Subgroups — C₄×C₅₃×#384 #27221284

Order 13568.

2094 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#384	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	6	yes	maximal
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×C₄⋊#95	C₂	1	6	yes	maximal
5	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
10	C₅₃×#384	C₄	1	16	yes	direct product factor
11	C₄×C₄⋊C₄×C₅₃	—	2	24	no	—
12	C₂×#95×C₅₃	C₂²	1	12	yes	—
13	C₄³×C₅₃	C₄	1	8	yes	complemented
14	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	48	yes	—
21	C₂×C₄²×C₅₃	—	2	24	no	—
22	C₂×C₄²×C₅₃	—	2	12	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
27	C₂×C₄²×C₅₃	D₄	1	12	yes	—
28	C₂×C₄²×C₅₃	Q₈	1	12	yes	—
29	#95×C₅₃	—	2	24	no	—
30	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	12	no	—
32	C₂²×C₄×C₅₃	—	2	24	no	—
33	C₂²×C₄×C₅₃	—	2	24	no	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
45	C₂²×C₄×C₅₃	#41	1	12	yes	—
46	C₄⋊C₄×C₅₃	—	2	96	no	—
47	C₄²×C₅₃	—	2	24	no	—
48	C₄²×C₅₃	—	4	16	no	—
49	C₄²×C₅₃	C₄²	1	16	yes	complemented
50	C₄²×C₅₃	C₄⋊C₄	1	48	yes	complemented
51	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
52	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
53	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
54	C₂³×C₅₃	C₂²×Q₈	1	3	yes	—
55	C₂³×C₅₃	C₂×#41	1	6	yes	—
56	C₂×C₄×C₅₃	—	2	12	no	—
57	C₂×C₄×C₅₃	—	2	12	no	—
58	C₂×C₄×C₅₃	—	2	12	no	—
59	C₂×C₄×C₅₃	—	2	48	no	—
60	C₂×C₄×C₅₃	—	4	8	no	—
61	C₂×C₄×C₅₃	—	4	24	no	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
66	C₂×C₄×C₅₃	#117	1	12	yes	—
67	C₂×C₄×C₅₃	C₄×D₄	1	24	yes	—
68	C₂×C₄×C₅₃	C₄×Q₈	1	24	yes	—
69	C₂×C₄×C₅₃	#127	1	4	yes	—
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
71	C₄×#384	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
74	C₂²×C₅₃	C₂×#117	1	3	yes	—
75	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
76	C₂²×C₅₃	C₂×C₄×Q₈	1	6	yes	—
77	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
79	C₂²×C₅₃	#532	1	6	yes	—
80	C₂²×C₅₃	#533	1	6	yes	—
81	C₄×C₅₃	—	2	12	no	—
82	C₄×C₅₃	—	4	16	no	—
83	C₄×C₅₃	C₄×C₄⋊C₄	1	24	yes	complemented
84	C₄×C₅₃	#384	1	8	yes	direct product factor
85	C₂×#384	C₂×C₅₃	1	2	yes	—
86	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	6	yes	—
87	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
88	C₄⋊#95	C₂×C₅₃	1	6	yes	—
89	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	3	yes	—
90	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
91	C₂×C₅₃	#2285	1	3	yes	—
92	C₂×C₅₃	C₄×#127	1	2	yes	—
93	C₂×C₅₃	C₄×C₄⋊Q₈	1	6	yes	—
94	C₂²×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
95	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
96	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
97	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
98	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
99	#384	C₄×C₅₃	1	16	yes	direct product factor
100	C₄×C₄⋊C₄	—	2	24	no	—
101	C₂×#95	C₂²×C₅₃	1	12	yes	—
102	C₄³	C₄×C₅₃	1	8	yes	complemented
103	C₅₃	C₄×#384	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
105	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
106	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
107	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	24	no	—
109	C₂×C₄⋊C₄	—	2	24	no	—
110	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	48	yes	—
