Finite Group Databasebeta

← #46812839

Subgroups — #46812839

Order 34432.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#31771407C211yesmaximal, characteristic, complemented
2C2×#22575394C211yesmaximal, characteristic, complemented
3#31770872C211yesmaximal, characteristic, complemented
4#31770419C211yesmaximal, characteristic, complemented
5#31770412C211yesmaximal, characteristic, complemented
6#31770390C211yesmaximal, characteristic, complemented
7#509×C269C211yesmaximal, characteristic, complemented
8#22575499—21no—
9#22575480—22no—
10#22575479—21no—
11#22575478—21no—
12#22575476—21no—
13C2×D2152C2211yescharacteristic
14D8×D269—21no—
15#22575459C2211yescharacteristic
16#22575394—22no—
17C2×#16864377C2211yescharacteristic, complemented
18#22575389C2211yescharacteristic, complemented
19#22575370—21no—
20#135×C269C2211yescharacteristic
21C2×D8×C269C2211yescharacteristic, complemented
22SD32×C269—21no—
23D16×C269—21no—
24M32×C269C2211yescharacteristic, complemented
25#16864403—22no—
26#16864402—41no—
27#16864402D411yescharacteristic
28C2×D1076—21no—
29C2×D1076—41no—
30D4×D269—22no—
31D4×D269—42no—
32#16864390—41no—
33#16864389—22no—
34D2152—21no—
35D2152D412yes—
36#16864387—22no—
37#16864387—41no—
38#16864386—21no—
39#16864386D411yescharacteristic
40#16864385—21no—
41C8×D269—21no—
42#16864377—21no—
43#41×C269—21no—
44C2×D4×C269—21no—
45Q16×C269D411yescharacteristic
46SD16×C269—21no—
47D8×C269—21no—
48D8×C269—21no—
49D8×C269D411yescharacteristic
50C2×C8×C269C2311yescharacteristic
51C16×C269—21no—
52C22×D269—42no—
53C22×D269—41no—
54Dic538D811yescharacteristic
55D1076—22no—
56D1076—42no—
57D1076—41no—
58D1076D811yescharacteristic
59C22⋊D269—21no—
60C22⋊D269—42no—
61C4×D269—21no—
62C4×D269—42no—
63#14524130—21no—
64#14524130—41no—
65C23×C269—41no—
66Q8×C269—21no—
67D4×C269—21no—
68D4×C269—41no—
69D4×C269—41no—
70D4×C269—41no—
71C2×C4×C269—41no—
72C2×C4×C269C2×D411yescharacteristic
73C8×C269C2×D411yescharacteristic
74C8×C269C2×D411yescharacteristic
75C22×C269—41no—
76C22×C269—81no—
77C22×C269—161no—
78C22×C269AGL(1,ℤ/8)11yescharacteristic
79C2×D269—21no—
80C2×D269—42no—
81C2×D269—162no—
82C2×D269—161no—
83C4×C269—41no—
84C4×C269#12011yescharacteristic
85C4×C269C2×D811yescharacteristic
86Dic269—21no—
87C2×C269—21no—
88C2×C269—81no—
89C2×C269—161no—
90C2×C269#44711yescharacteristic
91D269—41no—
92D269—82no—
93C269#219211yescharacteristic, complemented, Sylow
94#2192—2691nomaximal, Sylow
95#576—2691no—
96#509D26911yescharacteristic, complemented
97C2×D16—2691no—
98#472—2691no—
99#361—2691no—
100#359—2691no—
101#350—2691no—
102H(5,𝔽2)—5381no—
103AGL(1,ℤ/8)—5381no—
104AGL(1,ℤ/8)—5382no—
105AGL(1,ℤ/8)—5381no—
106#135—5381no—
107#135C2×D26911yescharacteristic
108C2×D8—2691no—
109C2×D8—5381no—
110C2×D8C2×D26911yescharacteristic, complemented
111#131—2691no—
112SD32—21no—
113D16—21no—
114D16—5382no—
115M32C2×D26911yescharacteristic, complemented
116C2×C16—2691no—
117#108—2691no—
118#100—5381no—
119#41—21no—
120#41—2691no—
121#41—5382no—
122#41—10761no—
123C2×D4—21no—
124C2×D4—5382no—
125C2×D4—5381no—
126C2×D4—10761no—
127C2×D4—10762no—
128Q16C22⋊D26911yescharacteristic
129SD16—21no—
130SD16—5382no—
131SD16—10761no—
132D8—21no—
133D8—21no—
134D8—2692no—
135D8—5382no—
136D8—5381no—
137D8—10761no—
138D8C22⋊D26911yescharacteristic
139M16—2691no—
140M16—5381no—
141M16—5381no—
142C2×C8—5381no—
143C2×C8C22×D26911yescharacteristic
144C16—21no—
145C16—5381no—
146C23—41no—
147C23—10762no—
148C23—10761no—
149Q8—21no—
150Q8—2691no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—2691no—
156D4—5381no—
157D4—5382no—
158D4—10762no—
159D4—10762no—
160D4—10761no—
161C2×C4—41no—
162C2×C4—5381no—
163C2×C4—10762no—
164C2×C4D4×D26911yescharacteristic
165C8—5381no—
166C8—10761no—
167C8D4×D26911yescharacteristic
168C8C2×C22⋊D26911yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—5381no—
173C22—10762no—
174C22—43041no—
175C22—43042no—
176C22#2257547711yescharacteristic
177C4—41no—
178C4—5381no—
179C4#2257542111yescharacteristic
180C4D8×D26911yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—10761no—
185C2—21522no—
186C2#3177073711yescharacteristic
subgroupG/H
centerC2#31770737
commutator, cosocleC2×C8×C269C23
FrattiniC2×C8C22×D269
Fitting#509×C269C2
socleC2×C269#447
hypercenter#509D269
hypocenterC269#2192
LCSG ▷ C2×C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31771407C2semidirect
2. C2×#22575394C2semidirect
3. #31770872C2semidirect
4. #31770419C2semidirect
5. #31770412C2semidirect
6. #31770390C2semidirect
7. #509×C269C2semidirect
17. C2×#16864377C22semidirect
18. #22575389C22semidirect
21. C2×D8×C269C22semidirect
24. M32×C269C22semidirect
93. C269#2192semidirect
96. #509D269semidirect
110. C2×D8C2×D269semidirect
115. M32C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714072
C2×#225753942
#317708722
#317704192
#317704122
#317703902
#509×C2692
#2192269