Finite Group Databasebeta

← H(3,𝔽4)×#6157 #28328988

Subgroups — H(3,𝔽4)×#6157 #28328988

Order 13568.

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#20291258C219yesmaximal, complemented
2#20291254C216yesmaximal, complemented
3#124×#6157C219yesmaximal, complemented
4#120×#6157C216yesmaximal, complemented
5H(3,𝔽4)×D53C211yesmaximal, characteristic
6C24×#6157C2212yescomplemented
7C2×Q8×#6157C2213yescomplemented
8C2×D4×#6157C2219yes—
9(C2×#6157)⋊Q8C2219yescomplemented
10#16811575C2219yes—
11#16811573C22136yescomplemented
12#16811571C22118yes—
13#16811570C22118yescomplemented
14#16811567C2219yescomplemented
15#16811564C2216yescomplemented
16C22⋊C4×#6157C22118yescomplemented
17C42×#6157C2213yescomplemented
18#16811366C416yescomplemented
19#16811365C419yescomplemented
20#124×D53C2219yes—
21#120×D53C2216yes—
22C53×H(3,𝔽4)C411yescharacteristic, complemented
23C23×#6157—412no—
24C23×#6157C2316yes—
25C23×#6157C2316yes—
26Q8×#6157—46no—
27D4×#6157—418no—
28C2×C4⋊#6157C2319yes—
29C2×C4⋊#6157C2319yes—
30C2×C22⋊#6157C23118yes—
31C2×C22⋊#6157C23118yes—
32C2×C4×#6157C2319yes—
33C2×C4×#6157C2319yes—
34(C2×#6157)⋊C4C23136yes—
35C24×D53C2312yes—
36C2×Q8×D53C2313yes—
37C2×D4×D53C2319yes—
38#13133206C2×C419yescomplemented
39#13133205C2×C419yescomplemented
40#13133204C2×C419yescomplemented
41#13133203C2×C4136yescomplemented
42#13133187C2×C4118yescomplemented
43#13133186C2×C416yescomplemented
44#13133185C2×C4118yescomplemented
45C22⋊C4×D53C23118yes—
46C42×D53C2313yes—
47#124×C53C2×C419yescomplemented
48#120×C53C2×C416yescomplemented
49C24×C53C22×C412yescomplemented
50C23×D53—46no—
51C23×D53—46no—
52C23×D53C22×C416yescomplemented
53C23×D53C2416yes—
54C22×#6157—236no—
55C22×#6157—418no—
56C22×#6157—432no—
57C22×#6157C2416yes—
58C22×#6157C2419yes—
59C22×#6157C2411yescharacteristic
60C2×Q8×C53C22×C413yescomplemented
61C2×D4×C53C22×C419yes—
62C2×C22⋊D53C22×C4118yes—
63Q8×D53—43no—
64Q8×D53—43no—
65D4×D53—49no—
66D4×D53—49no—
67C2×D212C22×C419yes—
68C2×C4×D53C2419yes—
69C2×Dic106C22×C419yescomplemented
70C22⋊#6157—418no—
71C22⋊#6157—418no—
72C4⋊#6157—49no—
73C4⋊#6157—49no—
74C4×#6157—218no—
75C4×#6157—49no—
76C22⋊C4×C53C22×C4118yescomplemented
77C42×C53C22×C413yescomplemented
78C22⋊Dic53C22×C4118yescomplemented
79#11719921C22×C4136yescomplemented
80C4×Dic53C22×C419yescomplemented
81C23×C53—46no—
82C23×C53C23×C416yes—
83C22×D53—218no—
84C22×D53—218no—
85C22×D53—418no—
86C22×D53—48no—
87C22×D53—424no—
88C22×D53C23×C416yes—
89C22×D53C2511yescharacteristic
90C2×#6157—29no—
91C2×#6157—218no—
92C2×#6157—218no—
93C2×#6157—424no—
94C2×#6157—424no—
95C2×#6157H(5,𝔽2)16yes—
96Q8×C53—43no—
97D4×C53—49no—
98C2×C4×C53C23×C419yes—
99C22⋊D53—418no—
100C2×Dic53C23×C419yes—
101D212—49no—
102C4×D53—29no—
103C4×D53—29no—
104Dic106—49no—
105C4×H(3,𝔽4)—531nomaximal, Sylow
106C22×C53—218no—
107C22×C53—48no—
108C22×C53C24×C411yescharacteristic
109C2×D53—218no—
110C2×D53—424no—
111C2×D53—46no—
112C2×D53—418no—
113C2×D53C2×H(5,𝔽2)13yes—
114C2×D53#58513yes—
115#6157—46no—
116#6157—49no—
117#6157H(3,𝔽4)14yesdirect product factor
118C4×C53—29no—
119Dic53—29no—
120C2×H(3,𝔽4)—531no—
121#2379—539no—
122#2375—536no—
123C4×#124—539no—
124C4×#120—536no—
125C2×C53—46no—
126C2×C53C4×H(5,𝔽2)13yes—
127D53—46no—
128D53C2×H(3,𝔽4)11yescharacteristic
129D53#352913yes—
130C24×C4—532no—
131H(3,𝔽4)—539no—
132H(3,𝔽4)—536no—
133H(3,𝔽4)#615711yescharacteristic, direct product factor
134C2×#124—539no—
135C2×#120—536no—
136C2×C4×Q8—533no—
137C2×C4×D4—539no—
138#391—539no—
139#390—539no—
140#388—5336no—
141#386—5318no—
