Subgroups — #20283278

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
3	#16810651	C₂	1	4	yes	maximal, complemented
4	#16810649	C₂	1	4	yes	maximal, complemented
5	#16810646	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719917	—	2	4	no	—
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	C₂²×#88322	C₂²	1	2	yes	—
11	#13133176	C₄	1	2	yes	—
12	#13133176	C₄	1	2	yes	—
13	#13133176	C₂²	1	2	yes	—
14	#13133176	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
16	C₂×C₂²⋊Dic₅₃	C₄	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
18	#13133146	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	4	no	—
21	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	4	yes	—
29	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
34	M₁₆×C₅₃	—	2	4	no	—
35	M₁₆×C₅₃	—	2	4	no	—
36	C₂×C₈×C₅₃	—	2	4	no	—
37	C₂×C₈×C₅₃	C₂³	1	4	yes	—
38	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
39	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
42	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
43	C₄×Dic₅₃	C₂³	1	4	yes	—
44	#11719917	—	2	4	no	—
45	#11719917	—	2	4	no	—
46	C₂×#88322	—	2	4	no	—
47	C₂×#88322	C₂³	1	4	yes	—
48	C₂³×C₅₃	—	4	2	no	—
49	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₅₃	—	4	4	no	—
67	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
68	C₈×C₅₃	—	2	4	no	—
69	#88322	—	2	4	no	—
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	—	4	4	no	—
74	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	Dic₅₃	—	2	8	no	—
84	#2905	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₅₃	#568	1	2	yes	—
90	C₂²×M₁₆	—	53	1	no	—
91	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
92	C₂×#117	—	53	1	no	—
93	#424	—	53	4	no	—
94	#424	—	53	4	no	—
95	#423	—	53	4	no	—
96	C₅₃	#2905	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	—	53	2	no	—
100	C₂×M₁₆	—	53	2	no	—
101	C₂×M₁₆	—	106	4	no	—
102	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
103	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
104	C₂²×C₈	—	53	2	no	—
105	C₂²×C₈	C₂×D₅₃	1	2	yes	—
106	#117	—	53	2	no	—
107	#117	—	53	2	no	—
108	#117	—	53	2	no	—
109	#117	—	53	2	no	—
110	C₂×C₄⋊C₄	—	53	2	no	—
111	C₂×C₂²⋊C₄	—	53	2	no	—
112	C₂×C₄²	—	53	2	no	—
113	C₄⋊C₈	—	53	16	no	—
114	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	—	53	4	no	—
116	C₂²×C₄	Dic₁₀₆	1	4	yes	—
117	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊D₅₃	1	4	yes	—
121	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	106	4	no	—
127	M₁₆	—	106	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	—	106	4	no	—
131	C₂×C₈	C₂²×D₅₃	1	4	yes	—
132	C₄⋊C₄	—	53	4	no	—
133	C₄⋊C₄	—	53	4	no	—
134	C₂²⋊C₄	—	53	4	no	—
135	C₂²⋊C₄	—	53	4	no	—
136	C₄²	—	53	4	no	—
137	C₄²	—	53	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	#11719919	1	4	yes	—
140	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	—	53	8	no	—
144	C₂×C₄	—	212	4	no	—
145	C₂×C₄	#11719919	1	4	yes	—
146	C₂×C₄	#11719919	1	4	yes	—
147	C₂×C₄	#11719919	1	4	yes	—
148	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
149	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
150	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
157	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	106	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×#11719919	1	2	yes	—
165	C₂²	C₂×#11719919	1	2	yes	—
166	C₂²	C₂×#11719919	1	2	yes	—
167	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
168	C₄	—	106	8	no	—
169	C₄	C₂×#11719919	1	2	yes	—
170	C₄	C₂×#11719919	1	2	yes	—
171	C₄	C₂×#11719919	1	2	yes	—
172	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×#11719919	1	1	yes	characteristic
178	C₂	#16811389	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂²×M₁₆×C₅₃	C₂
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	#2905
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16810651	C₂	semidirect
4. #16810649	C₂	semidirect
5. #16810646	C₂	semidirect
18. #13133146	C₂²	semidirect
96. C₅₃	#2905	semidirect