Subgroups — C₄²×C₂²⋊C₄×C₅₃ #27221259

Order 13568.

3342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄²×C₅₃	C₂²
commutator	C₂	C₂×C₄³×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄³×C₅₃	C₂	semidirect
4. C₄×#95×C₅₃	C₂	semidirect
8. C₂²×C₄²×C₅₃	C₄	semidirect
12. C₄×C₂²⋊C₄×C₅₃	C₄	direct (N × H)
30. C₂×C₄²×C₅₃	C₂×C₄	semidirect
32. #95×C₅₃	C₂×C₄	semidirect
45. C₂²×C₄×C₅₃	C₄²	semidirect
53. C₂²⋊C₄×C₅₃	C₄²	direct (N × H)
56. C₄²×C₅₃	C₂²⋊C₄	direct (N × H)
75. C₂×C₄×C₅₃	C₂×C₄²	semidirect
79. C₄²×C₂²⋊C₄	C₅₃	direct (N × H)
83. C₂²×C₅₃	C₄³	semidirect
92. C₄×C₅₃	C₄×C₂²⋊C₄	direct (N × H)
95. C₂×C₄³	C₂×C₅₃	semidirect
96. C₄×#95	C₂×C₅₃	semidirect
105. C₂²×C₄²	C₄×C₅₃	semidirect
109. C₄×C₂²⋊C₄	C₄×C₅₃	direct (N × H)
112. C₅₃	C₄²×C₂²⋊C₄	direct (N × H)
128. C₂×C₄²	C₂×C₄×C₅₃	semidirect
130. #95	C₂×C₄×C₅₃	semidirect
143. C₂²×C₄	C₄²×C₅₃	semidirect
151. C₂²⋊C₄	C₄²×C₅₃	direct (N × H)
154. C₄²	C₂²⋊C₄×C₅₃	direct (N × H)
173. C₂×C₄	C₂×C₄²×C₅₃	semidirect
180. C₂²	C₄³×C₅₃	semidirect
189. C₄	C₄×C₂²⋊C₄×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×C₄²×C₅₃	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×C₄³×C₅₃	2
C₄×#95×C₅₃	2
C₄²×C₂²⋊C₄	53

Subgroups — C42×C22⋊C4×C53 #27221259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₂²⋊C₄×C₅₃ #27221259