142#385—5318no—
143#382—539no—
144#379—536no—
145C4×C22⋊C4—5318no—
146C43—533no—
147C53C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
148C25—532no—
149C22×Q8—533no—
150C22×D4—539no—
151C23×C4—536no—
152C23×C4—536no—
153C23×C4—21212no—
154#124—539no—
155#124—539no—
156#124—5336no—
157#124—539no—
158#124C2×#615719yescomplemented
159#120—536no—
160#120—5318no—
161#120—5318no—
162#120C2×#615716yescomplemented
163C4×Q8—2126no—
164C4×D4—21218no—
165C2×C4⋊C4—539no—
166C2×C4⋊C4—539no—
167C2×C22⋊C4—5318no—
168C2×C22⋊C4—5318no—
169C2×C22⋊C4—5318no—
170C2×C42—533no—
171C2×C42—539no—
172C2×C42—539no—
173#95—5336no—
174C24—536no—
175C24—536no—
176C24—2126no—
177C24—2126no—
178C24C22×#615712yescomplemented
179C2×Q8—539no—
180C2×Q8—2123no—
181C2×Q8—2123no—
182C2×Q8C22×#615713yescomplemented
183C2×D4—5318no—
184C2×D4—539no—
185C2×D4—2129no—
186C2×D4—2129no—
187C2×D4C22×#615719yes—
188C22×C4—531no—
189C22×C4—539no—
190C22×C4—536no—
191C22×C4—539no—
192C22×C4—10636no—
193C22×C4—21218no—
194C22×C4—21232no—
195C4⋊C4—2129no—
196C4⋊C4—2129no—
197C22⋊C4—5318no—
198C22⋊C4—5336no—
199C22⋊C4—21218no—
200C22⋊C4—21218no—
201C22⋊C4C22×#6157118yescomplemented
202C42—539no—
203C42—10618no—
204C42—2129no—
205C42C22×#615713yescomplemented
206C23—46no—
207C23—531no—
208C23—536no—
209C23—10618no—
210C23—10618no—
211C23—21218no—
212C23—2128no—
213C23—21224no—
214C23C23×#615716yes—
215Q8—43no—
216Q8—2129no—
217D4—49no—
218D4—2129no—
219D4—21218no—
220C2×C4—536no—
221C2×C4—539no—
222C2×C4—1069no—
223C2×C4—1069no—
224C2×C4—1069no—
225C2×C4—10618no—
226C2×C4—10618no—
227C2×C4—21224no—
228C2×C4—21224no—
229C2×C4C23×#615719yes—
230C22—218no—
231C22—48no—
232C22—533no—
233C22—533no—
234C22—10618no—
235C22—2126no—
236C22—21218no—
237C22—21224no—
238C22C24×#615711yescharacteristic
239C4—29no—
240C4—534no—
241C4—1069no—
242C4—2126no—
243C4—2129no—
244C2—46no—
245C2—531no—
246C2—533no—
247C2—2126no—
248C2H(5,𝔽2)×#615713yes—
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×H(3,𝔽4)C4
prime radicalH(3,𝔽4)×D53C2
cosocleC22×D53C25
hypercenterH(3,𝔽4)#6157
hypocenterC53C4×H(3,𝔽4)
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20291258C2semidirect
2. #20291254C2semidirect
3. #124×#6157C2semidirect
4. #120×#6157C2semidirect
6. C24×#6157C22semidirect
7. C2×Q8×#6157C22semidirect
9. (C2×#6157)⋊Q8C22semidirect
11. #16811573C22semidirect
13. #16811570C22semidirect
14. #16811567C22semidirect
15. #16811564C22semidirect
16. C22⋊C4×#6157C22semidirect
17. C42×#6157C22semidirect
18. #16811366C4semidirect
19. #16811365C4semidirect
22. C53×H(3,𝔽4)C4semidirect
38. #13133206C2×C4semidirect
39. #13133205C2×C4semidirect
40. #13133204C2×C4semidirect
41. #13133203C2×C4semidirect
42. #13133187C2×C4semidirect
43. #13133186C2×C4semidirect
44. #13133185C2×C4semidirect
47. #124×C53C2×C4semidirect
48. #120×C53C2×C4semidirect
49. C24×C53C22×C4semidirect
52. C23×D53C22×C4semidirect
60. C2×Q8×C53C22×C4semidirect
69. C2×Dic106C22×C4semidirect
76. C22⋊C4×C53C22×C4semidirect
77. C42×C53C22×C4semidirect
78. C22⋊Dic53C22×C4semidirect
79. #11719921C22×C4semidirect
80. C4×Dic53C22×C4semidirect
117. #6157H(3,𝔽4)direct (N × H)
133. H(3,𝔽4)#6157direct (N × H)
147. C53C4×H(3,𝔽4)semidirect
158. #124C2×#6157semidirect
162. #120C2×#6157semidirect
178. C24C22×#6157semidirect
182. C2×Q8C22×#6157semidirect
201. C22⋊C4C22×#6157semidirect
205. C42C22×#6157semidirect

Maximal subgroups

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

subgroupindex
#202912582
#202912542
#124×#61572
#120×#61572
H(3,𝔽4)×D532
C4×H(3,𝔽4)53