Finite Group Databasebeta

← #44558423

Subgroups — #44558423

Order 32128.

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

normal subgroups only

This group has 430 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
1C2×#22548715C211yesmaximal, characteristic, complemented
2#30631626C212yesmaximal, complemented
3#30631623C212yesmaximal
4#30631617C212yesmaximal, complemented
5#30631612C212yesmaximal
6#30631607C212yesmaximal, complemented
7#30631599C212yesmaximal
8#30631555C211yesmaximal, characteristic, complemented
9C2×#126×C251C211yesmaximal, characteristic
10C22×Dic502C2212yes—
11#22548715—24no—
12C2×C4⋊Dic251C2212yes—
13C2×C4⋊Dic251C2212yes—
14C2×Dic251⋊C4C2214yes—
15C2×C22⋊Dic251C2214yes—
16C251⋊#95C2212yes—
17C251⋊#95C2212yes—
18C251⋊#95C2212yes—
19C251⋊#95C2214yes—
20C251⋊#95C2214yes—
21#126×C251—24no—
22C2×C4⋊C4×C251C2211yescharacteristic
23C2×C4⋊C4×C251C2211yescharacteristic
24C2×C4⋊C4×C251C2211yescharacteristic
25C2×C22⋊C4×C251C2211yescharacteristic
26C2×C22⋊C4×C251C2211yescharacteristic
27C2×C22⋊C4×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29C2×Dic502—24no—
30C2×Dic502—44no—
31C22×Dic251C2314yes—
32C22×Dic251C2314yes—
33C4⋊Dic251—24no—
34C4⋊Dic251—24no—
35Dic251⋊C4—44no—
36C22⋊Dic251—44no—
37C24×C251C2311yescharacteristic
38C22×C4×C251C2311yescharacteristic
39C22×C4×C251C2311yescharacteristic
40C22×C4×C251C2311yescharacteristic
41C22×C4×C251C2311yescharacteristic
42C22×C4×C251C2311yescharacteristic
43C22×C4×C251C2311yescharacteristic
44C4⋊C4×C251—22no—
45C4⋊C4×C251—22no—
46C4⋊C4×C251—22no—
47C22⋊C4×C251—22no—
48C22⋊C4×C251—22no—
49C22⋊C4×C251—22no—
50C42×C251D414yes—
51Dic502—48no—
52C2×Dic251—24no—
53C2×Dic251—24no—
54C2×Dic251—24no—
55C2×Dic251—24no—
56C2×Dic251—24no—
57C2×Dic251—24no—
58C23×C251—21no—
59C23×C251—22no—
60C23×C251—22no—
61C23×C251—22no—
62C23×C251C2411yescharacteristic
63C2×C4×C251—22no—
64C2×C4×C251—22no—
65C2×C4×C251—22no—
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—22no—
70C2×C4×C251—22no—
71C2×C4×C251—22no—
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D412yes—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251—41no—
78C22×C251—44no—
79C22×C251C22×D411yescharacteristic
80C22×C251H(5,𝔽2)12yes—
81C22×C2512−1+412yes—
82C22×C2512−1+412yes—
83C4×C251—22no—
84C4×C251—22no—
85C4×C251—22no—
86C4×C251—41no—
87C4×C251—41no—
88C4×C251—41no—
89Dic251—44no—
90Dic251—44no—
91C2×C251—81no—
92C2×C251#53511yescharacteristic
93C2×C251#53611yescharacteristic
94C2×C251#53611yescharacteristic
95C2×C251#56314yes—
96C251#282111yescharacteristic, complemented, Sylow
97#2821—2511nomaximal, Sylow
98C2×C4⋊Q8—2511no—
99C2×#126D25111yescharacteristic
100#400—2512no—
101#399—2512no—
102#398—2512no—
103#397—2512no—
104#395—2512no—
105#393—2512no—
106#383—2511no—
107C22×Q8—2512no—
108C4⋊Q8—5024no—
109#126—24no—
110C2×C4⋊C4—2514no—
111C2×C4⋊C4—2512no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4C2×D25111yescharacteristic
114C2×C4⋊C4C2×D25111yescharacteristic
115C2×C4⋊C4C2×D25111yescharacteristic
116C2×C22⋊C4—2514no—
117C2×C22⋊C4C2×D25111yescharacteristic
118C2×C22⋊C4C2×D25111yescharacteristic
119C2×C22⋊C4C2×D25111yescharacteristic
120C2×C42C2×D25111yescharacteristic
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2514no—
125#95—2514no—
126C24C22×D25111yescharacteristic
127C2×Q8—10044no—
128C2×Q8—5024no—
129C22×C4—2514no—
130C22×C4—2514no—
131C22×C4C22×D25111yescharacteristic
132C22×C4C22×D25111yescharacteristic
133C22×C4C22×D25111yescharacteristic
134C22×C4C22×D25111yescharacteristic
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—10044no—
141C4⋊C4—5024no—
142C4⋊C4—5024no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—10044no—
147C42C22⋊D25114yes—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23C23×D25111yescharacteristic
153Q8—10048no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—5024no—
164C2×C4—5024no—
165C2×C4—5024no—
166C2×C4—5024no—
167C2×C4—5024no—
168C2×C4—5024no—
169C2×C4C2×C22⋊D25112yes—
170C2×C4C2×C22⋊D25112yes—
171C2×C4C2×C22⋊D25112yes—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—44no—
176C22C22×C22⋊D25111yescharacteristic
177C22#2254876112yes—
178C22#2254876212yes—
179C22#2254876212yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—10044no—
187C4—10044no—
188C2—81no—
189C2#3063225111yescharacteristic
190C2#3063225811yescharacteristic
191C2#3063225811yescharacteristic
192C2#3063243914yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#126×C251C2
prime radicalC24×C251C23
hypercenterC2×#126D251
hypocenterC251#2821
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#22548715C2semidirect
2. #30631626C2semidirect
4. #30631617C2semidirect
6. #30631607C2semidirect
8. #30631555C2semidirect
96. C251#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#225487152
#306316262
#306316232
#306316172
#306316122
#306316072
#306315992
#306315552
C2×#126×C2512
#2821251