Subgroups — C₂²×C₅₃×#396 #27242048

Order 13568.

5838 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₅₃	C₂	1	3	yes	maximal, complemented
2	C₂³×C₂²⋊C₄×C₅₃	C₂	1	3	yes	maximal
3	C₂×C₅₃×#396	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
7	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
8	C₂²×C₄⋊C₄×C₅₃	—	2	24	no	—
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	36	yes	complemented
10	C₂²×C₂²⋊C₄×C₅₃	—	2	24	no	—
11	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	36	yes	—
12	C₅₃×#396	C₂²	1	64	yes	direct product factor
13	C₂×#95×C₅₃	C₂²	1	12	yes	complemented
14	C₂⁵×C₅₃	—	2	12	no	—
15	C₂⁵×C₅₃	—	2	12	no	—
16	C₂⁵×C₅₃	Q₈	1	8	yes	—
17	C₂⁵×C₅₃	C₂³	1	6	yes	—
18	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	—	2	12	no	—
20	C₂³×C₄×C₅₃	—	2	12	no	—
21	C₂³×C₄×C₅₃	—	2	12	no	—
22	C₂³×C₄×C₅₃	—	2	12	no	—
23	C₂³×C₄×C₅₃	—	2	12	no	—
24	C₂³×C₄×C₅₃	D₄	1	24	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	18	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	144	no	—
28	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	complemented
29	C₂×C₂²⋊C₄×C₅₃	—	2	144	no	—
30	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
31	#95×C₅₃	C₂³	1	16	yes	complemented
32	C₂⁴×C₅₃	—	2	96	no	—
33	C₂⁴×C₅₃	—	2	36	no	—
34	C₂⁴×C₅₃	—	2	36	no	—
35	C₂⁴×C₅₃	—	4	48	no	—
36	C₂⁴×C₅₃	—	4	16	no	—
37	C₂⁴×C₅₃	C₂×D₄	1	12	yes	—
38	C₂⁴×C₅₃	C₂×Q₈	1	24	yes	—
39	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
40	C₂⁴×C₅₃	#41	1	12	yes	—
41	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
42	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
43	C₂²×C₄×C₅₃	—	2	96	no	—
44	C₂²×C₄×C₅₃	—	2	36	no	—
45	C₂²×C₄×C₅₃	—	2	36	no	—
46	C₂²×C₄×C₅₃	—	2	36	no	—
47	C₂²×C₄×C₅₃	—	2	36	no	—
48	C₂²×C₄×C₅₃	—	2	36	no	—
49	C₂²×C₄×C₅₃	—	4	48	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	72	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
53	C₄⋊C₄×C₅₃	—	2	96	no	—
54	C₂²⋊C₄×C₅₃	—	2	96	no	—
55	C₂³×C₅₃	—	2	144	no	—
56	C₂³×C₅₃	—	2	12	no	—
57	C₂³×C₅₃	—	2	12	no	—
58	C₂³×C₅₃	—	4	72	no	—
59	C₂³×C₅₃	—	4	24	no	—
60	C₂³×C₅₃	—	4	128	no	—
61	C₂³×C₅₃	#121	1	48	yes	—
62	C₂³×C₅₃	C₂²⋊Q₈	1	48	yes	—
63	C₂³×C₅₃	#126	1	16	yes	—
64	C₂³×C₅₃	C₂²×D₄	1	18	yes	—
65	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
66	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
67	C₂³×C₅₃	C₂×#41	1	18	yes	—
68	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	144	no	—
70	C₂×C₄×C₅₃	—	2	12	no	—
71	C₂×C₄×C₅₃	—	2	12	no	—
72	C₂×C₄×C₅₃	—	2	12	no	—
73	C₂×C₄×C₅₃	—	2	12	no	—
74	C₂×C₄×C₅₃	—	2	12	no	—
75	C₂×C₄×C₅₃	—	4	72	no	—
76	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
77	C₂²×#396	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₅₃	—	2	24	no	—
79	C₂²×C₅₃	—	4	96	no	—
80	C₂²×C₅₃	—	4	12	no	—
81	C₂²×C₅₃	—	4	4	no	—
82	C₂²×C₅₃	#396	1	64	yes	direct product factor
83	C₂²×C₅₃	C₂×#121	1	36	yes	—
84	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	36	yes	—
85	C₂²×C₅₃	C₂×#126	1	12	yes	—
86	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
87	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
88	C₂²×C₅₃	C₂²×#41	1	3	yes	—
89	C₄×C₅₃	—	2	24	no	—
90	C₄×C₅₃	—	4	12	no	—
91	C₂³×C₄⋊C₄	C₂×C₅₃	1	3	yes	complemented
92	C₂³×C₂²⋊C₄	C₂×C₅₃	1	3	yes	—
93	C₂×#396	C₂×C₅₃	1	24	yes	direct product factor
94	C₂²×#95	C₂×C₅₃	1	1	yes	characteristic, complemented
95	C₂×C₅₃	—	4	8	no	—
96	C₂×C₅₃	C₂×#396	1	24	yes	direct product factor
97	C₂×C₅₃	C₂²×#121	1	3	yes	—
98	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	3	yes	—
99	C₂×C₅₃	C₂²×#126	1	1	yes	characteristic
100	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
101	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
102	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
103	C₂²×C₄⋊C₄	—	2	24	no	—
104	C₂²×C₄⋊C₄	C₂²×C₅₃	1	36	yes	complemented
105	C₂²×C₂²⋊C₄	—	2	24	no	—
106	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	36	yes	—
