Finite Group Databasebeta

← #44560017

Subgroups — #44560017

Order 32128.

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

normal subgroups only

This group has 460 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
1C22×#16860800C211yesmaximal, characteristic, complemented
2C2×#22548722C211yesmaximal, characteristic
3#30632258C211yesmaximal, characteristic
4#30631943C214yesmaximal, complemented
5#30631926C214yesmaximal, complemented
6#30631723C211yesmaximal, characteristic, complemented
7#30631696C211yesmaximal, characteristic, complemented
8#30631695C211yesmaximal, characteristic, complemented
9#536×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×#16860800—24no—
12C2×#16860800C2212yes—
13C2×#16860800C2212yes—
14#22548722—24no—
15C2×#16860794C2212yes—
16C2×#16860794C2211yescharacteristic
17C22×C251⋊C8C2212yes—
18C22×C251⋊C8C2211yescharacteristic
19#22548713—21no—
20#22548702—21no—
21#22548696—22no—
22#22548694C2214yes—
23#22548674C2211yescharacteristic
24#22548648C2212yes—
25#22548646C2212yes—
26#22548639C2214yes—
27#22548638C2214yes—
28C2×#41×C251C2211yescharacteristic
29C22×Q8×C251C2211yescharacteristic
30C4⋊Q8×C251—21no—
31#124×C251—21no—
32#123×C251—22no—
33C22⋊Q8×C251C2214yes—
34#117×C251C2211yescharacteristic
35C2×Dic502—21no—
36C2×Dic502—22no—
37C2×Dic502—22no—
38C2×Dic502C2312yes—
39C22×Dic251—21no—
40#16860800—24no—
41#16860800—24no—
42#16860794—24no—
43#16860794—22no—
44C2×C251⋊C8—24no—
45C2×C251⋊C8D414yes—
46C2×C251⋊C8C2312yes—
47C2×C251⋊C8C2311yescharacteristic
48C2×C251⋊C8C2311yescharacteristic
49C4⋊Dic251C2312yes—
50Dic251⋊C4—24no—
51C22⋊Dic251—21no—
52C22⋊Dic251—24no—
53C4×Dic251—21no—
54#41×C251—22no—
55#41×C251—22no—
56C2×Q8×C251—21no—
57C2×Q8×C251—22no—
58C2×Q8×C251—22no—
59C2×Q8×C251D411yescharacteristic
60C2×Q8×C251C2312yes—
61C2×D4×C251D412yes—
62C2×D4×C251D411yescharacteristic
63C22×C4×C251—21no—
64C22×C4×C251C2312yes—
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251—24no—
67C4⋊C4×C251C2312yes—
68C22⋊C4×C251—21no—
69C22⋊C4×C251—24no—
70C42×C251—21no—
71Dic502—22no—
72Dic502—22no—
73Dic502—42no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—42no—
77C251⋊C8—22no—
78C251⋊C8C2×D412yes—
79C251⋊C8C2×D412yes—
80C23×C251C2×D412yes—
81C23×C251C2×D411yescharacteristic
82Q8×C251—22no—
83Q8×C251—22no—
84Q8×C251—22no—
85Q8×C251—42no—
86D4×C251—24no—
87D4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—42no—
93C2×C4×C251C2×D412yes—
94C2×C4×C251C2×D412yes—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251#4112yes—
97C2×C4×C251C2411yescharacteristic
98C22×C251—22no—
99C22×C251—22no—
100C22×C251—24no—
101C22×C251#12112yes—
102C22×C251C22×D411yescharacteristic
103C4×C251—22no—
104C4×C251—42no—
105C4×C251—42no—
106C4×C251#12112yes—
107C4×C251C22×D411yescharacteristic
108C4×C251C2×#4111yescharacteristic
109Dic251—42no—
110Dic251—42no—
111C2×C251—22no—
112C2×C251—42no—
113C2×C251C2×#12111yescharacteristic
114C2×C251#57812yes—
115C251#305911yescharacteristic, complemented, Sylow
116#3059—2511nomaximal, Sylow
117C22×Q16—2511no—
118C2×#131—2511no—
119#536—2511no—
120#536D25111yescharacteristic
121#470—2514no—
122#467—2514no—
123#427—2511no—
124#419—2511no—
125#419—2511no—
126C2×#41C2×D25111yescharacteristic
127C22×Q8—2511no—
128C22×Q8C2×D25111yescharacteristic
129C2×Q16—5024no—
130C2×Q16—2512no—
131C2×Q16—2512no—
132#131—5024no—
133C2×M16—2511no—
134C2×M16—2512no—
135C22×C8—2512no—
136C22×C8—2511no—
137C4⋊Q8—21no—
138C4⋊Q8—5021no—
139#124—21no—
140#124—5021no—
141#123—22no—
142#123—5022no—
143C22⋊Q8—2514no—
144C22⋊Q8C2×D25114yes—
145#117—2511no—
146#117C2×D25111yescharacteristic
147#107—2512no—
148#106—2512no—
149#103—2514no—
150#103—2514no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—5022no—
157C2×Q8—5022no—
158C2×Q8—5021no—
159C2×Q8—2512no—
160C2×Q8C22⋊D25111yescharacteristic
161C2×Q8C22×D25112yes—
162C2×D4C22⋊D25112yes—
163C2×D4C22⋊D25111yescharacteristic
164C22×C4—21no—
165C22×C4—5021no—
166C22×C4C22×D25112yes—
167C22×C4C22×D25111yescharacteristic
168Q16—5024no—
169Q16—5024no—
170M16—5022no—
171M16—5024no—
172C2×C8—5024no—
173C2×C8—2512no—
174C2×C8—2514no—
175C2×C8—2511no—
176C2×C8—2511no—
177C4⋊C4—24no—
178C4⋊C4—5024no—
179C4⋊C4—2512no—
180C4⋊C4C22×D25112yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—5021no—
184C22⋊C4—5024no—
185C42—21no—
186C42—5021no—
187C23C2×C22⋊D25112yes—
188C23C2×C22⋊D25111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—10042no—
194Q8—5022no—
195Q8—5022no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—10042no—
204C2×C4—5022no—
205C2×C4—5022no—
206C2×C4D4×D25112yes—
207C2×C4C2×C22⋊D25112yes—
208C2×C4C2×C22⋊D25111yescharacteristic
209C2×C4#1686080912yes—
210C2×C4C23×D25111yescharacteristic
211C8—5022no—
212C8—2512no—
213C8—2512no—
214C22—22no—
215C22—24no—
216C22—22no—
217C22#2254868712yes—
218C22C22×C22⋊D25111yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—10042no—
223C4—10042no—
224C4#2254868712yes—
225C4C2×D4×D25111yescharacteristic
226C4C2×#1686080911yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#2254868711yescharacteristic
230C2#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#3059
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#16860800C2semidirect
4. #30631943C2semidirect
5. #30631926C2semidirect
6. #30631723C2semidirect
7. #30631696C2semidirect
8. #30631695C2semidirect
115. C251#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#168608002
C2×#225487222
#306322582
#306319432
#306319262
#306317232
#306316962
#306316952
#536×C2512
#3059251