Subgroups — C₂³×#13133264 #28297981

Order 13568.

59806 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	C₂²×#13133264	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#11719945	C₂	1	3	yes	maximal, complemented
3	C₂³×#11719947	C₂	1	6	yes	maximal, complemented
4	C₂⁴×Dic₁₀₆	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×D₅₃	C₂	1	2	yes	maximal, complemented
6	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133264	C₂²	1	1792	yes	direct product factor
8	C₂²×#11719945	C₂²	1	168	yes	complemented
9	C₂²×#11719947	C₂²	1	336	yes	complemented
10	C₂³×Dic₁₀₆	C₂²	1	6	yes	—
11	C₂³×Dic₁₀₆	C₂²	1	3	yes	—
12	C₂³×Dic₁₀₆	C₂²	1	168	yes	complemented
13	C₂²×Q₈×D₅₃	C₂²	1	112	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
15	C₂³×C₂²⋊D₅₃	C₂²	1	6	yes	—
16	C₂⁴×Dic₅₃	C₂²	1	6	yes	—
17	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
18	C₂²×#41×C₅₃	C₂²	1	56	yes	complemented
19	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
21	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
22	C₂⁴×D₅₃	C₂³	1	2	yes	—
23	#13133264	C₂³	1	4096	yes	direct product factor
24	C₂×#11719945	C₂³	1	1344	yes	complemented
25	C₂×#11719947	C₂³	1	2688	yes	complemented
26	C₂²×Dic₁₀₆	C₂³	1	168	yes	—
27	C₂²×Dic₁₀₆	C₂³	1	84	yes	—
28	C₂²×Dic₁₀₆	C₂³	1	1344	yes	complemented
29	C₂×Q₈×D₅₃	C₂³	1	896	yes	complemented
30	C₂²×D₂₁₂	C₂³	1	28	yes	—
31	C₂²×C₂²⋊D₅₃	C₂³	1	168	yes	—
32	C₂³×Dic₅₃	C₂³	1	6	yes	—
33	C₂³×Dic₅₃	C₂³	1	168	yes	—
34	C₂²×C₄×D₅₃	C₂³	1	168	yes	—
35	C₂⁵×C₅₃	C₂³	1	3	yes	—
36	C₂×#41×C₅₃	C₂³	1	448	yes	complemented
37	C₂²×Q₈×C₅₃	C₂³	1	28	yes	—
38	C₂²×D₄×C₅₃	C₂³	1	84	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	84	yes	—
42	C₂⁴×C₅₃	—	2	24	no	—
43	C₂⁴×C₅₃	C₂⁴	1	42	yes	—
44	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂³×D₅₃	—	2	16	no	—
46	C₂³×D₅₃	C₂⁴	1	28	yes	—
47	#41×C₅₃	C₂⁴	1	512	yes	complemented
48	C₂×Q₈×C₅₃	C₂⁴	1	112	yes	—
49	C₂×D₄×C₅₃	C₂⁴	1	336	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	14	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	42	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	336	yes	—
53	C₂×C₂²⋊D₅₃	C₂⁴	1	672	yes	—
54	C₂²×Dic₅₃	C₂⁴	1	84	yes	—
55	C₂²×Dic₅₃	C₂⁴	1	672	yes	—
56	Q₈×D₅₃	C₂⁴	1	1024	yes	complemented
57	#11719947	C₂⁴	1	3072	yes	complemented
58	#11719945	C₂⁴	1	1536	yes	complemented
59	C₂×D₂₁₂	C₂⁴	1	112	yes	—
60	C₂×C₄×D₅₃	C₂⁴	1	672	yes	—
61	C₂×Dic₁₀₆	C₂⁴	1	672	yes	—
62	C₂×Dic₁₀₆	C₂⁴	1	336	yes	—
63	C₂×Dic₁₀₆	C₂⁴	1	1536	yes	complemented
64	C₂³×C₅₃	—	2	168	no	—
65	C₂³×C₅₃	2₋¹⁺⁴	1	8	yes	complemented
