Subgroups — C₂×C₅₃×#2402 #27242062

Order 13568.

2078 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2402	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅₃×#400	C₂	1	4	yes	maximal
3	C₂×C₅₃×#398	C₂	1	4	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	4	yes	maximal
5	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
6	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#400	C₂²	1	32	yes	—
14	C₅₃×#398	C₂²	1	32	yes	—
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	16	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	32	yes	—
17	C₅₃×#376	C₂²	1	8	yes	—
18	C₂×#95×C₅₃	C₂²	1	4	yes	—
19	C₂×#95×C₅₃	C₂²	1	2	yes	—
20	C₂×#95×C₅₃	C₂²	1	4	yes	—
21	C₂×#95×C₅₃	C₂²	1	4	yes	—
22	C₂×#95×C₅₃	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
31	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
34	C₂×C₄²×C₅₃	—	2	16	no	—
35	C₂×C₄²×C₅₃	—	2	8	no	—
36	C₂×C₄²×C₅₃	Q₈	1	8	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	16	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
40	#95×C₅₃	C₂³	1	16	yes	—
41	#95×C₅₃	C₂³	1	8	yes	—
42	#95×C₅₃	C₂³	1	16	yes	—
43	#95×C₅₃	C₂³	1	16	yes	—
44	#95×C₅₃	C₂³	1	8	yes	—
45	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	—	2	8	no	—
48	C₂²×C₄×C₅₃	—	2	8	no	—
49	C₂²×C₄×C₅₃	—	2	8	no	—
50	C₂²×C₄×C₅₃	—	2	8	no	—
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	—	2	8	no	—
53	C₂²×C₄×C₅₃	—	2	8	no	—
54	C₂²×C₄×C₅₃	—	2	8	no	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₅₃	#41	1	16	yes	—
58	C₂²×C₄×C₅₃	#41	1	8	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	2	32	no	—
65	C₄⋊C₄×C₅₃	—	2	32	no	—
66	C₄⋊C₄×C₅₃	—	2	32	no	—
67	C₄²×C₅₃	—	2	32	no	—
68	C₄²×C₅₃	—	2	16	no	—
69	C₄²×C₅₃	C₂×Q₈	1	16	yes	—
70	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
71	C₂³×C₅₃	C₂×#41	1	4	yes	—
72	C₂³×C₅₃	C₂×#41	1	8	yes	—
73	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	32	no	—
75	C₂×C₄×C₅₃	—	2	16	no	—
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	2	8	no	—
78	C₂×C₄×C₅₃	—	2	8	no	—
79	C₂×C₄×C₅₃	—	2	8	no	—
80	C₂×C₄×C₅₃	—	2	8	no	—
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	2	8	no	—
85	C₂×C₄×C₅₃	—	4	16	no	—
86	C₂×C₄×C₅₃	—	4	16	no	—
87	C₂×C₄×C₅₃	#125	1	16	yes	—
88	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
89	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
90	C₂×C₄×C₅₃	C₂×#41	1	16	yes	—
91	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
92	C₂×#2402	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	C₂×#125	1	4	yes	—
94	C₂²×C₅₃	#529	1	16	yes	—
95	C₂²×C₅₃	#533	1	8	yes	—
96	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
97	C₂²×C₅₃	C₂²×#41	1	2	yes	—
98	C₂²×C₅₃	C₂²×#41	1	4	yes	—
99	C₄×C₅₃	—	2	16	no	—
100	C₄×C₅₃	—	2	8	no	—
101	C₄×C₅₃	—	4	8	no	—
102	C₄×C₅₃	—	4	8	no	—
103	C₄×C₅₃	C₂×#125	1	8	yes	—
104	#2402	C₂×C₅₃	1	16	yes	direct product factor
105	C₂×#400	C₂×C₅₃	1	4	yes	—
106	C₂×#398	C₂×C₅₃	1	4	yes	—
107	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	4	yes	—
108	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
109	C₂×#376	C₂×C₅₃	1	1	yes	characteristic
110	C₂×C₅₃	#2402	1	8	yes	direct product factor
111	C₂×C₅₃	C₂²×#125	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#529	1	4	yes	—
113	C₂×C₅₃	C₂×#533	1	2	yes	—
114	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
115	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
116	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
