Finite Group Databasebeta

← #44560183

Subgroups — #44560183

Order 32128.

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

normal subgroups only

This group has 384 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
1C4×#16860810C211yesmaximal, characteristic
2#30631991C214yesmaximal, complemented
3#30631984C214yesmaximal, complemented
4#30631727C212yesmaximal, complemented
5#30631711C211yesmaximal, characteristic, complemented
6#30631686C212yesmaximal
7C2×#125×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C2×#16860794C2212yes—
10C4×Dic502C2212yes—
11C4×D1004C2212yes—
12C4×C22⋊D251—22no—
13#22548674C2211yescharacteristic
14#22548672—21no—
15C42×D251—21no—
16#22548648C2214yes—
17#22548646C2214yes—
18C4⋊(C251⋊C8)C2214yes—
19#22548638C2214yes—
20#22548635C2214yes—
21#125×C251—21no—
22#125×C251—21no—
23#125×C251C2214yes—
24C2×C4⋊C4×C251—22no—
25C2×C4⋊C4×C251C2212yes—
26C2×C42×C251C2211yescharacteristic
27#16860810—22no—
28#16860810Q814yes—
29C2×Dic502C2311yescharacteristic
30C2×D1004C2311yescharacteristic
31C2×C22⋊D251—21no—
32C2×C4×D251—21no—
33C2×C4×D251—22no—
34#16860794—24no—
35C2×C251⋊C8C2314yes—
36C4⋊Dic251C2312yes—
37Dic251⋊C4—22no—
38C22⋊Dic251—21no—
39(C2×D251)⋊C4—22no—
40C4×Dic251—21no—
41C4×Dic251—22no—
42C22×C4×C251—22no—
43C22×C4×C251D412yes—
44C22×C4×C251C2311yescharacteristic
45C4⋊C4×C251—24no—
46C4⋊C4×C251—24no—
47C4⋊C4×C251—22no—
48C4⋊C4×C251C2314yes—
49C42×C251D411yescharacteristic
50C42×C251D411yescharacteristic
51C42×C251C2312yes—
52C22×D251—21no—
53Dic502—21no—
54Dic502C2×Q812yes—
55D1004—21no—
56D1004C2×Q812yes—
57C22⋊D251—24no—
58C2×Dic251—21no—
59C2×Dic251—22no—
60C4×D251—24no—
61C4×D251—44no—
62C251⋊C8—24no—
63C23×C251C2×D411yescharacteristic
64C2×C4×C251—24no—
65C2×C4×C251—42no—
66C2×C4×C251—44no—
67C2×C4×C251C2×D412yes—
68C2×C4×C251C2×D412yes—
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×Q812yes—
71C2×C4×C251#4112yes—
72C2×C4×C251C2411yescharacteristic
73C22×C251—22no—
74C22×C251C22⋊Q812yes—
75C22×C251C22×D411yescharacteristic
76C2×D251—22no—
77C2×D251—42no—
78C4×C251—22no—
79C4×C251—22no—
80C4×C251—44no—
81C4×C251C22⋊Q812yes—
82C4×C251C22×Q811yescharacteristic
83C4×C251C2×#4111yescharacteristic
84Dic251—22no—
85Dic251—42no—
86C2×C251—22no—
87C2×C251C2×C22⋊Q811yescharacteristic
88C2×C251#57512yes—
89D251—42no—
90C251#308011yescharacteristic, complemented, Sylow
91#3080—2511nomaximal, Sylow
92C2×#125D25111yescharacteristic, complemented
93C4×#41—2511no—
94#479—2514no—
95#478—2514no—
96#428—2512no—
97#423—2511no—
98#417—2512no—
99C2×#41—2511no—
100C2×M16—2512no—
101#125—21no—
102#125—21no—
103#125C2×D25114yes—
104C4×Q8—2512no—
105C4×D4—5022no—
106C4×D4—2512no—
107#117—5021no—
108#117—2511no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C2×D25112yes—
111C2×C42—5021no—
112C2×C42C2×D25111yescharacteristic
113#107—2514no—
114#106—2514no—
115C4⋊C8—2514no—
116#103—2514no—
117#102—2514no—
118#41—5022no—
119#41—2514no—
120C2×Q8—2511no—
121C2×D4—5021no—
122C2×D4—2511no—
123C22×C4—22no—
124C22×C4—5022no—
125C22×C4—5021no—
126C22×C4C22⋊D25112yes—
127C22×C4C22×D25111yescharacteristic
128M16—5024no—
129C2×C8—2514no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—5022no—
134C4⋊C4—2512no—
135C4⋊C4C22×D25114yes—
136C22⋊C4—5021no—
137C22⋊C4—5022no—
138C42—5021no—
139C42—5022no—
140C42C22⋊D25111yescharacteristic
141C42C22⋊D25111yescharacteristic
142C42C22×D25112yes—
143C23—5021no—
144C23C2×C22⋊D25111yescharacteristic
145Q8—5021no—
146Q8—2512no—
147D4—5024no—
148D4—5021no—
149D4—2512no—
150C2×C4—24no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—10044no—
154C2×C4—5022no—
155C2×C4—5021no—
156C2×C4—5024no—
157C2×C4C2×C22⋊D25112yes—
158C2×C4C2×C22⋊D25112yes—
159C2×C4C2×C22⋊D25111yescharacteristic
160C2×C4Q8×D25112yes—
161C2×C4#1686081112yes—
162C2×C4C23×D25111yescharacteristic
163C8—5024no—
164C22—22no—
165C22—10042no—
166C22—5022no—
167C22#2254869112yes—
168C22C22×C22⋊D25111yescharacteristic
169C4—22no—
170C4—22no—
171C4—44no—
172C4—10042no—
173C4—5022no—
174C4#2254869112yes—
175C4C2×Q8×D25111yescharacteristic
176C4C2×#1686081111yescharacteristic
177C2—22no—
178C2—10042no—
179C2C2×#2254869111yescharacteristic
180C2#3063248512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#125×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860810C22
hypercenterC2×#125D251
hypocenterC251#3080
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #30631991C2semidirect
3. #30631984C2semidirect
4. #30631727C2semidirect
5. #30631711C2semidirect
7. C2×#125×C251C2semidirect
90. C251#3080semidirect
92. C2×#125D251semidirect

Maximal subgroups

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

subgroupindex
C4×#168608102
#306319912
#306319842
#306317272
#306317112
#306316862
C2×#125×C2512
#3080251