66	C₂³×C₅₃	C₂⁵	1	84	yes	—
67	C₂³×C₅₃	C₂⁵	1	7	yes	—
68	C₂²×D₅₃	—	2	112	no	—
69	C₂²×D₅₃	C₂⁵	1	56	yes	—
70	Q₈×C₅₃	C₂⁵	1	64	yes	—
71	D₄×C₅₃	C₂⁵	1	192	yes	—
72	C₂×C₄×C₅₃	C₂⁵	1	28	yes	—
73	C₂×C₄×C₅₃	C₂⁵	1	84	yes	—
74	C₂×C₄×C₅₃	C₂⁵	1	192	yes	—
75	C₂²⋊D₅₃	C₂⁵	1	384	yes	—
76	C₂×Dic₅₃	C₂⁵	1	168	yes	—
77	C₂×Dic₅₃	C₂⁵	1	384	yes	—
78	D₂₁₂	C₂⁵	1	64	yes	—
79	C₄×D₅₃	C₂⁵	1	384	yes	—
80	Dic₁₀₆	C₂⁵	1	192	yes	—
81	Dic₁₀₆	C₂⁵	1	384	yes	—
82	C₂³×2₋¹⁺⁴	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	168	no	—
84	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	28	yes	complemented
85	C₂²×C₅₃	C₂⁶	1	24	yes	—
86	C₂²×C₅₃	C₂⁶	1	7	yes	—
87	C₂×D₅₃	—	2	112	no	—
88	C₂×D₅₃	C₂⁶	1	16	yes	—
89	C₄×C₅₃	C₂⁶	1	8	yes	—
90	C₄×C₅₃	C₂⁶	1	24	yes	—
91	Dic₅₃	C₂⁶	1	48	yes	—
92	C₂²×2₋¹⁺⁴	—	53	112	no	—
93	C₂³×#41	—	53	6	no	—
94	C₂³×#41	—	53	3	no	—
95	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
96	C₂⁴×Q₈	—	53	2	no	—
97	C₂⁴×Q₈	—	53	3	no	—
98	C₂×C₅₃	—	2	24	no	—
99	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	14	yes	complemented
100	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
101	D₅₃	—	2	16	no	—
102	C₂×2₋¹⁺⁴	—	53	1792	no	—
103	C₂²×#41	—	53	336	no	—
104	C₂²×#41	—	53	168	no	—
105	C₂²×#41	C₂×D₅₃	1	56	yes	complemented
106	C₂³×Q₈	—	53	112	no	—
107	C₂³×Q₈	—	53	6	no	—
108	C₂³×Q₈	—	53	3	no	—
109	C₂³×Q₈	—	53	168	no	—
110	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
111	C₂³×D₄	—	53	6	no	—
112	C₂³×D₄	—	53	1	no	—
113	C₂³×D₄	C₂×D₅₃	1	3	yes	—
114	C₂⁴×C₄	—	53	6	no	—
115	C₂⁴×C₄	—	53	6	no	—
116	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
117	C₅₃	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
118	C₂⁵	—	53	2	no	—
119	C₂⁵	C₂²×D₅₃	1	3	yes	—
120	2₋¹⁺⁴	—	53	4096	no	—
121	C₂×#41	—	53	2688	no	—
122	C₂×#41	—	53	1344	no	—
123	C₂×#41	C₂²×D₅₃	1	448	yes	complemented
124	C₂²×Q₈	—	53	896	no	—
125	C₂²×Q₈	—	53	168	no	—
126	C₂²×Q₈	—	53	84	no	—
127	C₂²×Q₈	—	53	1344	no	—
128	C₂²×Q₈	C₂²×D₅₃	1	28	yes	—
129	C₂²×D₄	—	53	168	no	—
130	C₂²×D₄	—	53	28	no	—
131	C₂²×D₄	C₂²×D₅₃	1	84	yes	—
132	C₂³×C₄	—	53	168	no	—
133	C₂³×C₄	—	53	6	no	—
134	C₂³×C₄	—	53	168	no	—
135	C₂³×C₄	C₂²×D₅₃	1	84	yes	—
136	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
137	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
138	C₂⁴	—	2	24	no	—
139	C₂⁴	—	53	28	no	—
140	C₂⁴	—	106	16	no	—
141	C₂⁴	C₂³×D₅₃	1	42	yes	—
142	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
143	#41	—	53	3072	no	—
144	#41	—	53	1536	no	—
145	#41	C₂³×D₅₃	1	512	yes	complemented
146	C₂×Q₈	—	53	1024	no	—