117	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
118	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
119	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
120	#400	C₂²×C₅₃	1	32	yes	—
121	#398	C₂²×C₅₃	1	32	yes	—
122	C₄×C₄⋊C₄	C₂²×C₅₃	1	16	yes	—
123	C₄×C₄⋊C₄	C₂²×C₅₃	1	32	yes	—
124	#376	C₂²×C₅₃	1	8	yes	—
125	C₂×#95	C₂²×C₅₃	1	2	yes	—
126	C₂×#95	C₂²×C₅₃	1	4	yes	—
127	C₂×#95	C₂²×C₅₃	1	4	yes	—
128	C₂×#95	C₂²×C₅₃	1	4	yes	—
129	C₂×#95	C₂²×C₅₃	1	2	yes	—
130	C₅₃	C₂×#2402	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
132	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
133	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
134	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
135	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	2	16	no	—
137	C₂×C₄⋊C₄	—	2	16	no	—
138	C₂×C₄⋊C₄	—	2	16	no	—
139	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
140	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
141	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
142	C₂×C₄²	—	2	16	no	—
143	C₂×C₄²	—	2	8	no	—
144	C₂×C₄²	Q₈×C₅₃	1	8	yes	—
145	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
146	C₂×C₄²	C₂³×C₅₃	1	16	yes	—
147	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
148	#95	C₂³×C₅₃	1	16	yes	—
149	#95	C₂³×C₅₃	1	8	yes	—
150	#95	C₂³×C₅₃	1	16	yes	—
151	#95	C₂³×C₅₃	1	16	yes	—
152	#95	C₂³×C₅₃	1	8	yes	—
153	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
164	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
165	C₂²×C₄	#41×C₅₃	1	16	yes	—
166	C₂²×C₄	#41×C₅₃	1	8	yes	—
167	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
168	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
169	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
170	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
171	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
172	C₄⋊C₄	—	2	32	no	—
173	C₄⋊C₄	—	2	32	no	—
174	C₄⋊C₄	—	2	32	no	—
175	C₄²	—	2	32	no	—
176	C₄²	—	2	16	no	—
177	C₄²	C₂×Q₈×C₅₃	1	16	yes	—
178	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
179	C₂³	C₂×#41×C₅₃	1	4	yes	—
180	C₂³	C₂×#41×C₅₃	1	8	yes	—
181	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
182	C₂×C₄	—	2	32	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	4	16	no	—
194	C₂×C₄	—	4	16	no	—
195	C₂×C₄	#125×C₅₃	1	16	yes	—
196	C₂×C₄	C₂²×Q₈×C₅₃	1	8	yes	—
197	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
198	C₂×C₄	C₂×#41×C₅₃	1	16	yes	—
199	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
200	C₂²	C₂×#125×C₅₃	1	4	yes	—
201	C₂²	C₅₃×#529	1	16	yes	—
202	C₂²	C₅₃×#533	1	8	yes	—
203	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
204	C₂²	C₂²×#41×C₅₃	1	2	yes	—
205	C₂²	C₂²×#41×C₅₃	1	4	yes	—
206	C₄	—	2	16	no	—
207	C₄	—	2	8	no	—
208	C₄	—	4	8	no	—
209	C₄	—	4	8	no	—
210	C₄	C₂×#125×C₅₃	1	8	yes	—
211	C₂	C₅₃×#2402	1	8	yes	direct product factor
212	C₂	C₂²×#125×C₅₃	1	1	yes	characteristic
213	C₂	C₂×C₅₃×#529	1	4	yes	—
214	C₂	C₂×C₅₃×#533	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2402	C₂	direct (N × H)
92. C₂×#2402	C₅₃	direct (N × H)
104. #2402	C₂×C₅₃	direct (N × H)
110. C₂×C₅₃	#2402	direct (N × H)
130. C₅₃	C₂×#2402	direct (N × H)
211. C₂	C₅₃×#2402	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2402	2
C₂×C₅₃×#400	2
C₂×C₅₃×#398	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₅₃×#376	2
C₂×#2402	53

Subgroups — C2×C53×#2402 #27242062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2402 #27242062