Finite Group Databasebeta

← #44560656

Subgroups — #44560656

Order 32128.

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

normal subgroups only

This group has 382 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#30632292C211yesmaximal, characteristic
2#30632011C212yesmaximal, complemented
3#30631994C212yesmaximal, complemented
4#30631887C212yesmaximal, complemented
5#30631826C212yesmaximal, complemented
6#30631785C212yesmaximal, complemented
7C4×#16860797C212yesmaximal, complemented
8#30631748C211yesmaximal, characteristic, complemented
9#519×C251C211yesmaximal, characteristic, complemented
10C2×#16860797C2212yes—
11C2×#16860797C2212yes—
12#22548712C2211yescharacteristic
13#22548707C2211yescharacteristic
14#22548700—22no—
15#22548689C2212yes—
16#22548688—22no—
17C4×D1004C2212yes—
18#22548648C2212yes—
19#22548646C2212yes—
20C4⋊(C251⋊C8)C2212yes—
21#22548639C2212yes—
22#22548639C2212yes—
23#22548635C2212yes—
24#22548635C2212yes—
25C22⋊(C251⋊C8)C2212yes—
26#22548626C2211yescharacteristic
27C4×C251⋊C8C2211yescharacteristic
28C22×Q8×C251C2211yescharacteristic
29C4×Q8×C251—22no—
30C4×Q8×C251C2212yes—
31C4×Q8×C251C2212yes—
32#117×C251—22no—
33#117×C251C2211yescharacteristic
34#117×C251C2211yescharacteristic
35C2×D1004—22no—
36C2×D1004C2312yes—
37C2×C22⋊D251—22no—
38C2×C4×D251—22no—
39#16860797—24no—
40#16860797—24no—
41C2×C251⋊C8C2312yes—
42C2×C251⋊C8C2312yes—
43C4⋊Dic251C2312yes—
44Dic251⋊C4—22no—
45(C2×D251)⋊C4—22no—
46(C2×D251)⋊C4—22no—
47C2×Q8×C251—24no—
48C2×Q8×C251—21no—
49C2×Q8×C251C2312yes—
50C22×C4×C251—21no—
51C22×C4×C251C2311yescharacteristic
52C4⋊C4×C251—22no—
53C4⋊C4×C251—22no—
54C4⋊C4×C251D412yes—
55C4⋊C4×C251C2312yes—
56C22⋊C4×C251—21no—
57C22⋊C4×C251D412yes—
58C42×C251—22no—
59C42×C251—22no—
60C42×C251—21no—
61C42×C251C2311yescharacteristic
62C42×C251C2311yescharacteristic
63C22×D251—22no—
64D1004—22no—
65D1004—22no—
66D1004—82no—
67C22⋊D251—82no—
68C2×Dic251—22no—
69C4×D251—82no—
70C251⋊C8—22no—
71C251⋊C8—22no—
72C23×C251C2×D411yescharacteristic
73Q8×C251—24no—
74Q8×C251—22no—
75Q8×C251#4114yes—
76C2×C4×C251—24no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251C2×D412yes—
81C2×C4×C251C2×D412yes—
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251C2411yescharacteristic
84C22×C251—21no—
85C22×C251—22no—
86C22×C251C22×D411yescharacteristic
87C2×D251—42no—
88C2×D251—42no—
89C2×D251—42no—
90C4×C251—24no—
91C4×C251—21no—
92C4×C251—22no—
93C4×C251—22no—
94C4×C251—42no—
95C4×C251C2×#4112yes—
96Dic251—42no—
97C2×C251—41no—
98C2×C251#52511yescharacteristic
99C2×C251#57712yes—
100D251—82no—
101C251#312611yescharacteristic, complemented, Sylow
102#3126—2511nomaximal, Sylow
103#540—2511no—
104#519D25111yescharacteristic, complemented
105#482—2512no—
106#479—2512no—
107#460—2512no—
108#448—2512no—
109#440—2512no—
110C4×SD16—2512no—
111#433—2511no—
112C22×Q8C2×D25111yescharacteristic
113C2×SD16—2512no—
114C2×SD16—2512no—
115#127—2511no—
116#125—2511no—
117#123—5022no—
118#121—5022no—
119#121—2512no—
120C4×Q8—22no—
121C4×Q8C2×D25112yes—
122C4×Q8C2×D25112yes—
123C4×D4—2512no—
124#117—22no—
125#117C2×D25111yescharacteristic
126#117C2×D25111yescharacteristic
127#107—2512no—
128#106—2512no—
129C4⋊C8—2512no—
130#103—2512no—
131#103—2512no—
132#102—2512no—
133#102—2512no—
134C22⋊C8—2512no—
135#97—2511no—
136C4×C8—2511no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D25112yes—
140C2×D4—5022no—
141C2×D4—5022no—
142C2×D4—2512no—
143C22×C4—21no—
144C22×C4—5022no—
145C22×C4C22×D25111yescharacteristic
146SD16—5024no—
147SD16—5024no—
148C2×C8—2512no—
149C2×C8—2512no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—5022no—
153C4⋊C4—2512no—
154C4⋊C4C22⋊D25112yes—
155C4⋊C4C22×D25112yes—
156C22⋊C4—21no—
157C22⋊C4—5022no—
158C22⋊C4—5022no—
159C22⋊C4C22⋊D25112yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D25111yescharacteristic
164C42C22×D25111yescharacteristic
165C23—5022no—
166C23C2×C22⋊D25111yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1686081014yes—
170D4—20082no—
171D4—20082no—
172D4—5022no—
173D4—5022no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—20082no—
179C2×C4—5022no—
180C2×C4C2×C22⋊D25112yes—
181C2×C4C2×C22⋊D25112yes—
182C2×C4C2×C22⋊D25111yescharacteristic
183C2×C4C23×D25111yescharacteristic
184C8—5022no—
185C8—5022no—
186C22—21no—
187C22—22no—
188C22—10042no—
189C22—10042no—
190C22—10042no—
191C22C22×C22⋊D25111yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—10042no—
198C4C2×#1686081012yes—
199C2—41no—
200C2—20082no—
201C2#3063219811yescharacteristic
202C2#3063249512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#519×C251C2
socleC22×C251C22×D4
prime radical#30632292C2
hypercenter#519D251
hypocenterC251#3126
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. #30632011C2semidirect
3. #30631994C2semidirect
4. #30631887C2semidirect
5. #30631826C2semidirect
6. #30631785C2semidirect
7. C4×#16860797C2semidirect
8. #30631748C2semidirect
9. #519×C251C2semidirect
101. C251#3126semidirect
104. #519D251semidirect

Maximal subgroups

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

subgroupindex
#306322922
#306320112
#306319942
#306318872
#306318262
#306317852
C4×#168607972
#306317482
#519×C2512
#3126251