147	C₂×Q₈	—	53	672	no	—
148	C₂×Q₈	—	53	336	no	—
149	C₂×Q₈	—	53	1536	no	—
150	C₂×Q₈	C₂³×D₅₃	1	112	yes	—
151	C₂×D₄	—	53	672	no	—
152	C₂×D₄	—	53	112	no	—
153	C₂×D₄	C₂³×D₅₃	1	336	yes	—
154	C₂²×C₄	—	53	84	no	—
155	C₂²×C₄	—	53	672	no	—
156	C₂²×C₄	—	53	672	no	—
157	C₂²×C₄	C₂³×D₅₃	1	42	yes	—
158	C₂²×C₄	C₂³×D₅₃	1	14	yes	—
159	C₂²×C₄	C₂³×D₅₃	1	336	yes	—
160	C₂³	—	2	168	no	—
161	C₂³	—	53	56	no	—
162	C₂³	—	106	112	no	—
163	C₂³	#13133264	1	8	yes	direct product factor
164	C₂³	C₂⁴×D₅₃	1	84	yes	—
165	C₂³	C₂⁴×D₅₃	1	7	yes	—
166	Q₈	—	53	384	no	—
167	Q₈	—	53	192	no	—
168	Q₈	C₂⁴×D₅₃	1	64	yes	—
169	D₄	—	53	384	no	—
170	D₄	—	53	64	no	—
171	D₄	C₂⁴×D₅₃	1	192	yes	—
172	C₂×C₄	—	53	168	no	—
173	C₂×C₄	—	53	384	no	—
174	C₂×C₄	—	53	384	no	—
175	C₂×C₄	C₂⁴×D₅₃	1	84	yes	—
176	C₂×C₄	C₂⁴×D₅₃	1	28	yes	—
177	C₂×C₄	C₂⁴×D₅₃	1	192	yes	—
178	C₂²	—	2	168	no	—
179	C₂²	—	53	16	no	—
180	C₂²	—	106	112	no	—
181	C₂²	C₂×#13133264	1	28	yes	direct product factor
182	C₂²	C₂⁵×D₅₃	1	24	yes	—
183	C₂²	C₂⁵×D₅₃	1	7	yes	—
184	C₄	—	53	48	no	—
185	C₄	C₂⁵×D₅₃	1	24	yes	—
186	C₄	C₂⁵×D₅₃	1	8	yes	—
187	C₂	—	2	24	no	—
188	C₂	—	106	16	no	—
189	C₂	C₂²×#13133264	1	14	yes	direct product factor
190	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133264	C₂	direct (N × H)
2. C₂³×#11719945	C₂	semidirect
3. C₂³×#11719947	C₂	semidirect
4. C₂⁴×Dic₁₀₆	C₂	semidirect
5. C₂³×Q₈×D₅₃	C₂	semidirect
6. C₂³×#41×C₅₃	C₂	semidirect
7. C₂×#13133264	C₂²	direct (N × H)
8. C₂²×#11719945	C₂²	semidirect
9. C₂²×#11719947	C₂²	semidirect
12. C₂³×Dic₁₀₆	C₂²	semidirect
13. C₂²×Q₈×D₅₃	C₂²	semidirect
18. C₂²×#41×C₅₃	C₂²	semidirect
23. #13133264	C₂³	direct (N × H)
24. C₂×#11719945	C₂³	semidirect
25. C₂×#11719947	C₂³	semidirect
28. C₂²×Dic₁₀₆	C₂³	semidirect
29. C₂×Q₈×D₅₃	C₂³	semidirect
36. C₂×#41×C₅₃	C₂³	semidirect
47. #41×C₅₃	C₂⁴	semidirect
56. Q₈×D₅₃	C₂⁴	semidirect
57. #11719947	C₂⁴	semidirect
58. #11719945	C₂⁴	semidirect
63. C₂×Dic₁₀₆	C₂⁴	semidirect
65. C₂³×C₅₃	2₋¹⁺⁴	semidirect
84. C₂²×C₅₃	C₂×2₋¹⁺⁴	semidirect
95. C₂³×#41	D₅₃	semidirect
99. C₂×C₅₃	C₂²×2₋¹⁺⁴	semidirect
105. C₂²×#41	C₂×D₅₃	semidirect
117. C₅₃	C₂³×2₋¹⁺⁴	semidirect
123. C₂×#41	C₂²×D₅₃	semidirect
145. #41	C₂³×D₅₃	semidirect
163. C₂³	#13133264	direct (N × H)
181. C₂²	C₂×#13133264	direct (N × H)
189. C₂	C₂²×#13133264	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×#13133264	2
C₂³×#11719945	2
C₂³×#11719947	2
C₂⁴×Dic₁₀₆	2
C₂³×Q₈×D₅₃	2
C₂³×#41×C₅₃	2
C₂³×2₋¹⁺⁴	53

Subgroups — C23×#13133264 #28297981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133264 #28297981