Subgroups — C₂³×#135×C₅₃ #27270592

Order 13568.

11422 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×#135×C₅₃	C₂	1	56	yes	maximal, direct product factor
3	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₅₃	—	2	28	no	—
8	C₂²×#41×C₅₃	C₂²	1	56	yes	complemented
9	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
10	C₂³×D₄×C₅₃	—	2	2	no	—
11	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₅₃	—	2	2	no	—
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#135×C₅₃	C₂²	1	448	yes	direct product factor
15	C₂²×Q₁₆×C₅₃	C₂²	1	28	yes	complemented
16	C₂²×SD₁₆×C₅₃	C₂²	1	56	yes	complemented
17	C₂²×D₈×C₅₃	C₂²	1	28	yes	complemented
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₅₃	C₂²	1	28	yes	complemented
21	C₂⁵×C₅₃	—	2	2	no	—
22	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
23	C₂×#41×C₅₃	—	2	336	no	—
24	C₂×#41×C₅₃	C₂³	1	224	yes	complemented
25	C₂²×Q₈×C₅₃	—	2	14	no	—
26	C₂²×Q₈×C₅₃	C₂³	1	28	yes	—
27	C₂²×D₄×C₅₃	—	2	56	no	—
28	C₂²×D₄×C₅₃	—	2	14	no	—
29	C₂²×D₄×C₅₃	C₂³	1	28	yes	—
30	C₂³×C₄×C₅₃	—	2	2	no	—
31	C₂³×C₄×C₅₃	—	2	56	no	—
32	C₂³×C₄×C₅₃	D₄	1	14	yes	—
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
36	#135×C₅₃	C₂³	1	512	yes	direct product factor
37	C₂×Q₁₆×C₅₃	C₂³	1	112	yes	complemented
38	C₂×SD₁₆×C₅₃	C₂³	1	224	yes	complemented
39	C₂×D₈×C₅₃	C₂³	1	112	yes	complemented
40	C₂²×C₈×C₅₃	C₂³	1	14	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	14	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	112	yes	complemented
43	C₂⁴×C₅₃	—	2	8	no	—
44	C₂⁴×C₅₃	—	2	28	no	—
45	C₂⁴×C₅₃	—	4	16	no	—
46	C₂⁴×C₅₃	C₂×D₄	1	14	yes	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	#41×C₅₃	—	2	448	no	—
49	#41×C₅₃	C₂⁴	1	128	yes	complemented
50	C₂×Q₈×C₅₃	—	2	84	no	—
51	C₂×Q₈×C₅₃	C₂⁴	1	56	yes	—
52	C₂×D₄×C₅₃	—	2	224	no	—
53	C₂×D₄×C₅₃	—	2	84	no	—
54	C₂×D₄×C₅₃	C₂⁴	1	56	yes	—
55	C₂²×C₄×C₅₃	—	2	28	no	—
56	C₂²×C₄×C₅₃	—	2	224	no	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	14	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	84	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
62	Q₁₆×C₅₃	C₂⁴	1	64	yes	complemented
63	SD₁₆×C₅₃	C₂⁴	1	128	yes	complemented
64	D₈×C₅₃	C₂⁴	1	64	yes	complemented
65	C₂×C₈×C₅₃	C₂⁴	1	28	yes	—
66	C₂×C₈×C₅₃	C₂⁴	1	28	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	64	yes	complemented
68	C₂³×C₅₃	—	2	56	no	—
69	C₂³×C₅₃	—	2	56	no	—
70	C₂³×C₅₃	—	4	112	no	—
71	C₂³×C₅₃	#135	1	8	yes	direct product factor
72	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
73	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
74	Q₈×C₅₃	—	2	56	no	—
75	Q₈×C₅₃	C₂⁵	1	16	yes	—
76	D₄×C₅₃	—	2	128	no	—
77	D₄×C₅₃	—	2	56	no	—
78	D₄×C₅₃	C₂⁵	1	16	yes	—
79	C₂×C₄×C₅₃	—	2	56	no	—
80	C₂×C₄×C₅₃	—	2	128	no	—
81	C₂×C₄×C₅₃	C₂²×D₄	1	28	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	21	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	56	yes	—
84	C₂×C₄×C₅₃	C₂⁵	1	7	yes	—
85	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
86	C₈×C₅₃	C₂⁵	1	8	yes	—
87	C₈×C₅₃	C₂⁵	1	8	yes	—
