Subgroups — C₄×C₅₃×H(3,𝔽₄) #27241865

Order 13568.

2664 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2379	C₂	1	9	yes	maximal, complemented
3	C₅₃×#2375	C₂	1	6	yes	maximal, complemented
4	C₄×#124×C₅₃	C₂	1	9	yes	maximal, complemented
5	C₄×#120×C₅₃	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
7	C₅₃×H(3,𝔽₄)	C₄	1	16	yes	direct product factor
8	C₂×#124×C₅₃	C₂²	1	9	yes	—
9	C₂×#120×C₅₃	C₂²	1	6	yes	—
10	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	complemented
11	C₂×C₄×D₄×C₅₃	C₂²	1	9	yes	—
12	C₅₃×#391	C₂²	1	9	yes	complemented
13	C₅₃×#390	C₂²	1	9	yes	—
14	C₅₃×#388	C₂²	1	36	yes	complemented
15	C₅₃×#386	C₂²	1	18	yes	—
16	C₅₃×#385	C₂²	1	18	yes	complemented
17	C₅₃×#382	C₂²	1	9	yes	complemented
18	C₅₃×#379	C₂²	1	6	yes	complemented
19	C₄×C₂²⋊C₄×C₅₃	C₂²	1	18	yes	complemented
20	C₄³×C₅₃	C₂²	1	3	yes	complemented
21	C₂⁵×C₅₃	C₂³	1	2	yes	—
22	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
23	C₂²×D₄×C₅₃	C₂³	1	9	yes	—
24	C₂³×C₄×C₅₃	—	4	12	no	—
25	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
27	#124×C₅₃	C₂×C₄	1	72	yes	complemented
28	#120×C₅₃	C₂×C₄	1	48	yes	complemented
29	C₄×Q₈×C₅₃	—	4	6	no	—
30	C₄×D₄×C₅₃	—	4	18	no	—
31	C₂×C₄⋊C₄×C₅₃	C₂³	1	9	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂³	1	9	yes	—
33	C₂×C₂²⋊C₄×C₅₃	C₂³	1	18	yes	—
34	C₂×C₂²⋊C₄×C₅₃	C₂³	1	18	yes	—
35	C₂×C₂²⋊C₄×C₅₃	C₂³	1	18	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
39	#95×C₅₃	C₂³	1	36	yes	—
40	C₂⁴×C₅₃	—	4	6	no	—
41	C₂⁴×C₅₃	—	4	6	no	—
42	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	complemented
43	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
44	C₂×Q₈×C₅₃	—	4	3	no	—
45	C₂×Q₈×C₅₃	—	4	3	no	—
46	C₂×Q₈×C₅₃	C₂²×C₄	1	12	yes	complemented
47	C₂×D₄×C₅₃	—	4	9	no	—
48	C₂×D₄×C₅₃	—	4	9	no	—
49	C₂×D₄×C₅₃	C₂²×C₄	1	36	yes	—
50	C₂²×C₄×C₅₃	—	2	36	no	—
51	C₂²×C₄×C₅₃	—	4	18	no	—
52	C₂²×C₄×C₅₃	—	4	32	no	—
53	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	4	9	no	—
58	C₄⋊C₄×C₅₃	—	4	9	no	—
59	C₂²⋊C₄×C₅₃	—	4	18	no	—
60	C₂²⋊C₄×C₅₃	—	4	18	no	—
61	C₂²⋊C₄×C₅₃	C₂²×C₄	1	72	yes	complemented
62	C₄²×C₅₃	—	2	18	no	—
63	C₄²×C₅₃	—	4	9	no	—
64	C₄²×C₅₃	C₂²×C₄	1	12	yes	complemented
65	C₂³×C₅₃	—	2	18	no	—
66	C₂³×C₅₃	—	2	18	no	—
67	C₂³×C₅₃	—	4	24	no	—
68	C₂³×C₅₃	—	4	8	no	—
69	C₂³×C₅₃	—	4	24	no	—
70	C₂³×C₅₃	C₂³×C₄	1	12	yes	—
71	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	12	no	—
73	D₄×C₅₃	—	4	36	no	—
74	C₂×C₄×C₅₃	—	2	9	no	—
75	C₂×C₄×C₅₃	—	2	9	no	—
76	C₂×C₄×C₅₃	—	2	18	no	—
77	C₂×C₄×C₅₃	—	2	9	no	—
78	C₂×C₄×C₅₃	—	2	18	no	—
79	C₂×C₄×C₅₃	—	4	24	no	—
80	C₂×C₄×C₅₃	—	4	24	no	—
81	C₂×C₄×C₅₃	C₂³×C₄	1	18	yes	—
82	C₂×C₄×C₅₃	H(5,𝔽₂)	1	6	yes	—
83	C₄×H(3,𝔽₄)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₅₃	—	2	36	no	—
85	C₂²×C₅₃	—	4	32	no	—
86	C₂²×C₅₃	—	4	6	no	—
87	C₂²×C₅₃	—	4	18	no	—
