Finite Group Databasebeta

← H(3,𝔽4)×#595 #17979220

Subgroups — H(3,𝔽4)×#595 #17979220

Order 4352.

2664 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 2664 conjugacy classes of subgroups — too many to draw. Only its 776 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#14545327C219yesmaximal, complemented
2#14545323C216yesmaximal, complemented
3#124×#595C219yesmaximal, complemented
4#120×#595C216yesmaximal, complemented
5D17×H(3,𝔽4)C211yesmaximal, characteristic
6C24×#595C2212yescomplemented
7C2×Q8×#595C2213yescomplemented
8C2×D4×#595C2219yes—
9(C2×#595)⋊Q8C2219yescomplemented
10#11763761C2219yes—
11#11763759C22136yescomplemented
12#11763757C22118yes—
13#11763756C22118yescomplemented
14#11763753C2219yescomplemented
15#11763750C2216yescomplemented
16C22⋊C4×#595C22118yescomplemented
17C42×#595C2213yescomplemented
18#11763552C416yescomplemented
19#11763551C419yescomplemented
20#124×D17C2219yes—
21#120×D17C2216yes—
22C17×H(3,𝔽4)C411yescharacteristic, complemented
23C24×D17C2312yes—
24C23×#595—412no—
25C23×#595C2316yes—
26C23×#595C2316yes—
27C2×Q8×D17C2313yes—
28C2×D4×D17C2319yes—
29C2×C22⋊#595C23118yes—
30C2×C22⋊#595C23118yes—
31Q8×#595—46no—
32D4×#595—418no—
33C2×C4⋊#595C2319yes—
34C2×C4⋊#595C2319yes—
35C2×C4×#595C2319yes—
36C2×C4×#595C2319yes—
37C17×#124C2×C419yescomplemented
38C17×#120C2×C416yescomplemented
39#10587639C2×C416yescomplemented
40#10587633C2×C419yescomplemented
41#10587623C2×C4118yescomplemented
42#10587622C2×C419yescomplemented
43#10587571C2×C4136yescomplemented
44#10587568C2×C4118yescomplemented
45C22⋊C4×D17C23118yes—
46#10587561C2×C419yescomplemented
47C42×D17C2313yes—
48(C2×#595)⋊C4C23136yes—
49C24×C17C22×C412yescomplemented
50C23×D17—46no—
51C23×D17—46no—
52C23×D17C22×C416yescomplemented
53C23×D17C2416yes—
54C22×#595—236no—
55C22×#595—418no—
56C22×#595—432no—
57C22×#595C2419yes—
58C22×#595C2416yes—
59C22×#595C2411yescharacteristic
60C2×Q8×C17C22×C413yescomplemented
61C2×D4×C17C22×C419yes—
62C2×C22⋊D17C22×C4118yes—
63Q8×D17—43no—
64Q8×D17—43no—
65D4×D17—49no—
66D4×D17—49no—
67C2×D68C22×C419yes—
68C2×C4×D17C2419yes—
69C2×Dic34C22×C419yescomplemented
70C22⋊#595—418no—
71C22⋊#595—418no—
72C4⋊#595—49no—
73C4⋊#595—49no—
74C4×#595—218no—
75C4×#595—49no—
76C22⋊C4×C17C22×C4118yescomplemented
77C42×C17C22×C413yescomplemented
78C22⋊Dic17C22×C4118yescomplemented
79#63227C22×C4136yescomplemented
80C4×Dic17C22×C419yescomplemented
81C4×H(3,𝔽4)—171nomaximal, Sylow
82C23×C17—46no—
83C23×C17C23×C416yes—
84C22×D17—218no—
85C22×D17—218no—
86C22×D17—418no—
87C22×D17—48no—
88C22×D17—424no—
89C22×D17C23×C416yes—
90C22×D17C2511yescharacteristic
91C2×#595—29no—
92C2×#595—218no—
93C2×#595—218no—
94C2×#595—424no—
95C2×#595—424no—
96C2×#595H(5,𝔽2)16yes—
97Q8×C17—43no—
98D4×C17—49no—
99C2×C4×C17C23×C419yes—
100C22⋊D17—418no—
101C2×Dic17C23×C419yes—
102D68—49no—
103C4×D17—29no—
104C4×D17—29no—
105Dic34—49no—
106C2×H(3,𝔽4)—171no—
107#2379—179no—
108#2375—176no—
109C4×#124—179no—
110C4×#120—176no—
111C22×C17—218no—
112C22×C17—48no—
113C22×C17C24×C411yescharacteristic
114C2×D17—218no—
115C2×D17—424no—
116C2×D17—46no—
117C2×D17—418no—
118C2×D17C2×H(5,𝔽2)13yes—
119C2×D17#58513yes—
120#595—46no—
121#595—49no—
122#595H(3,𝔽4)14yesdirect product factor
123C4×C17—29no—
124Dic17—29no—
125C24×C4—172no—
126H(3,𝔽4)—179no—
127H(3,𝔽4)—176no—
128H(3,𝔽4)#59511yescharacteristic, direct product factor
129C2×#124—179no—
130C2×#120—176no—
131C2×C4×Q8—173no—
132C2×C4×D4—179no—
133#391—179no—
134#390—179no—
135#388—1736no—
136#386—1718no—
137#385—1718no—
138#382—179no—
