Finite Group Databasebeta

← #44550821

Subgroups — #44550821

Order 32128.

378 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 89 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×#22548698C211yesmaximal, characteristic
2C2×#22548674C211yesmaximal, characteristic, complemented
3C2×#22548629C212yesmaximal
4#30631453C212yesmaximal, complemented
5#381×C251C211yesmaximal, characteristic, complemented
6C22×C22⋊D251C2211yescharacteristic
7#22548698—22no—
8#22548698—21no—
9#22548698—21no—
10#22548674—24no—
11C2×C4⋊Dic251—21no—
12C2×Dic251⋊C4C412yescomplemented
13C2×C22⋊Dic251C2211yescharacteristic
14C2×C22⋊Dic251C2212yes—
15C2×(C2×D251)⋊C4C412yescomplemented
16C2×C4×Dic251—21no—
17#22548629—24no—
18C23×C4×C251C2211yescharacteristic
19C2×C22⋊C4×C251C2212yes—
20#95×C251—22no—
21C23×D251C2×C411yescharacteristic
22C2×C22⋊D251—22no—
23C2×C22⋊D251—21no—
24C2×C22⋊D251—21no—
25C2×C22⋊D251—22no—
26C22×Dic251—22no—
27C22×Dic251C2×C411yescharacteristic
28C22×Dic251C2×C412yes—
29C4⋊Dic251—24no—
30Dic251⋊C4—22no—
31Dic251⋊C4—22no—
32C22⋊Dic251—22no—
33C22⋊Dic251—21no—
34C22⋊Dic251—21no—
35C22⋊Dic251D414yes—
36(C2×D251)⋊C4—22no—
37(C2×D251)⋊C4—22no—
38C4×Dic251—24no—
39C24×C251C2311yescharacteristic
40C22×C4×C251—22no—
41C22×C4×C251—22no—
42C22×C4×C251—22no—
43C22×C4×C251C2×C412yes—
44C22×C4×C251D412yes—
45C22×C4×C251D411yescharacteristic
46C22×C4×C251D411yescharacteristic
47C22⋊C4×C251—22no—
48C22⋊C4×C251—22no—
49C22×D251—21no—
50C22×D251—22no—
51C22×D251—42no—
52C22⋊D251—41no—
53C22⋊D251—42no—
54C22⋊D251—41no—
55C2×Dic251—22no—
56C2×Dic251—21no—
57C2×Dic251—22no—
58C2×Dic251—24no—
59C2×Dic251—44no—
60C2×Dic251—42no—
61C23×C251—42no—
62C23×C251C22×C411yescharacteristic
63C23×C251C2×D412yes—
64C23×C251C2×D411yescharacteristic
65C23×C251C2×D411yescharacteristic
66C23×C251#4112yes—
67C2×C4×C251—24no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—24no—
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—44no—
74C2×C4×C251—42no—
75C2×C4×C251C22⋊C412yes—
76C2×C4×C251C22⋊C412yes—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—22no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251C2×C22⋊C411yescharacteristic
83C22×C251C4×D412yes—
84C22×C251#12011yescharacteristic
85C22×C251#12112yes—
86C22×C251#12311yescharacteristic
87C2×D251—41no—
88C2×D251—42no—
89C2×D251—82no—
90C4×C251—22no—
91C4×C251—22no—
92C4×C251—82no—
93Dic251—44no—
94Dic251—41no—
95Dic251—42no—
96C2×C251—21no—
97C2×C251—22no—
98C2×C251—22no—
99C2×C251—21no—
100C2×C251#38611yescharacteristic
101C2×C251#41012yes—
102D251—81no—
103C251#189911yescharacteristic, complemented, Sylow
104#1899—2511nomaximal, Sylow
105C2×#123—2511no—
106C2×#117—2511no—
107C2×#99—2512no—
108#381D25111yescharacteristic, complemented
109#342—2512no—
110C22×D4—2511no—
111C23×C4C2×D25111yescharacteristic
112#123—5022no—
113#123—5021no—
114#123—5021no—
115#117—5024no—
116C2×C4⋊C4—5021no—
117C2×C4⋊C4—2512no—
118C2×C22⋊C4—2511no—
119C2×C22⋊C4—2512no—
120C2×C22⋊C4—2512no—
121C2×C22⋊C4C2×D25112yes—
122C2×C42—5021no—
123#99—5024no—
124#95—22no—
125C24—2511no—
126C24C22×D25111yescharacteristic
127C2×D4—5022no—
128C2×D4—5021no—
129C2×D4—5021no—
130C2×D4—5022no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—5022no—
135C22×C4—2512no—
136C22×C4—2511no—
137C22×C4C4×D25112yes—
138C22×C4C22⋊D25112yes—
139C22×C4D100411yescharacteristic
140C22×C4D100411yescharacteristic
141C4⋊C4—5024no—
142C4⋊C4—5022no—
143C4⋊C4—5022no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—5022no—
147C22⋊C4—5021no—
148C22⋊C4—5021no—
149C22⋊C4—5022no—
150C22⋊C4—5022no—
151C22⋊C4—2514no—
152C42—5024no—
153C23—42no—
154C23—10042no—
155C23—5022no—
156C23—5021no—
157C23C2×C4×D25111yescharacteristic
158C23D4×D25112yes—
159C23C2×C22⋊D25111yescharacteristic
160C23C2×D100411yescharacteristic
161C23#1686080912yes—
162D4—10041no—
163D4—10042no—
164D4—10041no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—10044no—
174C2×C4—10042no—
175C2×C4—5022no—
176C2×C4—5021no—
177C2×C4—5022no—
178C2×C4—5024no—
179C2×C4(C2×D251)⋊C412yes—
180C2×C4(C2×D251)⋊C412yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—20082no—
187C22—10041no—
188C22—10042no—
189C22C2×(C2×D251)⋊C411yescharacteristic
190C22(C22⋊D251)⋊C412yes—
191C22#2254868511yescharacteristic
192C22#2254868712yes—
193C22#2254869911yescharacteristic
194C4—22no—
195C4—22no—
196C4—82no—
197C4—10044no—
198C4—10041no—
199C4—10042no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—20081no—
205C2#3063156411yescharacteristic
206C2#3063166012yes—
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC24C22×D251
Fitting#381×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×C22⋊D251C22
cosocleC24×C251C23
hypercenter#381D251
hypocenterC251#1899
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#22548674C2semidirect
4. #30631453C2semidirect
5. #381×C251C2semidirect
12. C2×Dic251⋊C4C4semidirect
15. C2×(C2×D251)⋊C4C4semidirect
103. C251#1899semidirect
108. #381D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
C2×#225486742
C2×#225486292
#306314532
#381×C2512
#1899251