Subgroups — C₂×C₅₃×#2818 #27244197

Order 13568.

3666 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#126×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2818	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#394	C₂	1	6	yes	maximal, complemented
5	C₂×C₅₃×#392	C₂	1	6	yes	maximal, complemented
6	C₂×C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₅₃	C₂²	1	6	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
9	C₂×#127×C₅₃	—	2	8	no	—
10	C₂×#127×C₅₃	C₂²	1	8	yes	complemented
11	C₂×#126×C₅₃	—	2	8	no	—
12	C₂×#126×C₅₃	C₂²	1	8	yes	complemented
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
14	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
15	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#394	C₂²	1	48	yes	complemented
18	C₅₃×#392	C₂²	1	48	yes	complemented
19	C₅₃×#383	C₂²	1	8	yes	complemented
20	C₂×#95×C₅₃	C₂²	1	6	yes	—
21	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
22	C₂⁵×C₅₃	C₂³	1	4	yes	—
23	C₂²×D₄×C₅₃	—	2	24	no	—
24	C₂²×D₄×C₅₃	—	4	24	no	—
25	C₂²×D₄×C₅₃	—	4	24	no	—
26	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
27	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
31	#127×C₅₃	—	2	32	no	—
32	#126×C₅₃	—	2	32	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	12	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	4	24	no	—
37	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
38	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
39	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
40	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
41	#95×C₅₃	C₂³	1	24	yes	—
42	C₂⁴×C₅₃	—	2	2	no	—
43	C₂⁴×C₅₃	—	2	8	no	—
44	C₂⁴×C₅₃	—	2	24	no	—
45	C₂⁴×C₅₃	—	2	24	no	—
46	C₂⁴×C₅₃	—	2	12	no	—
47	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
48	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
49	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×D₄×C₅₃	—	2	48	no	—
51	C₂×D₄×C₅₃	—	4	48	no	—
52	C₂×D₄×C₅₃	—	4	48	no	—
53	C₂×D₄×C₅₃	—	4	96	no	—
54	C₂×D₄×C₅₃	—	4	64	no	—
55	C₂²×C₄×C₅₃	—	2	12	no	—
56	C₂²×C₄×C₅₃	—	2	6	no	—
57	C₂²×C₄×C₅₃	—	2	12	no	—
58	C₂²×C₄×C₅₃	—	2	24	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
63	C₄⋊C₄×C₅₃	—	2	24	no	—
64	C₂²⋊C₄×C₅₃	—	2	24	no	—
65	C₂²⋊C₄×C₅₃	—	4	48	no	—
66	C₄²×C₅₃	C₂×D₄	1	16	yes	complemented
67	C₂³×C₅₃	—	2	8	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	24	no	—
70	C₂³×C₅₃	—	2	12	no	—
71	C₂³×C₅₃	—	2	24	no	—
72	C₂³×C₅₃	—	4	48	no	—
73	C₂³×C₅₃	—	4	12	no	—
74	C₂³×C₅₃	—	4	16	no	—
75	C₂³×C₅₃	—	4	16	no	—
76	C₂³×C₅₃	—	4	48	no	—
77	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
78	C₂³×C₅₃	H(5,𝔽₂)	1	12	yes	—
79	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
80	D₄×C₅₃	—	4	96	no	—
81	D₄×C₅₃	—	4	64	no	—
82	C₂×C₄×C₅₃	—	2	24	no	—
83	C₂×C₄×C₅₃	—	2	12	no	—
84	C₂×C₄×C₅₃	—	2	6	no	—
85	C₂×C₄×C₅₃	—	2	24	no	—
86	C₂×C₄×C₅₃	—	2	12	no	—
87	C₂×C₄×C₅₃	—	4	12	no	—
88	C₂×C₄×C₅₃	—	4	16	no	—
89	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
90	C₂×#2818	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₅₃	—	4	24	no	—
92	C₂²×C₅₃	—	4	6	no	—
93	C₂²×C₅₃	—	4	8	no	—
94	C₂²×C₅₃	—	4	8	no	—
95	C₂²×C₅₃	—	4	24	no	—
96	C₂²×C₅₃	—	8	32	no	—
97	C₂²×C₅₃	—	8	8	no	—
98	C₂²×C₅₃	#535	1	12	yes	—
99	C₂²×C₅₃	#560	1	16	yes	—
100	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
102	C₄×C₅₃	—	2	12	no	—
103	C₄×C₅₃	—	4	6	no	—
104	C₄×C₅₃	—	4	8	no	—
105	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
106	C₂²×#126	C₂×C₅₃	1	1	yes	characteristic, complemented
107	#2818	C₂×C₅₃	1	16	yes	direct product factor
108	C₂×#394	C₂×C₅₃	1	6	yes	complemented
109	C₂×#392	C₂×C₅₃	1	6	yes	complemented
110	C₂×#383	C₂×C₅₃	1	1	yes	characteristic, complemented
111	C₂×C₅₃	—	8	8	no	—
112	C₂×C₅₃	—	8	2	no	—
113	C₂×C₅₃	#2818	1	8	yes	direct product factor
114	C₂×C₅₃	C₂×#535	1	3	yes	—
115	C₂×C₅₃	C₂×#560	1	4	yes	—
116	C₂³×D₄	C₂²×C₅₃	1	6	yes	—
117	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
118	C₂×#127	—	2	8	no	—
119	C₂×#127	C₂²×C₅₃	1	8	yes	complemented
120	C₂×#126	—	2	8	no	—
121	C₂×#126	C₂²×C₅₃	1	8	yes	complemented
122	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
123	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
124	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
125	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
126	#394	C₂²×C₅₃	1	48	yes	complemented
127	#392	C₂²×C₅₃	1	48	yes	complemented
128	#383	C₂²×C₅₃	1	8	yes	complemented
129	C₂×#95	C₂²×C₅₃	1	6	yes	—
130	C₅₃	C₂×#2818	1	1	yes	characteristic, direct product factor, Sylow
131	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
132	C₂⁵	C₂³×C₅₃	1	4	yes	—
133	C₂²×D₄	—	2	24	no	—
134	C₂²×D₄	—	4	24	no	—
135	C₂²×D₄	—	4	24	no	—
136	C₂²×D₄	C₂³×C₅₃	1	24	yes	—
137	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
138	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
139	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
140	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
141	#127	—	2	32	no	—
142	#126	—	2	32	no	—
143	C₂×C₄⋊C₄	—	2	12	no	—
144	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
145	C₂×C₂²⋊C₄	—	2	12	no	—
146	C₂×C₂²⋊C₄	—	4	24	no	—
147	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
148	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
149	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
150	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
151	#95	C₂³×C₅₃	1	24	yes	—
152	C₂⁴	—	2	2	no	—
153	C₂⁴	—	2	8	no	—
154	C₂⁴	—	2	24	no	—
155	C₂⁴	—	2	24	no	—
156	C₂⁴	—	2	12	no	—
157	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
158	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
159	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
160	C₂×D₄	—	2	48	no	—
161	C₂×D₄	—	4	48	no	—
162	C₂×D₄	—	4	48	no	—
163	C₂×D₄	—	4	96	no	—
164	C₂×D₄	—	4	64	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	2	24	no	—
169	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
170	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
171	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
173	C₄⋊C₄	—	2	24	no	—
174	C₂²⋊C₄	—	2	24	no	—
175	C₂²⋊C₄	—	4	48	no	—
176	C₄²	C₂×D₄×C₅₃	1	16	yes	complemented
177	C₂³	—	2	2	no	—
178	C₂³	—	2	8	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	2	24	no	—
182	C₂³	—	4	48	no	—
183	C₂³	—	4	12	no	—
184	C₂³	—	4	16	no	—
185	C₂³	—	4	16	no	—
186	C₂³	—	4	48	no	—
187	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
188	C₂³	H(5,𝔽₂)×C₅₃	1	12	yes	—
189	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
190	D₄	—	4	96	no	—
191	D₄	—	4	64	no	—
192	C₂×C₄	—	2	24	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	4	12	no	—
198	C₂×C₄	—	4	16	no	—
199	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
200	C₂²	—	4	8	no	—
201	C₂²	—	4	8	no	—
202	C₂²	—	4	24	no	—
203	C₂²	—	4	24	no	—
204	C₂²	—	4	6	no	—
205	C₂²	—	8	32	no	—
206	C₂²	—	8	8	no	—
207	C₂²	C₅₃×#535	1	12	yes	—
208	C₂²	C₅₃×#560	1	16	yes	—
209	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
210	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	6	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	4	8	no	—
213	C₄	—	4	6	no	—
214	C₂	—	8	2	no	—
215	C₂	—	8	8	no	—
216	C₂	C₅₃×#2818	1	8	yes	direct product factor
217	C₂	C₂×C₅₃×#535	1	3	yes	—
218	C₂	C₂×C₅₃×#560	1	4	yes	—