111	C₂×C₄²	—	2	12	no	—
112	C₂×C₄²	—	2	24	no	—
113	C₂×C₄²	C₂×C₄×C₅₃	1	12	yes	—
114	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
115	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
116	C₂×C₄²	C₂×C₄×C₅₃	1	12	yes	—
117	C₂×C₄²	D₄×C₅₃	1	12	yes	—
118	C₂×C₄²	Q₈×C₅₃	1	12	yes	—
119	#95	—	2	24	no	—
120	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
121	C₂²×C₄	—	2	12	no	—
122	C₂²×C₄	—	2	24	no	—
123	C₂²×C₄	—	2	24	no	—
124	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
125	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
126	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
127	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
128	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
129	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
130	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
131	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
132	C₂²×C₄	C₂×Q₈×C₅₃	1	6	yes	—
133	C₂²×C₄	C₂×Q₈×C₅₃	1	6	yes	—
134	C₂²×C₄	C₂×Q₈×C₅₃	1	6	yes	—
135	C₂²×C₄	#41×C₅₃	1	12	yes	—
136	C₄⋊C₄	—	2	96	no	—
137	C₄²	—	2	24	no	—
138	C₄²	—	4	16	no	—
139	C₄²	C₄²×C₅₃	1	16	yes	complemented
140	C₄²	C₄⋊C₄×C₅₃	1	48	yes	complemented
141	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
142	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
143	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
144	C₂³	C₂²×Q₈×C₅₃	1	3	yes	—
145	C₂³	C₂×#41×C₅₃	1	6	yes	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	—	2	12	no	—
148	C₂×C₄	—	2	12	no	—
149	C₂×C₄	—	2	48	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	4	24	no	—
152	C₂×C₄	C₂×C₄²×C₅₃	1	12	yes	—
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
155	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
156	C₂×C₄	#117×C₅₃	1	12	yes	—
157	C₂×C₄	C₄×D₄×C₅₃	1	24	yes	—
158	C₂×C₄	C₄×Q₈×C₅₃	1	24	yes	—
159	C₂×C₄	#127×C₅₃	1	4	yes	—
160	C₂×C₄	C₄⋊Q₈×C₅₃	1	12	yes	—
161	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
162	C₂²	C₂²×C₄⋊C₄×C₅₃	1	3	yes	—
163	C₂²	C₂×#117×C₅₃	1	3	yes	—
164	C₂²	C₂×C₄×D₄×C₅₃	1	6	yes	—
165	C₂²	C₂×C₄×Q₈×C₅₃	1	6	yes	—
166	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
167	C₂²	C₂×C₄⋊Q₈×C₅₃	1	3	yes	—
168	C₂²	C₅₃×#532	1	6	yes	—
169	C₂²	C₅₃×#533	1	6	yes	—
170	C₄	—	2	12	no	—
171	C₄	—	4	16	no	—
172	C₄	C₄×C₄⋊C₄×C₅₃	1	24	yes	complemented
173	C₄	C₅₃×#384	1	8	yes	direct product factor
174	C₂	C₂×C₄×C₄⋊C₄×C₅₃	1	3	yes	—
175	C₂	C₂×C₅₃×#384	1	1	yes	characteristic
176	C₂	C₅₃×#2285	1	3	yes	—
177	C₂	C₄×#127×C₅₃	1	2	yes	—
178	C₂	C₄×C₄⋊Q₈×C₅₃	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₅₃×#384	C₄	direct (N × H)
13. C₄³×C₅₃	C₄	semidirect
49. C₄²×C₅₃	C₄²	semidirect
50. C₄²×C₅₃	C₄⋊C₄	semidirect
71. C₄×#384	C₅₃	direct (N × H)
83. C₄×C₅₃	C₄×C₄⋊C₄	semidirect
84. C₄×C₅₃	#384	direct (N × H)
99. #384	C₄×C₅₃	direct (N × H)
102. C₄³	C₄×C₅₃	semidirect
103. C₅₃	C₄×#384	direct (N × H)
139. C₄²	C₄²×C₅₃	semidirect
140. C₄²	C₄⋊C₄×C₅₃	semidirect
172. C₄	C₄×C₄⋊C₄×C₅₃	semidirect
173. C₄	C₅₃×#384	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×#384	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₄³×C₅₃	2
C₅₃×C₄⋊#95	2
C₄×#384	53

Subgroups — C4×C53×#384 #27221284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#384 #27221284