88	C₂³×#135	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	—	2	56	no	—
90	C₂²×C₅₃	—	2	16	no	—
91	C₂²×C₅₃	—	4	112	no	—
92	C₂²×C₅₃	C₂×#135	1	28	yes	direct product factor
93	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
94	C₂²×C₅₃	C₂³×D₄	1	7	yes	—
95	C₄×C₅₃	—	2	16	no	—
96	C₄×C₅₃	C₂³×D₄	1	8	yes	—
97	C₄×C₅₃	C₂³×D₄	1	7	yes	—
98	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
99	C₂³×#41	C₂×C₅₃	1	2	yes	complemented
100	C₂²×#135	C₂×C₅₃	1	56	yes	direct product factor
101	C₂³×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
102	C₂³×SD₁₆	C₂×C₅₃	1	2	yes	complemented
103	C₂³×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
104	C₂⁴×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
105	C₂×C₅₃	—	2	8	no	—
106	C₂×C₅₃	—	4	16	no	—
107	C₂×C₅₃	C₂²×#135	1	14	yes	direct product factor
108	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
109	C₂²×#41	—	2	28	no	—
110	C₂²×#41	C₂²×C₅₃	1	56	yes	complemented
111	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
112	C₂³×D₄	—	2	2	no	—
113	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
114	C₂⁴×C₄	—	2	2	no	—
115	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
116	C₂×#135	C₂²×C₅₃	1	448	yes	direct product factor
117	C₂²×Q₁₆	C₂²×C₅₃	1	28	yes	complemented
118	C₂²×SD₁₆	C₂²×C₅₃	1	56	yes	complemented
119	C₂²×D₈	C₂²×C₅₃	1	28	yes	complemented
120	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
122	C₂³×C₈	C₂²×C₅₃	1	28	yes	complemented
123	C₅₃	C₂³×#135	1	1	yes	characteristic, direct product factor, Sylow
124	C₂⁵	—	2	2	no	—
125	C₂⁵	D₄×C₅₃	1	1	yes	characteristic
126	C₂×#41	—	2	336	no	—
127	C₂×#41	C₂³×C₅₃	1	224	yes	complemented
128	C₂²×Q₈	—	2	14	no	—
129	C₂²×Q₈	C₂³×C₅₃	1	28	yes	—
130	C₂²×D₄	—	2	56	no	—
131	C₂²×D₄	—	2	14	no	—
132	C₂²×D₄	C₂³×C₅₃	1	28	yes	—
133	C₂³×C₄	—	2	2	no	—
134	C₂³×C₄	—	2	56	no	—
135	C₂³×C₄	D₄×C₅₃	1	14	yes	—
136	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
137	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
138	C₂³×C₄	C₂³×C₅₃	1	14	yes	—
139	#135	C₂³×C₅₃	1	512	yes	direct product factor
140	C₂×Q₁₆	C₂³×C₅₃	1	112	yes	complemented
141	C₂×SD₁₆	C₂³×C₅₃	1	224	yes	complemented
142	C₂×D₈	C₂³×C₅₃	1	112	yes	complemented
143	C₂²×C₈	C₂³×C₅₃	1	14	yes	—
144	C₂²×C₈	C₂³×C₅₃	1	14	yes	—
145	C₂²×C₈	C₂³×C₅₃	1	112	yes	complemented
146	C₂⁴	—	2	8	no	—
147	C₂⁴	—	2	28	no	—
148	C₂⁴	—	4	16	no	—
149	C₂⁴	C₂×D₄×C₅₃	1	14	yes	—
150	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
151	#41	—	2	448	no	—
152	#41	C₂⁴×C₅₃	1	128	yes	complemented
153	C₂×Q₈	—	2	84	no	—
154	C₂×Q₈	C₂⁴×C₅₃	1	56	yes	—
155	C₂×D₄	—	2	224	no	—
156	C₂×D₄	—	2	84	no	—
157	C₂×D₄	C₂⁴×C₅₃	1	56	yes	—
158	C₂²×C₄	—	2	28	no	—
159	C₂²×C₄	—	2	224	no	—
160	C₂²×C₄	C₂×D₄×C₅₃	1	14	yes	—
161	C₂²×C₄	C₂×D₄×C₅₃	1	7	yes	—
162	C₂²×C₄	C₂×D₄×C₅₃	1	84	yes	—
163	C₂²×C₄	C₂⁴×C₅₃	1	7	yes	—
164	C₂²×C₄	C₂⁴×C₅₃	1	28	yes	—
165	Q₁₆	C₂⁴×C₅₃	1	64	yes	complemented
166	SD₁₆	C₂⁴×C₅₃	1	128	yes	complemented
167	D₈	C₂⁴×C₅₃	1	64	yes	complemented
168	C₂×C₈	C₂⁴×C₅₃	1	28	yes	—
169	C₂×C₈	C₂⁴×C₅₃	1	28	yes	—
170	C₂×C₈	C₂⁴×C₅₃	1	64	yes	complemented
171	C₂³	—	2	56	no	—
