Subgroups — C₂×C₄×#13133127 #27381415

Order 13568.

1790 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	C₂²×#13133127	C₂	1	4	yes	maximal
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×#13133127	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₅₃	C₄	1	4	yes	—
8	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
9	C₂×#13133127	C₄	1	32	yes	direct product factor
10	C₂×#13133127	C₂²	1	16	yes	—
11	C₂×C₄×#88322	C₄	1	8	yes	complemented
12	C₂×C₄×#88322	C₂²	1	4	yes	—
13	C₄²×Dic₅₃	C₄	1	4	yes	—
14	C₄²×Dic₅₃	C₂²	1	4	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
19	C₂³×Dic₅₃	C₂×C₄	1	4	yes	—
20	C₂²×#88322	C₂×C₄	1	8	yes	—
21	C₂²×#88322	C₂×C₄	1	16	yes	—
22	C₂²×#88322	C₂³	1	4	yes	—
23	C₂×C₄×Dic₅₃	—	2	16	no	—
24	C₂×C₄×Dic₅₃	C₂×C₄	1	16	yes	—
25	C₂×C₄×Dic₅₃	C₂×C₄	1	16	yes	—
26	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
28	#13133127	C₂×C₄	1	64	yes	direct product factor
29	C₄×#88322	C₂×C₄	1	16	yes	complemented
30	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	16	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	16	yes	complemented
40	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	16	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	8	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	16	no	—
50	C₂²×Dic₅₃	C₄²	1	16	yes	—
51	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
52	C₂×C₈×C₅₃	C₄²	1	32	yes	complemented
53	C₂×C₈×C₅₃	C₂²×C₄	1	16	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
55	C₄²×C₅₃	C₄²	1	8	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
58	C₄×Dic₅₃	—	2	32	no	—
59	C₄×Dic₅₃	C₄²	1	16	yes	—
60	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
61	C₂×#88322	C₄²	1	32	yes	complemented
62	C₂×#88322	C₂²×C₄	1	8	yes	—
63	C₂×#88322	C₂²×C₄	1	16	yes	—
64	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
65	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
66	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
67	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#97	1	32	yes	complemented
69	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	16	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	16	no	—
79	C₂×Dic₅₃	—	2	32	no	—
80	C₂×Dic₅₃	C₂×C₄²	1	16	yes	—
81	C₈×C₅₃	C₂×C₄²	1	16	yes	complemented
82	#88322	C₂×C₄²	1	16	yes	complemented
83	C₂×C₄×#97	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	C₄³	1	4	yes	—
85	C₂²×C₅₃	C₂×#97	1	8	yes	—
86	C₂²×C₅₃	C₄×M₁₆	1	16	yes	—
87	C₂²×C₅₃	C₂²×C₄²	1	2	yes	—
88	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
89	C₂²×C₅₃	C₂²×M₁₆	1	4	yes	—
90	C₄×C₅₃	C₄³	1	4	yes	—
91	C₄×C₅₃	C₂×#97	1	16	yes	complemented
92	C₄×C₅₃	C₂²×C₄²	1	2	yes	—
93	C₄×C₅₃	C₂²×C₄²	1	2	yes	—
94	Dic₅₃	—	2	16	no	—
95	C₂²×#97	—	53	4	no	—
96	C₂²×C₄×C₈	—	53	1	no	—
97	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
98	C₂×C₄³	—	53	1	no	—
99	C₄×#97	—	53	8	no	—
100	C₂×C₅₃	C₄×#97	1	8	yes	complemented
101	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
102	C₂×C₅₃	C₂²×#97	1	2	yes	—
103	C₂×C₅₃	C₂×C₄×M₁₆	1	4	yes	—
104	C₂³×C₈	—	53	2	no	—
105	C₂³×C₈	C₂×D₅₃	1	2	yes	—
106	C₂²×C₄²	—	53	2	no	—
107	C₂²×C₄²	—	53	4	no	—
108	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
109	C₂×#97	—	53	32	no	—
110	C₂×#97	—	53	16	no	—
111	C₂×C₄×C₈	—	53	8	no	—
112	C₂×C₄×C₈	—	53	4	no	—
113	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
114	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
115	C₄³	—	53	4	no	—
116	C₄³	—	53	4	no	—
117	C₅₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
118	C₂³×C₄	—	53	4	no	—
119	C₂³×C₄	C₄×D₅₃	1	2	yes	—
120	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂²×C₈	—	53	8	no	—
122	C₂²×C₈	—	53	16	no	—
