Finite Group Databasebeta

← #46812837

Subgroups — #46812837

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#31771402C211yesmaximal, characteristic
2#31771048C211yesmaximal, characteristic, complemented
3#31771045C211yesmaximal, characteristic, complemented
4#31770864C211yesmaximal, characteristic, complemented
5#31770414C211yesmaximal, characteristic, complemented
6#31770386C211yesmaximal, characteristic, complemented
7#360×C269C211yesmaximal, characteristic, complemented
8#41×D269—21no—
9#22575485—21no—
10#22575482—21no—
11#22575480—21no—
12#22575477—21no—
13#22575475C2211yescharacteristic
14#22575473—21no—
15C2×Dic1076C2211yescharacteristic, complemented
16C2×#16864385C2211yescharacteristic
17#22575433—21no—
18Dic2152—21no—
19#22575399—21no—
20#22575398—21no—
21#22575397—21no—
22#22575393C2211yescharacteristic, complemented
23#22575376—21no—
24#135×C269C2211yescharacteristic
25M32×C269C2211yescharacteristic, complemented
26#106×C269C2211yescharacteristic, complemented
27#16864403—21no—
28#16864402—21no—
29#16864402—41no—
30#16864401—21no—
31#16864401—41no—
32C2×Dic538—21no—
33Q8×D269—21no—
34D4×D269—21no—
35D4×D269—41no—
36C2×C4×D269—41no—
37C2×C4×D269D411yescharacteristic
38Dic1076—21no—
39Dic1076—21no—
40Dic1076D411yescharacteristic
41#16864392—21no—
42#16864391—21no—
43#16864390—21no—
44#16864389—21no—
45D2152D411yescharacteristic
46#16864387—21no—
47#16864385—21no—
48#16864385—21no—
49C2×#14524130D411yescharacteristic
50C4⋊Dic269—41no—
51#16864379—41no—
52#16864377—21no—
53#41×C269—21no—
54Q16×C269D411yescharacteristic
55SD16×C269—21no—
56D8×C269D411yescharacteristic
57C2×C8×C269C2311yescharacteristic
58C4⋊C4×C269—21no—
59C16×C269—21no—
60C22×D269—41no—
61C22×D269D811yescharacteristic
62Dic538—21no—
63Dic538—41no—
64Dic538—41no—
65Dic538—41no—
66D1076—21no—
67D1076—41no—
68C22⋊D269—41no—
69C22⋊D269—41no—
70C22⋊D269—41no—
71C2×Dic269—41no—
72C2×Dic269—41no—
73C2×Dic269D811yescharacteristic
74C4×D269—21no—
75C4×D269—21no—
76C4×D269—41no—
77C4×D269—41no—
78C4×D269—41no—
79#14524130—21no—
80Q8×C269—21no—
81D4×C269—21no—
82D4×C269—41no—
83C2×C4×C269—41no—
84C2×C4×C269—41no—
85C2×C4×C269C2×D411yescharacteristic
86C8×C269C2×D411yescharacteristic
87C8×C269C2×D411yescharacteristic
88C22×C269—41no—
89C22×C269C2×D811yescharacteristic
90C2×D269—21no—
91C2×D269—41no—
92C2×D269—41no—
93C2×D269—82no—
94C4×C269—41no—
95C4×C269—81no—
96C4×C269#12011yescharacteristic
97C4×C269AGL(1,ℤ/8)11yescharacteristic
98Dic269—21no—
99Dic269—41no—
100Dic269—81no—
101C2×C269—21no—
102C2×C269—81no—
103C2×C269#44711yescharacteristic
104D269—41no—
105D269—81no—
106C269#219111yescharacteristic, complemented, Sylow
107#2191—2691nomaximal, Sylow
108#575—2691no—
109#510—2691no—
110#510—2691no—
111#470—2691no—
112#360—2691no—
113#360D26911yescharacteristic, complemented
114#349—2691no—
115C2×#41—5381no—
116#137—5381no—
117#137—5381no—
118AGL(1,ℤ/8)—5381no—
119AGL(1,ℤ/8)—5381no—
120#135—2691no—
121#135—5381no—
122#135C2×D26911yescharacteristic
123C2×Q16—2691no—
124C2×M16—2691no—
125C22⋊Q8—5381no—
126Q32—5381no—
127Q32—5381no—
128SD32—5381no—
129SD32—5381no—
130M32—2691no—
131M32C2×D26911yescharacteristic, complemented
132#106C2×D26911yescharacteristic, complemented
133#103—5381no—
134#41—21no—
135#41—5381no—
136#41—5381no—
137#41—5381no—
138#41—10761no—
139#41—10761no—
140C2×Q8—5381no—
141C2×Q8—5381no—
142C2×D4—5381no—
143C2×D4—10761no—
144C22×C4—2691no—
145C22×C4—10761no—
146Q16—2691no—
147Q16—5381no—
148Q16—5381no—
149Q16—5381no—
150Q16D107611yescharacteristic
151SD16—21no—
152SD16—5381no—
153SD16—5381no—
154SD16—5381no—
155D8—2691no—
156D8—5381no—
157D8D107611yescharacteristic
158M16—5381no—
159M16—5381no—
160C2×C8—2691no—
161C2×C8C22×D26911yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—10761no—
164C22⋊C4—10761no—
165C16—21no—
166C16—5381no—
167C23—2691no—
168C23—10761no—
169Q8—21no—
170Q8—5381no—
171Q8—10761no—
172Q8—10761no—
173Q8—10761no—
174D4—21no—
175D4—41no—
176D4—5381no—
177D4—10761no—
178D4—10761no—
179D4—10761no—
180D4—10761no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—2691no—
184C2×C4—5381no—
185C2×C4—5381no—
186C2×C4—10761no—
187C2×C4—10761no—
188C2×C4—10761no—
189C2×C4—10761no—
190C2×C4—10761no—
191C2×C4D4×D26911yescharacteristic
192C8—5381no—
193C8D4×D26911yescharacteristic
194C8C2×D107611yescharacteristic
195C22—41no—
196C22—5381no—
197C22—10761no—
198C22—10761no—
199C22—21522no—
200C22D8×D26911yescharacteristic
201C4—41no—
202C4—81no—
203C4—5381no—
204C4—10761no—
205C4—21521no—
206C4#2257542311yescharacteristic
207C4#2257548011yescharacteristic
208C2—21no—
209C2—81no—
210C2—10761no—
211C2—21521no—
212C2#3177074011yescharacteristic
subgroupG/H
centerC2#31770740
commutator, cosocleC2×C8×C269C23
FrattiniC2×C8C22×D269
Fitting#360×C269C2
socleC2×C269#447
prime radical#31771402C2
hypercenter#360D269
hypocenterC269#2191
LCSG ▷ C2×C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #31771048C2semidirect
3. #31771045C2semidirect
4. #31770864C2semidirect
5. #31770414C2semidirect
6. #31770386C2semidirect
7. #360×C269C2semidirect
15. C2×Dic1076C22semidirect
22. #22575393C22semidirect
25. M32×C269C22semidirect
26. #106×C269C22semidirect
106. C269#2191semidirect
113. #360D269semidirect
131. M32C2×D269semidirect
132. #106C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714022
#317710482
#317710452
#317708642
#317704142
#317703862
#360×C2692
#2191269