	subgroup	G/H
center, socle	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
1. C₂²×#127×C₅₃	C₂	semidirect
2. C₂²×#126×C₅₃	C₂	semidirect
3. C₅₃×#2818	C₂	direct (N × H)
4. C₂×C₅₃×#394	C₂	semidirect
5. C₂×C₅₃×#392	C₂	semidirect
6. C₂×C₅₃×#383	C₂	semidirect
10. C₂×#127×C₅₃	C₂²	semidirect
12. C₂×#126×C₅₃	C₂²	semidirect
17. C₅₃×#394	C₂²	semidirect
18. C₅₃×#392	C₂²	semidirect
19. C₅₃×#383	C₂²	semidirect
39. C₂×C₄²×C₅₃	D₄	semidirect
66. C₄²×C₅₃	C₂×D₄	semidirect
90. C₂×#2818	C₅₃	direct (N × H)
105. C₂²×#127	C₂×C₅₃	semidirect
106. C₂²×#126	C₂×C₅₃	semidirect
107. #2818	C₂×C₅₃	direct (N × H)
108. C₂×#394	C₂×C₅₃	semidirect
109. C₂×#392	C₂×C₅₃	semidirect
110. C₂×#383	C₂×C₅₃	semidirect
113. C₂×C₅₃	#2818	direct (N × H)
119. C₂×#127	C₂²×C₅₃	semidirect
121. C₂×#126	C₂²×C₅₃	semidirect
126. #394	C₂²×C₅₃	semidirect
127. #392	C₂²×C₅₃	semidirect
128. #383	C₂²×C₅₃	semidirect
130. C₅₃	C₂×#2818	direct (N × H)
149. C₂×C₄²	D₄×C₅₃	semidirect
176. C₄²	C₂×D₄×C₅₃	semidirect
216. C₂	C₅₃×#2818	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×#127×C₅₃	2
C₂²×#126×C₅₃	2
C₅₃×#2818	2
C₂×C₅₃×#394	2
C₂×C₅₃×#392	2
C₂×C₅₃×#383	2
C₂×#2818	53

Subgroups — C2×C53×#2818 #27244197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2818 #27244197