Subgroups — #20283323

Order 6784.

440 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	#16811389	C₂	1	2	yes	maximal
2	C₂×#13133223	C₂	1	1	yes	maximal, characteristic
3	#16810671	C₂	1	2	yes	maximal, complemented
4	#16810670	C₂	1	4	yes	maximal, complemented
5	C₂×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719945	C₂²	1	1	yes	characteristic
9	#13133225	—	2	4	no	—
10	#13133223	—	2	4	no	—
11	#13133223	C₄	1	4	yes	—
12	#13133223	C₄	1	4	yes	—
13	C₂×#11719917	C₂²	1	2	yes	—
14	C₂×#11719917	C₂²	1	1	yes	characteristic
15	C₂×#11719912	C₂²	1	2	yes	—
16	M₁₆×D₅₃	C₂²	1	8	yes	—
17	C₂×C₈×D₅₃	C₂²	1	2	yes	—
18	#13133153	C₂²	1	2	yes	—
19	#13133153	C₂²	1	2	yes	—
20	#13133152	C₂²	1	8	yes	—
21	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	#108×C₅₃	C₂²	1	2	yes	—
24	#108×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₅₃	Q₈	1	2	yes	—
27	#11719945	C₂×C₄	1	2	yes	—
28	#11719945	C₂×C₄	1	2	yes	—
29	#11719945	D₄	1	2	yes	—
30	#11719945	D₄	1	2	yes	—
31	C₂×D₂₁₂	Q₈	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₂³	1	2	yes	—
33	C₂×Dic₁₀₆	Q₈	1	1	yes	characteristic
34	M₁₆×C₅₃	—	2	2	no	—
35	M₁₆×C₅₃	C₂³	1	4	yes	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#11719917	—	2	2	no	—
42	#11719917	C₂×C₄	1	2	yes	—
43	#11719917	C₂×C₄	1	2	yes	—
44	#11719917	C₂³	1	4	yes	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	C₂³	1	2	yes	—
47	#11719912	—	2	8	no	—
48	#11719912	C₂×C₄	1	8	yes	—
49	C₈×D₅₃	—	2	8	no	—
50	C₈×D₅₃	C₂×C₄	1	8	yes	—
51	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₅₃	C₄⋊C₄	1	4	yes	—
60	C₂²⋊D₅₃	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₅₃	C₂×Q₈	1	2	yes	—
62	D₂₁₂	C₄⋊C₄	1	2	yes	—
63	D₂₁₂	C₄⋊C₄	1	2	yes	—
64	C₄×D₅₃	C₂²×C₄	1	4	yes	—
65	C₄×D₅₃	C₂×D₄	1	4	yes	—
66	Dic₁₀₆	C₄⋊C₄	1	2	yes	—
67	Dic₁₀₆	C₄⋊C₄	1	2	yes	—
68	C₈×C₅₃	—	2	4	no	—
69	C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	#88322	—	2	4	no	—
72	#88322	C₂²×C₄	1	4	yes	—
73	C₂²×C₅₃	—	4	1	no	—
74	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×D₅₃	—	4	2	no	—
78	C₂×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
79	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
84	#2916	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	D₅₃	—	2	4	no	—
90	#568	—	53	2	no	—
91	C₂×#131	—	53	1	no	—
92	#430	—	53	2	no	—
93	#430	—	53	4	no	—
94	C₂×#108	—	53	1	no	—
95	C₂×#108	—	53	4	no	—
96	C₂×#108	D₅₃	1	1	yes	characteristic, complemented
97	C₅₃	#2916	1	1	yes	characteristic, complemented, Sylow
98	C₂×#41	—	53	1	no	—
99	#131	—	53	4	no	—
100	#131	—	53	4	no	—
101	#131	—	106	4	no	—
102	#131	—	106	4	no	—
103	C₂×M₁₆	—	53	1	no	—
104	C₂×M₁₆	—	53	2	no	—
105	C₂×M₁₆	—	53	2	no	—
106	C₂×M₁₆	—	53	8	no	—
107	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
108	C₂²×C₈	—	53	2	no	—
109	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#108	—	53	8	no	—
111	#108	—	53	2	no	—
112	#108	—	53	2	no	—
113	#108	C₂×D₅₃	1	2	yes	—
114	#108	C₂×D₅₃	1	2	yes	—
115	#41	—	53	2	no	—
116	#41	—	53	2	no	—
117	#41	—	53	2	no	—
118	#41	—	53	2	no	—
119	C₂×Q₈	—	53	1	no	—
120	C₂×D₄	—	53	2	no	—
121	C₂×D₄	—	53	1	no	—
122	C₂²×C₄	—	53	2	no	—
123	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	53	2	no	—
126	M₁₆	—	53	2	no	—
127	M₁₆	—	53	8	no	—
128	M₁₆	—	53	4	no	—
129	M₁₆	—	106	8	no	—
130	M₁₆	—	106	2	no	—
131	M₁₆	C₂²×D₅₃	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	53	8	no	—
135	C₂×C₈	—	106	8	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	C₄×D₅₃	1	2	yes	—
138	C₂×C₈	C₄×D₅₃	1	2	yes	—
139	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₂³	—	53	2	no	—
142	C₂³	Q₈×D₅₃	1	1	yes	characteristic
143	Q₈	—	53	2	no	—
144	Q₈	—	53	2	no	—
145	D₄	—	53	4	no	—
146	D₄	—	53	4	no	—
147	D₄	—	53	2	no	—
148	D₄	—	53	2	no	—
149	C₂×C₄	—	53	2	no	—
150	C₂×C₄	—	53	4	no	—
151	C₂×C₄	—	53	4	no	—
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	53	4	no	—
160	C₈	—	106	4	no	—
161	C₈	C₂×C₄×D₅₃	1	2	yes	—
162	C₈	C₂×C₄×D₅₃	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	53	4	no	—
165	C₂²	—	212	2	no	—
166	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
169	C₄	—	53	4	no	—
170	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
173	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	106	4	no	—
178	C₂	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂×C₅₃	C₂²×C₄⋊C₄
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#2916
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #16810671	C₂	semidirect
4. #16810670	C₂	semidirect
5. C₂×#13133153	C₂	semidirect
6. #108×D₅₃	C₂	semidirect
7. C₂×#108×C₅₃	C₂	semidirect
96. C₂×#108	D₅₃	semidirect
97. C₅₃	#2916	semidirect