Subgroups — C₂³×#13133193 #28290732

Order 13568.

14078 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	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133193	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11719919	C₂	1	2	yes	maximal, complemented
5	C₂³×#11719921	C₂	1	2	yes	maximal
6	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
9	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₅₃	—	2	16	no	—
11	C₂³×C₄×D₅₃	C₂²	1	28	yes	—
12	C₂×#13133193	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₅₃	C₂²	1	28	yes	complemented
14	C₂²×#11719919	C₂²	1	56	yes	complemented
15	C₂²×#11719921	C₂²	1	56	yes	—
16	C₂³×Q₈×C₅₃	—	2	16	no	—
17	C₂³×Q₈×C₅₃	C₂²	1	28	yes	complemented
18	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂⁴×D₅₃	—	2	8	no	—
21	C₂⁴×D₅₃	—	2	8	no	—
22	C₂⁴×D₅₃	Q₈	1	16	yes	complemented
23	C₂⁴×D₅₃	C₂³	1	14	yes	—
24	C₂³×Dic₅₃	—	2	8	no	—
25	C₂³×Dic₅₃	—	2	16	no	—
26	C₂³×Dic₅₃	C₂³	1	28	yes	—
27	C₂³×Dic₅₃	C₂³	1	14	yes	—
28	C₂²×C₄×D₅₃	—	2	224	no	—
29	C₂²×C₄×D₅₃	C₂³	1	112	yes	—
30	#13133193	C₂³	1	512	yes	direct product factor
31	C₂×C₄⋊Dic₅₃	C₂³	1	112	yes	complemented
32	C₂×#11719919	C₂³	1	224	yes	complemented
33	C₂×#11719921	C₂³	1	224	yes	—
34	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Q₈×C₅₃	—	2	224	no	—
36	C₂²×Q₈×C₅₃	C₂³	1	112	yes	complemented
37	C₂³×C₄×C₅₃	—	2	16	no	—
38	C₂³×C₄×C₅₃	D₄	1	16	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
40	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
41	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
42	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
43	C₂⁴×C₅₃	#41	1	8	yes	—
44	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
45	C₂³×D₅₃	—	2	128	no	—
46	C₂³×D₅₃	—	2	56	no	—
47	C₂³×D₅₃	—	2	56	no	—
48	C₂³×D₅₃	C₂×Q₈	1	112	yes	complemented
49	C₂³×D₅₃	C₂⁴	1	28	yes	—
50	C₂×Q₈×C₅₃	—	2	448	no	—
51	C₂×Q₈×C₅₃	C₂⁴	1	64	yes	complemented
52	C₂²×C₄×C₅₃	—	2	112	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	112	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	56	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
56	C₂²×Dic₅₃	—	2	56	no	—
57	C₂²×Dic₅₃	—	2	112	no	—
58	C₂²×Dic₅₃	C₂⁴	1	56	yes	—
59	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
60	C₂×C₄×D₅₃	—	2	448	no	—
61	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
62	#11719921	C₂⁴	1	128	yes	—
63	C₄⋊Dic₅₃	C₂⁴	1	64	yes	complemented
64	#11719919	C₂⁴	1	128	yes	complemented
65	C₂³×C₅₃	C₂²⋊Q₈	1	64	yes	complemented
66	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
67	C₂³×C₅₃	C₂²×Q₈	1	28	yes	—
68	C₂³×C₅₃	C₂×#41	1	28	yes	—
69	C₂³×C₅₃	C₂⁵	1	7	yes	—
70	C₂²×D₅₃	—	2	448	no	—
71	C₂²×D₅₃	—	2	56	no	—
72	C₂²×D₅₃	—	2	56	no	—
73	C₂²×D₅₃	C₂²×Q₈	1	112	yes	complemented
74	C₂²×D₅₃	C₂⁵	1	8	yes	—
75	Q₈×C₅₃	—	2	128	no	—
76	C₂×C₄×C₅₃	—	2	112	no	—
