Finite Group Databasebeta

← #44560296

Subgroups — #44560296

Order 32128.

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

normal subgroups only

This group has 414 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
1#30632198C211yesmaximal, characteristic
2#30632028C214yesmaximal, complemented
3#30632011C214yesmaximal, complemented
4#30631727C212yesmaximal, complemented
5#30631686C212yesmaximal, complemented
6#30631651C211yesmaximal, characteristic, complemented
7C2×#117×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C2×#16860794C2212yes—
10#22548698—21no—
11#22548694C2212yes—
12#22548689C2212yes—
13#22548684—21no—
14C4×C22⋊D251—22no—
15#22548674C2211yescharacteristic
16#22548648C2214yescomplemented
17#22548646C2214yescomplemented
18#22548638C2214yescomplemented
19#22548635C2214yescomplemented
20C22⋊(C251⋊C8)C2214yes—
21C23×C4×C251C2211yescharacteristic
22#117×C251—22no—
23#117×C251C2214yes—
24C2×C4⋊C4×C251C2212yes—
25C2×C22⋊C4×C251—21no—
26C2×C42×C251—21no—
27#16860810—24no—
28C2×Dic502C2311yescharacteristic
29C2×D1004C2311yescharacteristic
30C2×C22⋊D251—21no—
31C2×C22⋊D251—22no—
32C2×C4×D251—21no—
33#16860794—24no—
34C2×C251⋊C8C2314yes—
35C4⋊Dic251C2312yes—
36Dic251⋊C4—22no—
37C22⋊Dic251—21no—
38C22⋊Dic251—22no—
39(C2×D251)⋊C4—22no—
40C4×Dic251—21no—
41C24×C251D411yescharacteristic
42C22×C4×C251—24no—
43C22×C4×C251—22no—
44C22×C4×C251D411yescharacteristic
45C22×C4×C251D412yes—
46C22×C4×C251C2312yes—
47C22×C4×C251C2311yescharacteristic
48C4⋊C4×C251—22no—
49C4⋊C4×C251C2314yes—
50C22⋊C4×C251—24no—
51C42×C251—24no—
52C22×D251—21no—
53Dic502—22no—
54D1004—22no—
55C22⋊D251—44no—
56C22⋊D251—42no—
57C2×Dic251—22no—
58C2×Dic251—21no—
59C4×D251—42no—
60C251⋊C8—24no—
61C23×C251—22no—
62C23×C251—42no—
63C23×C251C2×D412yes—
64C23×C251C2×D411yescharacteristic
65C2×C4×C251—22no—
66C2×C4×C251—24no—
67C2×C4×C251—24no—
68C2×C4×C251—24no—
69C2×C4×C251—44no—
70C2×C4×C251C2×D412yes—
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251#4114yes—
73C2×C4×C251C2411yescharacteristic
74C22×C251—22no—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—44no—
79C22×C251#12312yes—
80C22×C251C22×D411yescharacteristic
81C2×D251—41no—
82C2×D251—42no—
83C4×C251—22no—
84C4×C251—44no—
85C4×C251#12312yes—
86C4×C251C2×#4112yes—
87Dic251—41no—
88Dic251—42no—
89C2×C251—22no—
90C2×C251—42no—
91C2×C251C2×#12311yescharacteristic
92C2×C251#57512yes—
93D251—81no—
94C251#309311yescharacteristic, complemented, Sylow
95#3093—2511nomaximal, Sylow
96#525—2511no—
97C2×#117D25111yescharacteristic, complemented
98#485—2514no—
99#482—2514no—
100#428—2512no—
101#417—2512no—
102#407—2511no—
103C2×#41—2511no—
104C23×C4C2×D25111yescharacteristic
105C2×M16—2512no—
106#123—5021no—
107C22⋊Q8—2512no—
108#121—2512no—
109#120—5021no—
110C4×D4—5022no—
111#117—22no—
112#117—2511no—
113#117C2×D25114yes—
114C2×C4⋊C4C2×D25112yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—2514no—
118#106—2514no—
119#103—2514no—
120#102—2514no—
121C22⋊C8—2514no—
122C24C22⋊D25111yescharacteristic
123#41—5024no—
124C2×Q8—2511no—
125C2×D4—5022no—
126C2×D4—5021no—
127C2×D4—2511no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—5021no—
131C22×C4C22⋊D25111yescharacteristic
132C22×C4C22⋊D25112yes—
133C22×C4C22×D25112yes—
134C22×C4C22×D25111yescharacteristic
135M16—5024no—
136C2×C8—2514no—
137C4⋊C4—22no—
138C4⋊C4—5022no—
139C4⋊C4—2512no—
140C4⋊C4C22×D25114yes—
141C22⋊C4—24no—
142C22⋊C4—5021no—
143C22⋊C4—5022no—
144C22⋊C4—5022no—
145C42—24no—
146C42—5021no—
147C23—22no—
148C23—42no—
149C23—5021no—
150C23C2×C22⋊D25112yes—
151C23C2×C22⋊D25111yescharacteristic
152Q8—5022no—
153D4—10044no—
154D4—10042no—
155D4—5022no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—10042no—
162C2×C4—5022no—
163C2×C4—5021no—
164C2×C4C2×C22⋊D25112yes—
165C2×C4C2×C22⋊D25111yescharacteristic
166C2×C4#1686081014yes—
167C2×C4C23×D25111yescharacteristic
168C8—5024no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—10042no—
175C22—10041no—
176C22#2254869812yes—
177C22C22×C22⋊D25111yescharacteristic
178C4—22no—
179C4—44no—
180C4—10042no—
181C4—10041no—
182C4#2254869812yes—
183C4C2×#1686081012yes—
184C2—22no—
185C2—42no—
186C2—20081no—
187C2C2×#2254869811yescharacteristic
188C2#3063248512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#117×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenterC2×#117D251
hypocenterC251#3093
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #30632028C2semidirect
3. #30632011C2semidirect
4. #30631727C2semidirect
5. #30631686C2semidirect
6. #30631651C2semidirect
7. C2×#117×C251C2semidirect
16. #22548648C22semidirect
17. #22548646C22semidirect
18. #22548638C22semidirect
19. #22548635C22semidirect
94. C251#3093semidirect
97. C2×#117D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
#306320282
#306320112
#306317272
#306316862
#306316512
C2×#117×C2512
#3093251