Finite Group Databasebeta

← #46812855

Subgroups — #46812855

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#31771046C211yesmaximal, characteristic, complemented
3#31770877C211yesmaximal, characteristic, complemented
4#31770418C211yesmaximal, characteristic, complemented
5#31770413C211yesmaximal, characteristic, complemented
6#31770388C211yesmaximal, characteristic, complemented
7C2×SD32×C269C211yesmaximal, characteristic, complemented
8#22575498—21no—
9#22575484—21no—
10#22575479—21no—
11#22575477—22no—
12#22575475C2211yescharacteristic
13#22575472—21no—
14D8×D269—21no—
15#22575459C2211yescharacteristic
16#22575399—21no—
17#22575396—21no—
18#22575393C2211yescharacteristic, complemented
19#22575389C2211yescharacteristic, complemented
20#22575372—21no—
21C2×Q16×C269C2211yescharacteristic, complemented
22C2×D8×C269C2211yescharacteristic
23SD32×C269—22no—
24C2×C16×C269C2211yescharacteristic, complemented
25#16864402—21no—
26#16864402D411yescharacteristic
27#16864401—22no—
28#16864401—41no—
29C2×C22⋊D269—42no—
30D4×D269—22no—
31D4×D269—41no—
32Dic1076D411yescharacteristic
33#16864392—41no—
34#16864391—21no—
35#16864390—22no—
36#16864389—41no—
37D2152D411yescharacteristic
38#16864387—22no—
39#16864386—21no—
40#16864386D411yescharacteristic
41#16864385—21no—
42C8×D269—21no—
43#16864377—21no—
44C2×Q8×C269—21no—
45C2×D4×C269—21no—
46Q16×C269—21no—
47Q16×C269—21no—
48D8×C269—21no—
49D8×C269D412yes—
50C2×C8×C269C2311yescharacteristic
51C16×C269—21no—
52C22×D269—42no—
53Dic538—21no—
54Dic538D811yescharacteristic
55D1076—21no—
56D1076D811yescharacteristic
57C22⋊D269—21no—
58C22⋊D269—42no—
59C22⋊D269—42no—
60C22⋊D269—41no—
61C2×Dic269—42no—
62C4×D269—21no—
63C4×D269—41no—
64#14524130—21no—
65#14524130—41no—
66C23×C269—41no—
67Q8×C269—41no—
68Q8×C269—41no—
69D4×C269—22no—
70D4×C269—41no—
71C2×C4×C269—41no—
72C2×C4×C269C2×D411yescharacteristic
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—81no—
82C4×C269#12011yescharacteristic
83C4×C269C2×D811yescharacteristic
84Dic269—21no—
85Dic269—41no—
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×SD32D26911yescharacteristic, complemented
96#473—2691no—
97#361—2691no—
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×Q16C2×D26911yescharacteristic, complemented
107C2×D8—5381no—
108C2×D8C2×D26911yescharacteristic
109#131—2691no—
110Q32—5381no—
111SD32—22no—
112SD32—5381no—
113M32—2691no—
114C2×C16C2×D26911yescharacteristic, complemented
115#108—2691no—
116#101—5381no—
117#41—2691no—
118#41—5382no—
119#41—5381no—
120#41—10761no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—5382no—
124C2×D4—10761no—
125C2×D4—10762no—
126Q16—21no—
127Q16—21no—
128Q16—2691no—
129Q16—10761no—
130SD16—5382no—
131SD16—5381no—
132SD16—10761no—
133D8—21no—
134D8—2691no—
135D8—5382no—
136D8C22⋊D26912yes—
137M16—2691no—
138M16—5381no—
139M16—5381no—
140C2×C8—5381no—
141C2×C8C22×D26911yescharacteristic
142C16—21no—
143C16—5381no—
144C23—41no—
145C23—10762no—
146Q8—41no—
147Q8—41no—
148Q8—2691no—
149Q8—5381no—
150D4—22no—
151D4—41no—
152D4—2691no—
153D4—5381no—
154D4—5381no—
155D4—10762no—
156D4—10762no—
157D4—10761no—
158C2×C4—41no—
159C2×C4—5381no—
160C2×C4—10762no—
161C2×C4—10761no—
162C2×C4D4×D26911yescharacteristic
163C8—5381no—
164C8—10761no—
165C8D4×D26911yescharacteristic
166C8C2×C22⋊D26911yescharacteristic
167C22—42no—
168C22—161no—
169C22—5381no—
170C22—10761no—
171C22—43042no—
172C22#2257547711yescharacteristic
173C4—81no—
174C4—5381no—
175C4—10761no—
176C4#2257542111yescharacteristic
177C4D8×D26911yescharacteristic
178C2—21no—
179C2—82no—
180C2—10761no—
181C2—21521no—
182C2#3177073711yescharacteristic
subgroupG/H
centerC2#31770737
commutator, cosocleC2×C8×C269C23
FrattiniC2×C8C22×D269
FittingC2×SD32×C269C2
socleC2×C269#447
prime radical#31771406C2
hypercenterC2×SD32D269
hypocenterC269#2194
LCSG ▷ C2×C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #31771046C2semidirect
3. #31770877C2semidirect
4. #31770418C2semidirect
5. #31770413C2semidirect
6. #31770388C2semidirect
7. C2×SD32×C269C2semidirect
18. #22575393C22semidirect
19. #22575389C22semidirect
21. C2×Q16×C269C22semidirect
24. C2×C16×C269C22semidirect
91. C269#2194semidirect
95. C2×SD32D269semidirect
106. C2×Q16C2×D269semidirect
114. C2×C16C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714062
#317710462
#317708772
#317704182
#317704132
#317703882
C2×SD32×C2692
#2194269