Finite Group Databasebeta

← #46812859

Subgroups — #46812859

Order 34432.

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

normal subgroups only

This group has 216 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#31771406C211yesmaximal, characteristic
2#31771048C211yesmaximal, characteristic, complemented
3C2×#22575397C211yesmaximal, characteristic, complemented
4#31770881C211yesmaximal, characteristic, complemented
5#31770411C211yesmaximal, characteristic, complemented
6#31770390C211yesmaximal, characteristic, complemented
7#361×C269C211yesmaximal, characteristic, complemented
8#22575498—21no—
9#22575486—21no—
10#22575479—21no—
11#22575477—22no—
12#22575475C2211yescharacteristic
13#22575472—21no—
14C2×Dic1076C2211yescharacteristic, complemented
15D8×D269—21no—
16#22575459C2211yescharacteristic
17Dic2152—21no—
18#22575398—21no—
19#22575397—22no—
20C2×#16864377C2211yescharacteristic, complemented
21#22575371—21no—
22C2×D8×C269C2211yescharacteristic
23M32×C269C2211yescharacteristic, complemented
24#108×C269C2211yescharacteristic, complemented
25#16864402—21no—
26#16864402D411yescharacteristic
27#16864401—22no—
28#16864401—41no—
29C2×Dic538—21no—
30C2×C22⋊D269—42no—
31D4×D269—22no—
32D4×D269—41no—
33Dic1076—21no—
34Dic1076—21no—
35Dic1076—41no—
36Dic1076D411yescharacteristic
37#16864391—21no—
38#16864391—41no—
39#16864390—22no—
40D2152D411yescharacteristic
41#16864387—22no—
42#16864386D411yescharacteristic
43#16864385—21no—
44C8×D269—21no—
45#16864377—21no—
46C2×D4×C269—21no—
47D8×C269—21no—
48D8×C269D412yes—
49M16×C269—21no—
50C2×C8×C269C2311yescharacteristic
51C16×C269—21no—
52C22×D269—42no—
53Dic538—21no—
54Dic538—41no—
55Dic538—41no—
56Dic538D811yescharacteristic
57D1076—21no—
58D1076D811yescharacteristic
59C22⋊D269—21no—
60C22⋊D269—42no—
61C22⋊D269—42no—
62C22⋊D269—41no—
63C2×Dic269—42no—
64C2×Dic269—41no—
65C4×D269—21no—
66C4×D269—41no—
67#14524130—21no—
68C23×C269—41no—
69D4×C269—22no—
70D4×C269—41no—
71C2×C4×C269C2×D411yescharacteristic
72C8×C269—41no—
73C8×C269C2×D411yescharacteristic
74C8×C269C2×D411yescharacteristic
75C22×C269—42no—
76C22×C269—161no—
77C22×C269AGL(1,ℤ/8)11yescharacteristic
78C2×D269—21no—
79C2×D269—41no—
80C2×D269—162no—
81C4×C269#12011yescharacteristic
82C4×C269C2×D811yescharacteristic
83Dic269—21no—
84Dic269—41no—
85Dic269—81no—
86C2×C269—21no—
87C2×C269—82no—
88C2×C269#44711yescharacteristic
89D269—41no—
90D269—81no—
91C269#219411yescharacteristic, complemented, Sylow
92#2194—2691nomaximal, Sylow
93#576—2691no—
94#510—2691no—
95C2×SD32—2691no—
96#473—2691no—
97#361D26911yescharacteristic, complemented
98#359—2691no—
99#350—2691no—
100H(5,𝔽2)—5381no—
101#137—5381no—
102AGL(1,ℤ/8)—5381no—
103AGL(1,ℤ/8)—5382no—
104#135—2691no—
105#135—5381no—
106C2×Q16—2691no—
107C2×D8—5381no—
108C2×D8C2×D26911yescharacteristic
109#131—2691no—
110Q32—5381no—
111SD32—5381no—
112SD32—5382no—
113M32C2×D26911yescharacteristic, complemented
114C2×C16—2691no—
115#108C2×D26911yescharacteristic, complemented
116#101—5381no—
117#41—2691no—
118#41—5382no—
119#41—5381no—
120#41—10761no—
121C2×Q8—5381no—
122C2×D4—21no—
123C2×D4—5382no—
124C2×D4—10761no—
125C2×D4—10762no—
126Q16—2691no—
127Q16—5381no—
128Q16—5381no—
129Q16—10761no—
130SD16—5382no—
131SD16—5381no—
132SD16—10761no—
133D8—21no—
134D8—2691no—
135D8—5382no—
136D8D107612yes—
137M16—21no—
138M16—2691no—
139M16—5381no—
140C2×C8—5381no—
141C2×C8C22×D26911yescharacteristic
142C16—21no—
143C16—5381no—
144C23—41no—
145C23—10762no—
146Q8—2691no—
147Q8—5381no—
148Q8—10761no—
149Q8—10761no—
150D4—22no—
151D4—41no—
152D4—2691no—
153D4—5381no—
154D4—5381no—
155D4—10762no—
156D4—10762no—
157D4—10761no—
158C2×C4—5381no—
159C2×C4—10762no—
160C2×C4—10761no—
161C2×C4—10761no—
162C2×C4D4×D26911yescharacteristic
163C8—41no—
164C8—5381no—
165C8D4×D26911yescharacteristic
166C8C2×D107611yescharacteristic
167C22—42no—
168C22—161no—
169C22—5381no—
170C22—10761no—
171C22—43042no—
172C22#2257548011yescharacteristic
173C4—5381no—
174C4—10761no—
175C4—21521no—
176C4#2257542311yescharacteristic
177C4D8×D26911yescharacteristic
178C2—21no—
179C2—82no—
180C2—10761no—
181C2—21521no—
182C2#3177074011yescharacteristic
subgroupG/H
centerC2#31770740
commutator, cosocleC2×C8×C269C23
FrattiniC2×C8C22×D269
Fitting#361×C269C2
socleC2×C269#447
prime radical#31771406C2
hypercenter#361D269
hypocenterC269#2194
LCSG ▷ C2×C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #31771048C2semidirect
3. C2×#22575397C2semidirect
4. #31770881C2semidirect
5. #31770411C2semidirect
6. #31770390C2semidirect
7. #361×C269C2semidirect
14. C2×Dic1076C22semidirect
20. C2×#16864377C22semidirect
23. M32×C269C22semidirect
24. #108×C269C22semidirect
91. C269#2194semidirect
97. #361D269semidirect
113. M32C2×D269semidirect
115. #108C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714062
#317710482
C2×#225753972
#317708812
#317704112
#317703902
#361×C2692
#2194269