Subgroups — C₂×C₅₃×#2642 #27242140

Order 13568.

2286 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2642	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#400	C₂	1	8	yes	maximal
5	C₂×C₅₃×#391	C₂	1	4	yes	maximal
6	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	—
10	C₂×#125×C₅₃	—	2	8	no	—
11	C₂×#125×C₅₃	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#400	C₂²	1	64	yes	—
18	C₅₃×#391	C₂²	1	32	yes	—
19	C₅₃×#376	C₂²	1	8	yes	—
20	C₂×#95×C₅₃	C₂²	1	4	yes	—
21	C₂×#95×C₅₃	C₂²	1	16	yes	—
22	C₂²×Q₈×C₅₃	—	2	8	no	—
23	C₂²×Q₈×C₅₃	—	2	16	no	—
24	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	—	2	32	no	—
31	#125×C₅₃	—	2	32	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
38	C₂×C₄²×C₅₃	—	2	8	no	—
39	C₂×C₄²×C₅₃	D₄	1	8	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
42	#95×C₅₃	C₂³	1	16	yes	—
43	#95×C₅₃	C₂³	1	64	yes	—
44	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	—	2	16	no	—
46	C₂×Q₈×C₅₃	—	2	32	no	—
47	C₂×Q₈×C₅₃	—	4	32	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	16	no	—
53	C₂²×C₄×C₅₃	—	2	16	no	—
54	C₂²×C₄×C₅₃	—	2	16	no	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₅₃	#41	1	8	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	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	2	yes	—
63	C₄⋊C₄×C₅₃	—	2	32	no	—
64	C₄⋊C₄×C₅₃	—	2	16	no	—
65	C₄⋊C₄×C₅₃	—	2	32	no	—
66	C₄²×C₅₃	—	2	16	no	—
67	C₄²×C₅₃	C₂×D₄	1	16	yes	—
68	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
69	C₂³×C₅₃	C₂×#41	1	4	yes	—
70	C₂³×C₅₃	2₋¹⁺⁴	1	8	yes	—
71	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	32	no	—
73	C₂×C₄×C₅₃	—	2	16	no	—
74	C₂×C₄×C₅₃	—	2	8	no	—
75	C₂×C₄×C₅₃	—	2	8	no	—
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	2	16	no	—
78	C₂×C₄×C₅₃	—	2	8	no	—
79	C₂×C₄×C₅₃	—	2	16	no	—
80	C₂×C₄×C₅₃	—	2	16	no	—
81	C₂×C₄×C₅₃	—	4	16	no	—
82	C₂×C₄×C₅₃	—	4	16	no	—
83	C₂×C₄×C₅₃	—	4	16	no	—
84	C₂×C₄×C₅₃	#124	1	16	yes	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
86	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
87	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
88	C₂×#2642	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	C₂×#124	1	4	yes	—
90	C₂²×C₅₃	#536	1	8	yes	—
91	C₂²×C₅₃	#541	1	16	yes	—
92	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₂²×#41	1	2	yes	—
94	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	4	yes	—
95	C₄×C₅₃	—	2	8	no	—
96	C₄×C₅₃	—	4	8	no	—
97	C₄×C₅₃	—	4	8	no	—
98	C₄×C₅₃	—	4	8	no	—
99	C₄×C₅₃	C₂×#124	1	8	yes	—
100	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
101	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic
102	#2642	C₂×C₅₃	1	16	yes	direct product factor
103	C₂×#400	C₂×C₅₃	1	8	yes	—
104	C₂×#391	C₂×C₅₃	1	4	yes	—
105	C₂×#376	C₂×C₅₃	1	1	yes	characteristic
106	C₂×C₅₃	#2642	1	8	yes	direct product factor
107	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
108	C₂×C₅₃	C₂×#536	1	2	yes	—
109	C₂×C₅₃	C₂×#541	1	4	yes	—
110	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
111	C₂×C₄⋊Q₈	—	2	8	no	—
112	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	—
113	C₂×#125	—	2	8	no	—
114	C₂×#125	C₂²×C₅₃	1	8	yes	—
115	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
116	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
117	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
118	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
119	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
120	#400	C₂²×C₅₃	1	64	yes	—
121	#391	C₂²×C₅₃	1	32	yes	—
122	#376	C₂²×C₅₃	1	8	yes	—
123	C₂×#95	C₂²×C₅₃	1	4	yes	—
124	C₂×#95	C₂²×C₅₃	1	16	yes	—
125	C₅₃	C₂×#2642	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×Q₈	—	2	8	no	—
127	C₂²×Q₈	—	2	16	no	—
128	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
129	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
130	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
131	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
132	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
133	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	32	no	—
135	#125	—	2	32	no	—
136	C₂×C₄⋊C₄	—	2	16	no	—
137	C₂×C₄⋊C₄	—	2	8	no	—
138	C₂×C₄⋊C₄	—	2	16	no	—
139	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
140	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
141	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
142	C₂×C₄²	—	2	8	no	—
143	C₂×C₄²	D₄×C₅₃	1	8	yes	—
144	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
145	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
146	#95	C₂³×C₅₃	1	16	yes	—
147	#95	C₂³×C₅₃	1	64	yes	—
148	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
149	C₂×Q₈	—	2	16	no	—
150	C₂×Q₈	—	2	32	no	—
151	C₂×Q₈	—	4	32	no	—
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	16	no	—
157	C₂²×C₄	—	2	16	no	—
158	C₂²×C₄	—	2	16	no	—
159	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
160	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
161	C₂²×C₄	#41×C₅₃	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
163	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
164	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
165	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
166	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
167	C₄⋊C₄	—	2	32	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₄⋊C₄	—	2	32	no	—
170	C₄²	—	2	16	no	—
171	C₄²	C₂×D₄×C₅₃	1	16	yes	—
172	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
173	C₂³	C₂×#41×C₅₃	1	4	yes	—
174	C₂³	2₋¹⁺⁴×C₅₃	1	8	yes	—
175	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
176	Q₈	—	4	32	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	—	4	16	no	—
187	C₂×C₄	—	4	16	no	—
188	C₂×C₄	#124×C₅₃	1	16	yes	—
189	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
190	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
191	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
192	C₂²	C₂×#124×C₅₃	1	4	yes	—
193	C₂²	C₅₃×#536	1	8	yes	—
194	C₂²	C₅₃×#541	1	16	yes	—
195	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
196	C₂²	C₂²×#41×C₅₃	1	2	yes	—
197	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	4	yes	—
198	C₄	—	2	8	no	—
199	C₄	—	4	8	no	—
200	C₄	—	4	8	no	—
201	C₄	—	4	8	no	—
202	C₄	C₂×#124×C₅₃	1	8	yes	—
203	C₂	C₅₃×#2642	1	8	yes	direct product factor
204	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
205	C₂	C₂×C₅₃×#536	1	2	yes	—
206	C₂	C₂×C₅₃×#541	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2642	C₂	direct (N × H)
88. C₂×#2642	C₅₃	direct (N × H)
102. #2642	C₂×C₅₃	direct (N × H)
106. C₂×C₅₃	#2642	direct (N × H)
125. C₅₃	C₂×#2642	direct (N × H)
203. C₂	C₅₃×#2642	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×C₄⋊Q₈×C₅₃	2
C₂²×#125×C₅₃	2
C₅₃×#2642	2
C₂×C₅₃×#400	2
C₂×C₅₃×#391	2
C₂×C₅₃×#376	2
C₂×#2642	53

Subgroups — C2×C53×#2642 #27242140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2642 #27242140