88	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	3	yes	—
90	C₂²×C₅₃	#585	1	3	yes	—
91	C₄×C₅₃	—	2	18	no	—
92	C₄×C₅₃	—	4	6	no	—
93	C₄×C₅₃	—	4	9	no	—
94	C₄×C₅₃	H(3,𝔽₄)	1	4	yes	direct product factor
95	C₂×H(3,𝔽₄)	C₂×C₅₃	1	1	yes	characteristic
96	#2379	C₂×C₅₃	1	9	yes	complemented
97	#2375	C₂×C₅₃	1	6	yes	complemented
98	C₄×#124	C₂×C₅₃	1	9	yes	complemented
99	C₄×#120	C₂×C₅₃	1	6	yes	complemented
100	C₂×C₅₃	—	4	12	no	—
101	C₂×C₅₃	C₄×H(5,𝔽₂)	1	3	yes	—
102	C₂×C₅₃	C₂×H(3,𝔽₄)	1	1	yes	characteristic
103	C₂×C₅₃	#3529	1	3	yes	—
104	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	complemented
105	H(3,𝔽₄)	C₄×C₅₃	1	16	yes	direct product factor
106	C₂×#124	C₂²×C₅₃	1	9	yes	—
107	C₂×#120	C₂²×C₅₃	1	6	yes	—
108	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	complemented
109	C₂×C₄×D₄	C₂²×C₅₃	1	9	yes	—
110	#391	C₂²×C₅₃	1	9	yes	complemented
111	#390	C₂²×C₅₃	1	9	yes	—
112	#388	C₂²×C₅₃	1	36	yes	complemented
113	#386	C₂²×C₅₃	1	18	yes	—
114	#385	C₂²×C₅₃	1	18	yes	complemented
115	#382	C₂²×C₅₃	1	9	yes	complemented
116	#379	C₂²×C₅₃	1	6	yes	complemented
117	C₄×C₂²⋊C₄	C₂²×C₅₃	1	18	yes	complemented
118	C₄³	C₂²×C₅₃	1	3	yes	complemented
119	C₅₃	C₄×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
120	C₂⁵	C₂³×C₅₃	1	2	yes	—
121	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
122	C₂²×D₄	C₂³×C₅₃	1	9	yes	—
123	C₂³×C₄	—	4	12	no	—
124	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
125	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
126	#124	C₂×C₄×C₅₃	1	72	yes	complemented
127	#120	C₂×C₄×C₅₃	1	48	yes	complemented
128	C₄×Q₈	—	4	6	no	—
129	C₄×D₄	—	4	18	no	—
130	C₂×C₄⋊C₄	C₂³×C₅₃	1	9	yes	—
131	C₂×C₄⋊C₄	C₂³×C₅₃	1	9	yes	—
132	C₂×C₂²⋊C₄	C₂³×C₅₃	1	18	yes	—
133	C₂×C₂²⋊C₄	C₂³×C₅₃	1	18	yes	—
134	C₂×C₂²⋊C₄	C₂³×C₅₃	1	18	yes	—
135	C₂×C₄²	C₂³×C₅₃	1	9	yes	—
136	C₂×C₄²	C₂³×C₅₃	1	9	yes	—
137	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
138	#95	C₂³×C₅₃	1	36	yes	—
139	C₂⁴	—	4	6	no	—
140	C₂⁴	—	4	6	no	—
141	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	complemented
142	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
143	C₂×Q₈	—	4	3	no	—
144	C₂×Q₈	—	4	3	no	—
145	C₂×Q₈	C₂²×C₄×C₅₃	1	12	yes	complemented
146	C₂×D₄	—	4	9	no	—
147	C₂×D₄	—	4	9	no	—
148	C₂×D₄	C₂²×C₄×C₅₃	1	36	yes	—
149	C₂²×C₄	—	2	36	no	—
150	C₂²×C₄	—	4	18	no	—
151	C₂²×C₄	—	4	32	no	—
152	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
153	C₂²×C₄	C₂⁴×C₅₃	1	9	yes	—
154	C₂²×C₄	C₂⁴×C₅₃	1	9	yes	—
155	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
156	C₄⋊C₄	—	4	9	no	—
157	C₄⋊C₄	—	4	9	no	—
158	C₂²⋊C₄	—	4	18	no	—
159	C₂²⋊C₄	—	4	18	no	—
160	C₂²⋊C₄	C₂²×C₄×C₅₃	1	72	yes	complemented
161	C₄²	—	2	18	no	—
162	C₄²	—	4	9	no	—
163	C₄²	C₂²×C₄×C₅₃	1	12	yes	complemented
164	C₂³	—	2	18	no	—
165	C₂³	—	2	18	no	—
