Finite Group Databasebeta

← #44552016

Subgroups — #44552016

Order 32128.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#30632251C211yesmaximal, characteristic
2#30631661C211yesmaximal, characteristic, complemented
3#30631475C214yesmaximal, complemented
4#413×C251C211yesmaximal, characteristic, complemented
5C22×D1004C411yescharacteristic, complemented
6#22548711—21no—
7#22548704—21no—
8#22548689—21no—
9#22548689—21no—
10#22548683C412yescomplemented
11#22548683C2212yes—
12#22548674—21no—
13#22548674C411yescharacteristic, complemented
14#22548630—21no—
15#22548630C2212yes—
16C2×#41×C251C2211yescharacteristic
17C2×M16×C251—21no—
18#101×C251—21no—
19#100×C251C2212yes—
20C23×D251C2×C412yes—
21C2×D1004—21no—
22C2×D1004—21no—
23C2×D1004—41no—
24C2×D1004—41no—
25C2×C22⋊D251—22no—
26C2×C22⋊D251—22no—
27C4⋊Dic251—21no—
28C4⋊Dic251—41no—
29C22⋊Dic251—22no—
30C22⋊Dic251C2×C412yes—
31(C2×D251)⋊C4—22no—
32(C2×D251)⋊C4—22no—
33C4×Dic251—21no—
34C4×Dic251—41no—
35#41×C251—41no—
36#41×C251—41no—
37C2×Q8×C251D411yescharacteristic
38C2×D4×C251C2×C411yescharacteristic
39C2×D4×C251D411yescharacteristic
40C2×D4×C251C2311yescharacteristic
41C22×C4×C251C2×C411yescharacteristic
42C22×C4×C251D411yescharacteristic
43C22×C4×C251D411yescharacteristic
44M16×C251—22no—
45M16×C251—41no—
46C2×C8×C251—41no—
47C22×D251—22no—
48C22×D251—42no—
49C22×D251—82no—
50D1004—42no—
51D1004—41no—
52D1004—41no—
53C22⋊D251—82no—
54C22⋊D251—82no—
55C2×Dic251—22no—
56C2×Dic251—42no—
57C23×C251C22⋊C411yescharacteristic
58C23×C251C22×C411yescharacteristic
59C23×C251C2×D411yescharacteristic
60Q8×C251—41no—
61D4×C251—41no—
62D4×C251—41no—
63D4×C251—41no—
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251C22⋊C411yescharacteristic
71C2×C4×C251C22⋊C411yescharacteristic
72C2×C4×C251C22⋊C411yescharacteristic
73C2×C4×C251C2×D411yescharacteristic
74C8×C251—42no—
75C22×C251—21no—
76C22×C251—21no—
77C22×C251—21no—
78C22×C251—41no—
79C22×C251—41no—
80C22×C251—41no—
81C22×C251C2×C22⋊C411yescharacteristic
82C2×D251—42no—
83C2×D251—42no—
84C2×D251—42no—
85C2×D251—84no—
86C4×C251—21no—
87C4×C251—21no—
88C4×C251—21no—
89C4×C251#9911yescharacteristic
90C4×C251#9911yescharacteristic
91Dic251—42no—
92Dic251—82no—
93C2×C251—21no—
94C2×C251—41no—
95C2×C251—41no—
96C2×C251—41no—
97C2×C251C2×#9911yescharacteristic
98D251—82no—
99C251#212211yescharacteristic, complemented, Sylow
100#2122—2511nomaximal, Sylow
101#535—2511no—
102#413D25111yescharacteristic, complemented
103#410—2511no—
104AΣL(1,𝔽16)—2514no—
105C2×#41C2×D25111yescharacteristic
106C22×D4—2511no—
107C2×M16—21no—
108#127—5021no—
109#124—5021no—
110#121—5021no—
111#121—5021no—
112#120—2512no—
113#120—2512no—
114#117—5021no—
115#117—2511no—
116#101—21no—
117#100C2×D25112yes—
118#99—5021no—
119#99—2512no—
120C24—2512no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D25111yescharacteristic
124C2×D4—10041no—
125C2×D4—10041no—
126C2×D4—5022no—
127C2×D4—5022no—
128C2×D4—5021no—
129C2×D4—5021no—
130C2×D4C4×D25111yescharacteristic
131C2×D4D100411yescharacteristic
132C2×D4C22×D25111yescharacteristic
133C22×C4C4×D25111yescharacteristic
134C22×C4C22⋊D25111yescharacteristic
135C22×C4D100411yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—10041no—
140C4⋊C4—5021no—
141C22⋊C4—5022no—
142C22⋊C4—5022no—
143C22⋊C4—5022no—
144C22⋊C4—2512no—
145C42—10041no—
146C42—5021no—
147C23—20082no—
148C23—10042no—
149C23—5022no—
150C23(C2×D251)⋊C411yescharacteristic
151C23C2×C4×D25111yescharacteristic
152C23C2×D100411yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—20082no—
158D4—20082no—
159D4—10042no—
160D4—10041no—
161D4—10041no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—10042no—
169C2×C4—5022no—
170C2×C4(C2×D251)⋊C411yescharacteristic
171C2×C4(C2×D251)⋊C411yescharacteristic
172C2×C4(C2×D251)⋊C411yescharacteristic
173C2×C4C2×C22⋊D25111yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—20084no—
182C22—10042no—
183C22—10042no—
184C22—10042no—
185C22C2×(C2×D251)⋊C411yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—20082no—
190C4—10042no—
191C4#2254862911yescharacteristic
192C4#2254862911yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—20082no—
198C2C2×#2254862911yescharacteristic
subgroupG/H
centerC2C2×#22548629
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#413×C251C2
socleC2×C251C2×#99
prime radical#30632251C2
cosocleC2×D4×C251C23
hypercenter#413D251
hypocenterC251#2122
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #30631661C2semidirect
3. #30631475C2semidirect
4. #413×C251C2semidirect
5. C22×D1004C4semidirect
10. #22548683C4semidirect
13. #22548674C4semidirect
99. C251#2122semidirect
102. #413D251semidirect

Maximal subgroups

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

subgroupindex
#306322512
#306316612
#306314752
#413×C2512
#2122251