77	C₂×C₄×C₅₃	C₂²×D₄	1	112	yes	—
78	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
79	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
80	C₂×Dic₅₃	—	2	56	no	—
81	C₂×Dic₅₃	—	2	112	no	—
82	C₂×Dic₅₃	C₂⁵	1	8	yes	—
83	C₂×Dic₅₃	C₂⁵	1	16	yes	—
84	C₄×D₅₃	—	2	128	no	—
85	C₂³×C₂²⋊Q₈	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	112	yes	complemented
87	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
88	C₂²×C₅₃	C₂³×Q₈	1	14	yes	—
89	C₂²×C₅₃	C₂²×#41	1	14	yes	—
90	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
91	C₂×D₅₃	—	2	8	no	—
92	C₂×D₅₃	—	2	8	no	—
93	C₂×D₅₃	—	2	224	no	—
94	C₂×D₅₃	C₂³×Q₈	1	16	yes	complemented
95	C₄×C₅₃	—	2	16	no	—
96	C₄×C₅₃	C₂³×D₄	1	16	yes	—
97	Dic₅₃	—	2	8	no	—
98	Dic₅₃	—	2	16	no	—
99	C₂⁴×Q₈	D₅₃	1	1	yes	characteristic, complemented
100	C₂⁵×C₄	—	53	1	no	—
101	C₂²×C₂²⋊Q₈	—	53	56	no	—
102	C₂³×C₄⋊C₄	—	53	2	no	—
103	C₂³×C₄⋊C₄	—	53	1	no	—
104	C₂³×C₂²⋊C₄	—	53	2	no	—
105	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	28	yes	complemented
106	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
107	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
108	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
109	D₅₃	—	2	16	no	—
110	C₂⁶	—	53	1	no	—
111	C₂³×Q₈	—	2	16	no	—
112	C₂³×Q₈	C₂×D₅₃	1	28	yes	complemented
113	C₂⁴×C₄	—	53	1	no	—
114	C₂⁴×C₄	—	53	2	no	—
115	C₂⁴×C₄	—	53	28	no	—
116	C₂⁴×C₄	—	106	16	no	—
117	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
118	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₂²⋊Q₈	—	53	448	no	—
120	C₂²×C₄⋊C₄	—	53	56	no	—
121	C₂²×C₄⋊C₄	—	53	28	no	—
122	C₂²×C₂²⋊C₄	—	53	56	no	—
123	C₅₃	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	53	16	no	—
125	C₂⁵	—	53	14	no	—
126	C₂⁵	—	106	8	no	—
127	C₂⁵	—	106	8	no	—
128	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
129	C₂²×Q₈	—	2	224	no	—
130	C₂²×Q₈	C₂²×D₅₃	1	112	yes	complemented
131	C₂³×C₄	—	2	16	no	—
132	C₂³×C₄	—	53	28	no	—
133	C₂³×C₄	—	53	14	no	—
134	C₂³×C₄	—	53	112	no	—
135	C₂³×C₄	—	106	16	no	—
136	C₂³×C₄	—	106	8	no	—
137	C₂³×C₄	—	106	224	no	—
138	C₂³×C₄	C₂²⋊D₅₃	1	16	yes	—
139	C₂³×C₄	C₂²×D₅₃	1	28	yes	—
140	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
141	C₂²⋊Q₈	—	53	512	no	—
142	C₂×C₄⋊C₄	—	53	224	no	—
143	C₂×C₄⋊C₄	—	53	112	no	—
144	C₂×C₂²⋊C₄	—	53	224	no	—
145	C₂⁴	—	53	112	no	—
146	C₂⁴	—	53	28	no	—
147	C₂⁴	—	106	128	no	—
148	C₂⁴	—	106	56	no	—
149	C₂⁴	—	106	56	no	—
150	C₂⁴	Q₈×D₅₃	1	8	yes	—
151	C₂⁴	#11719949	1	8	yes	—
152	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
153	C₂⁴	C₂³×D₅₃	1	7	yes	—
154	C₂×Q₈	—	2	448	no	—
155	C₂×Q₈	C₂³×D₅₃	1	64	yes	complemented
156	C₂²×C₄	—	2	112	no	—
157	C₂²×C₄	—	53	56	no	—
158	C₂²×C₄	—	53	28	no	—
159	C₂²×C₄	—	53	64	no	—
160	C₂²×C₄	—	106	56	no	—
161	C₂²×C₄	—	106	112	no	—
162	C₂²×C₄	—	106	448	no	—
