Finite Group Databasebeta

← H(3,𝔽4)×Dic17 #17197030

Subgroups — H(3,𝔽4)×Dic17 #17197030

Order 4352.

2664 conjugacy classes of subgroups; only the 1093 normal subgroups are shown.

normal subgroups only

This group has 2664 conjugacy classes of subgroups — too many to draw. Only its 1093 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#14531958C219yesmaximal, complemented
2#14531925C216yesmaximal, complemented
3#124×Dic17C219yesmaximal, complemented
4#120×Dic17C216yesmaximal, complemented
5C2×C17×H(3,𝔽4)C211yesmaximal, characteristic
6C24×Dic17C2212yescomplemented
7C2×Q8×Dic17C2213yescomplemented
8C2×D4×Dic17C2219yes—
9#11762716C2219yescomplemented
10#11762712C2219yes—
11#11762702C22136yescomplemented
12#11762689C22118yes—
13#11762684C22118yescomplemented
14#11762675C2219yescomplemented
15#11762665C2216yescomplemented
16C22⋊C4×Dic17C22118yescomplemented
17C42×Dic17C2213yescomplemented
18C17×H(3,𝔽4)C4116yescomplemented
19C2×C17×#124C2219yes—
20C2×C17×#120C2216yes—
21C25×C17C2312yes—
22C22×Q8×C17C2313yes—
23C22×D4×C17C2319yes—
24C23×Dic17—412no—
25C23×Dic17C2316yes—
26C23×Dic17C2316yes—
27C17×#124C2×C4172yescomplemented
28C17×#120C2×C4148yescomplemented
29C2×C22⋊C4×C17C23118yes—
30C2×C42×C17C2313yes—
31C2×C22⋊Dic17C23118yes—
32C2×C22⋊Dic17C23118yes—
33Q8×Dic17—46no—
34D4×Dic17—418no—
35C2×C4⋊Dic17C2319yes—
36C2×C4⋊Dic17C2319yes—
37C2×C4×Dic17C2319yes—
38C2×C4×Dic17C2319yes—
39#10587502C23136yes—
40C24×C17—46no—
41C24×C17—46no—
42C24×C17C22×C418yescomplemented
43C24×C17C2416yes—
44C2×Q8×C17—43no—
45C2×Q8×C17—43no—
46C2×Q8×C17C22×C4112yescomplemented
47C2×D4×C17—49no—
48C2×D4×C17—49no—
49C2×D4×C17C22×C4136yes—
50C22×C4×C17C2419yes—
51C22×Dic17—236no—
52C22×Dic17—418no—
53C22×Dic17—432no—
54C22×Dic17C2416yes—
55C22×Dic17C2419yes—
56C22×Dic17C2411yescharacteristic
57C22⋊C4×C17C22×C4172yescomplemented
58C42×C17C22×C4112yescomplemented
59C22⋊Dic17—418no—
60C22⋊Dic17—418no—
61C4⋊Dic17—49no—
62C4⋊Dic17—49no—
63C4×Dic17—218no—
64C4×Dic17—49no—
65C4×H(3,𝔽4)—171nomaximal, Sylow
66C23×C17—218no—
67C23×C17—218no—
68C23×C17—424no—
69C23×C17—48no—
70C23×C17—424no—
71C23×C17C23×C4112yes—
72C23×C17C2511yescharacteristic
73Q8×C17—412no—
74D4×C17—436no—
75C2×C4×C17—29no—
76C2×C4×C17—29no—
77C2×C4×C17C23×C4118yes—
78C2×Dic17—29no—
79C2×Dic17—218no—
80C2×Dic17—218no—
81C2×Dic17—424no—
82C2×Dic17—424no—
83C2×Dic17H(5,𝔽2)16yes—
84C2×H(3,𝔽4)D1711yescharacteristic
85#2379—179no—
86#2375—176no—
87C4×#124—179no—
88C4×#120—176no—
89C22×C17—236no—
90C22×C17—432no—
91C22×C17—46no—
92C22×C17—418no—
93C22×C17C24×C411yescharacteristic
94C22×C17C2×H(5,𝔽2)13yes—
95C22×C17#58513yes—
96C4×C17—218no—
97Dic17—46no—
98Dic17—49no—
99Dic17H(3,𝔽4)14yesdirect product factor
100C24×C4—172no—
101H(3,𝔽4)Dic17116yesdirect product factor
102C2×#124C2×D1719yes—
103C2×#120C2×D1716yes—
104C2×C4×Q8—173no—
105C2×C4×D4—179no—
106#391—179no—
107#390—179no—
108#388—1736no—
109#386—1718no—
110#385—1718no—