139#379—176no—
140C4×C22⋊C4—1718no—
141C43—173no—
142C2×C17—46no—
143C2×C17C4×H(5,𝔽2)13yes—
144D17—46no—
145D17C2×H(3,𝔽4)11yescharacteristic
146D17#352913yes—
147C25—172no—
148C22×Q8—173no—
149C22×D4—179no—
150C23×C4—176no—
151C23×C4—176no—
152C23×C4—6812no—
153#124—179no—
154#124—179no—
155#124—1736no—
156#124—179no—
157#124C2×#59519yescomplemented
158#120—176no—
159#120—1718no—
160#120—1718no—
161#120C2×#59516yescomplemented
162C4×Q8—686no—
163C4×D4—6818no—
164C2×C4⋊C4—179no—
165C2×C4⋊C4—179no—
166C2×C22⋊C4—1718no—
167C2×C22⋊C4—1718no—
168C2×C22⋊C4—1718no—
169C2×C42—173no—
170C2×C42—179no—
171C2×C42—179no—
172#95—1736no—
173C17C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
174C24—176no—
175C24—176no—
176C24—686no—
177C24—686no—
178C24C22×#59512yescomplemented
179C2×Q8—179no—
180C2×Q8—683no—
181C2×Q8—683no—
182C2×Q8C22×#59513yescomplemented
183C2×D4—1718no—
184C2×D4—179no—
185C2×D4—689no—
186C2×D4—689no—
187C2×D4C22×#59519yes—
188C22×C4—171no—
189C22×C4—179no—
190C22×C4—176no—
191C22×C4—179no—
192C22×C4—3436no—
193C22×C4—6818no—
194C22×C4—6832no—
195C4⋊C4—689no—
196C4⋊C4—689no—
197C22⋊C4—1718no—
198C22⋊C4—1736no—
199C22⋊C4—6818no—
200C22⋊C4—6818no—
201C22⋊C4C22×#595118yescomplemented
202C42—179no—
203C42—3418no—
204C42—689no—
205C42C22×#59513yescomplemented
206C23—46no—
207C23—171no—
208C23—176no—
209C23—3418no—
210C23—3418no—
211C23—6818no—
212C23—688no—
213C23—6824no—
214C23C23×#59516yes—
215Q8—43no—
216Q8—689no—
217D4—49no—
218D4—689no—
219D4—6818no—
220C2×C4—176no—
221C2×C4—179no—
222C2×C4—349no—
223C2×C4—349no—
224C2×C4—349no—
225C2×C4—3418no—
226C2×C4—3418no—
227C2×C4—6824no—
228C2×C4—6824no—
229C2×C4C23×#59519yes—
230C22—218no—
231C22—48no—
232C22—173no—
233C22—173no—
234C22—3418no—
235C22—686no—
236C22—6818no—
237C22—6824no—
238C22C24×#59511yescharacteristic
239C4—29no—
240C4—174no—
241C4—349no—
242C4—686no—
243C4—689no—
244C2—46no—
245C2—171no—
246C2—173no—
247C2—686no—
248C2H(5,𝔽2)×#59513yes—
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×H(3,𝔽4)C4
prime radicalD17×H(3,𝔽4)C2
cosocleC22×D17C25
hypercenterH(3,𝔽4)#595
hypocenterC17C4×H(3,𝔽4)
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #14545327C2semidirect
2. #14545323C2semidirect
3. #124×#595C2semidirect
4. #120×#595C2semidirect
6. C24×#595C22semidirect
7. C2×Q8×#595C22semidirect
9. (C2×#595)⋊Q8C22semidirect
11. #11763759C22semidirect
13. #11763756C22semidirect
14. #11763753C22semidirect
15. #11763750C22semidirect
16. C22⋊C4×#595C22semidirect
17. C42×#595C22semidirect
18. #11763552C4semidirect
19. #11763551C4semidirect
22. C17×H(3,𝔽4)C4semidirect
37. C17×#124C2×C4semidirect
38. C17×#120C2×C4semidirect
39. #10587639C2×C4semidirect
40. #10587633C2×C4semidirect
41. #10587623C2×C4semidirect
42. #10587622C2×C4semidirect
43. #10587571C2×C4semidirect
44. #10587568C2×C4semidirect
46. #10587561C2×C4semidirect
49. C24×C17C22×C4semidirect
52. C23×D17C22×C4semidirect
60. C2×Q8×C17C22×C4semidirect
69. C2×Dic34C22×C4semidirect
76. C22⋊C4×C17C22×C4semidirect
77. C42×C17C22×C4semidirect
78. C22⋊Dic17C22×C4semidirect
79. #63227C22×C4semidirect
80. C4×Dic17C22×C4semidirect
122. #595H(3,𝔽4)direct (N × H)
128. H(3,𝔽4)#595direct (N × H)
157. #124C2×#595semidirect
161. #120C2×#595semidirect
173. C17C4×H(3,𝔽4)semidirect
178. C24C22×#595semidirect
182. C2×Q8C22×#595semidirect
201. C22⋊C4C22×#595semidirect
205. C42C22×#595semidirect

Maximal subgroups

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

subgroupindex
#145453272
#145453232
#124×#5952
#120×#5952
D17×H(3,𝔽4)2
C4×H(3,𝔽4)17