Subgroups — C₄×C₅₃×H(3,ℤ/4) #27218209

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#117×C₅₃	C₂	1	2	yes	maximal
3	C₂×C₅₃×H(3,ℤ/4)	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
5	C₂×#117×C₅₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₅₃	—	2	4	no	—
8	C₂×#97×C₅₃	C₄	1	2	yes	complemented
9	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
10	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
11	C₄×C₂²⋊C₄×C₅₃	—	2	4	no	—
12	C₅₃×#376	C₄	1	4	yes	complemented
13	C₄³×C₅₃	C₄	1	4	yes	complemented
14	C₅₃×H(3,ℤ/4)	C₄	1	16	yes	direct product factor
15	C₂³×C₄×C₅₃	D₄	1	2	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×M₁₆×C₅₃	—	2	4	no	—
18	C₂×M₁₆×C₅₃	—	2	4	no	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	#117×C₅₃	—	2	8	no	—
22	#117×C₅₃	C₂×C₄	1	16	yes	—
23	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
25	C₂×C₄²×C₅₃	—	2	8	no	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	4	yes	—
32	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
33	#97×C₅₃	—	2	4	no	—
34	C₄×C₈×C₅₃	—	2	4	no	—
35	#95×C₅₃	—	2	8	no	—
36	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	8	no	—
38	C₂²×C₄×C₅₃	—	4	4	no	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	4	yes	—
49	M₁₆×C₅₃	—	2	16	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	C₄²	1	8	yes	complemented
53	C₄⋊C₄×C₅₃	—	2	16	no	—
54	C₂²⋊C₄×C₅₃	—	2	16	no	—
55	C₄²×C₅₃	—	2	8	no	—
56	C₄²×C₅₃	—	4	16	no	—
57	C₄²×C₅₃	C₄²	1	1	yes	characteristic
58	C₄²×C₅₃	C₄²	1	2	yes	—
59	C₄²×C₅₃	C₄²	1	1	yes	characteristic
60	C₄²×C₅₃	C₄²	1	16	yes	complemented
61	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂³×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₅₃	C₄×D₄	1	4	yes	—
69	C₂×C₄×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	16	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	16	no	—
75	C₂×C₄×C₅₃	#95	1	4	yes	—
76	C₂×C₄×C₅₃	#95	1	2	yes	—
77	C₂×C₄×C₅₃	#95	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₅₃	#117	1	2	yes	—
84	C₂×C₄×C₅₃	#117	1	4	yes	—
85	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
86	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
87	C₈×C₅₃	—	2	8	no	—
88	C₄×H(3,ℤ/4)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	—	2	4	no	—
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	—	4	4	no	—
92	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
93	C₂²×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
94	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	4	16	no	—
98	C₄×C₅₃	H(3,ℤ/4)	1	4	yes	direct product factor
99	C₄×C₅₃	C₄³	1	1	yes	characteristic
100	C₄×C₅₃	#376	1	1	yes	characteristic
101	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
102	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
104	C₄×#117	C₂×C₅₃	1	2	yes	—
105	C₂×H(3,ℤ/4)	C₂×C₅₃	1	4	yes	—
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	—	2	2	no	—
108	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
109	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
110	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
111	C₂×#117	C₂²×C₅₃	1	4	yes	—
112	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
113	C₄×M₁₆	—	2	4	no	—
114	C₂×#97	C₄×C₅₃	1	2	yes	complemented
115	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	complemented
116	C₄×C₄⋊C₄	—	2	4	no	—
117	C₄×C₂²⋊C₄	—	2	4	no	—
118	#376	C₄×C₅₃	1	4	yes	complemented
119	C₄³	C₄×C₅₃	1	4	yes	complemented
120	H(3,ℤ/4)	C₄×C₅₃	1	16	yes	direct product factor
121	C₅₃	C₄×H(3,ℤ/4)	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³×C₄	D₄×C₅₃	1	2	yes	—
123	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
127	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
128	#117	—	2	8	no	—
129	#117	C₂×C₄×C₅₃	1	16	yes	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₂²⋊C₄	—	2	4	no	—
132	C₂×C₄²	—	2	8	no	—
133	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
134	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
138	C₂×C₄²	D₄×C₅₃	1	4	yes	—
139	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
140	#97	—	2	4	no	—
141	C₄×C₈	—	2	4	no	—
142	#95	—	2	8	no	—
143	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
148	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
149	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
153	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
154	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	#41×C₅₃	1	4	yes	—
156	M₁₆	—	2	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₄²×C₅₃	1	8	yes	complemented
160	C₄⋊C₄	—	2	16	no	—
161	C₂²⋊C₄	—	2	16	no	—
162	C₄²	—	2	8	no	—
163	C₄²	—	4	16	no	—
164	C₄²	C₄²×C₅₃	1	1	yes	characteristic
165	C₄²	C₄²×C₅₃	1	2	yes	—
166	C₄²	C₄²×C₅₃	1	1	yes	characteristic
167	C₄²	C₄²×C₅₃	1	16	yes	complemented
168	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
169	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
170	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
171	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
172	C₂³	—	4	2	no	—
173	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
174	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂³	C₄×D₄×C₅₃	1	4	yes	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	16	no	—
182	C₂×C₄	#95×C₅₃	1	2	yes	—
183	C₂×C₄	#95×C₅₃	1	4	yes	—
184	C₂×C₄	#95×C₅₃	1	2	yes	—
185	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
187	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
188	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
190	C₂×C₄	#117×C₅₃	1	2	yes	—
191	C₂×C₄	#117×C₅₃	1	4	yes	—
192	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
193	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
199	C₂²	C₄×C₂²⋊C₄×C₅₃	1	4	yes	—
200	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	16	no	—
204	C₄	C₅₃×H(3,ℤ/4)	1	4	yes	direct product factor
205	C₄	C₄³×C₅₃	1	1	yes	characteristic
206	C₄	C₅₃×#376	1	1	yes	characteristic
207	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
208	C₄	C₄×C₄⋊C₄×C₅₃	1	4	yes	—
209	C₂	—	2	4	no	—
210	C₂	—	2	2	no	—
211	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
212	C₂	C₂×C₅₃×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₄²
commutator	C₄	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×#97×C₅₃	C₄	semidirect
9. C₂×C₄×C₈×C₅₃	C₄	semidirect
12. C₅₃×#376	C₄	semidirect
13. C₄³×C₅₃	C₄	semidirect
14. C₅₃×H(3,ℤ/4)	C₄	direct (N × H)
52. C₂×C₈×C₅₃	C₄²	semidirect
60. C₄²×C₅₃	C₄²	semidirect
88. C₄×H(3,ℤ/4)	C₅₃	direct (N × H)
98. C₄×C₅₃	H(3,ℤ/4)	direct (N × H)
114. C₂×#97	C₄×C₅₃	semidirect
115. C₂×C₄×C₈	C₄×C₅₃	semidirect
118. #376	C₄×C₅₃	semidirect
119. C₄³	C₄×C₅₃	semidirect
120. H(3,ℤ/4)	C₄×C₅₃	direct (N × H)
121. C₅₃	C₄×H(3,ℤ/4)	direct (N × H)
159. C₂×C₈	C₄²×C₅₃	semidirect
167. C₄²	C₄²×C₅₃	semidirect
204. C₄	C₅₃×H(3,ℤ/4)	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×C₄×M₁₆×C₅₃	2
C₄×#117×C₅₃	2
C₂×C₅₃×H(3,ℤ/4)	2
C₄×H(3,ℤ/4)	53

Subgroups — C4×C53×H(3,ℤ/4) #27218209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×H(3,ℤ/4) #27218209