166	C₂³	—	4	24	no	—
167	C₂³	—	4	8	no	—
168	C₂³	—	4	24	no	—
169	C₂³	C₂³×C₄×C₅₃	1	12	yes	—
170	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
171	Q₈	—	4	12	no	—
172	D₄	—	4	36	no	—
173	C₂×C₄	—	2	9	no	—
174	C₂×C₄	—	2	9	no	—
175	C₂×C₄	—	2	9	no	—
176	C₂×C₄	—	2	18	no	—
177	C₂×C₄	—	2	18	no	—
178	C₂×C₄	—	4	24	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	C₂³×C₄×C₅₃	1	18	yes	—
181	C₂×C₄	H(5,𝔽₂)×C₅₃	1	6	yes	—
182	C₂²	—	2	36	no	—
183	C₂²	—	4	32	no	—
184	C₂²	—	4	6	no	—
185	C₂²	—	4	18	no	—
186	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
187	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	3	yes	—
188	C₂²	C₅₃×#585	1	3	yes	—
189	C₄	—	2	18	no	—
190	C₄	—	4	6	no	—
191	C₄	—	4	9	no	—
192	C₄	C₅₃×H(3,𝔽₄)	1	4	yes	direct product factor
193	C₂	—	4	12	no	—
194	C₂	C₄×H(5,𝔽₂)×C₅₃	1	3	yes	—
195	C₂	C₂×C₅₃×H(3,𝔽₄)	1	1	yes	characteristic
196	C₂	C₅₃×#3529	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂×C₅₃×H(3,𝔽₄)	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2379	C₂	semidirect
3. C₅₃×#2375	C₂	semidirect
4. C₄×#124×C₅₃	C₂	semidirect
5. C₄×#120×C₅₃	C₂	semidirect
6. C₂⁴×C₄×C₅₃	C₂²	semidirect
7. C₅₃×H(3,𝔽₄)	C₄	direct (N × H)
10. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
12. C₅₃×#391	C₂²	semidirect
14. C₅₃×#388	C₂²	semidirect
16. C₅₃×#385	C₂²	semidirect
17. C₅₃×#382	C₂²	semidirect
18. C₅₃×#379	C₂²	semidirect
19. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
20. C₄³×C₅₃	C₂²	semidirect
27. #124×C₅₃	C₂×C₄	semidirect
28. #120×C₅₃	C₂×C₄	semidirect
42. C₂⁴×C₅₃	C₂²×C₄	semidirect
46. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
61. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
64. C₄²×C₅₃	C₂²×C₄	semidirect
83. C₄×H(3,𝔽₄)	C₅₃	direct (N × H)
94. C₄×C₅₃	H(3,𝔽₄)	direct (N × H)
96. #2379	C₂×C₅₃	semidirect
97. #2375	C₂×C₅₃	semidirect
98. C₄×#124	C₂×C₅₃	semidirect
99. C₄×#120	C₂×C₅₃	semidirect
104. C₂⁴×C₄	C₂²×C₅₃	semidirect
105. H(3,𝔽₄)	C₄×C₅₃	direct (N × H)
108. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
110. #391	C₂²×C₅₃	semidirect
112. #388	C₂²×C₅₃	semidirect
114. #385	C₂²×C₅₃	semidirect
115. #382	C₂²×C₅₃	semidirect
116. #379	C₂²×C₅₃	semidirect
117. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
118. C₄³	C₂²×C₅₃	semidirect
119. C₅₃	C₄×H(3,𝔽₄)	direct (N × H)
126. #124	C₂×C₄×C₅₃	semidirect
127. #120	C₂×C₄×C₅₃	semidirect
141. C₂⁴	C₂²×C₄×C₅₃	semidirect
145. C₂×Q₈	C₂²×C₄×C₅₃	semidirect
160. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
163. C₄²	C₂²×C₄×C₅₃	semidirect
192. C₄	C₅₃×H(3,𝔽₄)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×H(3,𝔽₄)	2
C₅₃×#2379	2
C₅₃×#2375	2
C₄×#124×C₅₃	2
C₄×#120×C₅₃	2
C₄×H(3,𝔽₄)	53

Subgroups — C4×C53×H(3,𝔽4) #27241865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×H(3,𝔽₄) #27241865