107	#396	C₂²×C₅₃	1	64	yes	direct product factor
108	C₂×#95	C₂²×C₅₃	1	12	yes	complemented
109	C₅₃	C₂²×#396	1	1	yes	characteristic, direct product factor, Sylow
110	C₂⁵	—	2	12	no	—
111	C₂⁵	—	2	12	no	—
112	C₂⁵	Q₈×C₅₃	1	8	yes	—
113	C₂⁵	C₂³×C₅₃	1	6	yes	—
114	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
115	C₂³×C₄	—	2	12	no	—
116	C₂³×C₄	—	2	12	no	—
117	C₂³×C₄	—	2	12	no	—
118	C₂³×C₄	—	2	12	no	—
119	C₂³×C₄	—	2	12	no	—
120	C₂³×C₄	D₄×C₅₃	1	24	yes	—
121	C₂³×C₄	C₂³×C₅₃	1	18	yes	—
122	C₂³×C₄	C₂³×C₅₃	1	18	yes	—
123	C₂×C₄⋊C₄	—	2	144	no	—
124	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	complemented
125	C₂×C₂²⋊C₄	—	2	144	no	—
126	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
127	#95	C₂³×C₅₃	1	16	yes	complemented
128	C₂⁴	—	2	96	no	—
129	C₂⁴	—	2	36	no	—
130	C₂⁴	—	2	36	no	—
131	C₂⁴	—	4	48	no	—
132	C₂⁴	—	4	16	no	—
133	C₂⁴	C₂×D₄×C₅₃	1	12	yes	—
134	C₂⁴	C₂×Q₈×C₅₃	1	24	yes	—
135	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
136	C₂⁴	#41×C₅₃	1	12	yes	—
137	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
138	C₂⁴	C₂⁴×C₅₃	1	3	yes	—
139	C₂²×C₄	—	2	96	no	—
140	C₂²×C₄	—	2	36	no	—
141	C₂²×C₄	—	2	36	no	—
142	C₂²×C₄	—	2	36	no	—
143	C₂²×C₄	—	2	36	no	—
144	C₂²×C₄	—	2	36	no	—
145	C₂²×C₄	—	4	48	no	—
146	C₂²×C₄	C₂×D₄×C₅₃	1	72	yes	—
147	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
148	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
149	C₄⋊C₄	—	2	96	no	—
150	C₂²⋊C₄	—	2	96	no	—
151	C₂³	—	2	144	no	—
152	C₂³	—	2	12	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	4	72	no	—
155	C₂³	—	4	24	no	—
156	C₂³	—	4	128	no	—
157	C₂³	#121×C₅₃	1	48	yes	—
158	C₂³	C₂²⋊Q₈×C₅₃	1	48	yes	—
159	C₂³	#126×C₅₃	1	16	yes	—
160	C₂³	C₂²×D₄×C₅₃	1	18	yes	—
161	C₂³	C₂²×Q₈×C₅₃	1	8	yes	—
162	C₂³	C₂²×Q₈×C₅₃	1	6	yes	—
163	C₂³	C₂×#41×C₅₃	1	18	yes	—
164	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
165	C₂×C₄	—	2	144	no	—
166	C₂×C₄	—	2	12	no	—
167	C₂×C₄	—	2	12	no	—
168	C₂×C₄	—	2	12	no	—
169	C₂×C₄	—	2	12	no	—
170	C₂×C₄	—	2	12	no	—
171	C₂×C₄	—	4	72	no	—
172	C₂×C₄	C₂²×D₄×C₅₃	1	24	yes	—
173	C₂²	—	2	24	no	—
174	C₂²	—	4	96	no	—
175	C₂²	—	4	12	no	—
176	C₂²	—	4	4	no	—
177	C₂²	C₅₃×#396	1	64	yes	direct product factor
178	C₂²	C₂×#121×C₅₃	1	36	yes	—
179	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	36	yes	—
180	C₂²	C₂×#126×C₅₃	1	12	yes	—
181	C₂²	C₂³×D₄×C₅₃	1	3	yes	—
182	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
183	C₂²	C₂²×#41×C₅₃	1	3	yes	—
184	C₄	—	2	24	no	—
185	C₄	—	4	12	no	—
186	C₂	—	4	8	no	—
187	C₂	C₂×C₅₃×#396	1	24	yes	direct product factor
188	C₂	C₂²×#121×C₅₃	1	3	yes	—
189	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	3	yes	—
190	C₂	C₂²×#126×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₅₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁶×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
3. C₂×C₅₃×#396	C₂	direct (N × H)
4. C₂²×#95×C₅₃	C₂	semidirect
9. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
12. C₅₃×#396	C₂²	direct (N × H)
13. C₂×#95×C₅₃	C₂²	semidirect
28. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
31. #95×C₅₃	C₂³	semidirect
77. C₂²×#396	C₅₃	direct (N × H)
82. C₂²×C₅₃	#396	direct (N × H)
91. C₂³×C₄⋊C₄	C₂×C₅₃	semidirect
93. C₂×#396	C₂×C₅₃	direct (N × H)
94. C₂²×#95	C₂×C₅₃	semidirect
96. C₂×C₅₃	C₂×#396	direct (N × H)
104. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
107. #396	C₂²×C₅₃	direct (N × H)
108. C₂×#95	C₂²×C₅₃	semidirect
109. C₅₃	C₂²×#396	direct (N × H)
124. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
127. #95	C₂³×C₅₃	semidirect
177. C₂²	C₅₃×#396	direct (N × H)
187. C₂	C₂×C₅₃×#396	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×C₄⋊C₄×C₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
C₂×C₅₃×#396	2
C₂²×#95×C₅₃	2
C₂²×#396	53

Subgroups — C22×C53×#396 #27242048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#396 #27242048