Finite Group Databasebeta

← #46814024

Subgroups — #46814024

Order 34432.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#22575422C211yesmaximal, characteristic
2#31770505C213yesmaximal, complemented
3#31770499C213yesmaximal, complemented
4#31770493C213yesmaximal, complemented
5#31770481C213yesmaximal, complemented
6#31770461C211yesmaximal, characteristic, complemented
7#379×C269C211yesmaximal, characteristic, complemented
8C22×C22⋊D269C2213yes—
9C22×C4×D269C2211yescharacteristic, complemented
10#22575422C418yescomplemented
11C2×#16864381C2213yescomplemented
12C2×C22⋊Dic269C2213yes—
13C2×C22⋊Dic269C2213yes—
14C2×C22⋊Dic269C2213yes—
15C2×#16864379C2213yescomplemented
16C2×C4×Dic269C2213yescomplemented
17#22575360C2213yescomplemented
18#22575360C2213yescomplemented
19C25×C269C2211yescharacteristic
20C2×C22⋊C4×C269C2213yes—
21C2×C22⋊C4×C269C2213yes—
22C23×D269C2311yescharacteristic
23C2×C22⋊D269—43no—
24C2×C22⋊D269—43no—
25C2×C22⋊D269C2×C4112yes—
26C22×Dic269C2311yescharacteristic
27C22×Dic269C2313yes—
28C22×Dic269C2313yes—
29C2×C4×D269—43no—
30#16864381—43no—
31C22⋊Dic269—43no—
32C22⋊Dic269—43no—
33C22⋊Dic269C2×C4112yescomplemented
34#16864379—43no—
35C4×Dic269—43no—
36C24×C269—43no—
37C24×C269—43no—
38C24×C269C2×C414yescomplemented
39C24×C269C2313yes—
40C22×C4×C269C2311yescharacteristic
41C22×C4×C269C2313yes—
42C22⋊C4×C269—43no—
43C22⋊C4×C269—43no—
44C22×D269—23no—
45C22×D269—23no—
46C22×D269C22×C412yes—
47C22⋊D269—412no—
48C2×Dic269—23no—
49C2×Dic269—23no—
50C2×Dic269—23no—
51C2×Dic269—23no—
52C2×Dic269—23no—
53C2×Dic269—23no—
54C2×Dic269C22×C416yes—
55C4×D269—44no—
56C23×C269—23no—
57C23×C269—23no—
58C23×C269—23no—
59C23×C269—23no—
60C23×C269—23no—
61C23×C269—23no—
62C23×C269—412no—
63C23×C269—412no—
64C23×C269—44no—
65C23×C269C22×C416yes—
66C23×C269C2411yescharacteristic
67C2×C4×C269—23no—
68C2×C4×C269—23no—
69C2×C4×C269—23no—
70C2×C4×C269—23no—
71C22×C269—26no—
72C22×C269—26no—
73C22×C269—26no—
74C22×C269—416no—
75C22×C269—43no—
76C22×C269—43no—
77C22×C269—43no—
78C22×C269—43no—
79C22×C269C23×C411yescharacteristic
80C22×C269H(5,𝔽2)13yes—
81C22×C269H(5,𝔽2)13yes—
82C2×D269—26no—
83C2×D269—43no—
84C2×D269—41no—
85C4×C269—43no—
86C4×C269—41no—
87Dic269—26no—
88Dic269—43no—
89Dic269—41no—
90C2×C269—46no—
91C2×C269#51813yes—
92C2×C269#56013yes—
93C2×C269H(3,𝔽4)11yescharacteristic
94D269—42no—
95C269#237511yescharacteristic, complemented, Sylow
96#2375—2691nomaximal, Sylow
97C2×#120—2691no—
98#390—2693no—
99#388—2693no—
100#388—2693no—
101#386—2693no—
102#379—2691no—
103#379D26911yescharacteristic, complemented
104C25C2×D26911yescharacteristic
105C22×D4—2693no—
106C23×C4—2691no—
107#120—2698no—
108C2×C4⋊C4—2693no—
109C2×C22⋊C4—2693no—
110C2×C22⋊C4—2693no—
111C2×C22⋊C4—2693no—
112C2×C22⋊C4—2693no—
113C2×C22⋊C4C2×D26913yes—
114C2×C22⋊C4C2×D26913yes—
115C2×C42—2693no—
116#95—2693no—
117#95—2693no—
118C24—43no—
119C24—43no—
120C24—2691no—
121C24C4×D26914yescomplemented
122C24C22×D26913yes—
123C2×D4—26912no—
124C2×D4—10763no—
125C2×D4—10763no—
126C22×C4—2693no—
127C22×C4—2691no—
128C22×C4—2693no—
129C22×C4—10763no—
130C22×C4C22×D26911yescharacteristic
131C22×C4C22×D26913yes—
132C4⋊C4—10763no—
133C22⋊C4—43no—
134C22⋊C4—43no—
135C22⋊C4—26912no—
136C22⋊C4—10763no—
137C22⋊C4—10763no—
138C22⋊C4—10763no—
139C42—10763no—
140C23—23no—
141C23—23no—
142C23—23no—
143C23—23no—
144C23—23no—
145C23—23no—
146C23—412no—
147C23—412no—
148C23—44no—
149C23—2692no—
150C23—5383no—
151C23—5383no—
152C23C2×C4×D26916yes—
153C23C23×D26911yescharacteristic
154D4—107612no—
155C2×C4—23no—
156C2×C4—23no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—2696no—
160C2×C4—5383no—
161C2×C4—5383no—
162C2×C4—5383no—
163C2×C4—5383no—
164C2×C4—5383no—
165C2×C4—5383no—
166C2×C4—10764no—
167C22—26no—
168C22—26no—
169C22—26no—
170C22—416no—
171C22—43no—
172C22—43no—
173C22—43no—
174C22—43no—
175C22—5386no—
176C22—10763no—
177C22—10761no—
178C22C22×C4×D26911yescharacteristic
179C22#2257549813yes—
180C22#2257549813yes—
181C4—43no—
182C4—41no—
183C4—5386no—
184C4—10763no—
185C4—10761no—
186C2—46no—
187C2—10762no—
188C2#3177107813yes—
189C2#3177134413yes—
190C2#3177134811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#379×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575422C2
hypercenter#379D269
hypocenterC269#2375
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #31770505C2semidirect
3. #31770499C2semidirect
4. #31770493C2semidirect
5. #31770481C2semidirect
6. #31770461C2semidirect
7. #379×C269C2semidirect
9. C22×C4×D269C22semidirect
10. #22575422C4semidirect
11. C2×#16864381C22semidirect
15. C2×#16864379C22semidirect
16. C2×C4×Dic269C22semidirect
17. #22575360C22semidirect
18. #22575360C22semidirect
33. C22⋊Dic269C2×C4semidirect
38. C24×C269C2×C4semidirect
95. C269#2375semidirect
103. #379D269semidirect
121. C24C4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754222
#317705052
#317704992
#317704932
#317704812
#317704612
#379×C2692
#2375269