2×M16C2×D25111yescharacteristic
135C22×C8—2512no—
136C22×C8—2511no—
137C4⋊Q8—5021no—
138C4⋊Q8—5021no—
139#124—5021no—
140#124—5021no—
141#123—5022no—
142#123—5022no—
143C22⋊Q8—2514no—
144C22⋊Q8—2514no—
145#117—2511no—
146#117C2×D25111yescharacteristic
147#107—2512no—
148#106—2512no—
149#103—2514no—
150#103C2×D25114yes—
151#41—5022no—
152#41—5022no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—5022no—
157C2×Q8—5022no—
158C2×Q8—5021no—
159C2×Q8—2512no—
160C2×Q8—2511no—
161C2×Q8C22×D25112yes—
162C2×D4—2512no—
163C2×D4—2511no—
164C22×C4—21no—
165C22×C4—5021no—
166C22×C4—2512no—
167C22×C4C22×D25111yescharacteristic
168Q16—5024no—
169Q16—5024no—
170M16—22no—
171M16—5024no—
172C2×C8—5024no—
173C2×C8—2514no—
174C2×C8—2511no—
175C2×C8—2511no—
176C2×C8C22×D25112yes—
177C4⋊C4—5024no—
178C4⋊C4—5024no—
179C4⋊C4—2512no—
180C4⋊C4C22×D25112yes—
181C22⋊C4—21no—
182C22⋊C4—5024no—
183C22⋊C4—5024no—
184C22⋊C4—5021no—
185C42—21no—
186C42—5021no—
187C23—2512no—
188C23D4×D25111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—10042no—
193Q8—5022no—
194Q8—5022no—
195Q8—5022no—
196D4—5024no—
197D4—5022no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—10042no—
202C2×C4—5022no—
203C2×C4—5022no—
204C2×C4—5022no—
205C2×C4—5022no—
206C2×C4—2512no—
207C2×C4D4×D25112yes—
208C2×C4D4×D25111yescharacteristic
209C2×C4#1686081012yes—
210C2×C4C23×D25111yescharacteristic
211C8—22no—
212C8—2512no—
213C8—2512no—
214C22—22no—
215C22—5024no—
216C22—5022no—
217C22#2254868812yes—
218C22C2×D4×D25111yescharacteristic
219C4—42no—
220C4—42no—
221C4—10042no—
222C4—10042no—
223C4—5022no—
224C4#2254868812yes—
225C4C2×D4×D25111yescharacteristic
226C4C2×#1686081011yescharacteristic
227C2—22no—
228C2—10042no—
229C2C2×#2254868811yescharacteristic
230C2#3063250312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#419×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860810C22
hypercenter#419D251
hypocenterC251#3059
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#16860800C2semidirect
5. #30631944C2semidirect
6. #30631927C2semidirect
7. #30631723C2semidirect
8. #30631697C2semidirect
9. #419×C251C2semidirect
115. C251#3059semidirect
125. #419D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168608002
C2×#225487232
#306322632
#306322612
#306319442
#306319272
#306317232
#306316972
#419×C2512
#3059251