123	C₂²×C₈	—	53	4	no	—
124	C₂²×C₈	C₄×D₅₃	1	16	yes	—
125	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
126	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
127	C₂×C₄²	—	53	4	no	—
128	C₂×C₄²	—	53	16	no	—
129	C₂×C₄²	—	53	4	no	—
130	C₂×C₄²	—	53	16	no	—
131	C₂×C₄²	—	106	16	no	—
132	C₂×C₄²	C₄×D₅₃	1	4	yes	—
133	C₂×C₄²	C₄×D₅₃	1	2	yes	—
134	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
135	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
136	#97	—	53	64	no	—
137	C₄×C₈	—	53	16	no	—
138	C₄×C₈	C₂×Dic₅₃	1	16	yes	complemented
139	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	—	53	16	no	—
141	C₂²×C₄	—	53	8	no	—
142	C₂²×C₄	—	106	16	no	—
143	C₂²×C₄	#11719912	1	16	yes	—
144	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
145	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
148	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
150	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
151	C₂×C₈	—	53	32	no	—
152	C₂×C₈	—	53	8	no	—
153	C₂×C₈	—	53	16	no	—
154	C₂×C₈	C₄×Dic₅₃	1	32	yes	complemented
155	C₂×C₈	C₂×C₄×D₅₃	1	16	yes	—
156	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
157	C₄²	—	53	16	no	—
158	C₄²	—	53	16	no	—
159	C₄²	—	106	32	no	—
160	C₄²	C₄×Dic₅₃	1	8	yes	—
161	C₄²	C₂×C₄×D₅₃	1	4	yes	—
162	C₄²	C₂²×Dic₅₃	1	4	yes	—
163	C₂³	C₄²×D₅₃	1	4	yes	—
164	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
165	C₂³	C₂×#11719912	1	8	yes	—
166	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
167	C₂×C₄	—	53	16	no	—
168	C₂×C₄	—	106	32	no	—
169	C₂×C₄	—	106	16	no	—
170	C₂×C₄	#13133127	1	32	yes	direct product factor
171	C₂×C₄	C₄²×D₅₃	1	8	yes	—
172	C₂×C₄	C₄²×D₅₃	1	4	yes	—
173	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
174	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
175	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
176	C₂×C₄	C₂×#11719912	1	16	yes	—
177	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
179	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
180	C₈	—	53	16	no	—
181	C₈	C₂×C₄×Dic₅₃	1	16	yes	complemented
182	C₂²	C₄²×Dic₅₃	1	4	yes	—
183	C₂²	C₂×#13133127	1	8	yes	—
184	C₂²	C₄×#11719912	1	16	yes	—
185	C₂²	C₂×C₄²×D₅₃	1	2	yes	—
186	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
187	C₂²	C₂²×#11719912	1	4	yes	—
188	C₄	—	106	16	no	—
189	C₄	C₄²×Dic₅₃	1	4	yes	—
190	C₄	C₂×#13133127	1	16	yes	direct product factor
191	C₄	C₂×C₄²×D₅₃	1	2	yes	—
192	C₄	C₂²×C₄×Dic₅₃	1	2	yes	—
193	C₂	C₄×#13133127	1	8	yes	direct product factor
194	C₂	C₂×C₄²×Dic₅₃	1	1	yes	characteristic
195	C₂	C₂²×#13133127	1	2	yes	—
196	C₂	C₂×C₄×#11719912	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×C₄×#97
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×#13133127	C₂	direct (N × H)
9. C₂×#13133127	C₄	direct (N × H)
11. C₂×C₄×#88322	C₄	semidirect
17. C₂×C₄×C₈×C₅₃	C₄	semidirect
28. #13133127	C₂×C₄	direct (N × H)
29. C₄×#88322	C₂×C₄	semidirect
39. C₄×C₈×C₅₃	C₂×C₄	semidirect
52. C₂×C₈×C₅₃	C₄²	semidirect
61. C₂×#88322	C₄²	semidirect
68. C₂×C₄×C₅₃	#97	semidirect
81. C₈×C₅₃	C₂×C₄²	semidirect
82. #88322	C₂×C₄²	semidirect
91. C₄×C₅₃	C₂×#97	semidirect
100. C₂×C₅₃	C₄×#97	semidirect
113. C₂×C₄×C₈	Dic₅₃	semidirect
117. C₅₃	C₂×C₄×#97	semidirect
138. C₄×C₈	C₂×Dic₅₃	semidirect
154. C₂×C₈	C₄×Dic₅₃	semidirect
170. C₂×C₄	#13133127	direct (N × H)
181. C₈	C₂×C₄×Dic₅₃	semidirect
190. C₄	C₂×#13133127	direct (N × H)
193. C₂	C₄×#13133127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133127	2
C₂²×C₄×#88322	2
C₂×C₄²×Dic₅₃	2
C₄×#13133127	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₄×#97	53

Subgroups — C2×C4×#13133127 #27381415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133127 #27381415