Finite Group Databasebeta

← #44558670

Subgroups — #44558670

Order 32128.

556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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×C42×D251C211yesmaximal, characteristic
2#30631583C212yesmaximal, complemented
3#30631568C214yesmaximal, complemented
4#30631558C214yesmaximal
5#30631553C211yesmaximal, characteristic, complemented
6#30631544C212yesmaximal
7C2×#125×C251C211yesmaximal, characteristic, complemented
8C22×C4×D251C2212yes—
9C22×C4×D251C2211yescharacteristic
10C2×C4⋊Dic251C2212yes—
11C2×Dic251⋊C4C2214yes—
12C2×(C2×D251)⋊C4C2214yes—
13C2×C4×Dic251C2212yes—
14C2×C4×Dic251C2211yescharacteristic
15C42×D251—24no—
16C251⋊#95C2214yes—
17C251⋊#95C2214yes—
18C251⋊#95C2214yes—
19#125×C251—24no—
20C2×C4⋊C4×C251C2212yes—
21C2×C4⋊C4×C251C2214yes—
22C2×C42×C251C2211yescharacteristic
23C23×D251C2311yescharacteristic
24C22×Dic251C2314yes—
25C22×Dic251C2311yescharacteristic
26C22×Dic251C2312yes—
27C2×C4×D251—24no—
28C2×C4×D251—24no—
29C2×C4×D251—24no—
30C2×C4×D251—22no—
31C2×C4×D251—22no—
32C2×C4×D251Q814yes—
33C4⋊Dic251—24no—
34Dic251⋊C4—28no—
35(C2×D251)⋊C4—28no—
36C4×Dic251—24no—
37C4×Dic251—22no—
38C22×C4×C251C2314yes—
39C22×C4×C251C2312yes—
40C22×C4×C251C2311yescharacteristic
41C4⋊C4×C251—28no—
42C4⋊C4×C251—24no—
43C42×C251D414yes—
44C22×D251—21no—
45C22×D251—22no—
46C22×D251—21no—
47C22×D251C2×Q812yes—
48C22×D251#4114yes—
49C2×Dic251—21no—
50C2×Dic251—22no—
51C2×Dic251—22no—
52C2×Dic251—21no—
53C2×Dic251—24no—
54C2×Dic251—24no—
55C2×Dic251—24no—
56C2×Dic251C2×Q812yes—
57C2×Dic251#4114yes—
58C4×D251—216no—
59C4×D251—28no—
60C23×C251C2411yescharacteristic
61C2×C4×C251—22no—
62C2×C4×C251—22no—
63C2×C4×C251—24no—
64C2×C4×C251—24no—
65C2×C4×C251—24no—
66C2×C4×C251C2×D414yes—
67C2×C4×C251C2×D412yes—
68C2×C4×C251C2×Q812yes—
69C2×C4×C251#4112yes—
70C22×C251C22×D411yescharacteristic
71C22×C251C22×Q811yescharacteristic
72C22×C251C2×#4112yes—
73C22×C251C2×#4112yes—
74C22×C251C2×#4111yescharacteristic
75C2×D251—22no—
76C2×D251—24no—
77C2×D251—24no—
78C2×D251—22no—
79C2×D251#12514yescomplemented
80C4×C251—24no—
81C4×C251—44no—
82C4×C251C22⋊Q814yes—
83Dic251—22no—
84Dic251—24no—
85Dic251—44no—
86C2×C251C2×C22⋊Q811yescharacteristic
87C2×C251#52512yes—
88C2×C251C2×#12511yescharacteristic
89C2×C251#52912yes—
90C2×C251#53311yescharacteristic
91D251—24no—
92C251#285111yescharacteristic, complemented, Sylow
93#2851—2511nomaximal, Sylow
94C2×#125D25111yescharacteristic, complemented
95C22×C42—2511no—
96#389—2512no—
97#387—2514no—
98#385—2514no—
99#382—2511no—
100#380—2512no—
101C23×C4—2512no—
102C23×C4—2511no—
103#125—24no—
104C2×C4⋊C4—2512no—
105C2×C4⋊C4—2514no—
106C2×C4⋊C4C2×D25112yes—
107C2×C4⋊C4C2×D25114yes—
108C2×C22⋊C4—2514no—
109C2×C42—5024no—
110C2×C42—2512no—
111C2×C42—2511no—
112C2×C42C2×D25111yescharacteristic
113#95—2514no—
114#95—2514no—
115#95—2514no—
116C24—2511no—
117C22×C4—5024no—
118C22×C4—5024no—
119C22×C4—5024no—
120C22×C4—5022no—
121C22×C4—5022no—
122C22×C4—2514no—
123C22×C4—2511no—
124C22×C4—2512no—
125C22×C4—2514no—
126C22×C4C22×D25114yes—
127C22×C4C22×D25112yes—
128C22×C4C22×D25111yescharacteristic
129C4⋊C4—28no—
130C4⋊C4—24no—
131C4⋊C4—5028no—
132C4⋊C4—5024no—
133C22⋊C4—5028no—
134C42—5024no—
135C42—5022no—
136C42C22⋊D25114yes—
137C23—5021no—
138C23—5022no—
139C23—5021no—
140C23—2512no—
141C23—2514no—
142C23C23×D25111yescharacteristic
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—50216no—
149C2×C4—5024no—
150C2×C4—5024no—
151C2×C4—5024no—
152C2×C4—5022no—
153C2×C4—5021no—
154C2×C4—5022no—
155C2×C4—5021no—
156C2×C4—5028no—
157C2×C4—2512no—
158C2×C4—2514no—
159C2×C4C2×C22⋊D25114yes—
160C2×C4C2×C22⋊D25112yes—
161C2×C4Q8×D25112yes—
162C2×C4#1686081112yes—
163C22—5024no—
164C22—5024no—
165C22—5022no—
166C22—5022no—
167C22—2514no—
168C22C22×C22⋊D25111yescharacteristic
169C22C2×Q8×D25111yescharacteristic
170C22#41×D25112yes—
171C22#41×D25112yes—
172C22C2×#1686081111yescharacteristic
173C4—24no—
174C4—44no—
175C4—10044no—
176C4—5024no—
177C4—5022no—
178C4#2254869114yes—
179C2—5024no—
180C2C2×#2254869111yescharacteristic
181C2#3063219412yes—
182C2#125×D25111yescharacteristic
183C2#3063222212yes—
184C2#3063224311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#125×C251C2
prime radicalC23×D251C23
hypercenterC2×#125D251
hypocenterC251#2851
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #30631583C2semidirect
3. #30631568C2semidirect
5. #30631553C2semidirect
7. C2×#125×C251C2semidirect
79. C2×D251#125semidirect
92. C251#2851semidirect
94. C2×#125D251semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D2512
#306315832
#306315682
#306315582
#306315532
#306315442
C2×#125×C2512
#2851251