Finite Group Databasebeta

← #44560749

Subgroups — #44560749

Order 32128.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#30632240C211yesmaximal, characteristic, complemented
2C2×#22548702C211yesmaximal, characteristic
3#30632045C214yesmaximal, complemented
4#30632040C214yesmaximal, complemented
5#30631697C211yesmaximal, characteristic, complemented
6#30631695C211yesmaximal, characteristic, complemented
7#30631692C211yesmaximal, characteristic
8M16×Dic251C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×#16860794C2211yescharacteristic
12#22548714C2212yescomplemented
13#22548706C2212yescomplemented
14#22548702—22no—
15#22548702C2212yes—
16#22548702C2212yes—
17#22548692—22no—
18C4×Dic502—22no—
19#22548674C2211yescharacteristic
20C2×C22⋊Dic251—22no—
21C2×C4×Dic251C2211yescharacteristic
22#22548640C2214yescomplemented
23#22548638C2214yescomplemented
24#22548636C2214yes—
25#22548625C2212yescomplemented
26C8×Dic251C2212yescomplemented
27C22×D4×C251C2211yescharacteristic
28C2×M16×C251C2211yescharacteristic
29#117×C251C2211yescharacteristic
30#102×C251C2214yes—
31C2×Dic502—22no—
32C2×Dic502—21no—
33C2×Dic502—22no—
34C2×Dic502—22no—
35C2×Dic502C2312yes—
36C22×Dic251—21no—
37C22×Dic251D412yes—
38#16860794—22no—
39C2×C251⋊C8C2312yes—
40C4⋊Dic251—22no—
41C4⋊Dic251C2312yes—
42Dic251⋊C4—22no—
43Dic251⋊C4—22no—
44C22⋊Dic251—24no—
45C22⋊Dic251—24no—
46C22⋊Dic251—21no—
47C4×Dic251—21no—
48C4×Dic251D412yes—
49C4×Dic251C2311yescharacteristic
50C4×Dic251C2311yescharacteristic
51C24×C251—21no—
52C2×D4×C251—21no—
53C2×D4×C251—22no—
54C2×D4×C251—22no—
55C2×D4×C251C2312yes—
56C22×C4×C251C2311yescharacteristic
57M16×C251—22no—
58C2×C8×C251C2312yes—
59C4⋊C4×C251C2312yes—
60C22⋊C4×C251—21no—
61C42×C251—21no—
62Dic502—22no—
63Dic502—22no—
64Dic502—44no—
65Dic502—42no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—42no—
69C2×Dic251—42no—
70C2×Dic251C2×D412yes—
71C2×Dic251C2×D412yes—
72C251⋊C8—22no—
73C23×C251—22no—
74C23×C251—44no—
75C23×C251—42no—
76C23×C251C2×D411yescharacteristic
77D4×C251—22no—
78D4×C251—22no—
79D4×C251—42no—
80C2×C4×C251—22no—
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251#4112yes—
83C2×C4×C251#4112yes—
84C2×C4×C251C2411yescharacteristic
85C8×C251—22no—
86C22×C251—22no—
87C22×C251—42no—
88C22×C251—44no—
89C22×C251—84no—
90C22×C251#12412yes—
91C22×C251C22×D411yescharacteristic
92C4×C251—42no—
93C4×C251#12412yes—
94C4×C251C2×#4111yescharacteristic
95C4×C251C2×#4111yescharacteristic
96Dic251—22no—
97Dic251—22no—
98Dic251—42no—
99Dic251—42no—
100C2×C251—22no—
101C2×C251—82no—
102C2×C251C2×#12411yescharacteristic
103C2×C251#57512yes—
104C251#314011yescharacteristic, complemented, Sylow
105#3140—2511nomaximal, Sylow
106#533—2511no—
107C2×#124—2511no—
108#489—2514no—
109#488—2514no—
110#419—2511no—
111#419—2511no—
112#418—2511no—
113#418D25111yescharacteristic
114C4×M16—2511no—
115C22×Q8—2511no—
116C22×D4C2×D25111yescharacteristic
117C2×M16—2511no—
118C2×M16C2×D25111yescharacteristic
119C4⋊Q8—2512no—
120#125—2512no—
121#124—5022no—
122#124—2512no—
123#124—2512no—
124C22⋊Q8—5022no—
125C4×Q8—5022no—
126#117—2511no—
127#117C2×D25111yescharacteristic
128C2×C22⋊C4—5022no—
129C2×C42—2511no—
130#103—2514no—
131#103—2514no—
132#102—2514no—
133#102C2×D25114yes—
134#97—2512no—
135C4×C8—2512no—
136C24—21no—
137C2×Q8—5022no—
138C2×Q8—5022no—
139C2×Q8—5022no—
140C2×Q8—5021no—
141C2×Q8—2512no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22×D25112yes—
146C22×C4—5021no—
147C22×C4—2512no—
148C22×C4C22×D25111yescharacteristic
149M16—22no—
150M16—5022no—
151C2×C8—2512no—
152C2×C8C22×D25112yes—
153C4⋊C4—5022no—
154C4⋊C4—5022no—
155C4⋊C4—5022no—
156C4⋊C4—2512no—
157C4⋊C4C22×D25112yes—
158C22⋊C4—21no—
159C22⋊C4—5021no—
160C22⋊C4—5024no—
161C22⋊C4—5024no—
162C42—21no—
163C42—5021no—
164C42—2512no—
165C42—2511no—
166C42—2511no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D25111yescharacteristic
171Q8—10044no—
172Q8—10042no—
173Q8—5022no—
174Q8—5022no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—10042no—
180C2×C4—10042no—
181C2×C4—5022no—
182C2×C4—5022no—
183C2×C4—2512no—
184C2×C4—2512no—
185C2×C4D4×D25111yescharacteristic
186C2×C4#1686080912yes—
187C2×C4#1686081012yes—
188C2×C4C23×D25111yescharacteristic
189C8—22no—
190C8—5022no—
191C22—22no—
192C22—42no—
193C22—44no—
194C22—84no—
195C22#2254870112yes—
196C22C2×D4×D25111yescharacteristic
197C4—42no—
198C4—10042no—
199C4—10042no—
200C4—5022no—
201C4—5022no—
202C4#2254870112yes—
203C4C2×#1686080911yescharacteristic
204C4C2×#1686081011yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#2254870111yescharacteristic
208C2#3063248312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×C251C22
hypercenter#418D251
hypocenterC251#3140
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #30632240C2semidirect
3. #30632045C2semidirect
4. #30632040C2semidirect
5. #30631697C2semidirect
6. #30631695C2semidirect
8. M16×Dic251C2semidirect
12. #22548714C22semidirect
13. #22548706C22semidirect
22. #22548640C22semidirect
23. #22548638C22semidirect
25. #22548625C22semidirect
26. C8×Dic251C22semidirect
104. C251#3140semidirect

Maximal subgroups

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

subgroupindex
#306322402
C2×#225487022
#306320452
#306320402
#306316972
#306316952
#306316922
M16×Dic2512
#418×C2512
#3140251