111#382—179no—
112#379—176no—
113C4×C22⋊C4—1718no—
114C43—173no—
115C2×C17—412no—
116C2×C17C4×H(5,𝔽2)13yes—
117C2×C17C2×H(3,𝔽4)11yescharacteristic
118C2×C17#352913yes—
119C25C22×D1712yes—
120C22×Q8C22×D1713yes—
121C22×D4C22×D1719yes—
122C23×C4—176no—
123C23×C4—176no—
124C23×C4—6812no—
125#124C2×Dic17172yescomplemented
126#120C2×Dic17148yescomplemented
127C4×Q8—686no—
128C4×D4—6818no—
129C2×C4⋊C4—179no—
130C2×C4⋊C4—179no—
131C2×C22⋊C4—1718no—
132C2×C22⋊C4—1718no—
133C2×C22⋊C4C22×D17118yes—
134C2×C42—179no—
135C2×C42—179no—
136C2×C42C22×D1713yes—
137#95—1736no—
138C17C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
139C24—46no—
140C24—46no—
141C24C22×Dic1718yescomplemented
142C24C23×D1716yes—
143C2×Q8—43no—
144C2×Q8—43no—
145C2×Q8C22×Dic17112yescomplemented
146C2×D4—49no—
147C2×D4—49no—
148C2×D4C22×Dic17136yes—
149C22×C4—171no—
150C22×C4—176no—
151C22×C4—179no—
152C22×C4—3436no—
153C22×C4—6818no—
154C22×C4—6832no—
155C22×C4C23×D1719yes—
156C4⋊C4—689no—
157C4⋊C4—689no—
158C22⋊C4—6818no—
159C22⋊C4—6818no—
160C22⋊C4C22×Dic17172yescomplemented
161C42—3418no—
162C42—689no—
163C42C22×Dic17112yescomplemented
164C23—218no—
165C23—218no—
166C23—424no—
167C23—48no—
168C23—424no—
169C23C23×Dic17112yes—
170C23C24×D1711yescharacteristic
171Q8—412no—
172D4—436no—
173C2×C4—29no—
174C2×C4—29no—
175C2×C4—176no—
176C2×C4—349no—
177C2×C4—3418no—
178C2×C4—3418no—
179C2×C4—6824no—
180C2×C4—6824no—
181C2×C4C23×Dic17118yes—
182C22—236no—
183C22—432no—
184C22—46no—
185C22—418no—
186C22C24×Dic1711yescharacteristic
187C22H(5,𝔽2)×D1713yes—
188C22#1176364413yes—
189C4—218no—
190C4—174no—
191C4—686no—
192C4—689no—
193C2—412no—
194C2H(5,𝔽2)×Dic1713yes—
195C2D17×H(3,𝔽4)11yescharacteristic
196C2#1454292413yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
Fitting, prime radicalC2×C17×H(3,𝔽4)C2
socle, cosocleC23×C17C25
hypercenterC2×H(3,𝔽4)D17
hypocenterC17C4×H(3,𝔽4)
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #14531958C2semidirect
2. #14531925C2semidirect
3. #124×Dic17C2semidirect
4. #120×Dic17C2semidirect
6. C24×Dic17C22semidirect
7. C2×Q8×Dic17C22semidirect
9. #11762716C22semidirect
11. #11762702C22semidirect
13. #11762684C22semidirect
14. #11762675C22semidirect
15. #11762665C22semidirect
16. C22⋊C4×Dic17C22semidirect
17. C42×Dic17C22semidirect
18. C17×H(3,𝔽4)C4semidirect
27. C17×#124C2×C4semidirect
28. C17×#120C2×C4semidirect
42. C24×C17C22×C4semidirect
46. C2×Q8×C17C22×C4semidirect
57. C22⋊C4×C17C22×C4semidirect
58. C42×C17C22×C4semidirect
99. Dic17H(3,𝔽4)direct (N × H)
101. H(3,𝔽4)Dic17direct (N × H)
125. #124C2×Dic17semidirect
126. #120C2×Dic17semidirect
138. C17C4×H(3,𝔽4)semidirect
141. C24C22×Dic17semidirect
145. C2×Q8C22×Dic17semidirect
160. C22⋊C4C22×Dic17semidirect
163. C42C22×Dic17semidirect

Maximal subgroups

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

subgroupindex
#145319582
#145319252
#124×Dic172
#120×Dic172
C2×C17×H(3,𝔽4)2
C4×H(3,𝔽4)17