163	C₂²×C₄	C₂×C₂²⋊D₅₃	1	112	yes	—
164	C₂²×C₄	C₂³×D₅₃	1	56	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
166	C₄⋊C₄	—	53	128	no	—
167	C₄⋊C₄	—	53	64	no	—
168	C₂²⋊C₄	—	53	128	no	—
169	C₂³	—	53	112	no	—
170	C₂³	—	53	8	no	—
171	C₂³	—	106	448	no	—
172	C₂³	—	106	56	no	—
173	C₂³	—	106	56	no	—
174	C₂³	#13133193	1	64	yes	direct product factor
175	C₂³	C₂²×C₂²⋊D₅₃	1	28	yes	—
176	C₂³	C₂×Q₈×D₅₃	1	28	yes	—
177	C₂³	C₂×#11719949	1	28	yes	—
178	C₂³	C₂⁴×D₅₃	1	7	yes	—
179	Q₈	—	2	128	no	—
180	C₂×C₄	—	2	112	no	—
181	C₂×C₄	—	53	16	no	—
182	C₂×C₄	—	53	8	no	—
183	C₂×C₄	—	106	56	no	—
184	C₂×C₄	—	106	112	no	—
185	C₂×C₄	—	106	128	no	—
186	C₂×C₄	C₂²×C₂²⋊D₅₃	1	112	yes	—
187	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
188	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
189	C₂²	—	53	16	no	—
190	C₂²	—	106	224	no	—
191	C₂²	—	106	8	no	—
192	C₂²	—	106	8	no	—
193	C₂²	C₂×#13133193	1	112	yes	direct product factor
194	C₂²	C₂³×C₂²⋊D₅₃	1	14	yes	—
195	C₂²	C₂²×Q₈×D₅₃	1	14	yes	—
196	C₂²	C₂²×#11719949	1	14	yes	—
197	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	106	8	no	—
200	C₄	—	106	16	no	—
201	C₄	C₂³×C₂²⋊D₅₃	1	16	yes	—
202	C₂	—	106	16	no	—
203	C₂	C₂²×#13133193	1	28	yes	direct product factor
204	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
206	C₂	C₂³×#11719949	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁴×Q₈×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁵×D₅₃	C₂²
hypercenter	C₂⁴×Q₈	D₅₃
hypocenter	C₅₃	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133193	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₅₃	C₂	semidirect
4. C₂³×#11719919	C₂	semidirect
6. C₂⁴×Q₈×C₅₃	C₂	semidirect
12. C₂×#13133193	C₂²	direct (N × H)
13. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
14. C₂²×#11719919	C₂²	semidirect
17. C₂³×Q₈×C₅₃	C₂²	semidirect
22. C₂⁴×D₅₃	Q₈	semidirect
30. #13133193	C₂³	direct (N × H)
31. C₂×C₄⋊Dic₅₃	C₂³	semidirect
32. C₂×#11719919	C₂³	semidirect
36. C₂²×Q₈×C₅₃	C₂³	semidirect
48. C₂³×D₅₃	C₂×Q₈	semidirect
51. C₂×Q₈×C₅₃	C₂⁴	semidirect
63. C₄⋊Dic₅₃	C₂⁴	semidirect
64. #11719919	C₂⁴	semidirect
65. C₂³×C₅₃	C₂²⋊Q₈	semidirect
73. C₂²×D₅₃	C₂²×Q₈	semidirect
86. C₂²×C₅₃	C₂×C₂²⋊Q₈	semidirect
94. C₂×D₅₃	C₂³×Q₈	semidirect
99. C₂⁴×Q₈	D₅₃	semidirect
105. C₂×C₅₃	C₂²×C₂²⋊Q₈	semidirect
112. C₂³×Q₈	C₂×D₅₃	semidirect
123. C₅₃	C₂³×C₂²⋊Q₈	semidirect
130. C₂²×Q₈	C₂²×D₅₃	semidirect
155. C₂×Q₈	C₂³×D₅₃	semidirect
174. C₂³	#13133193	direct (N × H)
193. C₂²	C₂×#13133193	direct (N × H)
203. C₂	C₂²×#13133193	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×D₅₃	2
C₂²×#13133193	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719919	2
C₂³×#11719921	2
C₂⁴×Q₈×C₅₃	2
C₂³×C₂²⋊Q₈	53

Subgroups — C23×#13133193 #28290732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133193 #28290732