172	C₂³	—	2	56	no	—
173	C₂³	—	4	112	no	—
174	C₂³	#135×C₅₃	1	8	yes	direct product factor
175	C₂³	C₂²×D₄×C₅₃	1	28	yes	—
176	C₂³	C₂²×D₄×C₅₃	1	7	yes	—
177	Q₈	—	2	56	no	—
178	Q₈	C₂⁵×C₅₃	1	16	yes	—
179	D₄	—	2	128	no	—
180	D₄	—	2	56	no	—
181	D₄	C₂⁵×C₅₃	1	16	yes	—
182	C₂×C₄	—	2	56	no	—
183	C₂×C₄	—	2	128	no	—
184	C₂×C₄	C₂²×D₄×C₅₃	1	28	yes	—
185	C₂×C₄	C₂²×D₄×C₅₃	1	21	yes	—
186	C₂×C₄	C₂²×D₄×C₅₃	1	56	yes	—
187	C₂×C₄	C₂⁵×C₅₃	1	7	yes	—
188	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
189	C₈	C₂⁵×C₅₃	1	8	yes	—
190	C₈	C₂⁵×C₅₃	1	8	yes	—
191	C₂²	—	2	56	no	—
192	C₂²	—	2	16	no	—
193	C₂²	—	4	112	no	—
194	C₂²	C₂×#135×C₅₃	1	28	yes	direct product factor
195	C₂²	C₂³×D₄×C₅₃	1	8	yes	—
196	C₂²	C₂³×D₄×C₅₃	1	7	yes	—
197	C₄	—	2	16	no	—
198	C₄	C₂³×D₄×C₅₃	1	8	yes	—
199	C₄	C₂³×D₄×C₅₃	1	7	yes	—
200	C₄	C₂⁶×C₅₃	1	1	yes	characteristic
201	C₂	—	2	8	no	—
202	C₂	—	4	16	no	—
203	C₂	C₂²×#135×C₅₃	1	14	yes	direct product factor
204	C₂	C₂⁴×D₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₅₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₅₃
socle	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₅₃	C₂	semidirect
2. C₂²×#135×C₅₃	C₂	direct (N × H)
3. C₂³×Q₁₆×C₅₃	C₂	semidirect
4. C₂³×SD₁₆×C₅₃	C₂	semidirect
5. C₂³×D₈×C₅₃	C₂	semidirect
6. C₂⁴×C₈×C₅₃	C₂	semidirect
8. C₂²×#41×C₅₃	C₂²	semidirect
14. C₂×#135×C₅₃	C₂²	direct (N × H)
15. C₂²×Q₁₆×C₅₃	C₂²	semidirect
16. C₂²×SD₁₆×C₅₃	C₂²	semidirect
17. C₂²×D₈×C₅₃	C₂²	semidirect
20. C₂³×C₈×C₅₃	C₂²	semidirect
24. C₂×#41×C₅₃	C₂³	semidirect
36. #135×C₅₃	C₂³	direct (N × H)
37. C₂×Q₁₆×C₅₃	C₂³	semidirect
38. C₂×SD₁₆×C₅₃	C₂³	semidirect
39. C₂×D₈×C₅₃	C₂³	semidirect
42. C₂²×C₈×C₅₃	C₂³	semidirect
49. #41×C₅₃	C₂⁴	semidirect
62. Q₁₆×C₅₃	C₂⁴	semidirect
63. SD₁₆×C₅₃	C₂⁴	semidirect
64. D₈×C₅₃	C₂⁴	semidirect
67. C₂×C₈×C₅₃	C₂⁴	semidirect
71. C₂³×C₅₃	#135	direct (N × H)
88. C₂³×#135	C₅₃	direct (N × H)
92. C₂²×C₅₃	C₂×#135	direct (N × H)
99. C₂³×#41	C₂×C₅₃	semidirect
100. C₂²×#135	C₂×C₅₃	direct (N × H)
101. C₂³×Q₁₆	C₂×C₅₃	semidirect
102. C₂³×SD₁₆	C₂×C₅₃	semidirect
103. C₂³×D₈	C₂×C₅₃	semidirect
104. C₂⁴×C₈	C₂×C₅₃	semidirect
107. C₂×C₅₃	C₂²×#135	direct (N × H)
110. C₂²×#41	C₂²×C₅₃	semidirect
116. C₂×#135	C₂²×C₅₃	direct (N × H)
117. C₂²×Q₁₆	C₂²×C₅₃	semidirect
118. C₂²×SD₁₆	C₂²×C₅₃	semidirect
119. C₂²×D₈	C₂²×C₅₃	semidirect
122. C₂³×C₈	C₂²×C₅₃	semidirect
123. C₅₃	C₂³×#135	direct (N × H)
127. C₂×#41	C₂³×C₅₃	semidirect
139. #135	C₂³×C₅₃	direct (N × H)
140. C₂×Q₁₆	C₂³×C₅₃	semidirect
141. C₂×SD₁₆	C₂³×C₅₃	semidirect
142. C₂×D₈	C₂³×C₅₃	semidirect
145. C₂²×C₈	C₂³×C₅₃	semidirect
152. #41	C₂⁴×C₅₃	semidirect
165. Q₁₆	C₂⁴×C₅₃	semidirect
166. SD₁₆	C₂⁴×C₅₃	semidirect
167. D₈	C₂⁴×C₅₃	semidirect
170. C₂×C₈	C₂⁴×C₅₃	semidirect
174. C₂³	#135×C₅₃	direct (N × H)
194. C₂²	C₂×#135×C₅₃	direct (N × H)
203. C₂	C₂²×#135×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₅₃	2
C₂²×#135×C₅₃	2
C₂³×Q₁₆×C₅₃	2
C₂³×SD₁₆×C₅₃	2
C₂³×D₈×C₅₃	2
C₂⁴×C₈×C₅₃	2
C₂³×#135	53

Subgroups — C23×#135×C53 #27270592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#135×C₅₃ #27270592